UBC Theses and Dissertations

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UBC Theses and Dissertations

A low cost three-dimensional vision system using space-encoded spot projection Yeung, Keith Kai 1985

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A LOW C O S T T H R E E - D I M E N S I O N A L V I S I O N S Y S T E M U S I N G S P A C E - E N C O D E D SPOT P R O J E C T I O N BY K E I T H K A I YEUNG B . A . S c . , T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , 19 A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E REQUIREMENTS FOR T H E D E G R E E OF M A S T E R OF A P P L I E D S C I E N C E i n T H E F A C U L T Y OF GRADUATE S T U D I E S D e p a r t m e n t o f E l e c t r i c a l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e _ r e q u i r e d s t a n d a r d T H E U N I V E R S I T Y OF B R I T I S H C O L U M B I A D e c e m b e r 1985 © K E I T H K A I Y E U N G , 1985 CP In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the Un i v e r s i t y of B r i t i s h Columbia, I agree that the Libr a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I further agree that permission f o r extensive copying of t h i s thesis for s c h o l a r l y purposes may be granted by the head of my department or by h i s or her representatives. I t i s understood that copying or pub l i c a t i o n of t h i s t h e s i s for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department Of ELECTRICAL ENGINEERING The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date Jan 16, 1986 i i A B S T R A C T V i s i o n s u p p o r t p l a y s an i m p o r t a n t r o l e i n i n t e l l i g e n t r o b o t i c s y s t e m s . I t i s o f t e n n e c e s s a r y to f i r s t r e c o g n i z e the l o c a t i o n and the s h a p e o f the w o r k p i e c e b e f o r e the r o b o t a r m c a n be d i r e c t e d to p e r f o r m v a r i o u s t a s k s on i t . A l o w c o s t t h r e e - d i m e n s i o n a l ( 3 - D ) v i s i o n s y s t e m b a s e d u p o n the s p a c e - e n c o d e d s p o t p r o j e c t i o n t e c h n i q u e has b e e n d e v e l o p e d to m e a s u r e the shape o f a s m o o t h , f e a t u r e l e s s c u r v e d s u r f a c e . The v i s i o n s y s t e m p r o j e c t s a 32x32 a r r a y o f s p o t s o n t o the m e a s u r e m e n t s u r f a c e and u s e s a b i n a r y s p a c e -e n c o d i n g scheme to i n c o r p o r a t e the c o l u m n a d d r e s s o f t h e s e p r o j e c t i o n s p o t s i n a s e r i e s of f i v e p r o j e c t i o n p a t t e r n s . B i n a r y t h r e s h o l d i n g i s i m p l e m e n t e d to e x t r a c t the s p o t f e a t u r e s i n the images o f the p r o j e c t i o n p a t t e r n s . The image c e n t r o i d p o s i t i o n s and the d e c o d e d c o l u m n a d d r e s s e s o f the s p o t f e a t u r e s , as we 11 as the t r a n s f o rma t i o n ma t r i c e s o f t h e c a m e r a and s l i d e p r o j e c t o r a r e u s e d to compute the s p a t i a l c o o r d i n a t e s o f the p r o j e c t i o n s p o t s . T h i s t h e s i s e s t a b l i s h e s a c a l i b r a t i o n p r o c e d u r e to d e r i v e the t r a n s f o r m a t i o n m a t r i x of the c a m e r a u s i n g the m e a s u r e d s p a t i a l and image c o o r d i n a t e s o f a s e t o f c a l i b r a t i o n image f e a t u r e p o i n t s . A s i m i l a r c a l i b r a t i o n p r o c e d u r e i s a l s o e s t a b l i s h e d f o r the d e r i v a t i o n of the t r a n s f o r m a t i o n m a t r i x o f the s l i d e p r o j e c t o r . The a c c u r a c y o f t h e s e m a t r i c e s a r e s u b s t a n t i a l l y i m p r o v e d due to the u t i l i z a t i o n o f the r a n d o m s a m p l e c o n s e n s u s ( R A N S A C ) a l g o r i t h m to e l i m i n a t e the g r o s s e r r o r d a t a p o i n t s i n the s a m p l e p o p u l a t i o n s and a l l o w o n l y the good s a m p l e s to be u s e d i n the d e r i v a t i o n o f the f i n a l t r a n s f o r m a t i o n m a t r i c e s . A s u r f a c e r e f l e c t a n c e m o d e l and a image c e n t r o i d s e p a r a t i o n m o d e l s a r e d e v e l o p e d to d e s c r i b e the image i n t e n s i t y and s p a c i n g s o f the p r o j e c t i o n s p o t s as a f u n c t i o n of the m e a s u r e m e n t s u r f a c e o r i e n t a t i o n . U s i n g t h e s e m o d e l s , the l i m i t i n g o r i e n t a t i o n s o f a m e a s u r e m e n t s u r f a c e w i t h r e s p e c t to the o p t i c a l c o n f i g u r a t i o n o f the v i s i o n s y s t e m we re f o u n d . T h r e e r e c o n s t r u c t i o n a l g o r i t h m s f o r f i t t i n g a s u r f a c e to the s c a t t e r e d a m p l i t u d e s a m p l e s o b t a i n e d by the s p o t p r o j e c t i o n s y s t e m a r e d e s c r i b e d i n the t h e s i s . The s u i t a b i l i t y o f t h e s e a l g o r i t h m s f o r the m e a s u r e m e n t of a i r c r a f t - w i n g s u r f a c e s i s a l s o d i s c u s s e d . T h e s h a p e m e a s u r e m e n t r e s u l t s on a c y l i n d r i c a l s u r f a c e s h owe d t h a t the a v e r a g e me a s u r erne n t e r r o r o f p o i n t s on a s m o o t h f e a t u r e l e s s s u r f a c e i s l e s s t h a n a t e n t h of an i n c h , and the maximum e r r o r i s l e s s t h a n a q u a r t e r o f an i n c h i n a t o t a l f i e l d o f 20x20 i n c h e s . i v TABLE OF CONTENTS Page A b s t r a c t i i Table of Contents i v L i s t of T a b l e s v i i L i s t of F i g u r e s v i i i Acknowledgements x i I n t r o d u c t i o n 1 Imaging System 10 2 . 1 G e n e r a l d e s c r i p t i o n o f the i m a g i n g s y s t e m 10 2 . 2 S p a c e - e n c o d e d s p o t p r o j e c t i o n scheme 12 2 . 3 B i n a r y s p a c e - e n c o d i n g scheme f o r the s p o t c o l u m n a d d r e s s e s 16 2 . 4 3 - D m e a s u r e m e n t o f the c y l i n d r i c a l t e s t s u r f ace 19 2 . 4 . 1 O p t i c a l c o f i g u r a t i o n o f the s p o t p r o j e c t i o n s y s t e m 19 2 . 4 . 2 E x p e r i m e n t a l p r o c e d u r e s 23 Camera and P r o j e c t Models 25 3 . 1 I n t r o d u c t i o n 25 3 . 2 R e v i e w o f Homogeneous t r a n s f o r m a t i o n s 26 3 . 3 D e s c r i p t i o n o f the camera m o d e l 29 3 . 4 D e s c r i p t i o n o f the s l i d e p r o j e c t o r mo d e l 36 3 . 5 M o d e l c a l i b r a t i o n p r o c e d u r e s 39 3 . 6 I m p l e m e n t a t i o n and e x p e r i m e n t a l r e s u l t s . . . . 43 3 . 6 . 1 C a m e r a m o d e l c a l i b r a t i o n 43 3 . 6 . 2 S l i d e p r o j e c t o r mo d e l c a l i b r a t i o n 47 3 . 7 D i s c u s s i o n 51 Experimental d e t e r m i n a t i o n of model parameters 52 4 . 1 P r o b l e m d e f i n i t i o n 52 4 . 2 D e s c r i p t i o n of the R a n d o m S a m p l e C o n s e n s u s A l g o r i t h m 53 V 4 . 3 I m p l e m e n t a t i o n of the Random S a m p l e C o n s e n s u s A l g o r i t h m 54 4 . 4 D e c o m p o s i t i o n o f the t r a n s f o r m a t i o n ma t r i x 58 4 . 5 R e s u l t s o f the mo d e l c a l i b r a t i o n p r o c e s s w i t h RAN SAC a l g o r i t h m 62 4 . 6 R e s u l t s o f m a t r i x d e c o m p o s i t i o n 6 7 4 . 7 E f f e c t i v e n e s s of the RAN SAC a l g o r i t h m 70 5 . Image Processing A l g o r i t h m 72 5 . 1 O v e r v i e w 72 5 . 2 Edge d e t e c t i o n 75 5 . 2 . 1 B i n a r y t h r e s h o l d i n g 76 5 . 3 Image c e n t r o i d e s t i m a t o r 7 8 5 . 3 . 1 P r o j e c t i o n s p o t e d g e - t r a c i n g 79 5 . 3 . 2 C a l c u l a t i o n of p r o j e c t i o n s p o t c e n t r o i d c o o r d i n a t e s 80 5 . 4 P r o j e c t i o n s p o t c o l u m n a d d r e s s d e c o d i n g 82 5 . 4 . 1 E f f e c t s of s u r f a c e d i s c o n t i n u i t y on a d d r e s s d e c o d i n g 83 5 . 4 . 2 N e a r e s t n e i g h b o u r s e a r c h a 1 gor i thm 85 5 . 5 D e t e m i n a t i o n of the 3 - D s p a t i a l c o o r d i n a t e s of the p r o j e c t i o n s p o t s 86 5 . 6 S o f t w a r e i m p l e m e n t a t i o n o f t h e image p r o c e s s i n g a l g o r i t h m 89 5 . 6 . 1 S o f t w a r e d e s i g n a p p r o a c h e s 89 5 . 6 . 2 D e s c r i p t i o n of s o f t w a r e 90 5 . 7 E x p e r i m e n t a l r e s u l t s and c o n c l u s i o n s 93 5 . 8 Summary 101 6. L i m i t a t i o n s of the Measurement Surface O r i e n t a t i o n 104 6 . 1 O v e r v i e w 104 6 . 2 R e f l e c t a n c e m o d e l of the m e a s u r e m e n t s u r f a c e 109 6 . 2 . 1 D i f f u s i v e r e f l e c t a n c e m o d e l 109 6 . 2 . 2 S p e c u l a r r e f l e c t a n c e mo d e l I l l 6 . 2 . 3 C o m b i n e d r e f l e c t a n c e m o d e l I l l 6 . 2 . 4 O p t i c a l g e o m e t r y o f the p r o j e c t i o n s y s t e m 112 6 . 2 . 5 R e f l e c t a n c e m o d e l c a l i b r a t i o n r e s u l t s 121 6 . 2 . 6 L i m i t i n g s u r f a c e o r i e n t a t i o n s 126 6 . 3 Image c e n t r o i d s p a c i n g m o d e l 128 6 . 3 . 1 A n a l y t i c a l r e p r e s e n t a t i o n o f p l a n a r s u r f a c e s 132 6 . 3 . 2 S p o t s p a c i n g s c a l c u l a t i o n p r o c e d u r e s and r e s u l t s 135 6 . 4 C o m b i n e d r e s u l t s o f the r e f l e c t a n c e a n d c e n t r o i d s p a c i n g m o d e l s 140 v i 7 . Surface R e c o n s t r u c t i o n 141 7 .1 O v e r v i e w 141 7 . 1 . 1 S u r f a c e c o n s t r a i n t s 143 7 . 2 N e a r e s t n e i g h b o u r r e c o n s t r u c t i o n a 1 g o r i t hm 144 7 . 2 . 1 I m p l e m e n t a t i o n 147 7 .3 T h i n p l a t e m o d e l r e c o n s t r u c t i o n 148 7 . 3 . 1 P h y s i c a l M o d e l 149 7 . 3 . 2 S u r f a c e c o n s t r a i n t s 150 7 . 3 . 3 I m p l e m e n t a t i o n 151 7 .4 W a r p i n g t r a n s f o r m a t i o n r e c o n s t r u c t i o n 157 7 . 4 . 1 S a m p l i n g t h e o r e m s 159 7 . 4 . 2 I m p l e m e n t a t i o n 165 7 . 5 S u r f a c e r e c o n s t r u c t i o n r e s u l t s 168 7 .6 D i s c u s s i o n 175 7 . 7 S e l e c t i o n o f r e c o n s t r u c t i o n a l g o r i t h m f o r a i r c r a f t - w i n g m e a s u r e m e n t d a t a 177 7 . 6 . 1 A c c u r a c y and S m o o t h n e s s 178 7 . 6 . 2 P i e c e - w i s e r e c o n s t r u c t i o n 179 7 . 6 . 3 C o m p u t a t i o n t i m e 180 7 . 6 . 4 D i s c u s s i o n 181 8. C o n c l u s i o n s and Future Recommendations 182 8 .1 Summary and c o n c l u s i o n s 182 8 .2 Recommendations f o r f u t u r e w o r k 186 References 189 Appendix 1 S p a t i a l Coordinates Meausrements of the P r o j e c t i o n Spots 194 v i i L I S T OF T A B L E S T a b l e s Page 3 . 1 T h e c ame r a t r a n s f o rma t i o n -ma t r i x 46 3 . 2 S p a t i a l and image c o o r d i n a t e s o f the c a l i b r a t i o n p o i n t s 46 3 . 3 The s l i d e p r o j e c t o r t r a n s f o r m a t i o n m a t r i x 49 3 . 4 The s p a t i a l and image c o o r d i n a t e s o f the c a l i b r a t i o n p o i n t s i n the s l i d e p r o j e c t o r s a m p l e p o p u l a t i o n 50 4 . 1 Image and s p a t i a l c o o r d i n a t e s of the s a m p l e p o i n t s u s e d i n c a m e r a m o d e l c a l i b r a t i o n 63 4 . 2 S p o t a d d r e s s e s and s p a t i a l c o o r d i n a t e s o f t h e s a m p l e p o i n t s u s e d i n s l i d e p r o j e c t o r m o d e l c a l i b r a t i o n 64 4 . 3 L i s t s of the 6 - t u p e s 65 4 . 4 The a v e r a g e e r r o r te rms u s i n g the i n i t i a l and f i n a l c ame r a mo d e l s 65 4 . 5 T h e f i n a l c a m e r a t r a n s f o r m a t i o n m a t r i x 65 4 . 6 The a v e r a g e e r r o r t e r m s u s i n g the i n i t i a l and f i n a l s l i d e p r o j e c t o r m o d e l s 66 4 . 7 The f i n a l s l i d e p r o j e c t o r t r a n s f o r m a t i o n m a t r i x 66 4 . 8 O p t i c a l p a r a m e t e r s o f the c a m e r a w i t h r e s p e c t to F 67 4 . 9 O p t i c a l p a r a m e t e r s of the s l i d e p r o j e c t o r w i t h r e s p e c t t o F 67 4 . 1 0 P o i n t n g V e c t o r s o f the c a m e r a and s l i d e p r o j e c t o r . . . 70 5 . 1 A r e a and moments c o n t r i b u t i o n o f the edge s e gme n t s 81 6 .1 Image i n t e n s i t i e s and r e l a t i v e v i e w i n g a n g l e s o f the p r o j e c t i o n s p o t s on the c y l i n d r i c a l s u r f a c e 124 6 . 2 Image i n t e n s i t i e s and r e l a t i v e v i e w i n g a n g l e s <j> o f t h e p r o j e c t i o n s p o t s on the c y l i n d r i c a l s u r f a c e . . . 125 7 .1 E r r o r t e r m s o f the t h r e e r e c o n s t r u c t i o n s u r f a c e s 170 7 .2 E r r o r t e r m s i n the i n t e r i o r r e g i o n o f the r e c o n s t r u c t e d c y l i n d r i c a l s u r f a c e s 171 7 .3 C o m p u t a t i o n t i m e s of the r e c o n s t r u c t i o n a l g o r i t h m o v e r an 80x80 s a m p l i n g l a t t i c e ( V A X - 1 1 / 7 5 0 ) 173 v i i i L I S T OF FIGURES F i g u r e s - Page 1 .1 C F - 1 8 m a t e r i a l s d i s t r i b u t i o n 2 ( C o u r t e s y o f M c D o n n e l l D o u g l a s , S t . L o u i s ) 2 . 1 B a s i c b l o c k d i a g r a m o f the s p o t p r o j e c t i o n s y s t e m . . . 11 2 . 2 T y p i c a l s c e n e o f the s p o t p r o j e c t i o n p a t t e r n 12 2 . 3 ( a ) R e f e r e n c e s p o t p r o j e c t i o n p a t t e r n 14 ( b ) - ( f ) B i n a r y s p a c e - e n c o d e d s p o t p a t t e r n 14 2 . 4 O p t i c a l g e o m e t r y o f the s p o t p r o j e c t i o n s y s t e m 15 2 . 5 P h y s i c a l d i m e n s i o n s of the c y l i n d r i c a l s u r f a c e 21 3 . 1 O p t i c a l g e o m e t r y o f the c a m e r a s y s t e m 27 3 . 2 S p a t i a l r e l a t i o n s h i p b e t w e e n the camera and the o b j e c t - c e n t e r e d r e f e r e n c e f r a m e F 29 3 . 3 S t e p by s t e p t r a n s f o r m a t i o n s f r o m F to F 30 3 . 4 S p a t i a l r e l a t i o n s h i p b e t w e e n the image p l a n e and c a m e r a c o o r d i n a t e s y s t e m F 32 3 . 5 S p a t i a l r e l a t i o n s h i p b e t w e e n F . and F 33 3 . 6 S p a t i a l r e l a t i o n s h i p b e t w e e n F and F 37 3 . 7 R e f e r e n c e o b j e c t u s e d i n earner I mode 1^ca 1 i b r a t i o n . . . 44 3 . 8 R e f e r e n c e o b j e c t u s e d i n s l i d e p r o j e c t o r mo d e l c a l i b r a t i o n 48 4 . 1 V e n n d i a g r a m r e p r e s e n t a t i o n o f the s a m p l e p o p u l a t i o n and the 6 - t u p l e s 55 4 . 2 P h y s i c a l r e p r e s e n t a t i o n o f the c a m e r a and s l i d e p r o j e c t o r o p t i c a l p a r ame t e r s 69 5 . 1 B l o c k d i a g r a m o f the image p r o c e s s i n g a l g o r i t h m . . . . 73 5 . 2 ( a ) - ( f ) G r e y - s c a l e images o f the p r o j e c t i o n p a t t e r n s 74 5 . 3 8 - c onne c t i v i t y o f an edge p i x e l 78 5 . 4 D i r e c t i o n v e c t o r s o f the edge s e g m e n t s 81 5 . 5 S e a r c h s e q u e n c e o f the n e a r e s t n e i g h b o r s e a r c h a 1 g o r i thm 85 5 . 6 ( a ) - ( f ) B i n a r y images o f the p r o j e c t i o n p a t t e r n s 95 5 . 7 ( a ) - ( f ) B i n a r y edge map o f the p r o j e c t i o n p a t t e r n s 96 5 . 8 ( a ) - ( f ) S p o t c e n t r o i d d i s t r i b u t i o n maps 97 6 . 1 S e a r c h a r e a s o f two a d j a c e n t r e f e r e n c e c e n t r o i d s . . . 107 6 . 2 O v e r l a p p i n g s e a r c h a r e a s o f two a d j a c e n t c e n t r o i d s . . 107 6 . 3 O p t i c a l g e o m e t r y o f a s p o t p r o j e c t i o n on a r e f l e c t i v e s u r f a c e 110 6 . 4 O p t i c a l c o n f i g u r a t i o n o f t h e c a m e r a and s l i d e p r o j e c t o r w i t h r e s p e c t to F 113 6 . 5 D i m e n s i o n s o f the r e f e r e n c e s p o t p a t t e r n 114 i x 6 . 6 The c y l i n d r i c a l s u r f a c e w i t h l o n g i t u d i n a l a x i s p a r a l l e l to the x - a x i s of F 115 6 . 7 O p t i c a l g e o m e t r y o f a p r o j e c t i o n s p o t on the c y l i n d r i c a l s u r f a c e w i t h n o r m a l v e c t o r s a l o n g the y - z p l a n e o f F 117 6 .8 T h e c y l i n d r i c a l s u r f a c e w i t h l o n g i t u d i n a l a x i s p a r a l l e l to the y - a x i s of F 119 6 . 9 O p t i c a l g e o m e t r y o f a p r o j e c t i o n s p o t on the c y l i n d r i c a l s u r f a c e w i t h n o r m a l v e c t o r s a l o n g the x - z p l a n e o f F 120 6 . 1 0 G r e y - s c a l e image o? the r e f e r e n c e p a t t e r n on the c y l i n d r i c a l s u r f a c e 122 6 . 1 1 Image i n t e n s i t y p r o f i l e of the p r o j e c t i o n s p o t s a c r o s s the c y l i n d r i c a l s u r f a c e 123 6 . 1 2 R e f l e c t a n c e m o d e l of the c y l i n d r i c a l s u r f a c e 127 6 . 1 3 The image i n t e n s i t i e s v e r s u s the s u r f a c e n o r m a l a n g l e s w i t h r e s p e c t to the y - a x i s o f F 129 6 . 1 4 T h e image i n t e n s i t i e s v e r s u s the s u r f a c e n o r m a l a n g l e s w i t h r e s p e c t to the x - a x i s of F 130 6 . 1 5 A p l a n a r s u r f a c e w i t h n o r m a l v e c t o r s o r i e n t a t e d a l o n g the y - z p l a n e o f F 131 6 . 1 6 A p l a n a r s u r f a c e w i t h n o r m a l v e c t o r s o r i e n t a t e d a l o n g t h e x - z p l a n e o f F 131 6 . 1 7 P l a n v i e w o f the s u r f a c e shown i n F i g u r e 6 . 1 5 v e c t o r a l o n g the x - z p l a n e of F 133 6 . 1 8 S i d e v i ew o f the s u r f a c e sh own i n F i g u r e 6 . 1 7 135 6 . 1 9 C e n t r o i d s p a c i n g s v e r s u s s u r f a c e i n c l i n a t i o n a n g l e s f o r s u r f a c e n o r m a l v e c t o r s a l o n g the y - z p l a n e 138 6 . 2 0 C e n t r o i d s p a c i n g s v e r s u s s u r f a c e i n c l i n a t i o n a n g l e s f o r s u r f a c e n o r m a l v e c o t r s a l o n g the x - z p l a n e 139 7 .1 S a m p l i n g g r i d p o i n t a s s i g n m e n t s 143 7 .2 T h e o r i g i n a l c y l i n d r i c a l s u r f a c e 145 7 .3 T h e s c a t t e r e d s u r f a c e s a m p l i n g p o i n t s 146 7 .4 T h e o r d e r of t h e s e a r c h s e q u e n c e i n the n e a r e s t n e i g h b o u r a l g o r i t h m 148 7 . 5 G r a p h i c a l r e p r e s e n t a t i o n o f the s u r f a c e d e p t h c o n s t r a i n t s 150 7 .6 C o m p u t a t i o n a l m o l e c u l e s f o r the i n t e r i o r g r i d p o i n t s ( f r o m T e r z o p o u l o s 1984) 154 7 . 7 C o m p u t a t i o n a l m o l e c u l e s f o r the l e f t c o r n e r a n d 1 ower edge g r i d p o i n t s ( f r o m T ' e r z o p o u l o s , 1 9 8 4 ) 155 7 .8 G r a p h i c a l r e p r e s e n t a t i o n o f the s a m p l e d g r i d p o i n t and t h e s a m p l i n g l a t t i c e 158 7 . 9 H e x a g o n a l s a m p l i n g l a t t i c e s i n the s p a t i a l and f r e q u e n c y d o m a i n s 162 7 . 1 0 Wa r p i n g t r a n s f o rma t i o n f r om the x - s p a c e to £ - s p a c e 166 7 . 1 1 N e a r e s t n e i g h b o u r r e c o n s t r u c t i o n s u r f a c e 169 7 . 1 2 E r r o r map o f the n e a r e s t n e i g h b o u r r e c o n s t r u c t i o n s u r f ace 170 X 7 . 1 3 T h i n p l a t e r e c o n s t r u c t i o n s u r f a c e 171 7 . 1 4 E r r o r map o f the t h i n p l a t e r e c o n s t r u c t i o n s u r f a c e . . 172 7 . 1 5 W a r p i n g t r a n s f o r m a t i o n r e c o n s t r u c t i o n s u r f a c e 173 7 . 1 6 E r r o r map o f the w a r p i n g t r a n s f o r m a t i o n r e c o n s t r u c t i o n s u r f a c e 174 7 . 1 7 The s u g g e s t e d b a c k l i g h t i n g p r o j e c t i o n s y s t e m 176 x i ACKNOWLEDGEMENTS I w o u l d l i k e to t h a n k my s u p e r v i s o r D r . P . D . L a w r e n c e f o r h i s s u p p o r t d u r i n g the c o u r s e o f t h i s t h e s i s I w i s h to t h a n k the C h i e f and the s t a f f at D e f e n c e R e s e a r c h E s t a b l i s h m e n t P a c i f i c (DREP) f o r t h e i r s u p p o r t and e n c o u r a g e m e n t . I w o u l d e s p e c i a l l y l i k e to t h a n k S t e v e H u g h e s f o r p r o o f - r e a d i n g the t h e s i s and a l l h i s a s s i s t a n c e t h r o u g h o u t the p r o j e c t . I a l s o w i s h to e x p r e s s my g r a t i t u d e t o w a r d s Bob C h a p p e l l f o r h i s c o n t r i b u t i o n o f the g r a p h i c s . I w o u l d l i k e to t h a n k J i m C l a r k f o r the p e r m i s s i o n to use h i s e f f i c i e n t s u r f a c e r e c o n s t r u c t i o n p r o g r a m . F i n a l l y , I am g r a t e f u l h e r s u p p o r t , e n c o u r a g e m e n t d u r i n g the c o u r s e o f t h i s the t h e s i s w o u l d n o t h a v e b e e n pos to my w i f e , G a y l e n e , f o r a l l , u n d e r s t a n d i n g and p a t i e n c e s i s . W i t h o u t h e r s u p p o r t , t h i s s i b 1 e . 1 CHAPTER 1 INTRODUCTION In r e c e n t y e a r s , i n t e l l i g e n t r o b o t i c s y s t e m s h a v e b e e n w i d e l y u s e d i n i n d u s t r i a l a p p l i c a t i o n s , s u c h as p i c k -a n d - p l a c e o p e r a t i o n s i n the a s s e m b l y l i n e s [ K e l l y et a l . 1981] and seam w e l d i n g [ S c h m i d t 1 9 8 4 ] , e t c . V i s i o n s u p p o r t p l a y s an i m p o r t a n t r o l e i n t h e s e s y s t e m s . I t i s o f t e n n e c e s s a r y to f i r s t r e c o g n i z e the l o c a t i o n and the shape of the w o r k p i e c e b e f o r e the r o b o t a r m c a n be d i r e c t e d to p e r f o r m v a r i o u s t a s k s on i t . S u c h s y s t e m s a r e i d e a l to r e p l a c e human o p e r a t i o n i n h a z a r d o u s w o r k i n g e n v i r o n m e n t s and i n t e d i o u s and r e p e t i t i o u s f u n c t i o n s . W i t h the v i s i o n -s e n s i n g c a p a b i l i t y , the r o b o t i c s y s t e m s c a n be p r o g r a m m e d to r e c o g n i z e c e r t a i n t y p e s of p r e - d e f i n e d a d v e r s e c o n d i t i o n s , l i k e the i n t r u s i o n o f a f o r e i g n o b j e c t i n t o the w o r k s p a c e , so the a p p r o p r i a t e r e s p o n s e s c a n be t a k e n . T h e n o n d e s t r u c t i v e e v a l u a t i o n (NDE) g r o u p at D e f e n c e R e s e a r c h E s t a b l i s h m e n t P a c i f i c (DREP) i n V i c t o r i a , B . C . i s a c t i v e l y i n v o l v e d i n a d e v e l o p m e n t p r o g r a m to a u t o m a t e the u l t r a s o n i c i n s p e c t i o n s on the C F - 1 8 a i r c r a f t . A s shown i n F i g u r e 1 . 1 , 40% o f the e x t e r n a l s k i n of the C F - 1 8 i s made o f g r a p h i t e - e p o x y m a t e r i a l . A f t e r the a i r c r a f t has b e e n i n s e r v i c e f o r a number of y e a r s , f l a w s , s u c h as d i s b o n d s , p o r o s i t y , d e l a m i n a t i o n , e t c . , may d e v e l o p on t h e s e m a t e r i a l s and the s t r u c t u r a l i n t e g r i t y o f the a i r c r a f t w o u l d be a f f e c t e d by t h e s e p r o b l e m s [ S t u r r o c k 1 9 8 3 ] . T h e s e f l a w s c a n 2 F-18 MATERIALS DISTRIBUTION Percent of Structural Weight | | Aluminum 50 4 YMk Steel 16.0 fe^3 Titanium 13 2 IV: j Graphite/Epoxy 9.0 Other 114 100 0 Percent of Structural Weight | | Aluminum 50 4 f ^ l Steel 16.0 Titanium 13.2 Graphite/Epoxy 9 0 \ 1 Other H « 100 0 F i g u r e 1.1 C F-18 m a t e r i a l s d i s t r i b u t i o n . ( C o u r t e s y of M c D o n n e l l D o u g l a s , S t . L o u i s ) 3 be d e t e c t e d by the c h a n g e s i n the s i g n a l r e f l e c t a n c e c h a r a c t e r i s t i c s of the m a t e r i a l s i n an u l t r a s o n i c i n s p e c t i o n p r o c e s s . H e n c e , a r o u t i n e u l t r a s o n i c i n s p e c t i o n p r o g r a m i s n e e d e d f o r d e t e c t i n g t h e s e p r o b l e m s at the e a r l y s t a g e b e f o r e a n y f u r t h e r damage d e v e l o p s . F u r t h e r m o r e , the g r a p h i t e - e p o x y m a t e r i a l c a n a l s o be damaged due to i m p a c t s f r o m s o u r c e s s u c h as g r o u n d s e r v i c i n g and m a i n t e n a n c e e q u i p m e n t or runway d e b r i s . T h e u l t r a s o n i c i n s p e c t i o n p r o c e d u r e i s a l s o u s e d to a s s e s s the e x t e n t of the damage and t o l o c a t e f l a w s w h i c h a r e n o t r e c o g n i z a b l e by v i s u a l i n s p e c t i o n s . P r e s e n t l y , the i n s p e c t i o n p r o c e s s i s done m a n u a l l y by an i n s p e c t o r who h o l d s an u l t r a s o n i c t r a n s d u c e r a g a i n s t the m a t e r i a l s and p e r f o r m s a r a s t e r s c a n o p e r a t i o n o v e r the p r o b l e m a t i c a r e a s . S i m u l t a n e o u s l y , he must m o n i t o r the c h a r a c t e r i s t i c s of the s i g n a l d i s p l a y e d on an o s c i l l o s c o p e f o r s i g n a l a n o m a l i e s . The i n s p e c t i o n t a s k i s v e r y t e d i o u s and r e q u i r e s a h i g h d e g r e e o f h a n d / e y e c o o r d i n a t i o n . I t p r o m p t e d a r e q u i r e m e n t f o r an a u t o m a t e d i n s p e c t i o n s y s t e m to be d e v e l o p e d f o r t h i s a p p l i c a t i o n . The f i r s t v e r s i o n of the r o b o t i n s p e c t i o n s y s t e m i s m a i n l y f o r the i n s p e c t i o n of the m a t e r i a l s on t h e w i n g and s t a b i l a t o r s u r f a c e s . In o r d e r f o r t h e r o b o t m a n i p u l a t o r to g u i d e the u l t r a s o n i c t r a n s d u c e r a l o n g a s u r f a c e , the t h r e e - d i m e n s i o n a l ( 3 - D ) shape of the s u r f a c e m u s t be a c q u i r e d p r i o r to the i n s p e c t i o n p r o c e s s . T h e o v e r a l l o b j e c t i v e of the t h e s i s i s to d e v e l o p a v i s i o n s y s t e m f o r m e a s u r i n g the 3 - D shape o f o b j e c t s u r f a c e s i n a l a b o r a t o r y e n v i r o n m e n t , and to a p p l y the e x p e r i e n c e and 4 i n s i g h t s g a i n e d f r o m t h i s p r o t o t y p i n g v i s i o n s y s t e m to the u p c o m i n g a i r c r a f t w i n g m e a s u r e m e n t a p p l i c a t i o n . W i t h the e x c e p t i o n o f a f e w f a s t e n i n g d e v i c e s and i d e n t i f i c a t i o n m a r k i n g s on the w i n g s and s t a b i l a t o r s , t h e s e s u r f a c e s a r e b a s i c a l l y c o n s i d e r e d to be s m o o t h and f e a t u r e l e s s . H a v i n g d e f i n e d the u n d e r l y i n g p r o p e r t i e s o f t h e s e s p e c i f i c m e a s u r e m e n t s u r f a c e s , the n e x t s t e p i s to f i n d t h e m o s t a p p r o p r i a t e t e c h n i q u e f o r m e a s u r i n g the 3 - D p r o f i l e o f a i r c r a f t - w i n g s h a p e d s u r f a c e s . R a n g e f i n d i n g t e c h n i q u e s u s e d i n 3 - D v i s i o n s y s t e m s a r e d i v i d e d i n t o two b r o a d c a t e g o r i e s , d i r e c t and image b a s e d m e t h o d s . The d i r e c t m e t h o d u s e s the t i m e o f f l i g h t [ P o l a r o i d ] or the p h a s e - r e 1 a t e d e f f e c t s [ N i t z a n et a l . 1977 ] t o m e a s u r e d i s t a n c e s to the s u r f a c e . The u l t r a s o n i c t r a n s d u c e r a v a i l a b l e f r o m P o l a r o i d f o r the t i m e o f f l i g h t m e a s u r e m e n t s y s t e m , e x h i b i t a beam p a t t e r n w i t h a m a j o r f o r w a r d l o b e o f a b o u t 3 0 ° s o l i d a n g l e w h i c h t r a n s l a t e s i n t o a 2 . 7 f e e t s q u a r e c r o s s s e c t i o n a l a r e a at a w o r k i n g d i s t a n c e o f 10 f e e t . N i t z a n et a l . [ 1977] u s e d the p h a s e s h i f t o f the l a s e r p u l s e s to o b t a i n the r a n g e i n f o r m a t i o n . T h e s e m e t h o d s a r e b o t h q u i t e t i m e c o n s u m i n g and the e q u i p m e n t c o s t t e n d s t o be e x p e n s i v e . B e c a u s e o f the c o a x i a l s o u r c e / d e t e c t o r a r r a n g e m e n t s , t h e s e m e t h o d s a r e n o t s u b j e c t to the p r o b l e m o f " m i s s i n g p a r t s " as i n the i m a g e - b a s e d t r i a n g u 1 a t i o n m e t h o d s . H o w e v e r , t h e y a r e n o t commonly u s e d b e c a u s e o f t h e s e o t h e r d i s a d v a n t a g e s . T h e i m a g e - b a s e d r a n g e f i n d i n g t e c h n i q u e s a r e f u r t h e r s u b d i v i d e d i n t o two g r o u p s , p a s s i v e and a c t i v e m e t h o d s . P a s s i v e t e c h n i q u e s c o n s i s t s o f p h o t o m e t r i c [Woodham 1981] and s t e r e o p s i s m e t h o d s [ B a k e r 1982 , G r i m s o n 1981 1 9 8 5 , C l a r k 1 9 8 5 ] . T h e p h o t o m e t r i c m e t h o d w o r k s b e s t i n f i n d i n g the s u r f a c e g r a d i e n t and i t does n o t m e a s u r e the a c t u a l d i s t a n c e s to the m e a s u r e m e n t s u r f a c e . S t e r e o p s i s m e t h o d i n v o l v e s a s t e r e o p a i r o f images and m a t c h i n g a l g o r i t h m s a r e u s e d t o i d e n t i f y the c o r r e s p o n d i n g s u r f a c e p o i n t s i n the s t e r e o i m a g e s . G e o m e t r i c t r i a n g u 1 a t i o n i s t h e n u s e d to d e t e r m i n e the r a n g e i n f o r m a t i o n o f t h e s e s u r f a c e p o i n t s f r o m t h e i r s t e r e o d i s p a r i t i e s . M o s t o f the p r o c e s s i n g t i m e i s u s e d i n the s t e r e o m a t c h i n g p r o c e s s . B a k e r [ 1 9 8 2 ] d e a l t w i t h the m a t c h i n g p r o b l e m by u s i n g l i n e by l i n e edge c o r r e l a t i o n p r o c e d u r e s on the edge d a t a o f the s t e r e o image p a i r to f i n d the b e s t c o r r e s p o n d e n c e s i n the l e f t and r i g h t i m a g e s of the s c e n e . G r i m s o n [1981] i m p l e m e n t e d the mu 1 1 i - r e s o 1 u t i o n m a t c h i n g a l g o r i t h m p r o p o s e d by M a r r and P o g g i o [ 1 9 7 9 ] , w h i c h 2 i n v o l v e s t h e m a t c h i n g o f the z e r o c r o s s i n g s o f the V G f i l t e r e d s t e r e o images at f o u r d i f f e r e n t s p a t i a l r e s o l u t i o n . T h e z e r o - c r o s s i n g m a t c h e s i n the l o w r e s o l u t i o n images a r e u s e d to g u i d e the m a t c h i n g s i n t h e f i n e r r e s o l u t i o n s . The z e r o - c r o s s i n g m a t c h i n g o p e r a t i o n was o n l y p e r f o r m e d a l o n g t h e e p i p o l a r l i n e s . In t h i s c a s e , the e p i p o l a r l i n e s c o i n c i d e w i t h the h o r i z o n t a l s c a n l i n e s o f the s t e r e o i m a g e s . I n a more r e c e n t p a p e r , G r i m s o n [ 1 9 8 5 ] s u g g e s t e d the m a t c h i n g s h o u l d be p e r f o r m a t the f i n e s t r e s o l u t i o n and u s e t h e d i s p a r i t y s c a n at the 1 owe r r e s o l u t i o n s f o r d i s a m b i g u a t i o n of the p o s s i b l e m a t c h p o i n t s . C l a r k [1985] 2 p r o p o s e d a n o t h e r m a t c h i n g scheme on the mu 1 1 i - r e s o 1 u t i o n V G 6 f i l t e r e d i m a g e s . T h i s scheme r e l i e s on the d i s p a r i t y f u n c t i o n o f one r e s o l u t i o n l e v e l to p r o v i d e a d i s p a r i t y e s t i m a t e to h e l p the m a t c h i n g at the n e x t f i n e r r e s o l u t i o n l e v e l . A l l t h e s e s t e r e o p s i s m e t h o d s have had g o o d s u c c e s s f o r s c e n e s w i t h c o m p l i c a t e d f e a t u r e s , h o w e v e r , due to the l a c k , o f edge image f e a t u r e s , t h e y w i l l have p o o r p e r f o r m a n c e f o r s c e n e of s m o o t h f e a t u r e l e s s c u r v e d s u r f a c e s . T h e o t h e r g r o u p o f i m a g e - b a s e d r a n g e f i n d i n g t e c h n i q u e i s known as a c t i v e s t e r e o or s t r u c t u r e d l i g h t m e t h o d s . T h i s a p p r o a c h i s the b a s i s f o r the m a j o r i t y of the 3 - D m a c h i n e v i s i o n s y s t e m s i n use t o d a y [ C o r b y 1 9 8 3 ] . A s u r v e y a r t i c l e by J a r v i s [1983] p r o v i d e s a g o o d summary of the p a p e r s p u b l i s h e d on the v a r i o u s a s p e c t s on t h i s t o p i c . I n t h i s a p p r o a c h , s t r u c t u r e d l i g h t p a t t e r n s , s u c h as s t r i p e l i g h t i n g [ A g i n and B i n f o r d 1 9 7 3 , S a t o et a l . 1 9 8 2 ] , g r i d l i g h t i n g [ H a l l e t a l . 1 9 8 1 , Le M o i q u e and Wa xman 1984] and s p o t p a t t e r n [ A l t s c h u l e r et a l . 1979 , P o s d a m e r and A l t s c h u l e r 1982] a r e p r o j e c t e d o n t o the s c e n e . T h e images o f the l i g h t p a t t e r n s c a n be p r o c e s s e d to y i e l d r a n g e i n f o r m a t i o n . A g i n and B i n f o r d [1973] u s e d a l a s e r beam l i g h t s o u r c e and a c y l i n d r i c a l l e n s to p r o j e c t a s h e e t o f l i g h t o n t o t h e s c e n e . T h e l i g h t s t r i p e was t h e n s c a n n e d a c r o s s the s c e n e i n a s t e p - w i s e m a n n e r by a r o t a t i n g m i r r o r . I n s t e a d o f h a v i n g a s c a n n i n g l i g h t s o u r c e , S a t o [ 1 9 8 2 ] p l a c e d the o b j e c t on a t u r n t a b l e and r o t a t e d the o b j e c t w i t h r e s p e c t to two s t a t i o n a r y s l i t - r a y l a s e r p r o j e c t i o n s y s t e m s l o c a t e d on e i t h e r s i d e s of t h e c a m e r a . A t e a c h i n c r e m e n t a l p o s i t i o n , the p r o j e c t i o n s y s t e m w i l l t a k e t u r n s to i l l u m i n a t e the 7 o b j e c t . I f t h e r e i s more t h a n one l i g h t s t r i p e i n the s c e n e , i t wo u l d be d i f f i c u l t f o r the p r o c e s s i n g a l g o r i t hm t o i d e n t i f y the o r i g i n a t i n g l i g h t s o u r c e of the image f e a t u r e s . H a l l et a l . [ 1 9 8 1 ] p r o j e c t e d a 5x6 g r i d o n t o the s u r f a c e o f an o b j e c t and t h e n t o o k the images o f the g r i d p a t t e r n f r o m two d i f f e r e n t p o s i t i o n s . U s i n g an i n t e r a c t i v e image d i s p l a y p r o g r a m , the image c o o r d i n a t e s o f e a c h g r i d v e r t e x w e r e l o c a t e d a n d r e c o r d e d m a n u a l l y . F r o m t h e image c o o r d i n a t e s o f the m a t c h i n g v e r t i c e s , the s p a t i a l c o o r d i n a t e s of the s u r f a c e p o i n t s w e r e t r i a n g u l a t e d . T h i s m e t h o d i s v e r y t e d i o u s a n d t i m e c o n s u m i n g , and i s o n l y s u i t a b l e f o r s m a l l g r i d l a t t i c e s . I t i s n e c e s s a r y to m a t c h the o r i g i n a t i n g l i g h t s o u r c e s t o the image p o s i t i o n s o f the f e a t u r e p o i n t s i n o r d e r t o t r i a n g u l a t e the s p a t i a l c o o r d i n a t e s o f t h e p o i n t s . P o s d a m e r et a l . [ 1982 ] p r o p o s e d a v i s i o n s y s t e m w h i c h p r o j e c t s a 128x128 a r r a y of l a s e r beams o n t o a s u r f a c e and u s e d a b i n a r y s p a c e - e n c o d i n g p r o j e c t i o n scheme to i n c o r p o r a t e the a d d r e s s o f the p r o j e c t i o n s p o t s i n t o a s e r i e s o f s p o t p r o j e c t i o n p a t t e r n s . I t a l l o w s 1 6 , 3 8 4 s u r f a c e s a m p l e s t o be a c q u i r e d by u s i n g o n l y e i g h t p r o j e c t i o n p a t t e r n s . I n a d d i t i o n to the s p e e d i m p r o v e m e n t , t h e r e a r e no m o v i n g p a r t s r e q u i r e d i n t h i s m e t h o d , t h u s , the c o n t r o l c i r c u i t r y and i n a c c u r a c i e s a s s o c i a t e d w i t h t h e m e c h a n i c a l s c a n n i n g d e v i c e s c a n a l s o be e l i m i n a t e d . I n o k u c h i [1984] i m p r o v e d t h e s p a c e - e n c o d i n g scheme by u s i n g G r a y - c o d e d s p o t p r o j e c t i o n p a t t e r n s . A l t h o u g h i m p l e m e n t a t i o n m e t h o d s w e r e d i s c u s s e d i n the c i t e d r e f e r e n c e s , m e a s u r e m e n t r e s u l t s o f 8 the m e t h o d s h a v e y e t emerged f r o m the l i t e r a t u r e . Spot projection method I t was P o s d a m e r ' s p r o p o s a l t h a t i n s p i r e d the i d e a o f u s i n g a s p a c e - e n c o d e d s p o t p r o j e c t i o n scheme f o r the 3 - D v i s i o n s y s t e m d e v e l o p e d i n t h i s t h e s i s . I n o r d e r to k e e p the e q u i p m e n t c o s t o f the v i s i o n s y s t e m l o w , a s t a n d a r d s l i d e p r o j e c t o r i s u s e d to p r o d u c e the s p o t p r o j e c t i o n p a t t e r n s i n s t e a d of the c u s t o m - m a d e l a s e r p r o j e c t i o n s y s t e m as s u g g e s t e d by P o s d a m e r . S i n c e the s p o t p r o j e c t i o n r a y s c r e a t e d by the s l i d e p r o j e c t o r a r e m o r e d i s p e r s i v e t h a n the f i n e c o l l i m a t e d l a s e r b e a m s , a s m a l l e r s i z e a r r a y o f 32x32 s p o t s i s u s e d i n the s l i d e p r o j e c t o r i m p l e m e n t a t i o n . By u s i n g t h e b i n a r y s p a c i n g - e n c o d i n g s c h e m e , the 3 - D s p a t i a l c o o r d i n a t e s o f a l l the 1024 p r o j e c t i o n s p o t s on the m e a s u r e m e n t s u r f a c e c a n be o b t a i n e d i n s i x p r o j e c t i o n s . I n a f i e l d o f v i ew o f 20x20 i n c h e s , the s p a t i a l r e s o l u t i o n o f a 32x32 s p o t p a t t e r n i s f i v e - e i g h t h s of an i n c h . The s p o t p r o j e c t i o n s t r u c t u r e d l i g h t t e c h n i q u e was c h o s e n as the m o s t s u i t a b l e r a n g e f i n d i n g m e t h o d f o r s m o o t h , f e a t u r e l e s s c u r v e d s u r f a c e s . Ob j ect ive s T h e p r i m a r y o b j e c t i v e o f t h i s t h e s i s l o w c o s t 3 - D v i s i o n s y s t e m b a s e d u p o n t h e t e c h n i q u e to m e a s u r e the shape of s m o o t h , f e a i s to d e v e l o p a s p o t p r o j e c t i o n t u r e 1 e s s c u r v e d 9 s u r f a c e s . B e c a u s e of the c o s t f a c t o r , o n l y s t a n d a r d c o m m e r c i a l l y a v a i l a b l e e q u i p m e n t i s u s e d i n the i m p l e m e n t a t i o n of the s y s t e m . A c y l i n d r i c a l s u r f a c e i n the s h a p e o f an a i r c r a f t w i n g was s p e c i a l l y m a n u f a c t u r e d by the m a c h i n e s h o p i n the E l e c t r i c a l E n g i n e e r i n g D e p a r t m e n t as the t e s t m e a s u r e m e n t s u r f a c e of the v i s i o n s y s t e m . The d e v e l o p m e n t a l p r o c e s s o f the s y s t e m was d i v i d e d i n t o a number o f i n t e r m e d i a t e s t a g e s . The o b j e c t i v e s o f t h e s e i n t e r m e d i a t e s t a g e s w e r e l i s t e d as f o l l o w s : 1) to a s s e m b l e the o p t i c a l e q u i p m e n t r e q u i r e d f o r the s p o t p r o j e c t i o n s y s t e m ; 2) t o d e r i v e c a m e r a and s l i d e p r o j e c t o r m o d e l s ; 3) to d e t e r m i n e the m o d e l p a r a m e t e r s e x p e r i m e n t a l l y ; 4) t o d e v e l o p an image p r o c e s s i n g a l g o r i t h m f o r d e t e r m i n i n g the s p a t i a l c o o r d i n a t e s o f the p r o j e c t i o n s p o t on the m e a s u r e m e n t s u r f a c e ; 5) t o i n v e s t i g a t e the l i m i t i n g m e a s u r e m e n t c o n f i g u r a t i o n of the s p o t p r o j e c t i o n s y s t e m ; 6) to s t u d y the s u r f a c e r e c o n s t r u c t i o n a l g o r i t h m s f o r f i t t i n g a s u r f a c e t o the s c a t t e r e d s amp 1e s o b t a i n e d f r o m the s p o t p r o j e c t i o n p a t t e r n s . T h e d e s c r i p t i o n of t h e s e i n t e r m e d i a t e s t a g e s a r e p r e s e n t e d i n C h a p t e r s 2 -7 o f t h i s t h e s i s i n the o r d e r as g i v e n i n t h e l i s t a b o v e . C h a p t e r 8 s u m m a r i z e s t h e r e s u l t s o f t h e s t u d y and o f f e r s some r e c o m m e n d a t i o n s f o r f u t u r e i m p r o v e m e n t s . 10 C H A P T E R 2 IMAGING S Y S T E M 2 . 1 G E N E R A L D E S C R I P T I O N OF T H E IMAGING S Y S T E M A c t i v e s t e r e o i m a g i n g , or s t r u c t u r e d l i g h t p r o j e c t i o n , i s a w i d e l y a d o p t e d t e c h n i q u e u s e d i n i n d u s t r i a l v i s i o n s y s t e m s f o r t h r e e - d i m e n s i o n a l shape m e a s u r e m e n t s . H i g h l y i l l u m i n a t e d l i g h t p a t t e r n s a r e p r o j e c t e d o n t o the o b j e c t o f i n t e r e s t . The l i g h t r a y s or p l a n e s f r o m the p r o j e c t e d p a t t e r n s f o r m b r i g h t s p o t s or c u r v e s on the s u r f a c e of the o b j e c t . The t h r e e - d i m e n s i o n a l s h a p e o f the o b j e c t c a n t h e n be t r i a n g u l a t e d f r o m the image p o s i t i o n s o f t h e s e i l l u m i n a t e d f e a t u r e s i n the s c e n e . T h i s i s b a s i c a l l y the i m a g i n g a c q u i s i t i o n scheme u t i l i z e d i n the s p a c e - e n c o d e d s p o t i m a g i n g s y s t e m . F i g u r e 2 . 1 shows the h a r d w a r e c o n f i g u r a t i o n o f the s p a c e - e n c o d e d s p o t p r o j e c t i o n s y s t e m . The i m a g i n g s y s t e m i s c o m p r i s e d of an i n d u s t r i a l g r a d e L e i t z s l i d e p r o j e c t o r and a MTI 512x512 p i x e l r e s o l u t i o n v i d e o c a m e r a . T h e v i d e o camera i s c o n n e c t e d to an I m a g i n g T e c h n o l o g y I n c . image d i g i t i z e r and f r a m e b u f f e r u n i t a t t a c h e d to a L S I - 1 1 / 2 3 m i c r o c o m p u t e r u s i n g the R T - 1 1 o p e r a t i n g s y s t e m . The d i g i t i z e d d a t a a r e t h e n t r a n s f e r r e d to the V A X - 1 1 / 7 5 0 c o m p u t e r f a c i l i t y on f l o p p y d i s k e t t e s f o r f u r t h e r image a n a l y s i s . A R a m t e k 512x512 p i x e l r e s o l u t i o n c o l o u r d i s p l a y m o n i t o r i n t e r f a c e d to the V A X c o m p u t e r i s a v a i l a b l e f o r image v i e w i n g . H a r d FRAME GRABBER I MAG ING TECHNOLOGY CORP AP-512 PRINTRONIX 9300 RAMTEK DISPLAY LE I TZ SLIDE PROJECTOR I LSI 11/23 RX-02 FLOPPY DRIVES RX-02 FLOPPY DRIVES B/W MONI TOR 1 DL-01 HARD DISC DAGE-MT VIDEO CAMERA F i g . 2.1 B a s i c b l o c k d logrom of the space—encoded Imaging s y s t e m . 12 c o p y o u t p u t o f b i n a r y image d a t a and 3 - D w i r e - f r a m e p l o t s a r e a v a i l a b l e on the P r i n t r o n i x p r i n t e r / p l o t t e r . 2 . 2 S P A C E - E N C O D E D SPOT P R O J E C T I O N SCHEME A r e f e r e n c e p a t t e r n of 3 2 x 3 2 • r e g u l a r l y s p a c e d l i g h t s p o t s as shown i n F i g u r e 2 . 2 , i s p r o j e c t e d f r o m the s l i d e p r o j e c t o r o n t o an o b j e c t l o c a t e d at the c e n t e r of the s c e n e . B r i g h t i l l u m i n a t e d s p o t s a r e c r e a t e d at the i n t e r s e c t i o n s o f the p r o j e c t e d beams and the o b j e c t s u r f a c e . 8 . 8 9 mm A i IIIIIIIIBIII688IIII8IIIIIIIIIIII I . . • i i i i i i i B i i i i i i i i i i i i i i i t D i i m o. 11 mm IIIBIIIIIIIBIISIIIIIIIIIIIIIBIII 0.28mm 8 . 8 9 mm _?_ ;;;;:;;;;!•:!;;••;;:;;;:;;;;;;;! F f g . 2.2 Reference p r o j e c t i o n p a t t e r n . I n o r d e r t o t r i a n g u l a t e the 3 - D s p a t i a l c o o r d i n a t e s of the s u r f a c e p o i n t s , the s y s t e m m u s t be a b l e to i d e n t i f y b o t h the p o s i t i o n o f the o r i g i n a t i n g t r a n s p a r e n t w i n d o w s on the s l i d e and the c a m e r a image l o c a t i o n of the i l l u m i n a t e d s p o t s . A 13 b i n a r y s p a c e - e n c o d i n g scheme i s i m p l e m e n t e d to i d e n t i f y the p o s i t i o n o f the t r a n s p a r e n t w i n d o w s on the s l i d e . A s i n g l e p r o j e c t i o n o f the r e f e r e n c e mask d o e s n o t c o n t a i n s u f f i c i e n t i n f o r m a t i o n to i d e n t i f y the c o l u m n a d d r e s s e s of the l i g h t b e a m s . T h e b i n a r y a d d r e s s e n c o d i n g scheme u s i n g a s e q u e n c e of s p a c e - e n c o d e d s p o t p r o j e c t i o n p a t t e r n s , as shown i n F i g u r e 2 . 3 , has b e e n i m p l e m e n t e d f o r t h i s a p p l i c a t i o n . The p r o j e c t i o n p a t t e r n on the r e f e r e n c e s l i d e c o n s i s t s o f 32 rows and 32 c o l u m n s of r e g u l a r l y s p a c e d t r a n s p a r e n t w i n d o w s i n an o p a q u e b a c k g r o u n d . A s shown i n F i g u r e 2 . 4 , the p r o j e c t o r l e n s c e n t e r and e a c h c o l u m n o f t r a n s p a r e n t w i n d o w s on the s l i d e d e f i n e s a l i g h t p l a n e w h i c h p r o d u c e s a t r a i l o f s p o t s on the o b j e c t s u r f a c e . The o p t i c a l p a t h of e a c h r e c e i v e d image s p o t on the c a m e r a d e f i n e s a r a y - l i n e w h i c h c o n n e c t s the i l l u m i n a t e d s p o t on the o b j e c t s u r f a c e t o the c a m e r a c e n t e r . The s p a t i a l c o o r d i n a t e s o f the s u r f a c e p o i n t c a n be u n i q u e l y l o c a t e d by the t r i a n g u 1 a t i o n m e t h o d . D e t a i l e d d e r i v a t i o n o f the s p a t i a l c o o r d i n a t e s f r o m the image c o o r d i n a t e s w i l l be g i v e n i n the s u b s e q u e n t c h a p t e r s . F o r an NxN s p o t p r o j e c t i o n a r r a y , l o g j C N ) s p a c e -e n c o d e d p r o j e c t i o n s t o g e t h e r w i t h the r e f e r e n c e p a t t e r n a r e n e e d e d t o p e r f o r m the n e c e s s a r y c o l u m n a d d r e s s d e c o d i n g f u n c t i o n . F o r i n s t a n c e , a 32X32 s p o t p r o j e c t i o n a r r a y r e q u i r e s f i v e s p a c e - e n c o d e d p a t t e r n s i n a d d i t i o n to the r e f e r e n c e p a t t e r n to i m p l e m e n t the b i n a r y c o d i n g s c h e m e , wh e r e a s , a 128x128 p r o j e c t i o n a r r a y r e q u i r e s o n l y e i g h t p r o j e c t i o n s f o r the d e c o d i n g p r o c e s s . T h u s , a s i g n i f i c a n t • • • • • • • • • • • • • • • • • • • • > • • • • • • • • B B B B B B B a s B a a a B B B B a a B B a B a B B a a a • D I I I I I B I I I I I K B I I I I I I I I I i a i l tiiiiiiiiiaBtiiiiintiiiiBaii •BBBBfBIBBBBBBSBBBIIBBIBBBBBt BBBBBBBBBBBIBBBBBBBBBBBIBBBII B B B I l B B B B B B I B t e i t l l l t B B B B I B S B BBISBBBSffBBBBBBBBBBBBBBBBBBBB BBSBBBBBBBBBBBBBBBIBBBBBBBBBB BBillilBBIBIttlBBBBSBBIBBBBBB BBBBgiBBBBBBBBBBBBBBBBBBBBBBB BBBBBIBBIBBBSIBEBBBBBBBBBBBBB BBgiBSIlBBBIBBtBBBBSBBBBBBBBB BBIBIBBBBBBBBBBBBBBBBBBBBBBBB BBBBlBBBSBBtBBtBBBBBBBBBBBBBB IBIBBBBBBBBIIBBBBBBBBBBBBBBBB BBBBBBfiBBBBBBBBBBBBBBBBBBBBBB BBBBISBBBBBBBBBBBBBBBBBBBBBBB BBIBBIIIIBBBBBIBBtBBBBBBBBBBB BBBBBaBBEBBBBBBBBBBBBBBBBBBBB •BBaBBBBBaBBBBBBBBBBBBBBBBBBB B B B B B B B B B B I S B B a B B B B S I B B i B B B B B BBaBBaBBBBBaafiaBBBBBBBBBBaBfiB BBBEiaBBaBBiaBaBBBBBBBBBBBBBB BBIBBBBBBBBBBBBBBBBBBBBDBBBBB BBBIBBBBBBBBBBBBBBBBBBBBBBSBB BBBBBBBBBBBBBBBBBBBBBBBBBBBBB BBBBSBBBBBXBBtBBBBBBBBBBBBBB BB8BBBC9BBBBBBBBBBBBBBBBBBSBB BBSCIBBBBBBB1BBBBBB8BBBBBIBBB IBIBBBBBBBBEBBBBBBBBBBBBBBBBB BBSBEBBaBBBBBBBBBBBBBBBBBBBBB (o) (b) a a a a mm mm mm mm mm mm mm mm mm ( c ) Bl BB • B B B B B B B a a a a a i BB • B B B B B B B B B B B a i • a BBBB BBBB a a a a Bl BB a a a a BBBB a a a a Bl IB B B B B BBBB B B B B B l IB a a a a BBBB a a a a Bl BB B B B B BBBB a a a a Bl BB a a a a BBBB a a a a B l IB a a a a a a a a B B B B B l • B a a a a a a a a B B B B B l IB B B B B BBBB B B B B B l IB a a a a • a n t a a a a B l IB a a a a BBBB B B B B B l IB a a a a B B D B B B B B B l IB B B B B • B O B B B B B B l • a BBBB • B O B B B B B B l IB B B B B B B B B a a a a B l IB B B B B • BBB B B B B aa IB B B B B a a a a a a a a B l IB B B B B a a a a B B B B B l IB a a a a B B I B B a a a a aa IB a a a a • B U B B B B B aa IB BBBB B B B B B B B B aa IS a a a a B B B B B B B B aa IB a a a a BBBB B B B B B l IB a a a a B B B B B B B B aa IB B B B B BBBB B B B B aa IB B B B B a a a a a a a a aa IB B B B B B B B B B B B B aa IB B B B B B B B B B B B B aa • a a a a a a a a a B B B B BB IB • B B B a a a a a a a a (<0 B B B B B B B B B B B B B B BBBBBBB B B B B B B B B B B B B B B BBBBBBB BBBBBBB BBBBBBB B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B BBBBBBB BBBBBBB B B B B B B B B B B B B B B BBBBBBB BBBBBBB B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B (e) B B B B B B B B B B B B B B B B B B f l B B B B B B B B B B B B B B B E B B B B B B B B B B B B B BBBBBBBBBBBBBBB* IBBBBBBBBBBBBBBB •BBBBBBBBBBBBBBB BBBEBBBBBEBBBBBB BBBBBBBIBIEBBBBB BBEBBBBBBBBBBBBB BBBBBBBBBBBBBBBB B t i i n i B i a a a i B B B BBBBBBBBBBBBBBBt •BBBBBBBBBBBBBBB IBaiBBBBBBBBBBBI BBBBBBBBBSBBBBBI •BBBBBBBBBBBBBBB B B B I B B B B B B B B B B B a B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B •BBBBBBBBBBBBBBB BBBEBEBBBBBBaBBB B B B B B B B B B B B B B B B B BBBBBBBBBBBBBBBB • B B B B B B B B B B B B B B a BBBBBBBBBBBBBBBB B B B B B B B B B B B B B B B B BBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBB •BBBBBBBBBBBBBBB B a B a a a a a a a a a a B B a BBBBBBBBBBB ( f ) F t g . 2.3 (o) Reference spot p r o j e c t i o n p a t t e r n . (b) - ( f ) BInory space-encoded spot pa t te rns . 15 Projection spots Cy I i ndr i coI test surface y ~* Imoge plane mage spots Projector opt i caI ax i s Camera lens center Projector ens center Fig. 2.4 Optical geometry of the spot projection system, i n c r e a s e i n . t h e s u r f a c e p o i n t r e s o l u t i o n can be a c h i e v e d w i t h o n l y a s m a l l i n c r e a s e i n the number of p r o j e c t i o n s . However, - i n c r e a s i n g the p r o j e c t i o n s p o t d e n s i t y w i l l l e a d to a s m a l l e r spot s i z e and c l o s e r s p a c i n g between the s p o t s . C o n s e q u e n t l y , the p r o b a b i l i t y of an a d d r e s s d e c o d i n g e r r o r i n the image a n a l y s i s p r o c e s s w i l l be h i g h e r . The maximum s p o t d e n s i t y i s d e t e r m i n e d by two f a c t o r s : ( l ) the c a p a b i l i t y of the i m a g i n g s y s t e m to r e s o l v e a d j a c e n t p r o j e c t i o n s p o t s i n the camera image; and (2) the s o p h i s t i c a t i o n of the e r r o r c o r r e c t i o n scheme i n the a d d r e s s d e c o d i n g p r o c e s s . The r e s o l u t i o n and l i g h t s e n s i t i v i t y of the camera a l s o p l a y an i m p o r t a n t r o l e i n the o v e r a l l p e r f o r m a n c e of the measurement s y s t e m . 16 2 . 3 B I N A R Y S P A C E - E N C O D I N G SCHEME FOR T H E S P O T COLUMN A D D R R E S S E S The p u r p o s e of s p a c e - e n c o d i n g p r o j e c t i o n i s to d e t e r m i n e the c o l u m n a d d r e s s o f e a c h i l l u m i n a t e d s p o t g e n e r a t e d on the r e f e r e n c e p r o j e c t i o n . The a d d r e s s d e c o d i n g p r o c e s s c a n be a c c o m p l i s h e d by m o n i t o r i n g the p r e s e n c e or a b s e n c e o f the s p o t on e a c h of the s p a c e - e n c o d e d p r o j e c t i o n s . U s i n g a 32x32 s p o t m a t r i x p a t t e r n as an e x a m p l e , the a d d r e s s of e a c h c o l u m n c a n be u n i q u e l y r e p r e s e n t e d by a 5 - b i t b i n a r y n u m b e r , a ^ a ^ a ^ ^ a ^ , w h e r e a^ i s the m o s t s i g n i f i c a n t b i t and a^ i s the l e a s t s i g n i f i c a n t b i t . T h u s , the r a n g e o f the a d d r e s s w o u l d be f r o m (OOOOO)^ to ( 1 1 1 1 1 ) 2 . T h e f i r s t s p a c e - e n c o d e d s p o t p a t t e r n i s f o r m e d by t u r n i n g on o n l y t h o s e c o l u m n s w h i c h have a " o n e " i n the l e a s t s i g n i f i c a n t b i t o f t h e i r a d d r e s s . T h e s e c o n d p a t t e r n i s f o r m e d by t u r n i n g on the c o l u m n s h a v i n g a " o n e " i n a d d r e s s b i t . T h i s a r r a n g e m e n t c o n t i n u e s up to the f i f t h p a t t e r n , w h i c h i s f o r m e d by o n l y t h o s e c o l u m n s w i t h a " o n e " i n the m o s t s i g n i f i c a n t b i t o f t h e i r a d d r e s s . A p i c t u r e o f the r e f e r e n c e p r o j e c t i o n p a t t e r n and the s p a c e - e n c o d e d p a t t e r n s a r e d e p i c t e d i n F i g u r e 2 . 3 . By e x a m i n i n g the p r o j e c t i o n p a t t e r n s on the o b j e c t i n the o r d e r s p e c i f i e d a b o v e , the p r e s e n c e or a b s e n c e o f an i l l u m i n a t e d s p o t d e t e r m i n e s the v a l u e o f the a s s o c i a t e d a d d r e s s b i t f o r t h a t p r o j e c t i o n p a t t e r n . A " o n e " i s a s s i g n e d to t h e a d d r e s s b i t 17 i f the s p o t i s p r e s e n t and a " z e r o " i s a s s i g n e d i f the s p o t i s a b s e n t . H e n c e , the p r e s e n c e / a b s e n c e c o n d i t i o n of the i l l u m i n a t e d s p o t c a n be t r a n s l a t e d d i r e c t l y i n t o the b i n a r y code of the s p o t c o l u m n a d d r e s s . Slide Product ion The s l i d e s t h a t g e n e r a t e the e n c o d e d p r o j e c t i o n p a t t e r n s w e r e g e n e r a t e d u s i n g the I n t e r n a t i o n a l I m a g i n g S y s t e m M o d e l 75 image p r o c e s s i n g f a c i l i t i e s at D e f e n c e R e s e a r c h E s t a b l i s h m e n t P a c i f i c . T h e image d a t a o f t h e s e p a t t e r n s a r e d o w n l o a d e d i n t o the Q u i c k C o l o u r R e c o r d e r (QCR) made by IMAPRO I n c . The QCR i s a h i g h r e s o l u t i o n ( 2 0 4 8 x 2 0 4 8 p i x e l s ) c o l o u r d i s p l a y u n i t e q u i p p e d w i t h a 35mm c a m e r a to c a p t u r e the image on the d i s p l a y u n i t . S t a n d a r d K o d a c h r o m e K R - 1 3 5 c o l o u r f i l m s w e r e u s e d to p r o d u c e the p r o j e c t i o n s l i d e s f o r the s p a c e - e n c o d e d i m a g i n g s y s t e m . Slide Alignment A m a j o r l i m i t a t i o n of the s l i d e p r o j e c t i o n m e t h o d i s t h e r e q u i r e m e n t o f p r e c i s e r e g i s t r a t i o n among a l l the s l i d e s . P a t t e r n m i s a l i g n m e n t w i l l s h i f t the p o s i t i o n s of the p r o j e c t e d s p o t s o f the s p a c e - e n c o d e d p r o j e c t i o n s on the o b j e c t s u r f a c e r e l a t i v e to the s p o t s g e n e r a t e d by the r e f e r e n c e p r o j e c t i o n . A s a r e s u l t , t h e w r o n g a d d r e s s may be d e c o d e d due to i n c o r r e c t i d e n t i f i c a t i o n o f the p r e s e n c e / a b s e n c e c o n d i t i o n of the s p a c e - e n c o d e d p r o j e c t i o n 18 at the s p o t l o c a t i o n s o b t a i n e d f r o m the r e f e r e n c e p r o j e c t i o n . S l i d e f i l m s a r e u s u a l l y n o t a c c u r a t e l y a l i g n e d when t h e y a r e m o u n t e d o n t o c a r d b o a r d h o l d e r s d u r i n g the p h o t o f i n i s h i n g p r o c e s s . F u r t h e r m o r e , the p r o b l e m i s c o m p o u n d e d by the s l o p p i n e s s of the s l i d e p o s i t i o n i n s i d e the s l o t o f the p r o j e c t o r . T h u s , m a n u a l a l i g n m e n t p r o c e d u r e s a r e u s u a l l y r e q u i r e d b e f o r e e a c h p r o j e c t i o n to a l i g n the t o p row o f t h e p r o j e c t e d s p o t w i t h a r e f e r e n c e m a r k e r i n the s c e n e . T h e f e a s i b i l i t y of i m p l e m e n t i n g an a u t o m a t i c a l i g n m e n t c o r r e c t i o n a l g o r i t h m was i n v e s t i g a t e d . I t was t h o u g h t t h a t t h e image o f f s e t o f the p r o j e c t i o n s p o t s due to the m i s a l i g n m e n t c o u l d be c o m p e n s a t e d by k n o w i n g the image o f f s e t o f a m a r k e r i n e a c h o f the s p a c e - e n c o d e d p r o j e c t i o n p a t t e r n s r e l a t i v e to a m a r k e r i n the r e f e r e n c e p r o j e c t i o n . S u b s e q u e n t l y , the o f f s e t of the r e s t o f the p r o j e c t i o n s p o t s c o u l d be c o m p u t e d f r o m the i n i t i a l o f f s e t . T h i s i d e a was f o u n d n o t to be f e a s i b l e b e c a u s e the amount of image o f f s e t e x p e r i e n c e d at the p r o j e c t i o n s p o t s i s g e n e r a l l y n o t a s i m p l e f u n c t i o n o f the i n i t i a l o f f s e t . I n f a c t , the amount o f o f f s e t i n t h e image p o s i t i o n i s a f u n c t i o n o f b o t h the s l i d e m i s a l i g n m e n t and the h e i g h t o f the p r o j e c t e d s p o t . S i n c e t h e h e i g h t of the p r o j e c t e d s p o t s v a r i e s w i t h the s h a p e o f the o b j e c t , i t w o u l d be d i f f i c u l t to e s t i m a t e the o f f s e t o f the image p o s i t i o n w i t h o u t k n o w i n g the z -c o o r d i n a t e o f t h e o b j e c t p o i n t . M a n u a l a l i g n m e n t p r o c e d u r e s w h i c h o n l y t a k e a f e w m i n u t e s to c o m p l e t e a p p e a r to be the 19 p r e s e n t s o l u t i o n o f t h i s p r o b l e m . T h i s s e t - u p p r o c e d u r e c a n be e l i m i n a t e d by u s i n g an e l e c t r o - o p t i c a l l i g h t s h u t t e r d e v i c e made o f n e m a t i c l i q u i d c r y s t a l . W i t h a L C D l i g h t s h u t t e r i n s t a l l e d i n the p r o j e c t o r , the b r i g h t n e s s of e a c h c o l u m n o f the s p o t p a t t e r n c a n be c o n t r o l l e d e l e c t r o n i c a l l y . The L C D s h u t t e r i m p l e m e n t a t i o n i s l e f t as a f u t u r e i m p r o v e m e n t o f the s p o t p r o j e c t i o n s y s t e m . 2 . 4 3 - D MEASUREMENT OF T H E C Y L I N D R I C A L T E S T S U R F A C E 2 . 4 . 1 O P T I C A L C O N F I G U R A T I O N OF T H E P R O J E C T I O N S Y S T E M Optical Equipment I n o r d e r t o k e e p the e q u i p m e n t c o s t o f the i m a g i n g s y s t e m l o w , t h e s y s t e m c o m p r i s e s o n l y o f e x i s t i n g o p t i c a l e q u i p m e n t w h i c h was r e a d i l y a v a i l a b l e at the E l e c t r i c a l E n g i n e e r i n g D e p a r t m e n t . T h e D a g e - M T I m o d e l 65 c a m e r a , e q u i p p e d w i t h a v i d i c o n s e n s o r and a 1 7 . 5 - 1 2 0 mm zoom l e n s , was u s e d to c a p t u r e the images o f the s p o t p r o j e c t i o n p a t t e r n s . T h e v i d e o s i g n a l f r o m the c a m e r a was d i g i t i z e d by the I m a g i n g T e c h n o l o g y C o r p o r a t i o n I P - 5 1 2 image p r o c e s s i n g u n i t t o f o r m a d i g i t a l image w i t h 512x480 p i x e l r e s o l u t i o n . T h e d i g i t i z e d g r e y - s c a l e i n t e n s i t i e s o f the p i x e l s have 8-b i t r e s o l u t i o n , o r an i n t e n s i t y r a n g e b e t w e e n 0 and 2 5 5 . The L e i t z s l i d e p r o j e c t o r a v a i l a b l e f o r the s p o t p r o j e c t i o n i s e q u i p p e d w i t h a 90 mm l e n s and a 250 w a t t p r o j e c t i o n l a m p . T h e p o w e r o f t h e p r o j e c t i o n lamp d e f i n e s the b r i g h t n e s s o f 2 0 the p r o j e c t i o n s p o t s , and the d e f i n e s the s i z e o f the s p o t d i s t a n c e . Lighting condition T h e s p o t p r o j e c t i o n i m a g i n g s y s t e m i s o p e r a t e d i n a d a r k , c o n t r o l l e d l i g h t i n g e n v i r o n m e n t w h e r e the a m b i e n t l i g h t i n t e n s i t y i s r e d u c e d to a m i n i m u m by e l i m i n a t i n g a l l the s u r r o u n d i n g l i g h t s o u r c e s . The a m b i e n t l i g h t i n t e n s i t y i s a t a l e v e l s u c h t h a t t h e o u t l i n e s o f the c y l i n d r i c a l t e s t s u r f a c e w e r e b a r e l y v i s i b l e w i t h the n a k e d e y e . The s p e c i a l l i g h t i n g c o n d i t i o n d e s c r i b e d a b o v e i s r e f e r r e d t o i n t h e t h e s i s as the l o w c o n t r o l l e d l i g h t i n g c o n d i t i o n o f the s p o t p r o j e c t i o n s y s t e m . I n the a b s e n c e o f the s p o t p r o j e c t i o n p a t t e r n , t h e r a n g e o f the a m b i e n t r e f l e c t i o n i n t e n s i t y v a l u e s o b t a i n e d by the D a g e - M T I c a m e r a e q u i p p e d w i t h a v i d i c o n s e n s o r was b e t w e e n 6 and 8 . W i t h t h e s p o t p a t t e r n p r o j e c t i o n on the t e s t s u r f a c e , the image i n t e n s i t y r a n g e o f the p r o j e c t i o n s p o t s wa s be t we en 5 5 a n d 9 5 . B e c a u s e o f the d i f f u s i v e r e f l e c t a n c e c h a r a c t e r i s t i c s o f the s u r f a c e , the a v e r a g e i n t e n s i t y v a l u e o f the n o n - i 1 l u m i n a t e d b a c k g r o u n d a r e a i n b e t w e e n the p r o j e c t i o n s p o t s was 2 2 . F u l l d e t a i l s o f t h e r e f l e c t a n c e c h a r a c t e r i s t i c s o f the c y l i n d r i c a l s u r f a c e w i l l be g i v e n i n C h a p t e r 6 . f o c a l l e n g t h of the l e n s p a t t e r n at a g i v e n wo r k i n g 21 Optical Geometry A c y l i n d r i c a l l y s h a p e d t e s t s u r f a c e wa s ma n u f a c t u r e d by t h e m a c h i n e s h o p a c c o r d i n g to the p h y s i c a l d i m e n s i o n as s p e c i f i e d i n F i g u r e 2 . 5 . I n the shape m e a s u r e m e n t p r o c e s s , the c y l i n d r i c a l s u r f a c e wa s p l a c e i n f r o n t o f a b a c k g r o u n d p l a n e l o c a t e d on the x - y p l a n e of the o b j e c t - c e n t e r e d r e f e r e n c e c o o r d i n a t e s y s t e m . I n o r d e r to p r o d u c e a s p o t 15 .5 " *1 h*~ 2 .8 " 12" 12" R a d i u s F i g . 2 . 5 P h y s i c a l d imens ions of the c y l i n d r i c a l t e s t s u r f a c e . p r o j e c t i o n p a t t e r n a b o u t two i n c h e s l a r g e r t h a n the c y l i n d r i c a l s u r f a c e , the w o r k i n g d i s t a n c e o f the L e i t z s l i d e p r o j e c t o r was a b o u t 21 f e e t f r o m the s u r f a c e . T h e a r e a on t h e b a c k g r o u n d p l a n e c o v e r e d by the s p o t p r o j e c t i o n p a t t e r n i s a b o u t 18x18 i n c h e s . H e n c e , the 32x32 s p o t p a t t e r n 22 p r o v i d e s a s p a t i a l r e s o l u t i o n of f i v e - e i g h t h s o f an i n c h . S a t o et a l . [ 1 9 8 2 ] f o u n d t h a t the m e a s u r e m e n t e r r o r o f a s l i t - r a y p r o j e c t i o n s y s t e m due to the q u a n t i z a t i o n r e s o l u t i o n o f the c a m e r a image p l a n e i s i n v e r s e l y p r o p o r t i o n a l to s i n ^ S , w h e r e /S i s the a n g l e b e t w e e n the i n c i d e n t l i g h t and t h e o p t i c a l a x i s o f the c a m e r a . S i n c e the o p t i c a l g e o m e t r y o f the s p o t p r o j e c t i o n s y s t e m i s i d e n t i c a l t o the s l i t - r a y p r o j e c t i o n s y s t e m d e v e l o p e d by S a t o , the e r r o r due to the q u a n t i z a t i o n r e s o l u t i o n o f the c a m e r a i n the s p o t p r o j e c t i o n s y s t e m c a n be w r i t t e n as [ S a t o et a l . 1982] : A r = Ae ( 2 . 1 ) f s i n ^ wh e r e L i s the wo r k i n g d i s t a n c e of the c ame r a , f i s the f o c a l l e n g t h o f the c a m e r a l e n s , and Ae i s the p i x e l s p a c i n g o f the c a m e r a . O b v i o u s l y , the q u a n t i z a t i o n e r r o r d e c r e a s e s as & a p p r o a c h e s 9 0 ° , h o w e v e r , i t a l s o i n c r e a s e s the a r e a of t h e s p o t p r o j e c t i o n p a t t e r n to be o c c l u d e d f r o m the v i s i b i l i t y o f the c a m e r a . A n a n g l e o f a p p r o x i m a t e l y 4 0 ° was f o u n d t o be the maximum a n g l e b e t w e e n t h e o p t i c a l axes o f t h e c a m e r a and s l i d e p r o j e c t o r , s u c h t h a t the e n t i r e s p o t p r o j e c t i o n p a t t e r n was s t i l l v i s i b l e by the c a m e r a . T h e w o r k i n g d i s t a n c e and the f o c a l l e n g t h o f the c a m e r a w e r e a d j u s t e d to h a v e the w h o l e c y l i n d r i c a l s u r f a c e j u s t i n c l u d e d i n t h e f i e l d o f v i e w of the c a m e r a a n d i t was m e a s u r e d to be 20x20 i n c h e s . The w o r k i n g d i s t a n c e of the 23 c a m e r a and the f o c a l l e n g t h of the l e n s w e r e m e a s u r e d to be 22 f e e t and 110 mm, r e s p e c t i v e l y . The p i x e l s p a c i n g of the v i d i c o n s e n s o r o f the camera was f o u n d to be a b o u t 0 . 0 0 1 i n c h . By s u b s t i t u t i n g t h e s e o p t i c a l p a r a m e t e r s i n t o E q u a t i o n ( 2 . 1 ) , t h e q u a n t i z a t i o n e r r o r of the c a m e r a s y s t e m was c a l c u l a t e d to be 0 . 0 9 i n c h e s . W i t h o u t s p e c i a l m e a s u r e m e n t d e v i c e s , the o r i e n t a t i o n a n g l e s o f the c a m e r a and s l i d e p r o j e c t o r w e r e n o t a v a i l a b l e . H o w e v e r , the c o a r s e p o s i t i o n m e a s u r e m e n t s o f t h e camera and p r o j e c t o r w i t h r e s p e c t to the r e f e r e n c e c o o r d i n a t e s y s t e m w e r e o b t a i n e d and a r e g i v e n as ( 0 . 1 f t , 9 f t , 2 0 f t ) and ( 0 . 5 f t , -5 f t , 2 1 f t ) , r e s p e c t i v e l y . 2 . 4 . 2 EXPER IMENTAL PROCEDURES T h e c y l i n d r i c a l t e s t s u r f a c e was p l a c e d i n f r o n t of a w h i t e b a c k g r o u n d p l a n e l o c a t e d on the x - y p l a n e of the r e f e r e n c e c o o r d i n a t e s y s t e m , as shown e a r l i e r i n F i g u r e 2 . 4 . The b o t t o m r i g h t c o r n e r o f the t e s t s u r f a c e was l o c a t e d at the p o i n t ( 0 . 0 , 0 . 8 7 , 0 . 0 ) w i t h r e s p e c t to the r e f e r e n c e c o o r d i n a t e s y s t e m . G r e y - s c a l e images o f t h e c y l i n d r i c a l s u r f a c e w i t h t h e r e f e r e n c e s p o t p r o j e c t i o n p a t t e r n and the f i v e s p a c e - e n c o d e d p r o j e c t i o n p a t t e r n s w e r e a c q u i r e d u n d e r the o p t i c a l c o n f i g u r a t i o n as s p e c i f i e d i n S e c t i o n 2 . 4 . 1 . T h i s s e t o f g r e y - s c a l e images was u s e d i n the d e v e l o p m e n t o f the image p r o c e s s i n g a l g o r i t h m and the r e f l e c t a n c e m o d e l of the c y l i n d r i c a l t e s t s u r f a c e to be d i s c u s s e d l a t e r i n t h i s t h e s i s . Due to t h e m i s a l i g n m e n t p r o b l e m o f t h e p r o j e c t i o n 24 s l i d e s , m a n u a l a d j u s t m e n t was p e r f o r m e d b e f o r e e a c h s p a c e -e n c o d e d s p o t p r o j e c t i o n to a l i g n the t o p r ow of the p r o j e c t e d s p o t w i t h the r e f e r e n c e m a r k e r on the b a c k g r o u n d p l a n e . T h e s e q u e n c e of g r e y - s c a l e i m a g e s w i l l be a n a l y z e d by the a l g o r i ' t h m s to be p r e s e n t e d i n t h i s t h e s i s f o r d e t e r m i n i n g the 3 - D shape of the c y l i n d r i c a l t e s t s u r f a c e . In o r d e r to r e l a t e the o b j e c t p o i n t s i n the 3 - D s p a c e to t h e i r c o r r e s p o n d i n g 2 - D image p o s i t i o n s , the c a m e r a and s l i d e p r o j e c t o r m o d e l s a r e r e q u i r e d t o e s t a b l i s h t h e c o n n e c t i o n s b e t w e e n the c o r r e s p o n d i n g o b j e c t and image p o i n t s . F u l l d e t a i l s of the m o d e l s and t h e i r c a l i b r a t i o n p r o c e d u r e s w i l l be p r e s e n t e d i n C h a p t e r s 3 and 4 . 25 CHAPTER 3 CAMERA A N D PROJECTOR MODELS 3.1 I N T R O D U C T I O N H o m o g e n e o u s c o o r d i n a t e s and t r a n s f o r m a t i o n a r e w i d e l y u s e d i n r o b o t i c s and c o m p u t e r g r a p h i c s a p p l i c a t i o n s to d e s c r i b e p o s i t i o n c o o r d i n a t e s of o b j e c t p o i n t s w i t h r e s p e c t to v a r i o u s c o o r d i n a t e r e f e r e n c e s y s t e m s [ P a u l 1 9 8 1 , F o l e y and V a n Dam 1 9 8 3 ] . The c a m e r a and s l i d e p r o j e c t o r t r a n s f o r m a t i o n m a t r i c e s d e s c r i b e the s p a t i a l r e l a t i o n s h i p o f the s u r f a c e p o i n t s i n an o b j e c t - c e n t e r e d c o o r d i n a t e s y s t e m w i t h r e s p e c t t o t h e c o r r e s p o n d i n g p o i n t s i n the image p l a n e of the c a m e r a , and a l s o the c o r r e s p o n d i n g t r a n s p a r e n t w i n d o w s on the p r o j e c t i o n s l i d e [ H a l l et a l . 1 9 8 1 , Duda and H a r t 1 9 7 3 ] . C a m e r a and s l i d e p r o j e c t o r t r a n s f o r m a t i o n m a t r i c e s h a v e b e e n u s e d i n n u m e r o u s c o m p u t e r v i s i o n a p p l i c a t i o n s , s u c h as s t e r e o r e c o n s t r u c t i o n [Hwa ng 1 9 8 0 ] , 3 - D s h a p e m e a s u r e m e n t s [ H a l l and M c P h e r s o n 1 9 8 3 ] , and p h o t o g r a m m e t r y [ G h o s h 1 9 7 9 ] . I f the l o c a t i o n s and o r i e n t a t i o n s o f the c a m e r a and s l i d e p r o j e c t o r w i t h r e s p e c t to an o b j e c t c e n t e r e d c o o r d i n a t e s y s t e m a r e k n o w n , the t r a n s f o r m a t i o n m a t r i c e s c a n t h e n be c o m p u t e d a n a l y t i c a l l y [ B a l l a r d and B r o w n 19 8 2 ] . H o w e v e r , s p e c i a l c a l i b r a t i o n d e v i c e s are r e q u i r e d t o o b t a i n a c c u r a t e r e s u l t s f o r t h e s e m e a s u r e m e n t s . 26 W i t h o u t s u c h d e v i c e s , the c a m e r a m o d e l c a n a l s o be d e r i v e d f r o m the s p a t i a l c o o r d i n a t e s o f a s e t o f known o b j e c t p o i n t s and t h e i r c o r r e s p o n d i n g image c o o r d i n a t e s [Hwang 1 9 8 0 ] . S i m i l a r l y , t h e s l i d e p r o j e c t o r m o d e l c a n be d e r i v e d f r o m the s p a t i a l c o o r d i n a t e s of a s e t o f known p r o j e c t i o n s p o t s and t h e i r c o r r e s p o n d i n g s p o t a d d r e s s e s i n the p r o j e c t i o n p a t t e r n [ P o s d a m e r and A l t s c h u l e r 1 9 8 2 ] . 3 . 2 R E V I E W O F HOMOGENEOUS TRANSFORMATIONS The c a m e r a m o d e l d e f i n e s t h e s p a t i a l t r a n s f o r m a t i o n of an o b j e c t p o i n t i n t h e 3 - D s p a c e o n t o a 2 - D p o i n t on the c a m e r a i m a g e . T h e p r o j e c t o r m o d e l d e f i n e s the l i g h t p r o j e c t i o n s t h r o u g h the t r a n s p a r e n t w i n d o w s on the s l i d e s o n t o i l l u m i n a t e d s p o t s on the o b j e c t s u r f a c e . T h e s e o b j e c t -image s p a t i a l r e l a t i o n s h i p s c a n be d e f i n e d by a h o m o g e n e o u s t r a n s f o r m a t i o n known as the p e r s p e c t i v e p r o j e c t i o n t r a n s f o r m a t i o n . F a m i l i a r i t y w i t h h o m o g e n e o u s c o o r d i n a t e s i s n e c e s s a r y to u n d e r s t a n d the d e r i v a t i o n of the t r a n s f o r m a t i o n m a t r i c e s f o r t h e c a m e r a and p r o j e c t o r m o d e l s as e x p l a i n e d i n s u b s e q u e n t s e c t i o n s o f t h i s c h a p t e r . F o r an i m a g i n g p r o c e s s as shown i n F i g u r e 3 . 1 , a p o i n t p i n t h e 3 - D C a r t e s i a n s p a c e w i t h c o o r d i n a t e s (x , y , z ) c a n be r e p r e s e n t e d i n t h e 4 - d i m e n s i o n a 1 o J o o r h o m o g e n e o u s c o o r d i n a t e s y s t e m by a l i n e w i t h c o o r d i n a t e s (x ,y ,z , 1 ) . S i m i l a r l y , an image p o i n t q l o c a t e d at (x , y , z ) i n the c a m e r a image c o o r d i n a t e s y s t e m c a n be c c c d e f i n e d i n h o m o g e n e o u s c o o r d i n a t e s as (x s , y s , z s , s ) , w h e r e c c c 27 s i s a l s o a non-zero a r b i t r a r y c o n s t a n t . For a g i v e n 4x4 t r a n s f o r m a t i o n m a t r i x A of the imaging p r o c e s s , the p o s i t i o n v e c t o r Q of the image p o i n t i s the r e s u l t of a p p l y i n g the m a t r i x A to the p o s i t i o n v e c t o r P. The image v e c t o r Q i s g i v e n by A P ( 3 . 1 ) o r xs ys z s (3 The homogeneous t o o l i n computer v i s i o n t r a n s f o r m a t i o n m a t r i x i s and r o b o t i c s a p p l i c a t i o n s . a u s e f u l The 4x4 28 h o m o g e n e o u s m a t r i x c a n r e p r e s e n t a n y t r a n s l a t i o n , r o t a t i o n or p e r s p e c t i v e t r a n s f o r m a t i o n s i n d i v i d u a l l y or i n any c o m b i n a t i o n . F o r e x a m p l e , a new c o o r d i n a t e s y s t e m i s f o r m e d by t r a n s l a t i o n o f a d i s p l a c e m e n t v e c t o r ( x { . y ^ , z ) i n the C a r t e s i a n s y s t e m , f o l l o w e d by r o t a t i o n of 8 d e g r e e s a b o u t the x - a x i s . T h e t r a n s l a t i o n t r a n s f o r m a t i o n m a t r i x A^ , and the r o t a t i o n m a t r i x A- a r e g i v e n as * T = wh e r e c8 = s6 = 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 c o s 8 s i n 8 0 c8 s8 0 ( 3 . 3 ) 0 0 - s 8 0 c6 0 0 1 ( 3 . 4 ) T h e r e s u l t a n t t r a n s f o r m a t i o n o f the t r a n s l a t i o n and r o t a t i o n c a n be c omb i n e d i n t o a s i n g l e ma t r i x o p e r a t i o n •ART — "^R * ^ T ( 3 . 5 ) T h e m a t r i x o f the c o m b i n e d o p e r a t i o n i s s i m p l y the p r o d u c t o f t h e two m a t r i c e s . T h i s i s one o f the u s e f u l p r o p e r t i e s o f h o m o g e n e o u s t r a n s f o r m a t i o n s a n d i s u s e d r e p e a t e d l y i n t h e d e r i v a t i o n of t h e c a m e r a and p r o j e c t o r mo d e l s . 29 3.3 D E S C R I P T I O N O F T H E CAMERA MODEL I t i s o f t e n i n c o n v e n i e n t to i d e n t i f y an o b j e c t p o i n t by i t s c o o r d i n a t e s w i t h r e s p e c t to the camera r e f e r e n c e system. Thus, one would l i k e to d e s c r i b e the p o s i t i o n of the o b j e c t p o i n t w i t h r e s p e c t to an o b j e c t - c e n t e r e d r e f e r e n c e frame F a d j a c e n t to the work p i e c e , as shown i n F i g u r e 3.2. F i g 3.2 Spo t l o l r e l a t i o n s h i p between the camera and the ob jec t - cen te red re fe rence frame, Fo. The new g l o b a l r e f e r e n c e frame i s d e f i n e d by the f o l l o w i n g -s e r i e s of t r a n s f o r m a t i o n s : l ) t r a n s l a t i o n of the o r i g i n f r om O to l o c a t i o n O ; 2) r o t a t i o n of the x-y plane about the c o z - a x i s by an angle 6, the pan a n g l e ; 3) r o t a t i o n of the new x-z p l a n e about the new x a x i s by an angle 0, the t i l t 30 a n g l e ; a n d 4) r o t a t i o n o f the new x - z p l a n e a r o u n d the new y a x i s by an a n g l e ^ , the s w i n g a n g l e . The s i g n o f the r o t a t i o n a n g l e s f o i l ows the r i g h t - h a n d e d r o t a t i o n c o n v e n t i o n . F i g u r e 3 . 3 shows a s t e p by s t e p t r a n s f o r m a t i o n o f t h e c o o r d i n a t e f r a m e s f r o m the c a m e r a to the g l o b a l r e f e r e n c e s y s t e m . Fig. 3.3 Step by step tronsformotion from the object-centered reference frame Fo to the camera reference frame Fc. T h e t r a n s l a t i o n t r a n s f o r m a t i o n m a t r i x A j . i s g I v e n as 1 0 0 0 0 1 0 0 0 0 1 0 ( 3 . 6 ) w h e r e ( x T > y T , z T ) i s the l o c a t i o n of the o r i g i n o f the 31 g l o b a l r e f e r e n c e frame w i t h r e s p e c t to the camera system. The combined r o t a t i o n m a t r i x i s determined by the pan, t i l t and swing o r i e n t a t i o n of the camera w i t h r e s p e c t to the g l o b a l r e f e r e n c e frame. The combined r o t a t i o n t r a n s f o r m a t i o n ma t r i x i s - RotCy.^O R o t(x,0) R o t(z,9) (3.7) whe re Rot(z.e) = R o t(x,0) R o t ( y . f ) = C 0 s 9 s i n 9 0 0 s i n 9 c o s 9 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 C 0 s <f> s i n<£ 0 0 - s i n$ c 0 s 0 0 0 0 0 1 CO s^ 0 - s i nf 0 0 1 0 0 s i n f 0 c o s f 0 0 0 0 1 (3.8) (3.9) (3.10) and 9,0, and f are the pan, t i l t and swing a n g l e s of the camera, r e s p e c t i v e l y . As shown i n F i g u r e 3.4, the o r i g i n O of the camera c r e f e r e n c e system F i s l o c a t e d at the c e n t e r of the camera c image p l a n e , and the y - a x i s i s p a r a l l e l to the o p t i c a l a x i s of the camera. The l e n s c e n t e r of the camera system i s p o s i t i o n e d at the p o i n t ( 0 , - f , 0 ) , where f i s the f o c a l l e n g t h of the camera l e n s . For an o b j e c t p o i n t l o c a t e d at ( x ^ . y ^ . z ^ ) w i t h r e s p e c t to the r o t a t e d camera c o o r d i n a t e s y s t em F , the c o o r d i n a t e s of i t s i ma ge p o i n t ( x . , y . , z . ) can C 1 1 1 be computed from the o p t i c a l geometry of the system. The 32 image c o o r d i n a t e s a r e g i v e n a s : x . •= 1 ( 3 . 1 1 ) y , o ( 3 . 1 2 ) ( 3 . 1 3 ) Comero lens center (Xj.O.Z.) OBJECT POINT (X 4 ,Y 4 ,Z 4 ) Fig. 3.4 CAMERA IMAGE PLANE Spotiol relationship between the Imoge plane of the camera and the camera — t * reference frame F„ . T h e image c o o r d i n a t e s , x . and z ^ , i n the a b o v e d i s c u s s i o n a r e the p h y s i c a l d i s t a n c e s o f t h e image p o i n t f r o m t h e c e n t e r o f t h e camera image p l a n e i n t h e v e r t i c a l and h o r i z o n t a l d i r e c t i o n s . U s u a l l y , the p o s i t i o n o f image p o i n t s a r e s p e c i f i e d i n p i x e l u n i t s , m e a s u r e d i n the h o r i z o n t a l and v e r t i c a l d i r e c t i o n s f r o m a c o r n e r o f a 33 p i c t u r e , s u c h as the u p p e r l e f t c o r n e r . The image c o o r d i n a t e s y s t e m w i t h the o r i g i n l o c a t e d at the u p p e r l e f t c o r n e r o f the image p l a n e i s shown i n F i g u r e 3 . 5 . The s p a t i a l c o o r d i n a t e s ( u , v ) i n p i x e l u n i t s , w i t h r e s p e c t to the new F i g . 3.5 Spa t i a l r e l a t i o n s h i p between the image plane coord ina te system F . and the camero re ference system F . s y s t e m F^ c a n t h e n be e x p r e s s e d i n t e r m s o f the c o o r d i n a t e s ( x . , 0 , z . ) o f t h e o r i g i n a l camera r e f e r e n c e f r a m e F . The new i i c image c o o r d i n a t e s , u and v a r e g i v e n b y : F CENTER OF THE IMAGE PLANE c z . i ( 3 . 1 4 ) u u k u 0 u 0 f 34 k ( 3 . 1 5 ) u J 4 where k and k are the h o r i z o n t a l and v e r t i c a l s p a c i n g s u V r b between the image p i x e l s , and u and v are the h o r i z o n t a l 6 • o o and v e r t i c a l d i s p l a c e m e n t s from the top l e f t hand co r n e r of the image c e n t e r , i n p i x e l u n i t s . L e t = - f / k u > and = - f / k , then the t r a n s f o r m a t i o n f rom the camera c o o r d i n a t e v system to the new image c o o r d i n a t e system can be r e p r e s e n t e d by the p e r s p e c t i v e t r a n s f o r m a t i o n m a t r i x A^ ,: ut v t t ( 3 . 1 6 ) whe r e *C 0 u 0 0 1 V 0 0 2 0 0 1 0 0 ( 3 . 1 7 ) For cameras w i t h unequal h o r i z o n t a l and v e r t i c a l p i x e l s p a c i n g or n o n - u n i t y aspect r a t i o , the m a t r i x A^ , has c o n v e n i e n t l y t a k e n t h i s f a c t o r i n t o account by p e r m i t t i n g unequal v a l u e s f o r k u and k v . C o n s e q u e n t l y , no s p e c i a l c a l i b r a t i o n p r o c e d u r e s are n e c e s s a r y to compensate f o r t h i s p r o p e r t y i n such c ame r a s . F i n a l l y , a l l the t r a n s f o r m a t i o n m a t r i c e s can be 35 c omb i n e d t o g e t h e r to d e s c r i b e the p e r s p e c t i v e p r o j e c t i o n of the p o s i t i o n v e c t o r P i n the 3 - D s p a c e o n t o an image v e c t o r Q on t h e 2 - D image p l a n e . The c o m b i n e d t r a n s f o r m a t i o n m a t r i x T i s g i v e n as the p r o d u c t o f a l l the i n d i v i d u a l c am 6 r t r a n s f o r m a t i o n m a t r i c e s : c am ^ V j \ a U a 1 2 a l 3 a 1 4 a 2 1 a 2 2 a 2 3 a 2 4 a 3 1 a 3 2 a 3 3 a 3 4 ( 3 . 1 8 ) w h e r e A ^ , , A ^ , and A^, a r e the p e r s p e c t i v e , r o t a t i o n and t r a n s l a t i o n t r a n s f o r m a t i o n m a t r i c e s , r e s p e c t i v e l y . S i n c e A ^ A p i s a 4x4 m a t r i x and A^. i s a 3x4 m a t r i x , h e n c e the p r o d u c t o f A ^ A j ^ A p becomes a 3x4 m a t r i x as g i v e n i n E q u a t i o n ( 3 . 1 8 ) . H e n c e , t h e r e l a t i o n s h i p b e t w e e n an o b j e c t p o i n t P w i t h r e s p e c t to the g l o b a l r e f e r e n c e s y s t e m and i t s c o r r e s p o n d i n g image p o i n t w i t h r e s p e c t to the image c o o r d i n a t e s y s t e m i s g i v e n b y : T P c am ( 3 . 1 9 ) whe r e Q = t u t v ( 3 . 2 0 ) 36 t - an a r b i t r a r y s c a l i n g f a c t o r , and ( 3 . 2 1 ) 3.4 D E S C R I P T I O N O F T H E P R O J E C T O R MODEL A c a m e r a s a m p l e s the l i g h t e n e r g y r e f l e c t e d f r o m a s u r f a c e to p r o d u c e an i m a g e . E a c h p i x e l o f the image i s g e n e r a t e d by a r a y e x t e n d i n g f r o m a p o i n t o r i g i n a t i n g on the o b j e c t s u r f a c e to the c a m e r a l e n s c e n t e r . C o n v e r s e l y , the p r o j e c t o r i l l u m i n a t e s s p o t s on the o b j e c t s u r f a c e by p r o j e c t i n g r a y s f r o m the l e n s c e n t e r t h r o u g h the t r a n s p a r e n t w i n d o w s on the s l i d e . In the c a s e o f the c a m e r a , the l i g h t p r o p a g a t i o n d i r e c t i o n i s f r o m an o b j e c t p o i n t to the l e n s c e n t e r . A l t h o u g h the p r o p a g a t i o n d i r e c t i o n i s r e v e r s e d f o r the s l i d e p r o j e c t o r , t h e ge ome t r y o f b o t h d e v i c e s i s i d e n t i c a l . H e n c e , the d e r i v a t i o n o f the t r a n s f o r m a t i o n m a t r i c e s f o r b o t h d e v i c e s i s the s a m e . S i m i l a r to the c a s e o f the c a m e r a , the t r a n s f o r m a t i o n m a t r i x o f the s l i d e p r o j e c t o r i s d e t e r m i n e d by the f o c a l l e n g t h o f the p r o j e c t o r l e n s , as w e l l as the d i s p l a c e m e n t , p a n , t i l t , and s w i n g a n g l e s o f the s l i d e p r o j e c t o r w i t h r e s p e c t t o the g l o b a l r e f e r e n c e f r a m e . H o w e v e r , t h e t r a n s f o r m a t i o n f r o m the r e f e r e n c e f rame F o f the r o t a t e d P s l i d e p r o j e c t o r t o the s l i d e c o o r d i n a t e r e f e r e n c e s y s t e m F g , as shown i s F i g u r e 3 . 6 , i s d i f f e r e n t f r o m the c a m e r a m o d e l . 37 F i g . 3.6 Spo t i o l r e l a t i o n s h i p between the s l i d e plane coord inate system F g and the p r o j e c t o r re ference system Fp . T h e o p t i c a l a x i s o f the p r o j e c t o r i s s t i l l p a r a l l e l to the y - a x i s o f F , b u t t h e o r i g i n . o f the s l i d e c o o r d i n a t e P r e f e r e n c e s y s t e m i s now at the b o t t o m s l e f t c o r n e r of the s l i d e p l a n e . T h e u and v axes o f the s l i d e p l a n e a r e p a r a l l e l to t h e x and z axes o f F , r e s p e c t i v e l y . T h e h o m o g e n e o u s m a t r i x A„ o f the t r a n s f o r m a t i o n f r o m F to F i s b P p s g i v e n a s : K l u * P 0 0 K 2 0 0 0 (3.22) 38 w h e r e K - - f / k , K - •= - f / k , f i s the f o c a l l e n g t h of the s l i d e p r o j e c t o r l e n s , and k and k a r e the h o r i z o n t a l and u v v e r t i c a l s p a c i n g s o f the t r a n s p a r e n t w i n d o w s o f the r e f e r e n c e p r o j e c t i o n p a t t e r n . " F r o m E q u a t i o n ( 3 . 6 ) , ( 3 . 7 ) and ( 3 . 2 2 ) , the c o m b i n e d t r a n s f o r m a t i o n m a t r i x o f t h e s l i d e p r o j e c t o r c a n be w r i t t e n a s : T p r o j - A P \ *T ( 3 - 2 3 ) G i v e n the c e n t r o i d c o o r d i n a t e s of the p r o j e c t e d s p o t ( x o , y o , z Q ) on the s u r f a c e of an o b j e c t , the c o l u m n and r o w a d d r e s s ( u , v ) o f the t r a n s p a r e n t w i n d o w i n t h e p r o j e c t i o n p a t t e r n on the s l i d e c a n be o b t a i n e d f r o m the m o d e l o f the s l i d e p r o j e c t o r as us P l l P 1 2 P 1 3 P l 4 X 0 v s = P 2 1 P 2 2 P 2 3 P 2 4 y o s P 3 1 P 3 2 P 3 3 P 3 4 . z 0 1 whe r e P l l P 1 2 P l 3 P l 4 T p r o j P 2 1 P 2 2 P 2 3 P24 P 3 1 P 3 2 P 3 3 P 3 4 . and s i s t h e a r b i t r a r y s c a l i n g f a c t o r ( 3 . 2 4 ) v e c t o r 39 3 . 5 MODEL CALIBRATION PROCEDURES T h e p u r p o s e o f the c a l i b r a t i o n p r o c e s s i s to d e t e r m i n e the c o e f f i c i e n t s o f the p e r s p e c t i v e p r o j e c t i o n t r a n s f o r m a t i o n m a t r i x w h i c h d e s c r i b e s the s p a t i a l r e l a t i o n s h i p o f an image p o i n t i n the c a m e r a or p r o j e c t o r c o o r d i n a t e s y s t e m t o i t s o b j e c t p o i n t i n t h e g l o b a l r e f e r e n c e f r a m e . S e v e r a l c a l i b r a t i o n m e t h o d s a r e a v a i l a b l e . One m e t h o d i s to c a r e f u l l y m e a s u r e a l l the o p t i c a l p a r a m e t e r s , s u c h a s , the f o c a l l e n g t h o f the l e n s , the d i s p l a c e m e n t b e t w e e n t h e g l o b a l r e f e r e n c e f r a m e and the i m a g i n g s y s t e m , and the p a n , t i l t and s w i n g a n g l e s o f the c a m e r a a n d p r o j e c t o r . T h e s e v a l u e s a r e t h e n s u b s t i t u t e d i n t o E q u a t i o n ( 3 . 1 8 ) t o compute the t r a n s f o r m a t i o n m a t r i x d i r e c t l y . In the a b s e n c e o f s p e c i a l h a r d w a r e and c a l i b r a t i o n d e v i c e s to p r o v i d e a c c u r a t e a n g u l a r m e a s u r e m e n t s , an a l t e r n a t i v e c a l i b r a t i o n p r o c e s s u s i n g the m e a s u r e d l o c a t i o n o f some known o b j e c t p o i n t s i n s p a c e and t h e i r c o r r e s p o n d i n g images i s more d e s i r a b l e . I f the p e r s p e c t i v e t r a n s f o r m a t i o n m a t r i x o f the i m a g i n g s y s t e m u n d e r c a l i b r a t i o n i s 11 a 12 a 13 a 1 4 21 a 22 a 23 a 2 4 31 a 32 a 33 a 3 4 ( 3 . 2 5 ) t h e n u s i n g E q u a t i o n ( 3 . 1 9 ) , the image c o o r d i n a t e s ( u , v ) of a 40 f e a t u r e p o i n t and i t s c o r r e s p o n d i n g c a l i b r a t i o n p o i n t (x , y , z ) c a n be w r i t t e n as o o o u t a l l v t a 2 1 t a 3 1 a l 2 a l 3 a 1 4 a 2 2 a 2 3 a 2 4 a 3 2 a 3 3 a 3 4 ( 3 . 2 6 ) w h e r e t i s the a r b i t r a r y s c a l i n g f a c t o r . S u b s t i t u t i n g f o r the v a r i a b l e t i n E q u a t i o n ( 3 . 2 6 ) y i e l d s , a U X o " a 3 1 U X o + a 1 2 y o " a 3 2 U y o + a 1 3 Z o " a 3 3 U Z o + a 1 4 - a 3 4 U - 0 ( 3 . 2 7 ) 21 O . . . v x ^ + a - . y ^ - a , . v y n + a . . z - a , , v z 31 o 22 o 32 o 23 o 33 o + a 24 a 3 4 V ( 3 . 2 8 ) Two l i n e a r e q u a t i o n s a r e f o r m e d f r o m e a c h c a l i b r a t i o n p o i n t . S i n c e t h e r e a r e t w e l v e unknown c o e f f i c i e n t s , i t r e q u i r e s at l e a s t s i x l i n e a r l y i n d e p e n d e n t c a l i b r a t i o n p o i n t s to s o l v e f o r t h e s e u n k n o w n s . I f the m i n i m u m number o f s i x f e a t u r e p o i n t s a r e u t i l i z e d i n t h i s c a l i b r a t i o n p r o c e s s , a s y s t e m o f t w e l v e l i n e a r e q u a t i o n s w i t h t w e l v e u n k n o w n v a r i a b l e s i s f o r m e d . T h e s y s t e m o f l i n e a r e q u a t i o n s i n m a t r i x r e p r e s e n t a t i o n i s g i v e n as C a = 0 whe r e ( 3 . 2 9 ) 41 [ a U a 1 2 a l 3 a 1 4 a 2 1 a 2 2 a 2 3 a 2 4 a 3 1 a 3 2 a 3 3 a 3 4 ] T ( 3 ' 3 0 ) and the i * * 1 and i - l S t r o w o f the m a t r i x C a r e c. = [ 0 0 0 0 x, y, z . 1 - v . x, - v . y. - v . z, - v . ] I k J k k k k k J k k k k ( 3 . 3 1 ) c i - l - [ x k y k z k 1 0 0 0 0 " u k y k " v k y k " u k z k - u k ] (3 . 32) f o r i = 2 , 4 , 6 , 8 , 1 0 , 1 2 k = 1 , 2 , 3 , 4 , 5 , 6 . S i n c e the e q u a t i o n C a = 0 i s h o m o g e n e o u s , i t i m p l i e s t h a t the s o l u t i o n f o r the unknown v a r i a b l e s a . . ' s w i l l a l l i J h a v e a common s c a l i n g f a c t o r . I f one of the u n k n o w n , s a y a j ^ , i s s e t to o n e , the s o l u t i o n s o f the e l e v e n r e m a i n i n g u n k n o w n v a r i a b l e s i n E q u a t i o n ( 3 . 2 9 ) a r e n o r m a l i z e d . A s e l e c t i o n o f s i x c a l i b r a t i o n p o i n t s w i t h no more t h a n f o u r c o p l a n a r p o i n t s among t h e m , s a t i s f i e s the l i n e a r i n d e p e n d e n c y r e q u i r e m e n t . T h e s e c a l i b r a t i o n p o i n t s f o r m an o v e r - d e t e r m i n e d l i n e a r s y s t e m of 12 e q u a t i o n s w i t h e l e v e n u n k n o w n v a r i a b l e s . The s o l u t i o n s f o r t h i s s y s t e m c a n be s o l v e d by u s i n g g e n e r a l i z e d l e a s t s q u a r e s r e g r e s s i o n m e t h o d [ P r a t t 1 9 7 8 ] . By r e a r r a n g i n g E q u a t i o n s ( 3 . 2 9 ) to ( 3 . 3 2 ) , the o v e r - d e t e r m i n e d s y s t e m o f l i n e a r e q u a t i o n s becomes 42 0 0 0 0 x y z 1 - v x X j y1 z1 1 O 1 O 1 O 1 0 - u j x j 0 0 0 0 x . y , z , 1 - v . x , x , y , z , 1 O 6 O 6 O 6 0 - u « * « 11 12 32 33 V l y l U l y l V 6 y 6 U 6 y 6 - v . •u. or , T A = R T h e l e a s t s q u a r e s p s e u d o i n v e r s e e s t i m a t o r T + E q u a t i o n ( 3 . 3 4 ) c a n be d e t e r m i n e d b y , [ P r a t t 1978] T + = ( T T T ) _ 1 T T T h e r e f o r e , t h e l e a s t s q u a r e e s t i m a t i o n o f A i A = T R o r , A = ( T T T ) _ 1 T T R 43 3 . 6 I M P L E M E N T A T I O N A N D E X P E R I M E N T A L R E S U L T S 3 . 6 . 1 CAMERA MODEL C A L I B R A T I O N Imp I ement at i on I n the c a m e r a m o d e l c a l i b r a t i o n p r o c e s s , a r e c t a n g u l a r b l o c k i s p l a c e d i n f r o n t o f a b a c k g r o u n d p l a n e l o c a t e d on the x - y p l a n e o f the r e f e r e n c e c o o r d i n a t e s y s t e m . T h e r e c t a n g u l a r b l o c k i s o r i e n t e d as shown i n F i g u r e 3 . 7 , so t h a t t h e two f r o n t f a c e s o f the b l o c k a r e v i s i b l e by the c a m e r a . A t o t a l o f f i f t y e v e n l y d i s t r i b u t e d c a l i b r a t i o n p o i n t s w e r e m a r k e d on the two f a c e s o f the b l o c k and on t h e b a c k g r o u n d p l a n e . The s p a t i a l c o o r d i n a t e s o f t h e s e c a l i b r a t i o n p o i n t s w i t h r e s p e c t to the o b j e c t - c e n t e r e d r e f e r e n c e c o o r d i n a t e s y s t e m w e r e m e a s u r e d m a n u a l l y . The i ma ge c o o r d i n a t e s o f t h e s e c a l i b r a t i o n p o i n t s we re r e c o r d e d m a n u a l l y by d i s p l a y i n g the image on the RAMTEK image d i s p l a y s y s t e m a n d p o s i t i o n i n g t h e image c u r s o r s e q u e n t i a l l y t h r o u g h t h e image p o s i t i o n s o f t h e s e c a l i b r a t i o n p o i n t s . By s u b s t i t u t i n g the image and s p a t i a l c o o r d i n a t e s o f the c a l i b r a t i o n p o i n t s i n t o E q u a t i o n s ( 3 . 2 9 ) t o ( 3 . 3 2 ) , an o v e r d e t e r m i n e d s y s t e m o f l i n e a r e q u a t i o n s i s f o r m e d . The l e a s t s q u a r e s e s t i m a t i o n o f the t w e l v e unknown c o e f f i c i e n t s o f the c a m e r a t r a n s f o r m a t i o n m a t r i x c a n be c a l c u l a t e d by u s i n g t h e p s e u d o i n v e r s e o p e r a t o r as d e s c r i b e d i n the p r e v i o u s s e c t i o n . 44 F i g . 3 . 7 R e f e r e n c e o b j e c t u s e d i n t h e c a m e r a m o d e l c a l i b r a t i o n p r o c e s s . U s i n g t h e c a l i b r a t i o n m a t r i x , a n y two o f t h e t h r e e s p a t i a l c o o r d i n a t e s , s a y x and y , c a n be c o m p u t e d f r o m t h e t h i r d s p a t i a l c o o r d i n a t e , z , and t h e image c o o r d i n a t e s o f t h e c a l i b r a t i o n s a m p l e s u s i n g E q u a t i o n s ( 3 . 2 7 ) a n d ( 3 . 2 8 ) . T h e a c c u r a c y o f t h e c a m e r a m o d e l c a n t h e n be d e t e r m i n e d by c o m p a r i n g t h e c a l c u l a t e d s p a t i a l c o o r d i n a t e s o f t h e s a m p l e w i t h t h e m e a s u r e d c o o r d i n a t e v a l u e s . T h e e r r o r t e r m s , A R , o f t h e c a l i b r a t i o n p o i n t s w i t h r e s p e c t t o t h e c a m e r a m o d e l a r e d e f i n e d a s : w h e r e x . a n d y o . a r e the c a l c u l a t e d s p a t i a l c o o r d i n a t e s c a l c a i o f t h e c a l i b r a t i o n p o i n t s , and x a n d y a r e t h e i r m e a s u r e d m m 45 c o o r d i n a t e v a l u e s . Experimental Results T h e t r a n s f o r m a t i o n m a t r i x o f the c a m e r a m o d e l o b t a i n e d by u s i n g a l l 50 d a t a p o i n t s i n the s a m p l e p o p u l a t i o n i s g i v e n i n T a b l e 3 . 1 . T h e e r r o r t e r m s A R , and the s p a t i a l a n d image c o o r d i n a t e s o f the c a l i b r a t i o n p o i n t s a r e t a b u l a t e d i n T a b l e 3 . 2 . A c a l i b r a t i o n p o i n t i s c o n s i d e r e d to be w i t h i n the e r r o r t o l e r a n c e o f the m o d e l i f i t s e r r o r t e r m i s l e s s t h a n the l a t e r a l s p a t i a l r e s o l u t i o n o f the s p o t p r o j e c t i o n p a t t e r n w h i c h i s f i v e - e i g h t h s o f an i n c h o r 3 .1% o f the c a m e r a ' s f i e l d o f v i e w . A s i n d i c a t e d i n T a b l e 3 . 2 , 28 out o f 50 c a l i b r a t i o n p o i n t s i n t h e s a m p l e p o p u l a t i o n h a v e e x c e e d e d the e r r o r t o l e r a n c e o f t h e m o d e l . S i n c e the number of bad s a m p l e s have e x c e e d e d 25% o f the s a m p l e p o p u l a t i o n , the m o d e l o b t a i n e d by t h i s p r o c e s s i s n o t a g o o d r e p r e s e n t a t i o n o f the s a m p l e p o p u l a t i o n . T h e r e s u l t s a l s o i m p l i e d t h a t some c a l i b r a t i o n p o i n t s m i g h t h a v e l a r g e m e a s u r e m e n t e r r o r s a s s o c i a t e d w i t h t h e m . B e c a u s e o f the bad s a m p l e p o i n t s , the l e a s t s q u a r e s a v e r a g i n g p r o c e s s on the e n t i r e d s a m p l e p o p u l a t i o n m i g h t n o t be t h e m o s t a p p r o p r i a t e t e c h n i q u e to g e n e r a t e the t r a n s f o r m a t i o n m a t r i x o f the c a m e r a m o d e l . H e n c e , i t i s n e c e s s a r y t o p r e p r o c e s s the d a t a p o i n t s i n the s amp 1e p o p u l a t i o n to d e t e r m i n e t h e i r c o m p a t i b i l i t y w i t h the r e s t of t h e p o p u l a t i o n . The i n c o m p a t i b l e s a m p l e s w i l l be d i s c a r d e d and w i l l n o t be u s e d f o r the d e r i v a t i o n of the c a m e r a m o d e l . 46 * * * * * * * CAMERA T R A N S F O R M A T I O N M A T R I X * * * * * * * - 3 . 9 6 9 0 - 1 9 . 2 0 5 3 1 . 1 9 4 8 4 5 6 . 6 2 1 1 - 2 2 . 5 3 4 3 - 1 . 5 6 1 5 - 4 . 0 8 6 9 3 6 3 . 4 1 8 0 - 0 . 0 0 9 1 - 0 . 0 1 0 7 - 0 . 0 1 4 0 1 . 0 0 0 0 T a b l e 3 . 1 T h e c a m e r a t r a n s f o r m a t i o n m a t r i x i n d e x I m a g e M e a s u r e d s p a t i a l C a l c u l a t e d E r r o r c o o r d . c o o r d i n a t e s c o o r d i n . t e r m I J X Y 2 • X Y AR ( p i x e l u n i t s ) ( i n c h e s ) ( i n c h e s ) ( i n c h e s ) 1 48 1 . 463 . 6 . 8 3 . 9 8 . 9 -6 . 9 3 . 2 1 . 8 2 465 . 462 . 6 . 8 4 . 9 9 . 5 -6 . 6 4 . 5 1 . . 4 3 439 . 462 . 8 . 8 6 . 4 1 8 . , 4 -6 . 3 6 . 5 1 . . 6 4 394 . 482 . 6 . 8 9 . 3 1 2 . . 1 6 . 3 9 . 6 8 . 5 5 351 . 482 . 6 . 8 1 1 . . 8 1 3 , . 8 8 . 8 12 . 2 8 . 5 6 473 . 34 1 . 2 . , 8 4 . 3 9 . 1 1 . , 9 3 . 9 e. 8 . . 9 7 445 . 341 . 2 . 8 6 . 8 1 8 . . 1 2 . 2 6 . 6 . 6 8 397 . 341 . 2 . 8 8 . , 9 12. . 8 2 . 6 9 . 3 8 . 5 9 366 . 341 . 2 . 8 1 8 . , 8 1 3 , . 3 2 . 8 1 1 . . 3 8 . 5 ie 324 . 341 . 2 . 8 13, , 4 1 4 , . 8 3 . 1 13 . 6 8 . 4 11 449 . 244 . 6 . 8 5 . 8 1 8 , . 6 5 . 8 5 . 6 8 . 8 2 12 468 . 243 . 6 . e 8 . 3 1 1 . 6 6 . 8 . 1 8 . 6 . 3 1 3 366 . 241 . 6 . 6 1 8 . . 8 1 3 . 3 6 1 1 . 1 8 . 4 1 4 484 . 155 . 9 . e 3 . 7 8. , 7 9 . , 1 3 . 3 8 . 4 1 5 424 . 1 52 . 9 . . 8 7 . 3 1 1 , . 8 9 . 6 7 . . 5 8 . 3 1 6 362 . 1 56 . 9 . 8 9 . . 8 1 2 . . 6 8 . 9 1 6 . 1 8 . 3 17 348 . 1 49 . 9 . 8 12 , . 3 1 4 , . 1 8 . 8 12 . 4 8 . 2 1 8 323 . 263 . 7 . 3 1 3 , . 4 1 4 . . 9 7 . 3 1 3 . 4 6 . 8 1 9 296 . 386 . 1 . 5 1 4 . . 6 1 4 . . 4 2 . 8 1 4 . 9 6 . 6 26 234 . 378 . 1 . 5 16 , . 8 12 . 1 2 . 1 1 6 . 3 8 . 7 21 1 86 . 377 . 1 . 5 1 7 . 5 9 . 8 2 . 2 1 7 . 8 6 . 8 22 129. 376 . 1 . 5 18 .9 7 . 6 2 . 3 1 9 . 3 8 . 9 23 276 . 317. 3 .5 1 5 , . 1 13, . 6 4 . 8 15 . 2 8 . 5 24 266 . 317. 3 .5 17 .8 18. . 6 4 . 1 1 7 .8 8 . 6 25 1 58 . 317. 3 . 5 18 . 3 8 . 5 4 . 2 1 8 . 4 8 . 7 26 36 1 . 233 . 6 .3 1 4 . .3 1 , . 8 6 . 8 9 .8 4 . 5 27 236 . 234 . 6 .3 1 6 .2 1 . 9 6 . 9 13 . 3 3 . 8 28 176. 235 . 6 . 3 1 7 . . 8 9 . 3 6 . 7 1 7 . 5 8 . 6 29 265 . 172. 8 .3 15 . 3 1 3 . 3 8 . 3 1 4 . 9 8 . 3 36 1 95 . 174. 8 . 3 1 7 . 1 18 . 4 8 . 5 1 6 . 6 8 1 . . 6 31 165. 1 75 . 8 . 3 18 . 2 8 . 7 8 . 6 1 7 . 3 . 8 32 296 . 133. 9 . 5 1 4 . 4 1 4 , . 6 9 . 2 1 4 . 1 6 . 4 33 2 14. 1 36 . 9 . 5 1 6 . 5 1 1 , . 3 9 . 4 1 6 . 8 8 . 6 34 1 39 . 137 . 9 . 5 18. . 5 8 . 8 9 . 7 1 7 . 9 6 . 7 35 428 . 166. 1 1 . 8 2 . 8 8 . 8 1 1 . . 9 1 . 9 8 1 . 9 36 319. 185. 1 1 .6 7 , . 8 e . 6 1 1 . . 8 7 . 9 . 2 37 192. 1 63 . 1 1 . 8 1 3 . 8 8 . 8 1 1 . . 8 1 3 . 9 1 . 2 38 165. 16 1. 1 1 . 6 1 7 .8 8 .8 1 1 . 7 1 7 . 5 6 . 9 39 165. 42 . 13 . 8 1 7 .8 8. . 8 13 . 6 1 7 . 2 8 . 7 46 192. 45 . 1 3 . 8 13, . 8 8 . 6 1 3 . . 7 1 3 . . 6 1 . . 6 4 1 277 . 47 . 13 . 8 9 . 8 8 . 8 13. . 8 9 . 8 1 . . 2 42 363 . 48 . 13 .8 5 . 8 8 . 8 1 4 , . 6 5 . 5 1 . 1 . , 1 43 495 . 56 . 13 .8 -1 , . 8 8 . 8 1 4 . . 3 -2 . 2 . 8 44 386 . 26 . 1 6 . 8 4 . 8 8 . 8 15. . 1 4 . 2 8 1 . . 9 45 1 27 . 1 5 . 16 .8 1 6 . 8 8 . 8 1 4 , . 5 16 . 2 . 5 46 61 . 1 2 . 16 . 8 19, . 8 8 . 8 1 4 , . 5 18 . 7 1 , . 6 47 66 . 129. 12 . 8 1 9 . 8 8. . 8 1 8 . 8 1 9 . . 4 1 . , 3 48 66 . 246 . 8 .8 1 9 . .8 8 . 8 6 .  8 28 . , 1 1 . , 6 49 256 . 1 9 . 1 6 . e 1 8 , 8 8 . , 8 1 4 . 8 1 8 . . 7 1 . . 4 56 36. 361 . 4 .8 28. 6 8 . 8 2. , 7 21 . 8 2 . 2 T a b l e 3 . 2 S p a t i a l and c a l i b r a t i o n image c o o r d i n a t e s o f the p o i n t s 47 T h i s i d e a l e a d s to the c o n c e p t o f r a n d o m s a m p l e c o n s e n s u s w h i c h w i l l be p r e s e n t e d i n the n e x t c h a p t e r . A s shown i n F i g u r e 3.7, the m a r k e r s o f some c a l i b r a t i o n p o i n t s a r e q u i t e f u z z y i n the p i c t u r e . I t i s d i f f i c u l t t o a c c u r a t e l y l o c a t e the image c o o r d i n a t e s o f t h e s e p o i n t s and c o n s e q u e n t l y g r o s s e r r o r p o i n t s o c c u r i n t h e s a m p l e p o p u l a t i o n . T h i s s i t u a t i o n c a n be a l l e v i a t e d by i n c r e a s i n g t h e amb i e n t l i g h t i n t e n s i t y i n the s c e n e and to i n c r e a s e the v i s i b i l i t y o f the m a r k i n g s f o r the c a l i b r a t i o n p o i n t s . 3 . 6 . 2 S L I D E P R O J E C T O R MODEL C A L I B R A T I O N T h e same r e c t a n g u l a r b l o c k i s u s e d as the r e f e r e n c e o b j e c t f o r t h e s l i d e p r o j e c t o r c a l i b r a t i o n p r o c e s s . The r e f e r e n c e s p o t p r o j e c t i o n p a t t e r n i s p r o j e c t e d o n t o t h i s r e c t a n g u l a r b l o c k . The b l o c k i s o r i e n t e d as shown i n F i g u r e 3.8, s u c h t h a t the two f r o n t f a c e s o f the b l o c k w i l l show the i l l u m i n a t e d s p o t s f r o m the s l i d e p r o j e c t o r . S i x t y - n i n e e v e n l y d i s t r i b u t e d p r o j e c t i o n s p o t s f r o m t h e r e f e r e n c e s p o t p a t t e r n w e r e s e l e c t e d i n a p s e u d o - r a n d o m m a n n e r as s a m p l e p o i n t s f o r t h e p r o j e c t o r m o d e l c a l i b r a t i o n p r o c e s s . The r o w and c o l u m n a d d r e s s e s o f t h e s e s e l e c t e d s p o t s and the s p a t i a l c o o r d i n a t e s a t t h e c e n t e r o f t h e s e s p o t s w i t h r e s p e c t to the r e f e r e n c e c o o r d i n a t e s y s t e m w e r e m e a s u r e d m a n u a l l y . U s i n g t h e c o l u m n a n d r o w a d d r e s s e s and the s p a t i a l c o o r d i n a t e s o f t h e s e c a l i b r a t i o n p o i n t s , an o v e r d e t e r m i n e d s y s t e m o f l i n e a r e q u a t i o n s i s f o r m e d . The c o e f f i c i e n t s o f t h e p r o j e c t o r 48 Y F i g . 3 . 8 The re fe rence object used in the s l i d e p r o j e c t o r mode c a l i b r a t i o n process . t r a n s f o r m a t i o n m a t r i x a r e d e t e r m i n e d by u s i n g the p s e u d o i n v e r s e o p e r a t o r as i n the c a s e o f the c a m e r a c a l i b r a t i o n p r o c e s s . S i m i l a r to the c a m e r a c a l i b r a t i o n p r o c e s s , t h e a c c u r a c y o f the s l i d e p r o j e c t o r m o d e l c a n be d e t e r m i n e d by c o m p a r i n g the c a l c u l a t e d s p a t i a l c o o r d i n a t e s w i t h t h e c a l i b r a t i o n p o i n t s w i t h the m e a s u r e d c o o r d i n a t e v a l u e s . T h e e r r o r t e r m s , A R , o f the c a l i b r a t i o n p o i n t s w i t h r e s p e c t t o t h e s l i d e p r o j e c t o r mo d e l a r e a l s o d e f i n e d a s : AR = [ ( x r a 1 - x ) 2 + ( v „ , - y ) 2 ] 1 / 2 c a l m c a l m (3.39) w h e r e x , and y , a r e t h e c a l c u l a t e d x and y c o o r d i n a t e s c a l • ' c a l J 49 o f the c a l i b r a t i o n p o i n t s , and x and y a r e the m e a s u r e d * m m c o o r d i n a t e v a l u e s . Experimental Results T h e t r a n s f o r m a t i o n m a t r i x o f the s l i d e p r o j e c t o r o b t a i n e d by u s i n g a l l 69 d a t a p o i n t s i n the s a m p l e p o p u l a t i o n i s g i v e n i n T a b l e 3 . 3 . The e r r o r t e r m s AR and the s p a t i a l a n d a d d r e s s c o o r d i n a t e s o f the c a l i b r a t i o n p o i n t s a r e l i s t e d i n T a b l e 3 . 4 . T h e r e a r e o n l y e i g h t s a m p l e s w i t h an e r r o r t e r m g r e a t e r t h a n the e r r o r t o l e r a n c e o f t h e m o d e l . T h i s i m p l i e d t h a t an a c c e p t a b l e m o d e l have b e e n f o u n d . T h e m o d e l c a n f u r t h e r be i m p r o v e d by d i s c a r d i n g t h e e i g h t b a d s a m p l e p o i n t s and be r e d e r i v e d b a s e d on the r e m a i n i n g g o o d d a t a p o i n t s . T h i s i d e a l e a d s to the c o n c e p t o f r a n d o m s a m p l e c o n s e n s u s t o be d i s c u s s e d i n the n e x t c h a p t e r f o r i m p r o v i n g t h e a c c u r a c y o f t h e m o d e l c a l i b r a t i o n p r o c e s s . * * * * * * * S L I D E P R O J E C T O R 0 . 0 2 0 6 - 1 . 6 5 2 4 - 1 . 7 4 0 7 - 0 . 0 1 0 5 0 . 0 0 0 0 0 . 0 0 0 9 TRANSFORMATION M A T R I X * * * * * * * - 0 . 4 1 4 5 2 9 . 7 8 3 6 - 0 . 1 3 7 0 2 9 . 2 7 8 4 - 0 . 0 0 6 7 1 . 0 0 0 0 T a b l e 3 . 3 S l i d e p r o j e c t o r t r a n s f o r m a t i o n m a t r i x 50 I n d e x l m o g e t l t o i u r t d s p a t i a l C o l c u l o t e d E r r o r c o o r d . c o o r d i n a t e s c o o r d i n . t e r m I J X Y 2 X Y 4 R ( p i x e l u n i t s ) ( I n c h e s ) ( i n c h e s ) ( i n c h e s ) 1 1 . 12. 9 . 5 1 5 2 4 . 12. 9 . 5 1 3 3 8 . 12. 9 . 4 1 0 4 1 2 . 1 2 . 9 . 4 8 5 1 4 . 12. 9 . 4 6 6 1 . 1 5 . 7 . 9 1 5 7 4 . 15. 7 . 9 13 e 8 . 15. 7 . 9 1 6 9 1 2 . 15. 7 . 8 8 i e 1 4 . 15. 7 . 9 6 11 1 . 28 . 5 . 0 15 1 2 4 . 28 . 5 . 0 13 1 3 8 . 20 . 5 , 1 16 i * 12. 20 . 5 , . 1 7 1 5 1 4 . 20 . 5 . 1 6 1 6 1 . 2 4 . 2 . 7 15 17 4 . 24 . 2 . 8 13 18 8 . 24 . 2 . 8 16 1 9 12 . 24 . 2 . 8 7 26 14. 24 . 2 . 9 6 21 1. 27 . 1 . .2 15 22 4 . 27 . 1 . . 2 13 23 8 . 27 . 1 , . 2 16 2 + 12 . 27 . 1 , . 3 7 25 1 4 . 27 . 1 . . 4 6 26 16 . 12 . 9 . 4 5 27 19 . 12. 9 . 4 4 28 22 . 12. 9 . 5 2 29 27 . 12. 9 . 6 0 30 1 6 . 21 . 4 . 6 5 31 1 9 . 21 . 4 . 6 4 32 22 . 21 . 4 . 6 2 33 27 . 21 . 4 . 6 0 34 1 6 . 25 . 2 . 4 5 35 1 9 . 25 . 2 . 4 4 36 22 . 25 . 2 . 4 2 37 27 . 2 5 . 2 . 4 6 38 1 6 . 36 . 0 . 8 5 39 1 9 . 36 . 0 . 8 4 46 22 . 36 . 0 , . 8 2 4 1 27 . 36 . 0 . 8 0 42 e . 6 . 1 6 . 9 1 7 43 4 . 6 . 16 . 9 1 5 44 i e . 0 . 1 6 . . 9 12 45 1 7 . 0 . 16 . 9 7 48 23 . 0 . 16. . 9 4 47 27 . 0 . 16 .9 1 48 31 . 0 . 16 .9 - 6 49 e. 3. 15 . 1 17 se 4 . 3. 1 5 . 1 15 51 1 e . 3 . 1 5 . . 1 12 52 1 7 . 3 . 15. . 1 7 53 23 . 3 . 15 . 1 4 54 27 . 3 . 1 5 . 1 1 55 31 . 3 . 15 . 1 - 0 56 e . 7 . 12 . 6 17 57 4 . 7 . 12 . 6 15 58 18 . 7 . 12 .6 12 59 1 7 . 7 . 12 . 6 7 66 2 3 . 7 . 12 .6 4 6 1 27 . 7 . 1 2 . 6 1 62 31 . 7 . 12 . 6 - 6 63 e . 10. 10 . 9 1 7 64 4 . 16. 1 0 . 9 1 5 65 1 e . 10. 10 .9 1 2 66 1 7 . 10. 10 . 9 7 67 23 . 10. 16 . 9 4 68 27 . 10. 16 . 9 1 69 31 . 10 . 10 . 9 - 6 4 8 . 3 9 . 5 1 5 . , 5 6 . 1 4 9 . 9 9 .  4 1 3 . . 4 0 . 1 6 12 . 1 9 .  4 16. . 6 0 . 1 6 1 4 , . 3 9 .  4 7 . 9 0 . 1 8 1 5 . , 4 9 . 3 6 . 6 0 . 1 5 8 . 2 7 , . 8 1 5 . , 5 0 . 1 4 9 . 9 7 . 8 13. , 3 0. . 1 6 1 2 . . 3 7 . 8 10. ,6 0 . 1 6 1 4 . . 4 7 . 8 7 . 9 0 . 1 7 1 5 . . 4 7 . 8 6 . 6 0 . 1 4 8 . 3 5 . , 1 1 5 . 4 0 . 1 4 9 . 9 5 . , 1 1 3 . . 3 0. . 1 6 12. . 3 5 . , 1 10. .5 0 . 1 9 1 4 . . 4 5 . 2 7 . 9 0 . 1 7 1 5 . . 5 5 , 2 6 . 6 6 . 2 5 8 . , 3 2 . 9 15, . 4 8 . 2 4 16. . 0 2 , . 9 13. 3 0 . 2 5 12. . 3 3 . 6 16, .5 6 . 2 8 1 4 . . 5 3 , . 1 7 , 8 0 .2 5 15. . 5 3 . 1 6 , . 5 6 .2 4 8 . 3 1 . 2 15 .  4 0 . 6 3 1 6 . . 6 1 . . 3 1 3 , . 2 0 . 1 5 1 2 . , 3 1 . , 4 10. , 5 0 . 2 8 1 4 . , 5 1 . 5 7 , . 8 0 . 2 5 1 5 . . 5 1 , . 6 6 , 5 0 . 2 5 1 5 . . 1 9 . , 4 5 , . 6 6 . 1 2 1 3 . . 5 9 .  4 4 . 3 0 . 1 8 1 1 . . 8 9 . , 5 2 . 9 6 . 1 4 8 . 9 9 . 6 6 . 5 0 . 2 5 1 5 . . 1 4 . . 7 5 . 5 6 . 1 1 1 3 . . 4 4 . . 7 4 . . 2 0 . 1 8 1 1 . . 8 4 . . 7 2 . 8 6 . 1 4 8 . 9 4 . . 8 6 . 5 6 . 2 4 1 5 . . 1 2 . 6 5 , 5 6 . 3 1 13 ,  4 2 , . 6 4 . , 2 6 . 2 8 1 1 . . 8 2 . . 6 2 . 8 0 . 2 4 8 . 9 2 . 6 6 . , 5 0 . 2 5 1 5 , , 1 6 . 6 5 , 5 0 . 7 1 1 3 , . 4 6 . 6 4 . , 2 0 . 8 8 1 1 , . 8 6 . 6 2 . 8 0. . 8 4 8 . 9 - 6 . . 1 6 .  4 6 . 8 9 6 . 6 1 6 , 7 1 8 , .2 0 . 4 4 6 . 6 1 6 . . 7 1 5 , . 8 6 .  4 9 0 . 6 1 6 . . 7 1 2 , . 1 0 .8 7 0 . 0 1 6 . . 8 7 , . 9 6 .2 2 0 . 0 1 6 . . 8 4 , 2 6 . 1 8 0 .0 1 6 . 8 1 , . 9 6 . 1 5 8 .0 16 . 8 - 0 , . 5 0 .0 9 8 .0 15 . 6 18 2 0 .  4 4 0 . 0 15 .0 1 5 , , 8 0 . 4 9 0 . 0 1 5 . . 0 1 2 . . 1 0 . 8 7 6 . 0 15. . 0 7 . 8 0 . 2 2 0 . 0 15. . 1 4 . . 2 0 . 6 8 6 . 6 15 . 1 1 , , 8 0 . 6 5 0 . 6 15, . 1 - 0 , .6 0 . 1 9 0 . 6 1 2 . . 6 1 8 . . 2 0 . 3 4 0 . 6 1 2 . . 6 1 5 , . 7 0 . 3 9 6 . 6 1 2 . 7 1 2 , . 1 6 . 8 7 6 . 6 12 . 7 7 , . 8 0 . 1 2 6 . 6 1 2 . . 8 4 . . 2 6 . 1 8 6 . 6 1 2 . 8 1 . . 8 6 . 1 5 0 . 6 12 . 8 - 0 . 6 6 .2 9 6 . 6 1 6 . . 9 18, . 2 6 . 3 4 6 . 6 16 . 9 15 , 7 6 . 3 9 6 . 6 16. . 9 12 . 6 0 . 8 7 6 , . 6 1 1 . . 0 7 , .8 6 . 1 2 6 . 6 1 1 , . 0 4 . .2 0 . 2 8 0 . 6 1 1 , . 1 1 . . 8 6 . 2 5 0 . 6 1 1 . . 1 - 0 . . 6 0 . 2 T a b l e 3.4 T h e s p a t i a l and image c o o r d i n a t e s of the c a l i b r a t i o n p o i n t s i n the s l i d e p r o j e c t o r s a m p l e p o p u l a t i o n 51 3.7 D I S C U S S I O N A l a r g e n umb er o f s p a t i a l and i ma ge c o o r d i n a t e m e a s u r e m e n t s a r e i n v o l v e d i n t h e s e m o d e l c a l i b r a t i o n p r o c e s s e s . H e n c e , t h e m e a s u r e m e n t r e s u l t s a r e l i a b l e t o have e r r o r s o c c a s i o n a l l y . A f e w g r o s s e r r o r s among t h e s e d a t a p o i n t s c a n h a v e d e v a s t a t i n g e f f e c t s on the a c c u r a c y o f the l e a s t s q u a r e s e s t i m a t i o n of the m o d e l . F o r a g i v e n s e t u p , the c a m e r a and p r o j e c t o r n e e d to be c a l i b r a t e d o n l y o n c e . T h e r e f o r e , i t w o u l d be w o r t h w h i l e to p r e - p r o c e s s the c o o r d i n a t e m e a s u r e m e n t s and d i s c a r d the s a m p l e s w h i c h a r e g r o s s l y i n e r r o r . F i n a l l y , o n l y the g o o d d a t a p o i n t s a r e u s e d t o d e r i v e the c a m e r a and s l i d e p r o j e c t o r m o d e l . T h i s l e a d s to t h e a p p l i c a t i o n o f the r a n d o m s a m p l e c o n s e n s u s ( R A N S A C ) a l g o r i t h m to the s a m p l i n g p o p u l a t i o n . C o m p l e t e d e t a i l s o f t h i s a l g o r i t h m w i l l be p r e s e n t e d i n t h e n e x t c h a p t e r . 52 CHAPTER 4 E X P E R I M E N T A L D E T E R M I N A T I O N OF MODEL P A R A M E T E R S 4 . 1 P R O B L E M D E F I N I T I O N G i v e n a camera image of a s e t of c a l i b r a t i o n p o i n t s , or the s c e n e o f a spot p a t t e r n p r o j e c t e d f r o m a s l i d e p r o j e c t o r , i t i s n e c e s s a r y to i n s t a n t i a t e a m a t h e m a t i c a l model to r e p r e s e n t the p e r s p e c t i v e t r a n s f o r m a t i o n of t h e s e i m a g i n g d e v i c e s . The d e r i v a t i o n of t h e s e m o d e l s was g i v e n p r e v i o u s l y i n C h a p t e r 3. T h e o r e t i c a l l y , e i t h e r the camera or p r o j e c t o r m o d e l c a n be d e r i v e d f r o m a s e t of s i x known l i n e a r l y i n d e p e n d e n t c a l i b r a t i o n p o i n t s . I f the m i n i m u m number of c a l i b r a t i o n p o i n t s i s u s e d , any measurement e r r o r s of the c a l i b r a t i o n p o i n t s w i l l be r e f l e c t e d i n the e s t i m a t e d m o d e l . T r a d i t i o n a l l y , a s e t of s a m p l i n g p o i n t s much l a r g e r t h a n the minimum r e q u i r e d i s c o l l e c t e d . A p p r o x i m a t i o n methods s u c h as the l e a s t s q u a r e s t e c h n i q u e , a r e t h e n u s e d to smooth out any g r o s s d e v i a t i o n s i n the m e a s u r e m e n t s . T h e r e a r e two m a j o r t y p e s of e r r o r s a s s o c i a t e d w i t h the l o c a t i o n measurement of each c a l i b r a t i o n p o i n t : l ) e r r o r s i n the a c t u a l d i s t a n c e measurements o f the f e a t u r e l o c a t i o n s ; and 2 ) e r r o r s i n the i d e n t i f i c a t i o n of the image c o o r d i n a t e s o f the f e a t u r e p o i n t s . Measurement e r r o r s 53 g e n e r a l l y f o l l o w the n o r m a l d i s t r i b u t i o n w i t h a z e r o m e a n , w h e r e a s , t h e i d e n t i f i c a t i o n e r r o r s a r e o f t e n s c e n e d e p e n d e n t and do n o t o b e y any s t a t i s t i c a l r u l e s . The i d e n t i f i c a t i o n e r r o r s may c a u s e g r o s s e r r o r s i n the m o d e l e s t i m a t i o n p r o c e s s . G e n e r a l l y , the t r a d i t i o n a l a p p r o x i m a t i o n m e t h o d s c a n p r o v i d e s a t i s f a c t o r y r e s u l t s . O c c a s i o n a l l y , t h e s e m e t h o d s may f a i l due to i n s u f f i c i e n t s i z e of the s a m p l e p o p u l a t i o n or g r o s s e r r o r s a s s o c i a t e d w i t h some s a m p l e p o i n t s . The R a n d o m S a m p l e C o n s e n s u s ( R A N S A C ) a l g o r i t h m , d e s i g n e d by F i s c h l e r and B o l l e s [ 1 9 8 1 ] , c a n be u s e d to i d e n t i f y the s a m p l e p o i n t s t h a t a r e g r o s s l y i n e r r o r w i t h i n a s a m p l e p o p u l a t i o n . A f t e r e l i m i n a t i n g t h e s e e r r o r s a m p l e s , the d e s i r e d m o d e l c a n t h e n be i n s t a n t i a t e d w i t h the r e m a i n i n g good s a m p l e p o i n t s u s i n g l e a s t s q u a r e s a p p r o x i m a t i o n . T h e R A N S A C a l g o r i t h m has b e e n i n c o r p o r a t e d i n t o the c a m e r a and p r o j e c t o r c a l i b r a t i o n p r o c e d u r e s f o r e l i m i n a t i n g the g r o s s e r r o r s among the s a m p l e p o i n t s . 4.2 RANDOM S A M P L E CONSENSUS A L G O R I T H M I n a s a m p l e p o p u l a t i o n P o f N d a t a p o i n t s , a m i n i m u m s u b s e t S o f n r a n d o m d a t a p o i n t s (n<N) a r e f i r s t s e l e c t e d to p r o d u c e an i n s t a n t i a t e d m o d e l . The a c c u r a c y o f t h i s m o d e l i s d e t e r m i n e d by c o m p a r i n g the m e a s u r e d s p a t i a l c o o r d i n a t e s a t e a c h s a m p l e p o i n t i n P w i t h the c o m p u t e d s a m p l e c o o r d i n a t e s o b t a i n e d f r om t h e mo d e l . A s amp 1e p o i n t i s c o n s i d e r e d t o be c o n s i s t e n t and w i l l be i n c l u d e d i n the 54 c o n s e n s u s s e t a s s o c i a t e d w i t h the m o d e l , i f the c o m p u t e d c o o r d i n a t e s at the s a m p l e p o i n t a r e w i t h i n a p r e a s s i g n e d e r r o r t o l e r a n c e l i m i t f r o m the m e a s u r e d v a l u e . The i n s t a n t i a t e d m o d e l i s a c c e p t e d as a v a l i d a p p r o x i m a t i o n o f the s a m p l e p o p u l a t i o n o n l y i f the number o f e l e m e n t s i n the c o n s e n s u s s e t i s g r e a t e r t h a n some t h r e s h o l d , t . T h i s p r e d e t e r m i n e d t h r e s h o l d v a l u e i s u s u a l l y e q u a l to a p e r c e n t a g e o f the t o t a l s a m p l e p o p u l a t i o n . O n l y t h e e l e m e n t s i n i t s a s s o c i a t e d c o n s e n s u s s e t a r e t h e n u s e d t o c o m p u t e the f i n a l m o d e l o f the s a m p l e p o p u l a t i o n . On the o t h e r h a n d , i f the s i z e o f the c o n s e n s u s s e t a s s o c i a t e d w i t h the m o d e l i s l e s s t h a n t , t h i s m o d e l w i l l be r e j e c t e d and the e n t i r e p r o c e d u r e i s r e p e a t e d w i t h a n o t h e r m i n i m u m s u b s e t u n t i l an a c c e p t a b l e c o n s e n s u s s e t i s f o u n d . I f an a c c e p t a b l e c o n s e n s u s s e t i s n o t f o u n d a f t e r a number o f t r i a l s , the p r o c e s s c a n e i t h e r t e r m i n a t e i n f a i l u r e or p r o c e e d t o compute t h e f i n a l m o d e l of the p o p u l a t i o n u s i n g t h e l a r g e s t c o n s e n s u s s e t a v a i l a b l e . 4 . 3 I M P L E M E N T A T I O N O F T H E RANDOM S A M P L E CONSENSUS A L G O R I T H M I n the c a m e r a m o d e l c a l i b r a t i o n s c e n a r i o , e a c h d a t a p o i n t i n the s a m p l e p o p u l a t i o n c o n s i s t s o f the s p a t i a l c o o r d i n a t e s o f the o b j e c t p o i n t and i t s c o r r e s p o n d i n g image c o o r d i n a t e s . F o r the s l i d e p r o j e c t o r c a l i b r a t i o n p r o c e s s , e a c h d a t a p o i n t i s d e f i n e d by t h e c e n t r o i d c o o r d i n a t e s o f the p r o j e c t e d s p o t , as w e l l as the r o w and c o l u m n number o f the c o r r e s p o n d i n g t r a n s p a r e n t w i n d o w on the s l i d e . T h e 55 a l g o r i t h m b e g i n s w i t h the s e l e c t i o n of a l i s t o f m 6 - t u p l e s , e a c h c o m p o s e d of s i x e v e n l y d i s t r i b u t e d and l i n e a r l y i n d e p e n d e n t f e a t u r e p o i n t s on the s c e n e . The r e s t r i c t i o n o f a l l o w i n g no more t h a n f o u r c o p l a n a r p o i n t s o f the s i x s e l e c t e d p o i n t s i s i m p o s e d t o f u l f i l l the l i n e a r i n d e p e n d e n c y r e q u i r e m e n t . A l l t h e p o i n t s i n the s a m p l e p o p u l a t i o n s a r e l o c a t e d on t h r e e p l a n a r s u r f a c e s : the two f a c e s o f the r e c t a n g u l a r b l o c k and the b a c k g r o u n d s u r f a c e on the x - y p l a n e of the r e f e r e n c e c o o r d i n a t e s y s t e m . E a c h 6-t u p l e s i s c o m p o s e d of two r a n d o m l y s e l e c t e d s a m p l e s f r o m e a c h o f t h e p l a n a r s u r f a c e s . T h e V e n n d i a g r a m r e p r e s e n t a t i o n o f t h e s a m p l e s e t and the 6 - t u p l e s i s shown i n F i g u r e 4 . 1 . The sample popu lat ion F i g . 4.1 Venn diogram repre sen ta t i on of the sample populat ion P and the subset of 6-TUPLES. 56 T h e m a t h e m a t i c a l m o d e l o f the d e v i c e t o be c a l i b r a t e d i s i n s t a n t i a t e d f o r t h e f i r s t 6 - t u p l e s u s i n g E q u a t i o n s ( 3 . 3 3 ) . F r o m t h i s m o d e l , a n y two of the t h r e e s p a t i a l c o o r d i n a t e s of an o b j e c t p o i n t c a n be c o m p u t e d f r o m the t h i r d s p a t i a l c o o r d i n a t e and the two image c o o r d i n a t e s u s i n g E q u a t i o n s ( 3 . 2 7 ) and ( 3 . 2 8 ) . F o r e x a m p l e , i f the u , v and z c o o r d i n a t e s of a p o i n t i n the s a m p l e s e t a r e k n o w n , the x and y c o o r d i n a t e s o f the o b j e c t p o i n t c a n t h e n be c a l c u l a t e d by u s i n g the i n s t a n t i a t e d t r a n s f o r m a t i o n m a t r i x . T h e d e v i a t i o n of the c a l c u l a t e d v a l u e s , x , and y . , f r o m the c.a 1 • ' c a l m e a s u r e d c o o r d i n a t e s , x a n d y , d e f i n e s the d e g r e e o f m m 6 a c c u r a c y of the e s t i m a t e d m o d e l at t h a t s a m p l e p o i n t . T h e e r r o r t e r m AR i s g i v e n as A n y s a m p l e p o i n t w i t h an e r r o r t e r m g r e a t e r t h a n 0 . 6 2 5 i n c h , the 1 a t e r a 1 s p a t i a 1 r e s o l u t i o n of the p r o j e c t i o n s y s t e m , i s c o n s i d e r e d t o be an i n c o n s i s t e n t p o i n t and i s e x c l u d e d f r o m the c o n s e n s u s s e t a s s o c i a t e d w i t h the 6 - t u p l e . E x p e r i m e n t a l r e s u l t s s h o w t h a t i f t h e r e a r e g r o s s e r r o r p o i n t s i n the 6 - t u p l e , t h e n the c o n s e n s u s s e t o f t h e i n s t a n t i a t e d m o d e l u s u a l l y c o n s i s t s o f l e s s t h a n 50 p e r c e n t o f the s a m p l e p o p u l a t i o n . T h u s , i n the c a m e r a and p r o j e c t o r c a l i b r a t i o n p r o c e s s , a 6 - t u p l e i s r e j e c t e d i f the s i z e o f the c o n s e n s u s s e t i s l e s s t h a n 75 p e r c e n t o f t h e s a m p l e p o p u l a t i o n , i n c l u d i n g a 25 p e r c e n t s a f e t y m a r g i n . T h e i n s t a n t i a t i o n p r o c e s s i s r e p e a t e d w i t h the n e x t 6 - t u p l e on 57 t h e l i s t u n t i l an a c c e p t a b l e m o d e l i s f o u n d . A f t e r a 6 - t u p l e i s a c c e p t e d , the i n c o n s i s t e n t p o i n t s a r e d i s c a r d e d f r o m the s a m p l e s e t . L e a s t s q u a r e s a p p r o x i m a t i o n i s a p p l i e d o n l y to the s a m p l e p o i n t s i n the c o n s e n s u s s e t f o r g e n e r a t i o n o f the f i n a l m o d e l of the c a l i b r a t i n g d e v i c e . I f the l i s t o f 6 - t u p l e s i s e x h a u s t e d b e f o r e an a c c e p t a b l e c o n s e n s u s s e t i s f o u n d , t h e p r o g r a m t e r m i n a t e s w i t h a f a i l u r e m e s s a g e . T h i s s i t u a t i o n i n d i c a t e s t h a t t h e r e a r e many g r o s s e r r o r p o i n t s i n the s a m p l e s e t and the c a l i b r a t i o n p r o c e s s m u s t be r e p e a t e d w i t h i m p r o v e d m e a s u r e m e n t a c c u r a c y . T h e r a n d o m s a m p l e c o n s e n s u s a l g o r i t h m i s s u m m a r i z e d i n t h e f o l l o w i n g p s e u d o h i g h - l e v e l l a n g u a g e p r o c e d u r e : B E G I N : C o l l e c t a s a m p l e s e t of N c a l i b r a t i o n p o i n t s C o m p i l e a l i s t o f m 6 - t u p l e s DO e a c h s e l e c t e d 6 - t u p l e , "WHILE f i n i s h = F A L S E I n s t a n t i a t e t h e m o d e l w i t h the 6 - t u p l e DO a l l the p o i n t s i n the samp 1e s e t P C o m p u t e t h e x , y c o o r d i n a t e s o f t h e s a m p l e p o i n t w i t h the i n s t a n t i a t e d m o d e l C o m p u t e t h e e r r o r t e r m AR I F AR < 0 . 6 2 5 i n c h T H E N A d d the c a l i b r a t i o n p o i n t i n t o t h e c o n s e n s u s s e t E L S E D i s c a r d the c a l i b r a t i o n p o i n t 58 E N D I F END DO I F the s i z e of the c o n s e n s u s s e t < . 7 5 * N T H E N Compute the f i n a l m o d e l u s i n g the e n t i r e c o n s e n s u s s e t S e t f i n i s h = T R U E E L S E T r y the n e x t 6 - t u p l e on t h e l i s t S e t f i n i s h = F A L S E E N D I F E N D DO I F f i n i s h = T R U E T H E N C a l i b r a t i o n p r o c e s s c o m p l e t e E L S E C a l i b r a t i o n p r o c e s s f a i l e d E N D I F E N D 4 . 4 D E C O M P O S I T I O N OF T H E T R A N S F O R M A T I O N M A T R I X N u m e r o u s p a p e r s h a v e b e e n p u b l i s h e d on the c a l i b r a t i o n p r o c e s s o f c a m e r a d e v i c e s , b u t m e t h o d s i n r e c o v e r i n g t h e p a r a m e t e r s o f t h e o p t i c a l c o n f i g u r a t i o n of t h e d e v i c e f r om a g i v e n t r a n s f o rma t i o n ma t r i x c a n o n l y be f o u n d i n a r e c e n t p a p e r by G a n a p a t h y [ 1 9 8 4 ] . He s u g g e s t e d an a l g e b r a i c m e t h o d w h i c h u t i l i z e s the v a l u e s of the t w e l v e known c o e f f i c i e n t s of the t r a n s f o r m a t i o n m a t r i x to d e t e r m i n e 59 the 10 u n k n o w n p h y s i c a l p a r a m e t e r s o f the o p t i c a l d e v i c e , , K 2 , U q , v , JCp, Yj, Zj, 0, <f>, and as d e f i n e d i n C h a p t e r 3. T h e c a l c u l a t e d p a r a m e t e r s c a n t h e n be u s e d to c o m p a r e w i t h t h e c o a r s e p h y s i c a l m e a s u r e m e n t s o f a f e w of t h e s e p a r a m e t e r s . T h i s p r o c e d u r e c a n be u s e d to c h e c k the v a l i d i t y o f the t r a n s f o r m a t i o n m a t r i c e s o b t a i n e d f r o m the m o d e l c a l i b r a t i o n p r o c e s s e s . A s d e f i n e d i n E q u a t i o n s (3.17) and ( 3 . 2 3 ) , the t r a n s f o r m a t i o n m a t r i c e s of the c a m e r a and s l i d e p r o j e c t o r a u g i v e n a s : and c am p r o j A C A R C A T C A p A j ^ p A p p (4.2) ( 4 . 3 ) w h e r e A^~., a n d ^-JQ a r e 1 ^ e p e r s p e c t i v e , r o t a t i o n and t r a n s l a t i o n t r a n s f o r m a t i o n m a t r i c e s o f t h e c a m e r a , and A p , A ^ p , a n d A p p a r e the p e r s p e c t i v e , r o t a t i o n and t r a n s l a t i o n t r a n s f o r m a t i o n m a t r i c e s of the s l i d e p r o j e c t o r . In the d e c o m p o s i t i o n p r o c e d u r e o f the c a m e r a m o d e l , we c a n r e p r e s e n t the r o t a t i o n m a t r i x A ^ a s : a b c 0 d e f 0 g h i 0 0 0 0 1 (4.4) and the - ^ r j ^ p Q m a t r i x i s r e p r e s e n t e d by 60 a b c P d e f q g h i r 0 0 0 1 (4.5) wh ere a c = e = g i P q r C c^ce-ses^S0 C9C0 = ces^+ses0Cf = b Y T + c Z T = d X T + e Y T + f Z T = g X T + h Y T + i Z T = c o s i n e b = c^se+cestfs^ d = - S 8 C 0 f = S0 h = s e s f - c e s ^ c f s i n e L e t us r e p r e s e n t the c o e f f i c i e n t s of the camera m a t r i x by ° 1 1 ' ° 1 2 ' '•'» c 3 3 a n < * °34" From E q u a t i o n s ( 4 . 5 ) and ( 3 . 1 7 ) , the unknown p a r a m e t e r s a, b i , p, q and r can be e x p r e s s e d i n terms of the t r a n s f o r m a t i o n m a t r i x c o e f f i c i e n t s and g i v e n a s : 11 13 21 23 31 33 K,g + u d l 6 o K , i + u f 1 o K_a + v d 2 o K . i + v f 2 0 = d 12 14 '22 24 32 34 K „ h + u e 1 o r + u q 1 o n K-b + v e 2 o K 2 p + V Q q e q ( 4 . 6 ) 61 S i n c e i s an o r t h o r n o r m a 1 r o t a t i o n m a t r i x [ G o l d s t e i n 1 9 5 0 ] , t h e t r a n s p o s e o f the m a t r i x i s the i n v e r s e of the m a t r i x and the d e t e r m i n a n t of the m a t r i x i s u n i t y . H e n c e , the f o l l o w i n g c o n s t r a i n t s c a n be i m p o s e d on the e l e m e n t s o f 2 v.2 a + b 2 + c ,2 2 d + e + f 2 2 = g + h2 + i 2 = 1 . i = a e -- bd , g = b f -• c e , h = cd - a f , a = e i • - h f , b = f g • - d i , c = dh - eg . d = he • - b i , e = a i • - gc , f = bg - a h , ad + be + c f = dg + eg + f i = ag + bh + c i = 0 ( 4 . 7 ) U s i n g the c o n s t r a i n t e q u a t i o n s s p e c i f i e d i n E q u a t i o n ( 4 . 7 ) a n d the d e f i n i t i o n s of the m a t r i x c o e f f i c i e n t s g i v e n i n E q u a t i o n ( 4 . 6 ) , the t e n unknown o p t i c a l p a r a m e t e r s o f t h e c a m e r a c o n f i g u r a t i o n c a n be s o l v e d by f o l l o w i n g t h e a l g e b r a i c p r o c e d u r e s as s u g g e s t e d by G a n a p a t h y . S i n c e s q u a r e r o o t o p e r a t i o n s w e r e i n v o l v e d i n the c a l c u l a t i o n s of the u n k n o w n v a r i a b l e s q , , K . , u and v , i t i s n e c e s s a r y t o n 1 2 o o J d e t e r m i n e t h e s i g n s o f t h e s e v a r i a b l e s as w e l l as t h e i r m a g n i t u d e s . I t c a n be p r o v e n t h a t the s i g n s o f any two among the t h r e e p a r a m e t e r s . K ^ and q a r e p o s i t i v e . I n o u r c a s e , K j a n d q a r e d e f i n e d to be p o s i t i v e . The r e d u n d a n t c o n s t r a i n t c o n d i t i o n s g i v e n i n E q u a t i o n ( 4 . 6 ) and ( 4 . 7 ) c a n t h e n be u s e d to d e t e r m i n e the s i g n s o f t h e s e unknown p a r a m e t e r s K » , u and v . 2 o o I n o u r s i t u a t i o n , t h i s ma t r i x dec omp o s i t i o n p r o c e s s 62 i s u s e d as an e r r o r c h e c k i n g t o o l on the v a l i d i t y of the t r a n s f o r m a t i o n m a t r i c e s p r o d u c e d by the m o d e l c a l i b r a t i o n p r o c e d u r e s . T h e s o l u t i o n s of the t e n o p t i c a l p a r a m e t e r s a r e c a l c u l a t e d f o r e a c h p o s s i b l e s e t o f v a l u e s i n K „ , u and v . 2 o o The r e s u l t s o f t h e s e o p t i c a l p a r a m e t e r s w e r e c h e c k e d a g a i n s t the c o a r s e m e a s u r e m e n t s of the c a m e r a and s l i d e p r o j e c t o r p o s i t i o n s . O n l y one s e t of K - , u , and v v a l u e s c o u l d r 2 o o g e n e r a t e t h e d i s p l a c e m e n t v e c t o r f o r the c a m e r a t h a t r e s e m b l e d the v a l u e s g i v e n i n the c o a r s e m e a s u r e m e n t s o f the o p t i c a l p a r a m e t e r s . 4.5 R E S U L T S OF T H E MODEL C A L I B R A T I O N PROCESS T h e image and s p a t i a l c o o r d i n a t e s o f the s a m p l e s u s e d i n the c a m e r a m o d e l c a l i b r a t i o n p r o c e s s a r e l i s t e d i n T a b l e 4 . 1 . T h e r o w and c o l u m n a d d r e s s and the s p a t i a l c o o r d i n a t e s o f the s amp l e s u s e d i n the s l i d e p r o j e c t o r mo d e l c a l i b r a t i o n p r o c e s s a r e g i v e n i n T a b l e 4 . 2 . T h e l i s t s o f the e i g h t 6-t u p l e s c h o s e n f o r the RANSAC a l g o r i t h m u s e d i n t h e s e two p r o c e s s e s a r e g i v e n i n T a b l e 4 . 3 . The n u m b e r s a s s o c i a t e d w i t h e a c h 6 - t u p l e a r e the i n d e x number o f the s a m p l e as g i v e n i n T a b l e s 4 . 1 and 4 . 2 . The s a m p l e p o p u l a t i o n s i z e s of the c a m e r a and s l i d e p r o j e c t o r c a s e s a r e 50 and 6 9 , and t h e i r m i n i m u m c o n s e n s u s s e t s i z e s u s e d i n t h e RANSAC a l g o r i t h m a r e 37 and 5 1 , r e s p e c t i v e l y . I n t h e camera c a l i b r a t i o n c a s e , o n l y the 6 - t u p l e s n u m b e r e d 4 and 5 on the l i s t g i v e n i n T a b l e 4 . 3 c a n n o t f o r m 63 Number o f C a l i b r a t i o n p o i n t s - 50 1 nd«x S p a t i a l C o o r d l n o t * * 1 moge Co o r d 11 X ( i n . ) Y ( l n . ) Z( In. ) 1 J 1 .75 3 .875 8 . 8 5 48 1 483 2 .75 4 . 9 9 . 5 465 482 3 .75 6 . 4 1 8 .4 439 482 4 .75 9 . 3 1 2 . 1 5 394 482 5 .75 11 .75 1 3 . 7 5 351 482 6 2 .75 4 . 25 9 .18 473 341 7 2 .75 6 . 1 8 . 1 5 445 341 8 2 .75 8 . 9 12 . 397 341 » 2.75 18.8 1 3 . 2 5 366 341 1 8 2 . 75 13 .48 14 .8 324 34 1 1 1 6 . 5 .75 18 . 449 244 12 6 . 6 .25 1 1 . 6 5 488 243 1 3 6 . 18 .8 1 3 . 2 5 366 241 1 4 9. 3 . 7 8 . 7 484 155 1 5 9 . 7 .25 1 1 . 424 152 1 6 9 . 9 .75 1 2 . 5 5 382 158 1 7 9 . 12 .25 1 4 . 1 5 348 1 49 1 8 1 9 7 . 25 13 .35 1 4 . 8 5 323 283 1 . 5 14 .6 1 4 . 4 298 388 28 1 .5 16 .85 12.1 234 378 21 1 . 5 17 .5 9 .75 188 377 22 1 . 5 18 .85 7 . 625 129 376 23 3 . 5 15 .15 1 3 . 5 5 278 317 24 3 . 5 16 .95 1 8 . 6 288 317 25 3.5 18 .27 8 . 5 1 58 317 26 6 . 25 14 .3 1 . 8 381 233 27 6 . 25 16 .2 1 . 875 238 234 28 6 . 25 17 .75 9 . 3 1 78 235 29 8 . 25 15 .25 1 3 . 3 2 5 265 172 38 8 .25 17 .125 1 8 . 3 7 5 1 95 174 31 8 . 25 18.2 8 . 7 165 1 75 32 9 . 5 14 .45 1 4 . 6 296 133 33 9 . 5 16 .55 1 1 . 2 5 214 136 34 9 . 5 18 .55 8 . 8 139 137 35 1 1 . 2 . 8 . 428 1 86 36 1 1 . 7 . 8 . 319 185 37 1 1 . 1 3 . 8 . 1 92 1 83 38 1 1 . 17 .8 8 . 185 1 8 1 39 13. 1 7 . 8 . 1 85 42 48 13. 13. 8 . 1 92 45 4 1 13. 9 . 8 . 277 47 42 13. 5 . 8 . 363 48 43 13. -1 . 8 . 495 58 44 1 6 . 4 . 8 . 386 28 45 1 6 . 1 6 . 8 127 1 5 46 1 6 . 1 9 . 8 . 61 1 2 47 12. 1 9 . 8 . 68 1 29 48 8 . 1 9 . 8 . 68 246 49 1 6 . 18 . 8 . 256 1 9 58 4 . 28 . 8 . 36 361 T a b l e 4.1 S p a t i a l and image c o o r d i n a t e s of the s a m p l e p o i n t s u s e d i n c a m e r a m o d e l c a l i b r a t i o n . an a c c e p t a b l e c o n s e n s u s s e t . A l l the o t h e r 6 - t u p l e s h a v e c o l l e c t e d the same 41 p o i n t s f r o m the s a m p l e p o p u l a t i o n to f o r m t h e i r c o n s e n s u s s e t . I n o t h e r w o r d s , n i n e p o i n t s i n the s a m p l e p o p u l a t i o n a r e g r o s s l y i n e r r o r . The a v e r a g e o f the e r r o r t e r m s A R , d e f i n e d i n E q u a t i o n ( 4 . 2 0 ) , a s s o c i a t e d w i t h t h e i n i t i a l m o d e l s d e r i v e d f r o m a l l the 6 - t u p l e s , and the f i n a l m o d e l s d e r i v e d f r o m t h e common c o n s e n s u s s e t o f the a c c e p t a b l e 6 - t u p l e s , a r e l i s t e d i n T a b l e 4.4. T h e a v e r a g e s o f t h e e r r o r t e r m s h a v e d e c r e a s e d f r o m o v e r h a l f an i n c h to 64 Number o f c a l i b r a t i o n p o i n t s = 69 1 ndex S p a t i a l C o o r d i n a t e s Spot a d d r e s s * * X( I n. ) Y( In. ) Z ( I n . ) Co 1umn Ro< 1 9 . 5 15.45 8 . 25 1 12 2 9 . 5 13.45 9.875 4 1 2 3 9 . 45 16.65 12.15 8 1 2 4 9 . 4 8 . 8 14.3 12 1 2 5 9 . 35 6 . 75 15.35 1 4 1 2 7 7 . 875 15.5 8 . 2 1 1 5 7 7 . 875 13.45 9 . 9 4 1 5 8 7 . 875 18.65 12.25 8 1 5 9 7 . 88 8.8 14.375 12 1 5 i e 7 . 865 6 . 7 15.375 1 4 15 11 5 . 15.45 8 . 25 1 28 12 5 . 13.4 9 . 9 4 28 1 3 5 . 1 18.6 12.25 8 28 1 4 5 . 125 7.9 14.375 12 28 1 5 5 . 125 6 . 7 15.5 1 4 28 16 2 . 675 15.5 8 . 25 1 24 1 7 2 . 75 13.35 1 8 . 4 24 18 2 . 825 18.5 12.25 8 24 1 9 2 . 858 7 . 8 14.5 12 24 28 2 . 875 6 . 5 15.5 1 4 24 21 1 . 1 55 15.375 8 . 3 1 27 22 1 . 175 13.3 1 8 . 4 27 23 1 . 2 18.5 12.25 8 27 24 1 . 3 7 .75 14.5 12 27 25 1 . 37 6 . 5 15.5 1 4 27 26 9 . 375 5 . 5 15.125 1 6 1 2 27 9 . 48 4.175 13.5 1 9 1 2 28 9 . 5 2 .75 11.75 22 12 29 9 . 55 . 375 8 .875 27 i : 38 4 . 625 5 . 5 15.128 1 6 21 3 1 4 . 625 4.125 13.375 1 9 21 32 4 . 625 2 .75 11.75 22 21 33 4 . 625 .375 8 .875 27 2 1 34 2 . 375 5 . 45 15.128 1 6 25 35 2 . 375 4.125 13.375 1 9 25 36 2 . 375 2 .75 11.75 22 25 37 2 . 375 .375 8 . 875 27 25 38 . 75 5 . 5 15.12 1 6 38 39 .75 4 . 1 13.375 1 9 38 48 .75 2.75 11.75 22 38 4 1 .75 . 375 8 . 875 27 38 8 42 1 6 . 85 17.85 8 . 8 43 1 6 . 85 15.4 e. 4 8 44 1 6 . 85 12.87 8 . 1 8 8 45 1 6 . 85 7 . 7 8 . 1 7 8 46 1 6 . 85 4 . 2 8 . 23 8 47 1 6 . 85 1 . 8 8 . 27 8 48 1 6 . 85 -.5 8 . 31 8 49 1 5 . 1 17.85 8 . 8 3 58 1 5 . 1 8 15.4 8 . 4 3 51 1 5 . 1 12.87 8 . 18 3 52 1 5 . 1 7 . 7 8 . 1 7 3 53 1 5 . 1 4.2 8 . 23 3 54 15 . 1 1 . 8 8 . 27 3 55 15 . 1 - . 5 8 . 31 3 56 12 . 65 17.85 8 . 8 7 57 12 . 65 15.4 8 . 4 7 58 12 . 65 12.87 8 . 1 8 7 59 1 2 . 65 7 . 7 6. 1 7 7 68 1 2 . 65 4 . 2 8. 23 7 61 12 . 65 1 . 8 8. 27 7 62 12 .65 -.5 8 . 31 7 63 18 .875 17.85 8 . 8 1 6 64 18 . 875 15.4 8 . 4 1 8 65 18 . 875 12.87 8 . 18 1 8 66 1 8 .875 7 . 7 8 1 7 1 8 67 18 .875 4 . 2 8 . 23 1 8 68 1 8 .875 1 . 8 8 . 27 1 8 69 1 8 .875 -.5 8 . 31 18 T a b l e 4.2 S p o t a d d r e s s e s and s p a t i a l c o o r d i n a t e s o f the s a m p l e p o i n t s u s e d i n s l i d e p r o j e c t o r mo d e l c a l i b r a t i o n . a b o u t 0 . 0 6 i n c h e s w h e n the f i n a l m o d e l was u s e d . T h e f i n a l m o d e l o f the c a m e r a i s g i v e n i n T a b l e 4.5. S i m i l a r e n c o u r a g i n g r e s u l t s w e r e a l s o e x p e r i e n c e d i n the s l i d e p r o j e c t o r c a l i b r a t i o n p r o c e s s . A g a i n , o n l y two out of e i g h t 6 - t u p l e s f a i l e d to acc umu l a t e an a c c e p t a b l e 65 6-TUPLES used i n the Camera Model C a l i b r a t i o n Process 1 8 12 28 23 42 37 2 1 12 28 23 42 37 3 3 12 28 23 42 37 4 1 12 28 23 44 37 5 5 15 20 34 36 45 6 8 13 30 20 40 41 7 16 7 33 29 38 35 8 9 11 34 23 35 39 6-TUPLES used i n the S l i d e Projec tor Model C a l i b r a t i o n Process 1 32 28 4 17 57 45 2 32 31 9 17 57 45 3 38 31 12 8 59 68 4 43 36 12 8 59 68 5 42 50 31 32 17 22 6 17 9 28 32 50 61 7 17 9 35 28 50 61 8 7 19 24 33 68 57 T a b l e 4 . 3 L i s t s o f the 6 - t u p l e s 6 - 1 up 1 e i n d e x n umb e r A v e r a g e o f the e r r o r t e r m s AR ( i n c h e s ) I n i t i a l Mo d e l F i n a l Mo d e l 1 2 3 4 5 6 7 8 T a b l e 4 . 4 . 544 . 527 .525 1 . 205 1 . 5 1 8 .549 .611 .635 . 060 .060 .060 F a i l e d RANSAC A l g o r i t h m F a i l e d RANSAC A l g o r i t h m .060 . 060 .060 The a v e r a g e e r r o r t e r m s u s i n g the i n i t a l and the f i n a l c a m e r a m o d e l . T h e f i n a l c ame r a mo d e l 0 . 1 5 3 7 2 8 . 6 8 0 9 0 . 0 0 0 9 - 2 1 . 8 0 3 7 - 0 . 2 4 8 3 - 0 . 0 0 0 1 8 . 0 7 2 9 1 . 4 0 0 1 •0.0043 4 7 5 . 0 4 1 2 422 . 1 2 2 9 1 . 0 0 0 0 T a b l e 4 . 5 T h e f i n a l c a m e r a t r a n s f o r m a t i o n m a t r i x 66 c o n s e n s u s s e t . F i v e o u t o f the s i x a c c e p t e d 6 - t u p l e s f o r m e d the same c o n s e n s u s s e t w i t h the i d e n t i c a l 61 d a t a p o i n t s and the s i x t h one f o r m e d a s m a l l e r c o n s e n s u s s e t w i t h 57 d a t a p o i n t s . The l a r g e r c o n s e n s u s s e t was u s e d to d e r i v e the f i n a l m o d e l o f the s l i d e p r o j e c t o r . The a v e r a g e o f the e r r o r t e r m s AR a s s o c i a t e d w i t h the i n i t i a l m o d e l d e r i v e d f r o m the 6 - t u p l e s and t h e f i n a l m o d e l d e r i v e d f r o m the c o n s e n s u s s e t , a r e l i s t e d i n T a b l e 4 . 6 . I t was f o u n d t h a t the a v e r a g e e r r o r d e c r e a s e d f r o m .21 i n c h e s to .07 i n c h e s w h e n the f i n a l m o d e l o f the s l i d e p r o j e c t o r was u s e d . The f i n a l m o d e l o f the s l i d e p r o j e c t o r i s g i v e n i n T a b l e 4 . 7 . 6 - t u p l e A v e r a g e of the e r r o r t e r m s AR ( i n c h e s ) i n d e x number I n i t i a l M o d e l F i n a l M o d e l 1 .213 .060 2 .219 .060 3 .619 F a i l e d RANSAC 4 . 209 .073 5 . 407 F a i l e d RANSAC 6 .224 . 0 7 3 7 .224 . 0 7 3 8 .288 .073 T a b l e 4 . 6 T h e a v e r a g e e r r o r t e r m s u s i n g t h e i n i t a l t h e f i n a l s l i d e p r o j e c t o r m o d e l . The f i n a l s l i d e p r o j e c t o r m o d e l 0 . 0 2 3 3 - 1 . 6 9 7 9 - 0 . 3 6 4 0 3 0 . 0 4 3 0 - 1 . 7 3 2 2 0 . 0 0 4 3 - 0 . 0 8 0 7 2 9 . 0 6 5 8 0 . 0 0 0 7 0 . 0 0 1 0 - 0 . 0 0 3 7 1 . 0 0 0 0 T a b l e 4 . 7 F i n a l s l i d e p r o j e c t o r t r a n s f o r m a t i o n m a t r i x 67 I f t h e p r o b a b i l i t y i n s e l e c t i n g a good d a t a p o i n t f r o m the s a m p l e p o p u l a t i o n i s w , t h e n the e x p e c t e d number of t r i e s t o p i c k . N good d a t a p o i n t s f r o m the s a m p l e p o p u l a t i o n i s w ^ [ F i s c h l e r and B o l l e s 1 9 8 1 ] . In the c a m e r a and p r o j e c t o r m o d e l c a l i b r a t i o n p r o c e s s e s , w i s f o u n d to be e q u a l t o 4 1 / 5 0 and 6 1 / 6 9 , r e s p e c t i v e l y . T h e r e f o r e , the e x p e c t e d number o f t r i e s to p i c k 6 g o o d d a t a p o i n t s f r o m the c a m e r a a n d p r o j e c t o r s a m p l e p o p u l a t i o n s a r e 3 . 2 9 and 2 . 6 5 , r e s p e c t i v e l y . 4 . 6 R E S U L T S O F M A T R I X D E C O M P O S I T I O N T h e v a l u e s of the o p t i c a l p a r a m e t e r s o b t a i n e d by u s i n g t h i s s e t o f v a l u e s a r e g i v e n i n T a b l e 4 . 8 . The i d e n t i c a l p r o c e d u r e has b e e n r e p e a t e d f o r the dec omp o s i t i o n o f the s l i d e p r o j e c t o r t r a n s f o r m a t i o n m a t r i x to c a l c u l a t e the o p t i c a l p a r a m e t e r s i n the s l i d e p r o j e c t o r c o n f i g u r a t i o n . The r e s u l t s a r e l i s t e d i n T a b l e 4 . 9 . <j> = - 7 8 . 1 ° 9 = 9 6 . 3 ° f = - 6 . 9 ° X p = 2 . 4 1 i n c h e s Y T = 1 0 6 . 9 i n c h e s Z T = 2 3 0 . 5 i n c h e s T a b l e 4 . 8 O p t i c a l p a r a m e t e r s o f the c a m e r a * = - 7 1 . 7 4 ° 8 •= 3 5 . 0 ° V> = 5 5 . 8 2 X p = 4 . 7 7 i n c h e s Y T = - 6 2 . 7 2 i n c h e s Z T = 2 5 4 . 2 2 i n c h e s T a b l e 4 . 9 O p t i c a l p a r a m e t e r s o f the s l i d e p r o j e c t o r 68 F r o m the p a r a m e t e r s g i v e n i n T a b l e s 4 . 8 and 4 . 9 , the p o s i t i o n s o f the c a m e r a and s l i d e p r o j e c t o r w i t h r e s p e c t to the r e f e r e n c e c o o r d i n a t e s y s t e m c a n be o b t a i n e d f r o m X^,, Y^, and Zj., and the r o t a t i o n t r a n s f o r m a t i o n m a t r i c e s ^ R C a n C * "^RP c a n be d e t e r m i n e d f r o m 9 , <t> and U s i n g t h e s e r o t a t i o n t r a n s f o r m a t i o n m a t r i c e s , the o r i e n t a t i o n of the c a m e r a and p r o j e c t o r c a n t h e n be o b t a i n e d . S i n c e the c a m e r a o p t i c a l a x i s i s a l o n g the y - a x i s o f the c a m e r a c o o r d i n a t e s y s t e m , the u n i t v e c t o r o f the o p t i c a l a x i s i s ( 0 , 1 , 0 ) . T h e p r o j e c t i o n o f t h i s u n i t v e c t o r o n t o the r e f e r e n c e c o o r d i n a t e s y s t e m c a n be r e p r e s e n t e d by the f o l l o w i n g h o m o g e n e o u s t r a n s f o r m a t i o n : [0 1 0 1 ] T = * [ x c o y c o z c o 1 ] T ( 4 . 8 ) T T w h e r e [0 1 0 l ] and [x y z l ] a r e the u n i t v e c t o r c o J c o c o of the c a m e r a o p t i c a l a x i s w i t h r e s p e c t to the c a m e r a a n d the r e f e r e n c e c o o r d i n a t e s y s t e m s , r e s p e c t i v e l y , [x y z J r J co J co co T l ] i s d e f i n e d as the p o i n t i n g v e c t o r o f the c a m e r a . S i m i l a r l y , the p o i n t i n g v e c t o r o f the s l i d e p r o j e c t o r c a n be d e f i n e d by the f o l l o w i n g e x p r e s s i o n : [0 1 0 l ] 1 = A_„ * [x y z l ] 1 ( 4 . 9 ) KP po J p o po The p o i n t i n g v e c t o r s and the p h y s i c a l r e p r e s e n t a t i o n o f t h e o p t i c a l p a r a m e t e r s f o r the c a m e r a and s l i d e p r o j e c t o r a r e g i v e n i n T a b l e 4 . 1 0 and F i g u r e 4 . 2 , r e s p e c t i v e l y . I f no p r i o r k n o w l e d g e o f the o p t i c a l c o n f i g u r a t i o n i s F i g . 4.2 Phys ica l representa t ion of the camera and s l i d e p ro jec to r o p t i c a l c o n f i g u r a t i o n . 70 T h e p o i n t i n g v e c t o r of the c a m e r a i n h o m o n g e n e o u s r e p r e s e n t a t i o n [ 0 . 0 2 , - . 4 0 , - 0 . 9 8 , 1] T h e p o i n t i n g v e c t o r o f the s l i d e p r o j e c t o r i n h o m o g e n e o u s r e p r e s e n t a t i o n [ 0 . 1 8 , 0 . 2 6 , - 0 . 9 5 , 1] T a b l e 4 . 1 0 The p o i n t i n g v e c t o r s o f the c a m e r a and s l i d e p r o j e c t o r a v a i l a b l e , one c a n u s e the r e d u n d a n t c o n s t r a i n t c o n d i t i o n s i n E q u a t i o n ( 4 . 6 ) and ( 4 . 7 ) to d e t e r m i n e t h e p r o p e r s i g n v a l u e s f o r K » , u and v . The m a t r i x d e c o m p o s i t i o n m e t h o d i s 2 o o s u c c e s s f u l i n s o l v i n g the o p t i c a l p a r a m e t e r s o f the c a m e r a and s l i d e p r o j e c t i o n c o n f i g u r a t i o n f o r the s p o t p r o j e c t i o n i m a g i n g s y s t e m . H o w e v e r , i t i s an a l g e b r a i c m e t h o d t h a t p r o v i d e s no i n s i g h t i n t o the u n d e r l y i n g ge ome t r y o f the o p t i c a l e q u i p m e n t . I t i s more s u i t e d f o r c h e c k i n g the v a l i d i t y o f a g i v e n t r a n s f o r m a t i o n m a t r i x . 4 . 7 E F F E C T I V E N E S S OF T H E R A N S A C A L G O R I T H M A s a c o n t r o l e x p e r i m e n t , the t r a n s f o r m a t i o n m a t r i c e s o f t h e c a m e r a and s l i d e p r o j e c t o r w e r e d e r i v e d f r o m the e n t i r e s a m p l e p o p u l a t i o n o f e a c h c a l i b r a t i o n p r o c e s s w i t h o u t u s i n g t h e R A N S A C a l g o r i t h m to e l i m i n a t e t h e g r o s s e r r o r e n t r i e s . T h e y r e l i e d s o l e l y on the l e a s t s q u a r e s a p p r o x i m a t i o n t o r e d u c e the m e a s u r e m e n t e r r o r s . When the R A N S A C a g l o r i t h m w a s i n c o r p o r a t e d , the a v e r a g e o f the e r r o r 71 t e r m s a s s o c i a t e d w i t h the c a m e r a m o d e l was r e d u c e d f r o m o v e r h a l f of an i n c h to .06 i n c h . S i m i l a r l y , the a v e r a g e of the e r r o r t e r m s f o r the p r o j e c t o r m o d e l was r e d u c e d f r o m 0 . 2 1 i n c h to 0 . 0 7 i n c h . When m a t r i x d e c o m p o s i t i o n was p e r f o r m e d on the m o d e l s o b t a i n e d w i t h o u t u s i n g R A N S A C , the o p t i c a l p a r a m e t e r s r e c o v e r e d f r o m t h e s e t r a n s f o r m a t i o n m a t r i c e s d i d n o t r e s e m b l e any of the m e a s u r e d p a r a m e t e r v a l u e s . M o r e i m p o r t a n t l y , as d e s c r i b e d l a t e r i n C h a p t e r 5 , the s p a t i a l c o o r d i n a t e s of the p r o j e c t i o n s p o t s c a l c u l a t e d by u s i n g t h e s e m o d e l d i d n o t l o o k l i k e the o r i g i n a l s u r f a c e . On the c o n t r a r y , f a v o u r a b l e r e s u l t s w e r e o b t a i n e d f r o m the t r a n s f o r m a t i o n m a t r i c e s g e n e r a t e d by u s i n g the RANSAC a 1gor i t h m . 72 C H A P T E R 5 IMAGE P R O C E S S I N G A L G O R I T H M 5 . 1 O V E R V I E W A d e t a i l e d d e s c r i p t i o n o f the image p r o c e s s i n g a l g o r i t h m s u s e d i n d e t e r m i n i n g the 3 - D s p a t i a l c o o r d i n a t e s of the i l l u m i n a t e d s p o t s on t h e c y l i n d r i c a l t e s t s u r f a c e , w i t h r e s p e c t t o the r e f e r e n c e c o o r d i n a t e s y s t e m , i s p r e s e n t e d i n t h i s c h a p t e r . The image a n a l y s i s p r o c e s s , d e p i c t e d i n F i g u r e 5 . 1 , c o n s i s t s o f the f o l l o w i n g t a s k s : 1) l o c a t i o n o f the image o u t l i n e s of the p r o j e c t i o n s p o t s ; 2) e s t i m a t i o n o f the image c e n t r o i d s of the p r o j e c t i o n s p o t s ; 3) d e t e r m i n a t i o n o f the c o l u m n a d d r e s s o f the p r o j e c t i o n s p o t s ; 4) d e t e r m i n a t i o n o f t h e 3 - D s p a t i a l c o o r d i n a t e s o f the p r o j e c t i o n s p o t s . The d e s c r i p t i o n o f t h e s e t a s k s i s p r e s e n t e d i n s e c t i o n 2 to s e c t i o n 5 o f t h i s c h a p t e r i n the o r d e r p r e s c r i b e d i n t h e l i s t a b o v e . T h e s i x g r e y - s c a l e images of the s p a c e - e n c o d e d s p o t p r o j e c t i o n p a t t e r n s on a c y l i n d r i c a l s u r f a c e w e r e a c q u i r e d and u s e d as the i n p u t i n f o r m a t i o n to the image p r o c e s s i n g 73 a l g o r i t h m ; p i c t u r e s o f t h e s e T h e s e g r e y - s c a l e images o f a c q u i r e d u n d e r the o p t i c a l s e c t i o n 2.4. images a r e shown i n F i g u r e 5 . 2 . the p r o j e c t i o n p a t t e r n s we re c o n f i g u r a t i o n as s p e c i f i e d i n 6 g roy s c a l e Imoges ^ of the p r o j e c t I o n p a t t e r n s EDGE DETECTION B i n a r y E d g e ^ Mops CENTROID ESTIMATION C e n t r o i d D i s t r i bu 11 on Mops ADDRESS Co 1umn SPATIAL 3D S p o t i a l COORDINATES DECOD1NG A d d r e s s CALCULATION C o o r d i n o t e s F i g . 5.1 B l o c k d i a g ram of the image p r o c e s s i n g a l g o r i t h m . When o p e r a t i n g the s p o t p r o j e c t i o n i m a g i n g s y s t e m i n the 3 - D me a s u r e m e n t o f the c y l i n d r i c a l s u r f a c e , the a m b i e n t l i g h t r e f l e c t i o n f r om the o b j e c t s u r f a c e wa s s u b s t a n t i a l l y r e d u c e d to e n h a n c e the image i n t e n s i t y c o n t r a s t b e t w e e n the s p o t p r o j e c t i o n p a t t e r n and the n o n - i 1 1 u m i n a t e d a r e a s on the c y l i n d r i c a l s u r f a c e . The h i g h i n t e n s i t y c o n t r a s t o f the s p o t p a t t e r n f r o m t h e b a c k g r o u n d e n a b l e s the s p o t f e a t u r e s to be d e t e c t e d by s i m p l e image a n a l y s i s t e c h n i q u e s , s u c h as b i n a r y t h r e s h o l d i n g [ B o l l e s and C a i n 1983 , H o r a u d 1 9 8 1 , C h a n 1 9 8 5 ] . 74 75 5.2 EDGE DETECTION Edges are commonly • d e f i n e d as abrupt i n t e n s i t y changes i n an image. I n t e n s i t y changes i n an image of a scene can be caused by: l ) i l l u m i n a t i o n changes, such as shadows or l i g h t p r o j e c t i o n p a t t e r n s ; 2) changes i n s u r f a c e o r i e n t a t i o n of the o b j e c t i n the scene w i t h r e s p e c t to the v i e w e r ; 3) changes i n the d i s t a n c e between the v i s i b l e s u r f a c e s and the v i e w e r ; and 4) changes i n r e f l e c t i o n c h a r a c t e r i s t i c s of the s u r f a c e . The edge f e a t u r e s i n the images of the space-encoded spot p a t t e r n s p r o j e c t e d onto a smooth and f e a t u r e l e s s curve s u r f a c e are m a i n l y due to the i l l u m i n a t i o n changes between the spot p a t t e r n s and the background. I f the s i z e of the p r o j e c t i o n spot on the c y l i n d r i c a l s u r f a c e i s about h a l f an i n c h square and the s u r f a c e c u r v a t u r e w i t h i n the spot i s almost f l a t , then the image i n t e n s i t y of the p i x e l s i n s i d e the spot i s c o n s i d e r e d to be c o n s t a n t and much h i g h e r than the i n t e n s i t y l e v e l s i n the s u r r o u n d i n g background. However, the image i r r a d i a n c e of the p r o j e c t i o n spot on d i f f e r e n t r e g i o n s of the o b j e c t s u r f a c e may v a r y s u b s t a n t i a l l y depending on the o r i e n t a t i o n of the s u r f a c e w i t h r e s p e c t to the o p t i c a l geometry of the imaging sy s t em. The p r i n c i p l e f u n c t i o n of the edge d e t e c t i o n process i s to l o c a t e the o u t l i n e of the i l l u m i n a t e d spots i n the images of the p r o j e c t i o n p a t t e r n s . 76 5 . 2 . 1 B I N A R Y T H R E S H O L D I N G I n t h e b i n a r y t h r e s h o l d i n g p r o c e s s , the image p i x e l s w i t h an i n t e n s i t y v a l u e h i g h e r t h a n some p r e d e t e r m i n e d t h r e s h o l d v a l u e , T , a r e i d e n t i f i e d as f e a t u r e e l e m e n t s o f the s p o t p r o j e c t i o n p a t t e r n , and t h e i r i n t e n s i t y v a l u e s a r e s e t to o n e . T h e p i x e l s w i t h an i n t e n s i t y l e v e l b e l o w the t h r e s h o l d v a l u e a r e known as b a c k g r o u n d e l e m e n t s , and the image i n t e n s i t i e s o f t h e s e p i x e l s a r e s e t to z e r o . B i n a r y t h r e s h o l d i n g o p e r a t i o n on g r e y - s c a l e images c a n e a s i l y be i m p l e m e n t e d i n s o f t w a r e or i n the image d i g i t i z e r h a r d w a r e . The h a r d w a r e i m p l e m e n t a t i o n t a k e s a d v a n t a g e o f t h e l o o k - u p - t a b l e f e a t u r e i n the image d i g i t i z e r to p e r f o r m the t h r e s h o l d i n g f u n c t i o n . The r e s u l t i n g b i n a r y i m a g e s o f t h e s p o t p a t t e r n s a r e d i s p l a y e d on the o u t p u t m o n i t o r o f the image a c q u i s i t i o n s y s t e m . In the h a r d w a r e i m p l e m e n t a t i o n , o n l y the b i n a r y images o f the s p o t p a t t e r n s w i l l be r e c o r d e d . In the s o f t w a r e i m p l e m e n t a t i o n , the g r e y - s c a l e images of the p a t t e r n s a r e a c q u i r e d f r o m t h e image d i g i t i z e r u n i t and t h e b i n a r y v e r s i o n s o f t h e p r o j e c t i o n p a t t e r n s a r e t h e n g e n e r a t e d . T h e r e f o r e , t h e r e s p o n s e t i m e o f the s o f t w a r e i m p l e m e n t a t i o n i s l o n g e r . I f t h e b i n a r y t h r e s h o l d i n g t e c h n i q u e f a i l s to p r o v i d e s a t i s f a c t o r y r e s u l t s , t h e n o t h e r m o r e s o p h i s t i c a t e d g r e y - s c a l e edge o p e r a t o r s w i l l be r e q u i r e d to p e r f o r m the edge d e t e c t i o n f u n c t i o n . T h e r e f o r e , i t i s a d v a n t a g e o u s to r e c o r d t h e g r e y - s c a l e images of the p r o j e c t i o n p a t t e r n s 77 d u r i n g the d e v e l o p m e n t s t a g e o f the i m a g i n g s y s t e m . S i n c e the i r r a d i a n c e i n t e n s i t i e s o f the p r o j e c t i o n s p o t s f r o m the c y l i n d r i c a l s u r f a c e v a r y as a f u n c t i o n of the s u r f a c e o r i e n t a t i o n , s e v e r a l t h r e s h o l d l e v e l s may need to be s e l e c t e d b e f o r e a s a t i s f a c t o r y b i n a r y image of the s p o t p a t t e r n i s o b t a i n e d . As g i v e n i n S e c t i o n 2 . 4 . 1 , the image i n t e n s i t y r ange o f the p r o j e c t i o n s p o t s was b e t w e e n 55 and 9 5 , the a v e r a g e i n t e n s i t y v a l u e o f the n o n - i 11umi n a t e d b a c k g r o u n d a r e a i n be tween the p r o j e c t i o n s p o t s was 22 . I f the t h r e s h o l d l e v e l i s s e t above 5 5 , the some p r o j e c t i o n s p o t s w i l l be d e l e t e d by the p r o c e s s . On the o t h e r h a n d , i f the t h r e s h o l d l e v e l i s s e t b e l o w 2 2 , some p i x e l s i n the b a c k g r o u n d n o n - i 1 1 u m i n a t e d a r e a s may be mi sc 1 a s s i f i e d as s p o t f e a t u r e e l e m e n t s . A b i n a r y image o f the s p o t p a t t e r n i s c o n s i d e r e d to be a c c e p t a b l e i f a l l the v i s i b l e spo t f e a t u r e s i n the g r e y - s c a l e images a re i d e n t i f i e d and no p i x e l f r o m the n o n - i 1 l u m i n a t e d s u r f a c e a r e a i s c l a s s i f i e d as f e a t u r e e l e m e n t by the b i n a r y t h r e s h o l d i n g a l g o r i t h m . F r o m the b i n a r y images o f the p r o j e c t i o n p a t t e r n s , t he edge p i x e l s o f the s p o t s c a n be i d e n t i f i e d by a t r a n s i t i o n o f image i n t e n s i t y w i t h any one o f t h e i r i m m e d i a t e n e i g h b o u r s i n the n o r t h , s o u t h , e a s t or w e s t d i r e c t i o n s . A n edge segment can be f o r m e d by an edge p i x e l any one o f i t s e i g h t n e i g h b o u r p i x e l s . The 8 - c o n n e c t i v i t y o f an edge p i x e l i s d e m o n s t r a t e d i n F i g u r e 5 . 3 . T h i s p r o p e r t y o f the edge p i x e l s i s u t i l i z e d by the e d g e - t r a c i n g a l g o r i t h m i n the image c e n t r o i d e s t i m a t i o n p r o c e s s to i d e n t i f y the b o r d e r o u t l i n e s o f the p r o j e c t i o n s p o t s . 78 \ 1 i\ c "**• 1 r F i g . 5.3 8 - C o n n e c t i v i t y of on edge p i x e l . 5.3 IMAGE CENTROID ESTIMATION T h e t a s k o f t h e i m a g e c e n t r o i d e s t i m a t i o n p r o c e s s i s t o l o c a t e t h e i m a g e c e n t r o i d s o f t h e i l l u m i n a t e d s p o t s p r o d u c e d b y t h e p r o j e c t i o n o f t h e s p a c e - e n c o d e d s p o t p a t t e r n s o n t h e c y l i n d r i c a l s u r f a c e . T h e i m a g e c e n t r o i d p o s i t i o n s c a n be d e t e r m i n e d f r o m t h e e d g e o u t l i n e s o f t h e s p o t s . 79 5 . 3 . 1 P R O J E C T I O N S P O T E D G E - T R A C I N G The e d g e - t r a c i n g p r o c e s s i s an o p e r a t i o n w h i c h l o c a t e s the edge p i x e l s of the p r o j e c t i o n s p o t s g i v e n i n the b i n a r y edge map of a s p o t p r o j e c t i o n p a t t e r n . The o p e r a t i o n b e g i n s by r a s t e r s c a n n i n g the b i n a r y edge map f o r an edge p i x e l . A f t e r an edge p i x e l i s f o u n d , a n e i g h b o u r h o o d edge p i x e l t r a c i n g p r o c e d u r e i s i n i t i a t e d to f i n d a l l the s u c c e s s i v e l y c o n n e c t e d edge p i x e l s w h i c h f o r m the o u t l i n e of the p r o j e c t i o n s p o t . As the o u t l i n e of the s p o t i s b e i n g t r a c e d , the c o n t r i b u t i o n f r o m each edge p i x e l to the c o o r d i n a t e v a l u e s of the c e n t r o i d i s c a l c u l a t e d . When the n e i g h b o u r h o o d edge t r a c i n g p r o c e d u r e i s c o m p l e t e , the c e n t r o i d c o o r d i n a t e s of the p r o j e c t i o n s p o t w i l l be a v a i l a b l e . The image v a l u e s o f t h e s e edge p i x e l s i n the b i n a r y edge map can t h e n be r e s e t to z e r o b e f o r e the s e a r c h of a new edge c u r v e b e g i n s . T h i s w i l l p r e v e n t the same edge c u r v e f r o m b e i n g t r a c e d a g a i n . The r a s t e r s c a n o p e r a t i o n c o n t i n u e s u n t i l a l l the s p o t o u t l i n e s i n the b i n a r y edge map have been t r a c e d . A t s u c h t i m e , the image v a l u e of a l l the e l e m e n t s i n the b i n a r y edge map s h o u l d be z e r o . S i n c e the o u t l i n e s of the i l l u m i n a t e d s p o t s on the o b j e c t s u r f a c e a r e c l o s e d , the image o u t l i n e s o f the p r o j e c t i o n s p o t s must a l s o be c l o s e d . T h e r e f o r e , any edge o u t l i n e i n a b i n a r y edge map w h i c h does not f o r m a c l o s e d p a t h i s c o n s i d e r e d to be i n v a l i d . T h u s , the c e n t r o i d p o s i t i o n of the image f e a t u r e i n s i d e an open b o r d e r c u r v e 80 w i l l n o t be c a l c u l a t e d . 5 . 3 . 2 C A L C U L A T I O N O F P R O J E C T I O N SPOT C E N T R O I D C O O R D I N A T E S T h e F r e e m a n c h a i n c o d i n g m e t h o d [ F r e e m a n 1961] was u s e d to e s t i m a t e the c e n t r o i d p o s i t i o n s o f the p r o j e c t i o n s p o t s . T h i s m e t h o d a l l o w s the c e n t r o i d c a l c u l a t i o n to be p e r f o r m e d as e a c h edge p o i n t i s t r a c e d . H e n c e , i t e l i m i n a t e s the e x t r a p r o c e d u r e o f s t o r i n g the image c o o r d i n a t e s i n a memory b u f f e r as i n the m e t h o d o f c e n t r o i d c a l c u l a t i o n b a s e d u p o n the G r e e n ' s I n t e g r a l T h e o r e m [ E l g a z z a r et a l . 1984 , H o r a u d et a l . 1 9 8 1 ] . U s i n g the c h a i n code r e p r e s e n t a t i o n , an edge c u r v e i n the b i n a r y edge map c a n be u n i q u e l y d e f i n e d by t h e p o s i t i o n o f an edge p i x e l on the c u r v e , f o l l o w e d by an o r d e r e d s e q u e n c e o f n u m b e r s d e s c r i b i n g the d i r e c t i o n v e c t o r s w h i c h p o i n t t o t h e n e x t c o n n e c t e d edge p o i n t a l o n g t h e c u r v e . The o r d e r e d s e q u e n c e w i l l t e r m i n a t e when the i n i t i a l edge p o i n t i s r e a c h e d a g a i n . The d i r e c t i o n v e c t o r s o f the e i g h t n e i g h b o u r s w i t h r e s p e c t to an edge p i x e l , P , i s shown i n F i g u r e 5 . 4 . The g e o m e t r i c p a r a m e t e r s of a p r o j e c t i o n s p o t i m a g e , s u c h as the a r e a of t h e s p o t and the moments a b o u t the v e r t i c a l and h o r i z o n t a l axes o f the image p l a n e , c a n be c a l c u l a t e d f r o m the sum of a l l the i n c r e m e n t a l c o n t r i b u t i o n s f r o m e a c h s u c c e s s i v e edge segment on the s p o t o u t l i n e . T h e c o n t r i b u t i o n s f r o m the edge s e g m e n t s o r i e n t e d i n t h e v a r i o u s d i r e c t i o n s a r e g i v e n i n T a b l e 5 . 1 [ F r e e m a n 1 9 6 1 ] . I f an edge o u t l i n e o f a p r o j e c t i o n s p o t c o n s i s t s of N 81 Ve c t o r D i r e c t i o n s A A , AM x 1 AM 8 7 6 5 4 3 2 1 Y 1 / 2 Y * Y - l / 2 Y 2 / 2 - Y / 2 + l / 6 0 0 - Y + l / 2 - Y 2 / 2 + Y / 2 - l / 6 - Y - Y 2 / 2 • Y - l / 2 - Y 2 / 2 + Y / 2 + l / 6 0 0 Y + l / 2 Y 2 / 2 + Y / 2 + l / 6 0 X 2 / 2 + X / 2 + l / 6 X 2 / 2 X 2 / 2 - X / 2 + l / 6 0 - X 2 / 2 + X / 2 - l / 6 - X 2 / 2 - X 2 / 2 - X / 2 - l / 6 T a b l e 5 . 1 A r e a a n d moments c o n t r i b u t i o n o f the edge s e gme n t s . 82 p i x e l s , the t o t a l a r e a , and the x and y moments o f the e n c l o s e d image a r e g i v e n by N A r e a Z A A . , ( 5 . 1 ) i = l 1 N M x = I A M x . , ( 5 . 2 ) i = 1 N M I AM . , , v y i m l y i ( 5 . 3 ) w h e r e A A . , AM . , and AM . a r e t h e i n c r e m e n t a l c o n t r i b u t i o n s l x i y l t o the a r e a , x -moment and y -moment of t h e p r o j e c t i o n s p o t f r o m the i * * 1 edge segment o f the b o u n d a r y c u r v e , as g i v e n i n T a b l e 5 . 1 . T h e image c e n t r o i d c o o r d i n a t e s (X , Y ) o f the c c p r o j e c t i o n s p o t a r e g i v e n by X = M / A r e a , and ( 5 . 4 ) C X Y = M / A r e a . ( 5 . 5 ) c y 5 . 4 P R O J E C T I O N S P O T COLUMN A D D R E S S D E C O D I N G T h e p r o c e d u r e s u s e d i n d e c o d i n g the c o l u m n a d d r e s s o f t h e i l l u m i n a t e d s p o t s on the o b j e c t s u r f a c e a r e p r e s e n t e d i n t h i s s e c t i o n . Due to the m i s a l i g n m e n t o f the p r o j e c t i o n s l i d e s , t h e image c e n t r o i d s o f the p r o j e c t i o n s p o t s p r o d u c e d by the s p a c e - e n c o d e d p r o j e c t i o n p a t t e r n s do n o t u s u a l l y l i n e 83 up e x a c t l y w i t h the c o r r e s p o n d i n g s p o t s g e n e r a t e d by the r e f e r e n c e p r o j e c t i o n p a t t e r n . A l t h o u g h the m i s a l i g n m e n t has b e e n m i n i m i z e d by m a n u a l a d j u s t m e n t s , the d i s p l a c e m e n t s among t h e c o r r e s p o n d i n g s p o t c e n t r o i d s f r o m t h e v a r i o u s p r o j e c t i o n p a t t e r n s a r e s t i l l n o t i c e a b l e . H o w e v e r , the s p a t i a l v a r i a t i o n s among t h e s e s p o t c e n t r o i d s h a v e b e e n r e s t r i c t e d to a f o u r p i x e l r a d i u s n e i g h b o u r h o o d f r o m the s p o t c e n t r o i d s o f the r e f e r e n c e p a t t e r n . S i n c e t h i s i s an i n h e r e n t p r o b l e m i n the p r o j e c t i o n s y s t e m , a scheme m u s t be i n c o r p o r a t e d to c o m p e n s a t e f o r the e f f e c t s c a u s e d by t h e s p a t i a l v a r i a t i o n s of t h e s e p r o j e c t i o n s p o t s . T o d e t e r m i n e the p r e s e n c e / a b s e n c e of the p r o j e c t i o n s p o t s i n t h e image of a s p a c e - e n c o d e d s p o t p a t t e r n , a n e a r e s t n e i g h b o u r s e a r c h o p e r a t i o n m u s t be p e r f o r m e d a b o u t e a c h o f the c e n t r o i d s f o u n d i n the image of the r e f e r e n c e p r o j e c t i o n p a t t e r n . I f a c e n t r o i d i s f o u n d w i t h i n t h e s e a r c h a r e a a b o u t a r e f e r e n c e c e n t r o i d , the c o l u m n a d d r e s s b i t o f the s p o t , w h i c h i s a s s o c i a t e d w i t h the s p a c e - e n c o d e d p r o j e c t i o n p a t t e r n , i s s e t to o n e . I f a c e n t r o i d i s n o t f o u n d , t h e a s s o c i a t e d c o l u m n a d d r e s s b i t o f the s p o t i s s e t to z e r o . T h e a d d r e s s w o r d s o f the r e f e r e n c e c e n t r o i d s a r e d e c o d e d by r e p e a t e d a p p l i c a t i o n s of the above p r o c e d u r e on the i m a g e s o f a l l the s p a c e - e n c o d e d p r o j e c t i o n p a t t e r n s . 5.4 . 1 E F F E C T S O F S U R F A C E D I S C O N T I N U I T Y ON A D D R E S S D E C O D I N G I f t h e r e a r e d e p t h d i s c o n t i n u i t i e s on the m e a s u r e m e n t s u r f a c e , s u c h as the edges o f the c y l i n d r i c a l s u r f a c e , the 84 p r o j e c t i o n s p o t s o n t o t h e s e a r e a s may p r o d u c e i l l u m i n a t e d s p o t s on b o t h s i d e s of the d i s c o n t i n u i t i e s . S i n c e the image p o s i t i o n o f the p r o j e c t i o n s p o t s i s a f u n c t i o n of the s u r f a c e d e p t h , the s p o t s on b o t h s i d e s o f a d e p t h d i s c o n t i n u i t y may a p p e a r as two s e p a r a t e d s p o t s on the c a m e r a i m a g e . B e s i d e s c a u s i n g s p l i t i m a g e s , d e p t h d i s c o n t i n u i t i e s c a n s o m e t i m e s j o i n the images o f d i f f e r e n t p r o j e c t i o n s p o t s t o g e t h e r . T h e c o l u m n a d d r e s s d e c o d i n g a 1 g o r i t h m w i 11 u s u a l l y f a i l to g i v e the r i g h t c o l u m n a d d r e s s f o r t h e s e p r o j e c t i o n s p o t s . F u r t h e r m o r e , b e c a u s e o f the m i s a l i g n m e n t p r o b l e m on t h e p r o j e c t i o n s l i d e s , the p r o j e c t i o n s p o t s f r o m one p r o j e c t i o n p a t t e r n may be on one s i d e of the d i s c o n t i n u i t y w h i l e t h e c o r r e s p o n d i n g s p o t s f r o m a n o t h e r p r o j e c t i o n p a t t e r n may be on the o p p o s i t e s i d e . The image p o s i t i o n s o f t h e s e c o r r e s p o n d i n g s p o t s may v a r y q u i t e s u b s t a n t i a l l y d e p e n d i n g on the m a g n i t u d e o f the d e p t h d i s c o n t i n u i t y . T h e s e s i t u a t i o n s w i l l a l s o r e s u l t i n w r o n g a d d r e s s b i t a s s i g n m e n t f o r t h e s e p r o j e c t i o n s p o t s . T h e b i n a r y e n c o d i n g scheme c u r r e n t l y u s e d i n the s p a c e - e n c o d e d p r o j e c t i o n p a t t e r n l a c k s e r r o r d e t e c t i o n c a p a b i l i t y . H o w e v e r , o t h e r e n c o d i n g s c h e m e s , s u c h as t h e G r a y code or the Hamming c o d e , c a n be u s e d to d e t e c t i n c o r r e c t l y d e c o d e d a d d r e s s v a l u e s . In t h o s e c a s e s , t h e p r o j e c t i o n s p o t s w i t h i n c o r r e c t l y d e c o d e d c o l u m n a d d r e s s w i l l be d i s c a r d e d f r o m the s u r f a c e s a m p l e p o p u l a t i o n . The i m p l e m e n t a t i o n o f s u c h e r r o r d e t e c t i o n schemes i s l e f t as a f u t u r e i m p r o v e m e n t to the i m a g i n g s y s t e m . 85 5 . 4 . 2 N E A R E S T NEIGHBOUR S E A R C H A L G O R I T H M T h e n e a r e s t n e i g h b o u r s e a r c h a l g o r i t h m i m p l e m e n t e d was d e v e l o p e d by H a l l [ 1 9 8 2 ] , and l a t e r m o d i f i e d by C l a r k [ l 9 8 5 ] f o r p e r f o r m i n g a d i a m o n d s e a r c h p a t t e r n o v e r a r e c t a n g u l a r g r i d . The g o a l o f the s e a r c h o p e r a t i o n i s to f i n d a c e n t r o i d m a r k e r w i t h i n a r a d i u s of f o u r p i x e l u n i t s f r o m t h e c e n t e r o f t h e s e a r c h a r e a . T h e s e a r c h p a t h g e n e r a t e d by t h e C l a r k a l g o r i t h m i s an o u t w a r d e x p a n d i n g d i a m o n d s h a p e d s p i r a l p a t t e r n , as shown i n F i g u r e 5 . 5 . T h e s e a r c h p r o c e d u r e c o n s i s t s o f two s t a g e s : l ) f o u r r o u n d s o f d i a m o n d s h a p e s p i r a l s e a r c h o p e r a t i o n ; a n d 2) i n d i v i d u a l t e s t i n g o f t h e e i g h t o u t s i d e d i a g o n a l g r i d p o i n t . A f t e r the 33 © 34 19 32 © © 35 20 9 18 31 © 36 21 10 3 8 17 30 37 22 11 4 1 2 7 16 29 38 23 12 5 6 15 28 © 39 24 13 14 27 © 40 25 26 © 41 F i g . 5 .5 neighbour search a l go r i thm. 86 c o m p l e t i o n o f f o u r r o u n d s of s p i r a l o p e r a t i o n s , the s e a r c h w i l l s t o p at the g r i d p o i n t m a r k e d number 41 i n the d i a g r a m . I n d i v i d u a l t e s t s a r e t h e n u s e d to e x a m i n e the e i g h t d i a g o n a l g r i d p o i n t s , n u m b e r e d 42 t h r o u g h 4 9 , w h i c h a r e a l s o w i t h i n the f o u r p i x e l r a d i u s s e a r c h a r e a . The c o m p l e x i t y o f the s e a r c h a l g o r i t h m i n c r e a s e s when the s e a r c h p o i n t r e a c h e s the b o u n d a r y o f a s e a r c h r e g i o n . A c o m p l i c a t e d scheme was d e v e l o p e d t o d e t e r m i n e t h e l o c a t i o n o f the s u b s e q u e n t s e a r c h p o i n t s a t the b o u n d a r y . The d e t a i l e d d e s c r i p t i o n of the i m p l e m e n t a t i o n o f t h i s s e a r c h a l g o r i t h m c a n be f o u n d i n C l a r k [ 1 9 8 5 ] . 5 . 5 D E T E R M I N A T I O N O F T H E 3 - D S P A T I A L C O O R D I N A T E S O F T H E P R O J E C T I O N SPOTS T h e 3 - D s p a t i a l c o o r d i n a t e s o f a p r o j e c t i o n s p o t on the o b j e c t s u r f a c e , w i t h r e s p e c t to the g l o b a l r e f e r e n c e c o o r d i n a t e s y s t e m , c a n be c a l c u l a t e d f r o m the image c e n t r o i d c o o r d i n a t e s and the c o l u m n a d d r e s s o f the s p o t . L e t the s p a t i a l c o o r d i n a t e s at t h e c e n t e r o f a p r o j e c t i o n s p o t on the c y l i n d r i c a l s u r f a c e be ( x , y , z ) , and t h e c o l u m n and r o w a d d r e s s e s o f t h i s p r o j e c t i o n s p o t i n the p r o j e c t i o n p a t t e r n be ( u , v ) . The h o m o g e n e o u s r e l a t i o n s h i p o f t h e p o s i t i o n v e c t o r o f t h i s s u r f a c e p o i n t and t h e a d d r e s s v e c t o r of the s p o t c a n be g i v e n b y : 87 ut vt t P l l P12 P13 P14 P21 P22 P23 P24 P31 P32 P33 P34 (5.6) wh ere {p.^} i s the p e r s p e c t i v e t r a n s f o rma t i o n m a t r i x of the s l i d e p r o j e c t o r , and t i s an a r b i t r a r y s c a l i n g c o n s t a n t . S i m i l a r l y , the homogeneous r e l a t i o n s h i p between the p o i n t ( x , y , z ) i n the r e f e r e n c e c o o r d i n a t e s y s t e m , and the image c e n t r o i d of the s p o t , (x ,y ) i n the image p l a n e , can be c c r e p r e s e n t e d by: X s c c s C l l C12 °13 C14 C21 C22 C23 C24 C31 C32 C33 C34 (5.7) where { c ^ j ) i s the p e r s p e c t i v e t r a n s f o r m a t i o n m a t r i x of the camera, and s i s an a r b i t r a r y s c a l i n g c o n s t a n t . S u b s t i t u t i n g f o r t and s i n E q u a t i o n s ( 5 . 6 ) and ( 5 . 7 ) y i e l d s , ( c H - C 3 1 X c ) x + ( c 1 2 - C 3 2 X c ) y + ( c 1 3 - C 3 3 X c ) z ( c 2 r C 3 1 y c ) x + ( c 2 2 - C 3 2 y c ) y + ( c 2 3 - C 3 3 y c ) z ( P n - P 3 1 u ) x + ( P 1 2 - P 3 2 u ) y + C p 1 2 - p 3 3 u ) z 34 c 14 C34 Xc- C24 P 3 4 U " P14 ( p 2 1 - p 3 1 v ) x + ( P 2 2 - P 3 2 v ) y + ( p 2 3 " P 3 3 u ) z P 3 4 V " P24 (5.8) 88 I t i s s u f f i c i e n t to s o l v e f o r the unknown v a r i a b l e x, y and z by u s i n g the f i r s t t h r e e e q u a t i o n s i n ( 5 . 8 ) . A 3x3 s y s t e m o f l i n e a r e q u a t i o n s can t h e n be f o r m e d and g i v e n i n m a t r i x r e p r e s e n t a t i o n a s : C l l " C 3 1 X c C12" C32 Xc c13" C33 Xc C 2 l " C 3 1 y c C22" C32 yc C23~ C33 yc P l l - p 3 1 U P 1 2 - p 3 2 U Pl3-P33 U or , A X = P x y z C, .X - c , . 34 c 14 c. x -c 34 c 24 P 3 4 U "Pl4 (5.9) ( 5.10) By c o m p u t i n g the i n v e r s e of m a t r i x A, the s p a t i a l c o o r d i n a t e s at the c e n t e r of a p r o j e c t i o n s p o t on the c y l i n d r i c a l s u r f a c e can be e x p r e s s e d a s : A" 1 P ( 5 .11) The c y l i n d r i c a l s u r f a c e was m a n u f a c t u r e d a c c o r d i n g to the f o l l o w i n g a m p l i t u d e f u n c t i o n of z ( i n c h e s ) : [12 2 - ( y - 8 . 6 2 5 ) 2 ] 1 / 2 - 9.20 , 0.0£x£l2.0 and 0.875^16.375 0 , otherwise (5.12) U s i n g t h i s a n a l y t i c a l f u n c t i o n , s u r f a c e a m p l i t u d e v a l u e , z, of the measurement e r r o r each p r o j e c t i o n spot of on the the 89 c y l i n d r i c a l s u r f a c e c a n t h e n be c a l c u l a t e d . 5.6 SOFTWARE I M P L E M E N T A T I O N OF T H E IMAGE P R O C E S S I N G A L G O R I T H M A l l the image p r o c e s s i n g c o n c e p t s d e s c r i b e d i n t h i s c h a p t e r a r e i m p l e m e n t e d i n a F o r t r a n s o f t w a r e p a c k a g e c a l l e d P O S . T h e s o f t w a r e p a c k a g e was d e v e l o p e d on the V A X 1 1 / 7 5 0 c o m p u t e r a t E l e c t r i c a l E n g i n e e r i n g , U B C and D e f e n c e R e s e a r c h E s t a b l i s h m e n t P a c i f i c (DREP) i n V i c t o r i a , B . C . The VMS o p e r a t i n g s y s t e m i s u s e d i n the V A X c o m p u t e r i n b o t h l o c a t i o n s . T h e RAMTEK 9300 and t h e I n t e r n a t i o n a l I m a g i n g s y s t e m m o d e l 75 image p r o c e s s o r a r e a v a i l a b l e at the UBC and DREP c o m p u t e r f a c i l i t i e s , r e s p e c t i v e l y , f o r image v i e w i n g . S i n c e two d i f f e r e n t c o m p u t e r f a c i l i t i e s w e r e i n v o l v e d i n the p r o g r a m d e v e l o p m e n t p r o c e s s , i t i s e s s e n t i a l t h a t p r o g r a m and d a t a f i l e c o m p a t i b i l i t y b e t w e e n t h e two s y s t e m s must be m a i n t a i n e d . A g e n e r a l p u r p o s e g r a p h i c s p a c k a g e c a l l e d M A G I C i s a v a i l a b l e at DREP to g e n e r a t e 2 - D or 3 - D t e r m i n a l g r a p h i c s d i s p l a y s . H a r d c o p y v e r s i o n s o f the g r a p h i c s a r e a v a i l a b l e on t h e P r i n t r o n i x Mo d e l 300 p r i n t e r / p l o t t e r or on t h e QMS l a s e r p r i n t e r . 5.6.1 SOFTWARE D E S I G N APPROACHES T h e s o f t w a r e p a c k a g e POS was d e v e l o p e d u s i n g s t r u c t u r a l p r o g r a m m i n g m e t h o d o l o g y a n d t o p - d o w n p r o g r a m m i n g t e c h n i q u e s . T h e image p r o c e s s i n g s o f t w a r e p a c k a g e i s 90 d e s i g n e d to be a p o r t a b l e s t a n d a l o n e p a c k a g e to be r u n on d i f f e r e n t c o m p u t e r s y s t e m s . In o r d e r to o b t a i n c o m p a t i b i l i t y among c o m p u t e r s y s t e m s , o n l y s t a n d a r d F o r t r a n 77 i n s t r u c t i o n s w e r e u s e d i n the p r o g r a m . O b v i o u s l y , s y s t e m l i b r a r y f u n c t i o n s o f the VMS o p e r a t i n g s y s t e m and e x t e r n a l l y r e f e r e n c e d s u b r o u t i n e s c o u l d n o t be u t i l i z e d . T h e m o d u l a r i t y c o n c e p t was a l s o i n c o r p o r a t e d i n t o the s o f t w a r e p a c k a g e i n a n t i c i p a t i o n o f f u t u r e e x p a n s i o n . A n o p e r a t i o n a l v e r s i o n of the s p o t p r o j e c t i o n i m a g i n g s y s t e m i s s c h e d u l e d t o be i n s t a l l e d at DREP n e x t y e a r . In m o d u l a r -s t r u c t u r e d p r o g r a m s , f u t u r e t a s k s t h a t a r e d e s i g n e d s p e c i a l l y f o r the a p p l i c a t i o n s a t DREP c a n be r e a d i l y i n c o r p o r a t e d i n t o the s y s t e m . In t h i s s o f t w a r e p a c k a g e , e a c h image p r o c e s s i n g t a s k i s i m p l e m e n t e d as a f u n c t i o n m o d u l e . F u t u r e t a s k s c a n be a d d e d to the s y s t e m i n the f o r m of new f u n c t i o n m o d u l e s . A n e x e c u t i v e d r i v e r r o u t i n e i s r e s p o n s i b l e f o r i n v o k i n g the r e q u i r e d f u n c t i o n a l m o d u l e s i n t h e p r e s c r i b e d s e q u e n c e to compute the s p a t i a l c o o r d i n a t e s o f the p r o j e c t i o n s p o t s on the m e a s u r e m e n t s u r f a c e . 5 . 6 . 2 D E S C R I P T I O N O F SOFTWARE T h e e x e c u t i v e d r i v e r p r o g r a m POS i s the n u c l e c u s o f the s o f t w a r e p a c k a g e . I t f e t c h e s t h e image d a t a f r o m the d a t a f i l e s a n d a c t i v a t e s t h e f u n c t i o n a l m o d u l e s to p e r f o r m image p r o c e s s i n g t a s k s , s u c h as b i n a r y t h r e s h o l d i n g , s p o t o u t l i n e t r a c i n g , s p o t c e n t r o i d e s t i m a t i o n , s p o t c o l u m n a d d r e s s d e c o d i n g and s p a t i a l c o o r d i n a t e c a l c u l a t i o n s . T h e 91 i n p u t i n f o r m a t i o n f o r the image p r o c e s s i n g p r o g r a m a r e the s i x g r e y - s c a l e i m a g e s o f the s p a c e - e n c o d e d s p o t p a t t e r n s , and the t r a n s f o r m a t i o n m a t r i c e s o f the c a m e r a and s l i d e p r o j e c t o r . T h e p r o g r a m c o m p u t e s the s p a t i a l c o o r d i n a t e s o f t h e p r o j e c t i o n s p o t s on the m e a s u r e m e n t s u r f a c e f r o m t h e s e i n p u t i n f o r m a t i o n . A l i s t i n g o f the e x e c u t i v e d r i v e p r o g r a m i n p s e u d o h i g h - l e v e l l a n g u a g e i s g i v e n as f o l l o w s : B E G I N : {PROGRAM ' P O S ' } G e t the image d a t a o f the r e f e r e n c e p r o j e c t i o n p a t t e r n C A L L e d g e_f i 11 e r C A L L t r a c e I = 2 D O 1 U N T I L I > 6 G e t the image d a t a o f t h e I** 1 s pa c e - enc od e d p a t t e r n C A L L e d g e _ f i l t e r C A L L t r a c e C A L L ma r k c e n t r o i d DO 2 J = 1 , N {N i s t h e number o f s p o t f e a t u r e s f o u n d i n t h e r e f e r e n c e p a t t e r n } C A L L s e a r c h I F (a c e n t r o i d i s f o u n d ) T H E N a d d r e s s b i t " 1 - 2 " i s s e t to one E L S E a d d r e s s b i t " 1 - 2 " i s s e t to z e r o E N D I F 92 I - I + 1 2 C O N T I N U E 1 C O N T I N U E G e t the c a m e r a and p r o j e c t o r t r a n s f o r m a t i o n m a t r i c e s DO 3 J - 1 , N C A L L p o s i t i o n 3D 3 C O N T I N U E END PROGRAM: A number o f s u b r o u t i n e s a r e c a l l e d f r o m t h i s p r o g r a m . T h e d e s c r i p t i o n o f t h e s e s u b r o u t i n e s a r e g i v e n as f o l l o w s : 1) e d g e _ J l i t e r T h i s s u b r o u t i n e p e r f o r m s b i n a r y t h r e s h o l d i n g o p e r a t i o n on the i n p u t g r e y s c a l e image and g e n e r a t e s a b i n a r y o u t p u t i m a g e . 2) trace T h i s s u b r o u t i n e t r a c e s the edge o u t l i n e s o f t h e s p o t f e a t u r e s f o u n d i n the b i n a r y i m a g e . The c e n t r o i d p o s i t i o n s a r e c a l c u l a t e d f r o m t h e image c o o r d i n a t e s of the edge p o i n t s . I t a l s o r e t u r n s t h e t o t a l number o f s p o t f e a t u r e s f o u n d i n the i m a g e . 3) mark centroid T h i s s u b r o u t i n e m a r k s the c e n t r o i d s o f the s p o t f e a t u r e s on a 512x512 image m a t r i x w i t h the i n t e n s i t y v a l u e o f 255 ( f u l l s c a l e ) . 93 4) search T h i s s u b r o u t i n e p e r f o r m s a s p i r a l s e a r c h o p e r a t i o n o v e r a f o u r p i x e l r a d i u s n e i g h b o u r h o o d a b o u t a s p e c i f i e d image p o s i t i o n t o s e a r c h f o r a c e n t r o i d m a r k e r , i . e . a p i x e l w i t h a v a l u e o f 2 5 5 . T h e l o g i c a l v a r i a b l e FOUND i s s e t to T R U E i f a c e n t r o i d m a r k e r i s f o u n d , o t h e r w i s e , i t i s s e t to F A L S E . 5) po s i t ion_3D T h i s s u b r o u t i n e c o m p u t e s the 3 - D s p a t i a l c o o r d i n a t e s o f the p r o j e c t i o n s p o t s f r o m the g i v e n t r a n s f o r m a t i o n m a t r i c e s f o r the c a m e r a and s l i d e p r o j e c t o r , a n d the s p o t c o l u m n a d d r e s s e s and c e n t r o i d c o o r d i n a t e s . 5 .7 E X P E R I M E N T A L R E S U L T S A b i n a r y image o f the s p o t p a t t e r n i s c o n s i d e r e d to be a c c e p t a b l e i f a l l t h e v i s i b l e s p o t f e a t u r e s i n t h e g r e y -s c a l e image a r e i d e n t i f i e d and no p i x e l o f the n o n -i l l u m i n a t e d a r e a s i s c l a s s i f i e d as a f e a t u r e e l e m e n t . A f t e r a n a l y z i n g t h e g r e y - s c a l e images o f t h e s p o t p r o j e c t i o n p a t t e r n s a c q u i r e d i n the shape m e a s u r e m e n t o f the c y l i n d r i c a l s u r f a c e , a t h r e s h o l d l e v e l o f 45 was e m p i r i c a l l y d e t e r m i n e d to p r o d u c e a b i n a r y image o f the s p o t p a t t e r n s t h a t s a t i s f i e d t h e c r i t e r i a s u g g e s t e d a b o v e . I t mu s t be e m p h a s i z e d t h a t t h i s t h r e s h o l d l e v e l i s s y s t e m d e p e n d e n t . T h e v a l u e o f the t h r e s h o l d l e v e l s h o u l d v a r y a c c o r d i n g to 94 the a m b i e n t l i g h t c o n d i t i o n , the s e n s i t i v i t y o f the camera and the b r i g h t n e s s of the s p o t p r o j e c t i o n p a t t e r n s . In t h i s a p p l i c a t i o n , the p r o j e c t i o n s p o t s we re the o n l y v i s i b l e f e a t u r e s i n the i m a g e s and an a c c e p t a b l e t h r e s h o l d l e v e l was o b t a i n e d q u i t e r e a d i l y . Two o t h e r edge d e t e c t i o n a l g o r i t h m s , the S o b e l g r a d i e n t o p e r a t o r [ R o s e n f e l d and K a k , 1 9 8 2 ] , and the V G o p e r a t o r [ M a r r and H i l d r e t h 1 9 8 0 ] , w e r e a l s o u s e d f o r d e t e c t i n g the s p o t f e a t u r e s i n the i m a g e s o f the s p o t p a t t e r n s . S a t i s f a c t o r y r e s u l t s w e r e • a l s o r e p o r t e d f o r t h e s e two a l g o r i t h m s . By t a k i n g a d v a n t a g e s o f t h e c o n t r o l l e d l i g h t i n g o p e r a t i o n c o n d i t i o n o f the p r o j e c t i o n s y s t e m , c omp a r a b l e r e s u l t s we re o b t a i n e d u s i n g the b i n a r y t h r e s h o l d i n g m e t h o d w i t h s u b s t a n t i a l l y l e s s c o m p u t a t i o n t i m e . T h u s , b i n a r y t h r e s h o l d i n g was i m p l e m e n t e d i n t o the image p r o c e s s i n g a l g o r i t h m . T h e b i n a r y images of the s p a c e - e n c o d e d p r o j e c t i o n p a t t e r n s f o r m e d by u s i n g a t h r e s h o l d l e v e l o f 45 a r e shown i n F i g u r e 5 . 6 . F r o m t h e s e b i n a r y images o f t h e s p o t p a t t e r n , the edge p i x e l s o f the s p o t s w e r e i d e n t i f i e d . The b i n a r y edge maps o f t h e p r o j e c t i o n s p o t s a r e shown i n F i g u r e 5 . 7 . The image c e n t r o i d c o o r d i n a t e s o f the p r o j e c t i o n s p o t s w e r e e s t i m a t e d u s i n g t h e F r e e m a n c h a i n c o d i n g m e t h o d and the s p o t c e n t r o i d l o c a t i o n s a r e ma r k e d as " + " i n the c e n t r o i d d i s t r i b u t i o n maps as g i v e n i n F i g u r e 5 . 8 . T h e l i s t o f the c e n t r o i d c o o r d i n a t e s and the d e c o d e d c o l u m n a d d r e s s o f the p r o j e c t i o n s p o t s c a n be f o u n d i n A p p e n d i x 1 . T h e e d g e s o f the c y l i n d r i c a l s u r f a c e h a v e c r e a t e d 95 • IMIIMMIMiHIIMtMllMMMi • • t %r m « w # f> w # # i • • • • a a a a a • a • • a> • • • • • • • • • • » • • » # • » • • • • » • « a a a • • a a> • • » i i » i i l i i i i M M i i i • • • • • • a a • < • • • at I • at a> i • • a • i a a a • i aaa a> < • a* » aaa a> • •» •> • •» • • •• • • • * • • • a> a> • a * • • •••••I D l l l t l l l l M l O • g • • • • • • I I I I I I I M t t l l l l M I • • oaa»i»»»»»»"»»«»»»»»,»»» • ••••••••MIMM , B aiattiitiiiiiiiiii • • • aitiniiiiiiiMiiitiiiiiiii lailDIIIIUHMIM mi , , • • • Mini KtllllMIMMM <'••••• , . • • ••ItlllHMI >»M ..IIIIIIIMI M I I M X M l i .••••IIMMIMM aaa i m • 9 ****',',*t^"^^^ , . ••• ••• » a ••••••••••••.»»»»«»n»«»aa • • • • • • • • • • • • • a a • • a • a a • • • • • • • • • • • • a • a • • a • • • a • a • • • • • a • a • • • • a a a K a • • • • a a • t • a i • • • a a • • a • • • • • • • • • » • • • • • a a a • • • ••••• • • a a a • • • a a a » • • a a a • • • • • • • • • a a a * • • a a • • • a a a • • • • » • • » • • a a a • • • • • > • * • • a a » • • • a a a • • • t a * • • • • a a • a) • • a a a a a a mm a a • • • • a a a a a a a a • • • • • • • • a a • • a a a a mm • • a a • • • • • • a a a a a a a V. • • • . • • • # • . a * . #• a a a a • a> • • • • • • a a aa • • a a • a» a a • • • • a a aa a a • • • • • • • • a t a t a a • a • • a a a a at a a a a> • • • • • • a a a* at a a • • • • • • a a a* at a a • m mm mm a a a a aa at a a • a • • • • » a aa •• a a » a mm mm mm a a aa mm m, m mm mm mm a a aa at a a • a • • • • • • a a aa i t a a a a • • • • a a at » . a a si m mm mm mm a t a t a a b) a a a a • • • a a a a a a a a a • a a a a a a a a a o a • • • • a a a a a a o a * • »>••> • • - • • » . . - - .. a a a a • a a a a a a a • • • • a a a a a a a a • • • • • • » • a a a a • • • • • • • • a a a a • • • • • a a a a • • • • • • • • a a a a a a a a • • • • a a a a a a a a a a a a • • • • • a a a a • a> • • » a a a a a a a a> a a • • • • • a a a a • • a • • • • • a a a a a • • • • • • • a a a a a a a a • • • • a a a a a a a a • • • • a a a a a a a a a a a a a a a a • • • • a a a a a a a a • a a a a a a a a a a a • • a a a a a a a a a a • • • • a a a a a a a a a a a a a a a a t a a a • a a a • a a a t a a a • a a a • a a a t a a a • a a a • a a a • a a a t a a a • a a a • a a a • a a a t a a a • a a a • a a • a a • a a • a a c) • • • • • a a a ' * w a r a a • » . •••••••• • • • • • • • a •••••••a ••••••a* d) • •••ta a a •••••a a a • ••••a* a • ••••aaa • ••••a a a • ••••aaa u m i a a ••••ta a a M I I I I a a t i m i t a aatt* a a a • attt a a a ••••• a a a I I M I I I a • ••••aa a • •nt a a a ••••< a a a a t e i t a a a a t t i t a a a aatt i • • • e) f ) F i g u r e 5.6 ( a ) - ( f ) B i n a r y images o f the p r o j e c t i o n p a t t e r n s 96 u u u u u u i i c O O C O O o o r j O O O O O O O O n o c o c n o o I O O O O O O O O O O O D O O D O O O D D O O D e o c o r o o o o ' O O O O O O D t i C O O O O D D O O O D O O D O U O O O n c D O C i ! G D P D O O P O O D O O O O D O O O D O O O O D O C D O o n r . r> j 0 0 0 < a = O <Wc? O ^ W D O f J O O O O O i 5 * ' j C e o o o r, o ' C O O O o O O D D O O D O O O O D D O O O D O O o i o i i d O O i O O O O O O D O O O O D D O O O O D O H O d b C o o u i U D O O O O O O D O O O O O O O O O O O O O D O O U O O o o o x l D O D D O o O D O O O O D O O D D O O D O D D O O O o c i c > o o o o o O O O O O O O O O O O O O D O O O O D O O O D O o o o o o n o c i O O O O O O O O O O O O O O O O O D O O O O O C O O U O e O O O O O O O O O O O O O O D O O O O O O O O O O O O O O o n o O o o O O O O O O O O O D D O O D D O D O O O D D O D D D O i i D O O O O O O D O O G O D O O O D O O D O O O O O O O O O o e o o o o D O O O O O D O O O O D O O O O O O O O O O O O O O o o o G O O O O O O O O D D O O O O O O O D O O O O O O O O O O o n c i O O O O O O O O Q O O O O D O O O O O O O O O O P O O O o o o o o o O D D O O O O O O D D O O O O D O O O D O 0 0 0 0 0 0 0 0 0 0 0 O O O D O D D O O O O O D D O O O O O O O O O O O o o o o O O O O O O o O D D D O O O O O O O O O O O O O O O O o e o f U i O O D O O O O O O D O OOOOOOOOOOOOOOOOO000 O O O O O D O O O O O O O O D O O O O O O O O O O O O O O C O O O O D O D D D D O O O O D D O 0 O 0 0 0 O 0 0 0 D 0 0 00D0IG0O n O D D O O O O D O D O O O D O O O O D O O O u o o o o t o u O O O C t D O O O O O O O O O D D O D O O O O O O O O O O O O O O O ^ < ^ U U O O D 0 O « > O 0 C > c . c o O O O O O O O O O O C O D ^ O o O O O O O O O D O P O O O O o o D O O O O O O O O O O O O P D O O t j . e _ c D < t - c - . - « , « > . » . o o c O o o o O O O O o o c o * * o o O O O O O O O O O O f > » o » n p O O O O O O D O O 0 0 0 0 . 0 O O O O D O O O O O O O O . . O O O O O O O O O O O O O O o a o o O O O O D O O O 0 0 D 0 l > 0 O O O D O O O O O O O O O D . O O O D D D O O D O O O O O O . O O O O D O O O O O O O O D O ' O O O O O O O O O O O O O O O o o o O D O O O O O O O O O O O I O O O O O O O O O O O O O e o o o o O D O 0 0 0 0 0 0 0 0 0 0 * 0 0 O O O O O 0 0 0 0 0 0 0 1 » O 0 O D D D O O O 0 0 0 0 0 0 * 0 o 0 O D O O O O O O O O O O O O O O o O O O O 0 O 0 0 0 0 0 » O O O D O O O O O O O O O O D . O O 0 O O O O ' O O O O D O O O O D O O D O O D O O D O O O O O O O O a) u u O 0 0 0 0 0 0 0 0 o a o 0 0 o o O 0 o • o o O 0 o o D o o o O 0 o o 0 0 0 0 O 0 o o D 0 o o 0 0 O D DO 0 0 0 0 O Ci O O O O . o o O 0 ° ° o O " e 0 " .O" o o o o 0 o o o O 0 0 0 0 0 0 0 • 0 o o 0 o o o o 0 O D 0 0 o o DP O 0 o o o o O 0 0 0 0 0 0 0 0 * o 0 o <=> o o O D 0 0 0 0 oo » r 0 0 0 O o o 0 0 0 0 0 0 0 0 o> o o O O o o O O o o 0 0 0 0 • • 0 O 0 O 0 o o o 0 0 0 0 oo 0 0 o o 0 D o o o o 0 0 0 0 0 0 • 0 O 0 D O D O 0 0 0 0 0 0 0 0 CO o o O O o o o o 0 0 0 0 oo • 0 o o O O o o O 0 0 0 0 0 0 0 oo o o D O o o O O 0 0 DO 0 0 DO o 0 D O o o O 0 0 D 0 0 0 0 oo o o O O D O o o D O 0 0 oo <• 0 0 O O o o D 0 0 0 0 0 0 0 DO o o O O D O 0 0 o o 0 0 DO DO O 0 D D O O o o o o o o oo DO o 0 O D • o o o o o 0 0 0 0 oo 0 0 O O o o o o 0 0 0 0 O D DO o o o i o o o o o o o 0 0 0 0 DO 0 o b) U 0 0 0 O 0 D 0 o o o o 0 D o D 0 0 0 0 0 0 0 0 o o o o O O D O 0 0 D O 0 D 0 0 o o o o o o o o 0 0 D O O O O O o o o o O O O O • O t r f - o o • - * O o o o O O D O O O O O O O O O o o o o • O O O O O O O 0 0 0 0 o o o o •O D 0 O O O O O O O O o o o o DO 0 0 O O O O 0 0 0 0 o o o o DO o o O O D O O O O O o o o o • O O O O O O O o o o o o o o o D O D O O O O O O O O O o o o o • O O O O O O O O O O O o o o o • O O O O O O O O O O O o o o o D D O O O O O O O O O O 0 D 0 0 DO 0 0 O O O O O O O O o o o o DO o o O O O O D D O O o o o o • O O D O O O O D D O O o o o o DO o o O O O O O O O O o o o o • O O O O O O O o o o o o o o o • 0 0 0 O O O O O O O O o o o o • O O O D O D D D O D O o o o o • O O D O O O O O O O O ODOO • O O O O O O O O O O O o o o o • O O O O O O O O O D O o o o o i D C i o c) O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O • • O O o o o O O O O O O O O O O O O O O O O O D O O O O D O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O aooooooo OOOOOOOO O O O O O O D O O O O O O O O O O O O O O O O O o o o o o o o o O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O • O O O O O O O O O O D D O D O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O P V I D O O O O O O O O O O 0 0 0 0 I O o o 0 0 0 0 * 0 0 o ooooio o o t g i u o o o t m i t o r j D OOODO O O D O O O O O O O O O D I I I O O O ooooe o o o 0 0 0 0 0 O O D O O 0 0 I o o o O O D 0 I O O O O D 0 O 0 O O O 00001 o o o t o i t t o o o OOOOOOOO oottt o o o C 0 0 0 I O O O e) d) O O D O D O O O D O O O D O O D O O O O O O D O O O O O O O D O 0 0 0 0 0 O 0 0 O 0 O 0 0 O O 0 O O O O O O O O O O O O O O O O *> ° ° 0 D O P A Q O P O O D D O D D D O O O O D O D t t D D O D OODOOOOOODOOOOOO 0 0 D 0 O D O 0 0 0 0 0 O D O C OOOOOOOODOOOODO O 0 0 0 0 0 D 0 0 D 0 0 OOODO OOOOOOOOOOOOIOOO OOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOO o o o o D o o o o o o o o n o o OOOOOOOOOOOOOOOO ODODDOOODODDOODO 0 0 OOOOOOOO OOOO 0 0 OOOOOOOOOOOOOOOO 0 0 0 0 0 0 OO OOOOOOO o OOO 0 0 0 0 0 0 0 D D 0 O o o OOOOOOODDOODOOOO O 0 0 O D 0 0 0 O O O O O O O o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 D 0 OOOOOOOOOOOOOOOO f ) F i g u r e 5.7 ( a ) - ( f ) B i n a r y edge maps of the p r o j e c t i o n p a t t e r n s . 97 • •> *>• • •**••••>• < a) b) c) d) e) f ) F i g u r e 5.8 ( a ) - ( f ) S p o t c e n t r o i d s d i s t r i b u t i o n map! 98 d e p t h d i s c o n t i n u i t i e s i n the s c e n e . I t i s e v i d e n t f r o m row number f i v e of the p r o j e c t i o n p a t t e r n shown i n F i g u r e s 5 . 6 ( a ) and 5 . 7 ( a ) t h a t the d e p t h d i s c o n t i n u i t i e s have c a u s e d some p r o j e c t i o n s p o t s to a p p e a r as s p l i t i m a g e s and some o t h e r s p o t s i m a g e s to be j o i n e d t o g e t h e r . F u r t h e r m o r e , b e c a u s e o f t h e m i s a l i g n m e n t p r o b l e m of the p r o j e c t i o n s l i d e s , some s p o t s f r o m one p r o j e c t i o n p a t t e r n a r e on one s i d e o f the d i s c o n t i n u i t y w h i l e the c o r r e s p o n d i n g s p o t s f r o m a n o t h e r p r o j e c t i o n p a t t e r n a r e on the o p p o s i t e s i d e . H e n c e , t h e image c e n t r o i d p o s i t i o n s of t h e s e s p o t s f r o m the d i f f e r e n t p r o j e c t i o n p a t t e r n s v a r y more t h a n 4 p i x e l s w i t h r e s p e c t to t h e i r r e f e r e n c e c e n t r o i d p o s i t i o n s . T h e s e s i t u a t i o n s a r e w e l l i l l u s t r a t e d by the s p o t s i n r o w number f i v e o f t h e p r o j e c t i o n p a t t e r n s i n F i g u r e s 5 . 6 ( a ) - f ) and 5 . 7 ( a ) - f ) . T h e f i r s t e r r o r c o n d i t i o n due to d e p t h d i s c o n t i n u i t y i s an i n h e r e n t p r o b l e m w i t h s p o t p r o j e c t i o n s c h e m e , and i s u n a v o i d a b l e u n l e s s the s y s t e m i s o n l y u s e d w i t h s m o o t h , c u r v e d s u r f a c e s . H o w e v e r , the s e c o n d e r r o r c o n d i t i o n due t o d e p t h d i s c o n t i n u i t y and s l i d e m i s a l i g n m e n t c a n be r e s o l v e d by u s i n g an e 1 e c t r o - o p t i c a 1 L C D l i g h t s h u t t e r i n s t a l l e d i n the s l i d e p r o j e c t o r t o g e n e r a t e the d i f f e r e n t s p a c e - e n c o d e d p a t t e r n s u n d e r e l e c t r o n i c c o n t r o l . B y e x a m i n i n g the d e c o d e d c o l u m n a d d r e s s of the p r o j e c t i o n s p o t s l i s t e d i n A p p e n d i x 1, i t i s f o u n d t h a t the o n l y p r o j e c t i o n , s p o t s w h i c h have i n c o r r e c t l y d e c o d e d c o l u m n a d d r e s s e s a r e t h e ones i n the i m m e d i a t e v i c i n i t y of the d e p t h d i s c o n t i n u i t y . I t i s a l s o e v i d e n t t h a t the a d d r e s s d e c o d i n g e r r o r s a r e c a u s e d by the two s i t u a t i o n s d i s c u s s e d 99 a b o v e . F r o m the r e s u l t s o f the o t h e r p r o j e c t i o n s p o t s , we c a n c o n c l u d e t h a t the m a n u a l a l i g n m e n t p r o c e d u r e p e r f o r m e d b e f o r e t h e p r o j e c t i o n o f e a c h s p o t p a t t e r n has managed to r e d u c e t h e a l i g n m e n t e r r o r s among the p r o j e c t i o n s l i d e s to a m a x i m u m o f f o u r p i x e l u n i t s . M o r e o v e r , t h e y have a l s o i n d i c a t e d t h a t the n e a r e s t n e i g h b o u r s e a r c h a l g o r i t h m has a c c u r a t e l y d e t e c t e d the a p p r o p r i a t e c e n t r o i d m a r k e r s i n the c e n t r o i d d i s t r i b u t i o n maps of the s p a c e - e n c o d e d p r o j e c t i o n p a t t e r n s d u r i n g the s p o t a d d r e s s d e c o d i n g p r o c e s s . T h e s p a t i a l c o o r d i n a t e s o f t h e p r o j e c t i o n s p o t s o b t a i n e d by the i m a g i n g p r o c e s s i n g a l g o r i t h m and the m e a s u r e m e n t e r r o r s o f the s u r f a c e a m p l i t u d e v a l u e s are a l s o l i s t e d i n A p p e n d i x 1. F r o m the shape m e a s u r e m e n t r e s u l t s o f the c y l i n d r i c a l t e s t s u r f a c e , i t i s e v i d e n t t h a t the o v e r a l l a c c u r a c y o f the s u r f a c e a m p l i t u d e m e a s u r e m e n t r e s u l t s i s b e t t e r t h a n a q u a r t e r o f an i n c h , e x c e p t f o r the s a m p l e s t h a t a r e i n t h e i m m e d i a t e v i c i n i t y o f the e d g e s on the c y l i n d r i c a l s u r f a c e . I t i s b e c a u s e the a d d r e s s d e c o d i n g p r o c e s s f a i l e d t o p r o v i d e the c o r r e c t c o l u m n a d d r e s s o f t h e s e s a m p l e p o i n t s . L a r g e s u r f a c e a m p l i t u d e e r r o r s at t h e s e s a m p l e s a r e c a u s e d by the i n c o r r e c t c o l u m n a d d r e s s v a l u e s b e i n g u s e d i n t h e s p a t i a l c o o r d i n a t e s c a l c u l a t i o n p r o c e s s . I f t h e s e s p e c i a l s a m p l e p o i n t s a r e i g n o r e d , and o n l y the i n t e r i o r p r o j e c t i o n s p o t s on the c y l i n d r i c a l s u r f a c e a r e c o n s i d e r e d , t h e a v e r a g e of the m e a s u r e m e n t e r r o r m a g n i t u d e s o f t h e i n t e r i o r p r o j e c t i o n s p o t s i s l e s s t h a n o n e - t e n t h o f an i n c h . T h i s r e s u l t c a n t h e n be u s e d to d e s c r i b e the a c c u r a c y o f the s p o t p r o j e c t i o n i m a g i n g s y s t e m . 100 A s d e s c r i b e d e a r l i e r i n S e c t i o n 2 . 4 , a 32x32 s p o t p a t t e r n was u s e d i n the s h a p e m e a s u r e m e n t o f the c y l i n d r i c a l s u r f a c e . T h e c o v e r a g e a r e a o f the r e f e r e n c e s p o t p a t t e r n on t h e b a c k g r o u n d p l a n e l o c a t e d on the x - y p l a n e of the r e f e r e n c e c o o r d i n a t e s y s t e m was a b o u t 20 i n c h e s s q u a r e . T h u s , t h e l a t e r a l s p a t i a l r e s o l u t i o n o f the 32x32 s p o t p r o j e c t i o n p a t t e r n i s o n l y 0 . 6 5 i n c h . H o w e v e r , the s p a t i a l r e s o l u t i o n w i l l i n c r e a s e w i t h the number of the s p o t s i n the p r o j e c t i o n p a t t e r n . T h e i m m e d i a t e r a m i f i c a t i o n s o f the i n c r e a s e i n the s p o t a r r a y s i z e a r e the e x t r a s p a c e - e n c o d e d p a t t e r n s r e q u i r e d f o r the s p o t c o l u m n d e f i n i t i o n , and the r e d u c t i o n i n the s i z e o f the p r o j e c t i o n s p o t s , as we 11 as the s p a c i n g s i n b e t w e e n t h e m . The f i r s t c o n s e q u e n c e c a n e a s i l y be h a n d l e d by u s i n g an a d d i t i o n a l s p a c e - e n c o d e d p r o j e c t i o n f o r the c a s e o f t h e 64x64 p a t t e r n , or two a d d i t i o n a l p r o j e c t i o n s f o r t h e c a s e o f t h e 128x128 p a t t e r n , e t c . H o w e v e r , the s e c o n d r a m i f i c a t i o n may impose s e v e r a l s e r i o u s p r o b l e m s i n t h e image a n a l y s i s p r o c e s s . F i r s t l y , due to the r e d u c t i o n i n the s i z e o f t h e p r o j e c t i o n s p o t s and t h e i r s e p a r a t i o n , the image i n t e n s i t y c o n t r a s t b e t w e e n t h e i l l u m i n a t e d a r e a s and t h e b a c k g r o u n d a r e a on the t e s t s u r f a c e w i l l d e c r e a s e . S e c o n d l y , due t o t h e r e d u c t i o n i n the s p o t s e p a r a t i o n s p a c i n g s , t h e f o u r p i x e l r a d i u s s e a r c h a r e a s o f s ome a d j a c e n t s p o t c e n t r o i d s u s e d i n the a d d r e s s d e c o d i n g p r o c e s s may o v e r l a p . T h r i d l y , m i s a l i g n m e n t p r o b l e m o f t h e p r o j e c t i o n s l i d e s w i l l be m o r e c r i t i c a l on the s p o t c o l u m n a d d r e s s d e c o d i n g p r o c e s s . 101 A s e r i e s o f 64x64 s p a c e - e n c o d e d s p o t p r o j e c t i o n s was e x p e r i m e n t e d w i t h on the c y l i n d r i c a l t e s t s u r f a c e u s i n g the same o p t i c a l c o n f i g u r a t i o n as s p e c i f i e d i n S e c t i o n 2 . 4 . I t was f o u n d t h a t the b i n a r y t h r e s h o l d i n g p r o c e s s c o u l d n o t i n d i v i d u a l l y i d e n t i f y 30% o f the p r o j e c t i o n s p o t s on the c y l i n d r i c a l s u r f a c e , e s p e c i a l l y f o r the ones on the s i d e o r i e n t a t e d away f r o m the c a m e r a . F u r t h e r m o r e , b e c a u s e o f the s m a l l e r s p o t s p a c i n g s and the m i s a l i g n m e n t p r o b l e m o f the p r o j e c t i o n s l i d e s , i n c o r r e c t l y d e c o d e d c o l u m n a d d r e s s e s w e r e r e p o r t e d i n o v e r 50% o f the p r o j e c t i o n s p o t s . T h e s e r e s u l t s h a v e i n d i c a t e d t h a t the i m p l e m e n t a t i o n o f a 64x64 s p o t p a t t e r n i s n o t f e a s i b l e u n d e r the c u r r e n t s y s t e m c o n f i g u r a t i o n . I f the s p a t i a l r e s o l u t i o n o f a 64x64 s p o t p a t t e r n i s r e q u i r e d , the L C D l i g h t s h u t t e r s h o u l d be u s e d to g e n e r a t e the s p a c e - e n c o d e d p r o j e c t i o n p a t t e r n s , and a more l i g h t s e n s i t i v e c a m e r a m i g h t be more p r e f e r a b l e . 5.8 SUMMARY I n t h i s c h a p t e r , we h a v e p r e s e n t e d an image p r o c e s s i n g a l g o r i t h m f o r d e t e r m i n i n g the s p a t i a l c o o r d i n a t e s o f t h e p r o j e c t i o n s p o t s on the c y l i n d r i c a l t e s t s u r f a c e . The a l g o r i t h m u s e s t h e b i n a r y t h r e s h o l d i n g t e c h n i q u e to d e t e c t t h e s p o t f e a t u r e s i n the g r e y - s c a l e images o f the s p a c e -e n c o d e d p r o j e c t i o n p a t t e r n s . The c e n t r o i d c o o r d i n a t e s o f the s p o t s a r e t h e n c o m p u t e d by u s i n g the image o u t l i n e s o f the s p o t f e a t u r e s . By e x a m i n i n g the p r e s e n c e / a b s e n c e c o n d i t i o n s o f t h e s p o t s i n t h e s e i m a g e s , the c o l u m n a d d r e s s e s o f the 102 s p o t s a r e d e c o d e d and the s p a t i a l c o o r d i n a t e s o f the s p o t s a r e t h e n o b t a i n e d . I t i s c l e a r f r om the r e s u l t s o f the s p o t a d d r e s s d e c o d i n g p r o c e s s t h a t the s u r f a c e d i s c o n t i n u i t i e s at the e d g e s of the c y l i n d r i c a l s u r f a c e h a v e c a u s e d some p r o b l e m s i n d e c o d i n g the a d d r e s s e s o f the p r o j e c t i o n s p o t s n e a r the d i s c o n t i n u i t y . Two p o s s i b l e c a u s e s f o r t h e s e a d d r e s s d e c o d i n g e r r o r s w e r e s u g g e s t e d . T h e f i r s t i s due to an i n h e r e n t p r o b l e m w i t h the s p o t p r o j e c t i o n m e t h o d , w h i l e the s e c o n d i s r e l a t e d t o the m i s a l i g n m e n t p r o b l e m of the p r o j e c t i o n s l i d e s . The s l i d e m i s a l i g n m e n t p r o b l e m c o u l d be e l i m i n a t e d by u s i n g an e 1 e c t r o - o p t i c a 1 L C D l i g h t s h u t t e r to p r o d u c e the s p o t p r o j e c t i o n p a t t e r n s . T h e r e l i a b i l i t y o f the d e c o d e d s p o t a d d r e s s e s w o u l d be i m p r o v e d by i n t r o d u c i n g an e r r o r d e t e c t i o n s p a c e - e n c o d i n g scheme to d e t e c t the i n c o r r e c t l y d e c o d e d a d d r e s s e s . B o t h s u g g e s t i o n s a r e l e f t to f u t u r e wo r k . A 64x64 s p o t p a t t e r n was e x p e r i m e n t e d w i t h on the c y l i n d r i c a l t e s t s u r f a c e . Due to the l o w c o n t r a s t b e t w e e n some o f the p r o j e c t i o n s p o t s and the b a c k g r o u n d , and t h e m i s a l i g n m e n t p r o b l e m o f the p r o j e c t i o n s l i d e s , the e r r o r r a t e o f t h e a d d r e s s d e c o d i n g p r o c e s s i n t h i s c a s e i s above 50%. T h i s makes the 64x64 s p o t p a t t e r n i n o p e r a t i v e i n the p r e s e n t s y s t e m c o n f i g u r a t i o n . H o w e v e r , e n c o u r a g i n g r e s u l t s w e r e o b t a i n e d by u s i n g t h e 32x32 s p o t p a t t e r n . T h e a v e r a g e a n d m a x i m u m s u r f a c e a m p l i t u d e m e a s u r e m e n t e r r o r s on the s m o o t h i n t e r i o r r e g i o n o f the c y l i n d r i c a l s u r f a c e a r e f o u n d to be l e s s t h a n .1 and .25 i n c h , r e s p e c t i v e l y . A f e w 103 l i m i t a t i o n s o f t h e 32x32 s p o t p r o j e c t i o n scheme w i l l be d i s c u s s e d i n the n e x t c h a p t e r . 104 C H A P T E R 6 L I M I T A T I O N S O F T H E MEASUREMENT S U R F A C E O R I E N T A T I O N 6.1 O V E R V I E W The p e r f o r m a n c e o f the s p o t p r o j e c t i o n i m a g i n g s y s t e m d e p e n d s on the l i g h t s e n s i t i v i t y o f t h e c a m e r a , the b r i g h t n e s s o f the s p o t p a t t e r n s p r o d u c e d by the s l i d e p r o j e c t o r , t h e r e f l e c t a n c e c h a r a c t e r i s t i c s o f the m e a s u r e m e n t s u r f a c e , and the o p t i c a l g e o m e t r y o f the i m a g i n g d e v i c e s w i t h r e s p e c t to the s u r f a c e o r i e n t a t i o n o f the m e a s u r e m e n t s u r f a c e . U n f o r t u n a t e l y , many o f t h e s e d e s i g n f a c t o r s o f the i m a g i n g s y s t e m a r e r e s t r i c t e d by t e c h n i c a l , e n v i r o n m e n t a l and e q u i p m e n t l i m i t a t i o n s . F o r e x a m p l e , the w o r k i n g d i s t a n c e b e t w e e n t h e s l i d e p r o j e c t o r or c a m e r a to the m e a s u r e m e n t s u r f a c e i s l i m i t e d to t h e p h y s i c a l s i z e o f the r o o m a v a i l a b l e t o a h o r i z o n t a l s p o t p r o j e c t i o n s y s t e m . I n an o v e r h e a d p r o j e c t i o n c o n f i g u r a t i o n , t h e wo r k i n g d i s t a n c e i s r e s t r i c t e d by the h e i g h t o f the c e i l i n g . In o r d e r to k e e p the e q u i p m e n t c o s t o f the i m a g i n g s y s t e m l o w , the s y s t e m c o m p r i s e s o n l y o f e x i s t i n g e q u i p m e n t w h i c h i s r e a d i l y a v a i l a b l e at the E l e c t r i c a l E n g i n e e r i n g D e p a r t m e n t . A v i d i c o n s e n s o r e q u i p p e d D A G E - M T I M o d e l 65 c a m e r a i s a v a i l a b l e f o r t h e s p o t p r o j e c t i o n e x p e r i m e n t . T h i s camera was o r i g i n a l l y p u r c h a s e d f o r an o u t d o o r a p p l i c a t i o n and i n t e n d e d to be u s e d i n d a y l i g h t i n t e n s i t y c o n d i t i o n s . F o r a s p o t p r o j e c t i o n s y s t e m o p e r a t i n g i n l o w l i g h t i n g c o n d i t i o n , 105 a more s e n s i t i v e image sensor s u c h ' as the Newicon s e n s o r , w o u l d be more s u i t a b l e f o r t h i s a p p l i c a t i o n . The L e i t z s l i d e p r o j e c t o r a v a i l a b l e f o r the p r o j e c t i o n of spot p a t t e r n s i s e q u i p p e d w i t h a 90 ran l e n s and a 250 watt p r o j e c t i o n lamp. The power of the p r o j e c t i o n lamp determines the b r i g h t n e s s of the i l l u m i n a t e d s p o t s , and the f o c a l l e n g t h of the p r o j e c t o r lens d e f i n e s the s i z e of the p r o j e c t i o n p a t t e r n at a g i v e n w o r k i n g d i s t a n c e . The o p t i c a l c o n f i g u r a t i o n of the spot p r o j e c t i o n imaging sys tem was s p e c i f i e d i n S e c t i o n 2 . 4 . For a measurement s u r f a c e w i t h u n i f o r m r e f l e c t a n c e c h a r a c t e r i s t i c s , as i n the case of an a i r c r a f t w i n g s u r f a c e , i t i s i m p o r t a n t to know the r e l a t i o n s h i p of the image i n t e n s i t y w i t h r e s p e c t to the measurement s u r f a c e o r i e n t a t i o n . S u b s e q u e n t l y , t h i s w i l l a f f e c t the a c c u r a c y of the s p a t i a l c o o r d i n a t e s measurements p r o v i d e d by the imaging s y s t e m . It has been shown f r o m the shape measurements u s i n g the c y l i n d r i c a l s u r f a c e , that the s p a t i a l c o o r d i n a t e e r r o r s of the p r o j e c t i o n spots w h i c h have c o r r e c t l y decoded column a d d r e s s e s are l e s s than one q u a r t e r of an' i n c h . L a r g e measurement e r r o r s a r e ' caused by u s i n g i n c o r r e c t decoded column addresses i n the s p a t i a l c o o r d i n a t e c a l c u l a t i o n s . T h u s , we would l i k e to determine the e f f e c t s of the measurement s u r f a c e o r i e n t a t i o n on the spot column address d e c o d i n g p r o c e s s . It i s c r i t i c a l f o r the image p r o c e s s i n g a l g o r i t h m to i d e n t i f y a l l the p r o j e c t i o n s spot i n the images of the p r o j e c t i o n p a t t e r n . M i s i d e n t i f i c a t i o n of spot images w i l l r e s u l t i n erroneous column addresses being a s s i g n e d to these 106 p r o j e c t i o n s p o t s . S p o t i d e n t i f i c a t i o n e r r o r i s c a u s e d by t h e f o i l ow f a c t o r s : 1) the image i r r a d i a n c e i n t e n s i t y o f the p r o j e c t i o n s p o t b e i n g b e l o w the t h r e s h o l d l e v e l u s e d i n t h e b i n a r y t h r e s h o l d i n g p r o c e s s ; 2) the p r o j e c t i o n s p o t s b e i n g t o o c l o s e t o g e t h e r f o r the d e t e c t i o n a l g o r i t h m to s e p a r a t e the i n d i v i d u a l edge o u t l i n e s o f t h e i r i m a g e s ; 3) t h e image c e n t r o i d s o f a d j a c e n t p r o j e c t i o n s p o t s b e i n g l e s s t h a n n i n e p i x e l s a p a r t . ' T h e c a u s e s o f e r r o r i n f e a t u r e i d e n t i f i c a t i o n due t o t h e f i r s t two c a s e s a r e o b v i o u s . I n the t h i r d c a s e , b e c a u s e o f t h e s l i d e m i s a l i g n m e n t p r o b l e m , the image c e n t r o i d s o f t h e p r o j e c t i o n s p o t s f r o m t h e d i f f e r e n t s p a c e - e n c o d e d p a t t e r n s c a n be l o c a t e d a n y w h e r e w i t h i n a f o u r p i x e l r a d i u s f r o m t h e c o r r e s p o n d i n g c e n t r o i d s i n t h e image o f t h e r e f e r e n c e p r o j e c t i o n p a t t e r n . N e a r e s t n e i g h b o u r s e a r c h o p e r a t i o n s a r e i m p l e m e n t e d t o f i n d t h e image c e n t r o i d s o f t h e s p a c e - e n c o d e d s p o t p a t t e r n s . A s i l l u s t r a t e d i n F i g u r e 6 . 1 , t h e s e a r c h a r e a s a b o u t t h e two a d j a c e n t r e f e r e n c e c e n t r o i d s i n the image o f a s p a c e - e n c o d e d p a t t e r n may o v e r l a p i f t h e y a r e l e s s t h a n n i n e p i x e l s a p a r t . I f t h e c o r r e s p o n d i n g c e n t r o i d of C I , f o r e x a m p l e , i s p r e s e n t i n t h e o v e r l a p p i n g s e a r c h a r e a , as shown i n F i g u r e 6 . 2 , t h e n the r e s u l t s o f b o t h s e a r c h o p e r a t i o n s w i l l be p o s i t i v e . T h e r e f o r e , t h e c o l u m n a d d r e s s b i t s a s s o c i a t e d w i t h t h e 107 Fig. 6.1 Search areas of two adjacent reference centroids. S P O T C E N T R O I D Fig. 6.2 Overlapping search areas of two adjacent centroids. 108 s p a c e - e n c o d e d p a t t e r n f o r t h e s e two s p o t s , C I and C 2 , w i l l b o t h be s e t to o n e . C o n s e q u e n t l y , the a d d r e s s b i t f o r the s p o t C2 i s i n c o r r e c t l y d e c o d e d to be a o n e . H e n c e , i n o r d e r to p r e v e n t t h i s s i t u a t i o n f r o m h a p p e n i n g , the image s p a c i n g b e t we e n two r e f e r e n c e c e n t r o i d s mu st be g r e a t e r t h a n n i n e p i x e l u n i t s . I f t h e s i z e o f the s p o t images a r e l e s s t h a n 8 p i x e l s s q u a r e and t h e c e n t r o i d s o f two a d j a c e n t s p o t s a r e l e s s t h a n n i n e p i x e l s a p a r t , t h e n the image p r o c e s s i n g a l g o r i t h m c a n l o c a t e the s e p a r a t e o u t l i n e s o f a d j a c e n t s p o t s . A r e f l e c t a n c e m o d e l w h i c h d e s c r i b e s the image i r r a d i a n c e m e a s u r e m e n t s u r f a c e i s p r e s e n t e d i n the n e x t s e c t i o n . T h i s m o d e l i s u s e f u l i n d e t e r m i n i n g the l i m i t i n g o r i e n t a t i o n s o f t h e m e a s u r e m e n t s u r f a c e w h i c h c a u s e the image i r r a d i a n c e i n t e n s i t y of the s p o t to d r o p b e l o w the b i n a r y t h r e s h o l d i n g l e v e l . P l a n a r s u r f a c e s i n v a r i o u s o r i e n t a t i o n s w i t h r e s p e c t to the r e f e r e n c e c o o r d i n a t e s y s t e m c a n be r e p r e s e n t e d by a n a l y t i c a l f u n c t i o n s . The a n a l y t i c a l f u n c t i o n s a r e u s e d t o d e v e l o p the r e l a t i o n s h i p o f the image c e n t r o i d s p a c i n g s w i t h r e s p e c t to the m e a s u r e m e n t s u r f a c e o r i e n t a t i o n . W i t h t h i s r e l a t i o n s h i p , the l i m i t i n g o r i e n t a t i o n s o f t h e m e a s u r e m e n t s u r f a c e w h i c h c a u s e the image c e n t r o i d s e p a r a t i o n s to be c l o s e r t h a n n i n e p i x e l s c a n be d e t e r m i n e d . T h e p r o c e d u r e s f o r c a l c u l a t i n g the c e n t r o i d s p a c i n g s o f a d j a c e n t p r o j e c t i o n s p o t s on t h e p l a n a r s u r f a c e s , a n d t h e i r r e l a t i o n s h i p w i t h t h e s u r f a c e o r i e n t a t i o n a r e p r e s e n t e d i n S e c t i o n 6 . 3 . 109 6.2 REFLECTANCE MODEL OF THE MEASUREMENT SURFACE T h e image i r r a d i a n c e o f a p r o j e c t i o n s p o t f r o m a m e a s u r e m e n t s u r f a c e i s a f u n c t i o n o f the l i g h t r e f l e c t a n c e p r o p e r t i e s o f the s u r f a c e m a t e r i a l , the b r i g h t n e s s of the s p o t p r o j e c t i o n p a t t e r n , and t h e o r i e n t a t i o n of t h e s u r f a c e e l e m e n t s w i t h r e s p e c t to the i m a g i n g s y s t e m . I f the r e f l e c t a n c e c h a r a c t e r i s t i c s o f the s u r f a c e m a t e r i a l i s h o m o g e n e o u s , and the b r i g h t n e s s of t h e p r o j e c t i o n p a t t e r n i s u n i f o r m , t h e n the c h a n g e s i n image i r r a d i a n c e f r o m the p r o j e c t i o n s p o t s a r e o n l y due t o the v a r i a t i o n o f the s u r f a c e o r i e n t a t i o n . The image i r r a d i a n c e f r o m any r e f l e c t i v e s u r f a c e c a n be dec omp o s e d i n t o t h r e e c omp o n e n t s : t h e d i f f u s i v e , s p e c u l a r , and a m b i e n t t e r m s . S i n c e the s p o t p r o j e c t i o n s y s t e m i s o p e r a t i n g i n a c o n t r o l l e d l i g h t i n g e n v i r o n m e n t w i t h v e r y l o w a m b i e n t l i g h t i n t e n s i t y , the a m b i e n t r e f l e c t i v e t e r m f r o m t h e s u r f a c e i s n e g l i g i b l e c o m p a r e d to the two o t h e r t e r m s . A r e f l e c t a n c e m o d e l i s s u g g e s t e d by H a l l and M c P h e r s o n [1983] to d e s c r i b e the d i f f u s i v e and s p e c u l a r t e r m s o f a r e f l e c t i v e s u r f a c e . 6 . 2 . 1 D I F F U S I V E REFLECTANCE MODEL A s shown i n F i g u r e 6 . 3 , an i n c i d e n t l i g h t v e c t o r i i s p r o j e c t e d o n t o a s u r f a c e e l e m e n t o f the r e f l e c t i v e s u r f a c e S , a n d t h e s u r f a c e n o r m a l v e c t o r n i s at an a n g l e 6 w i t h n r e s p e c t t o t h e y - a x i s o f the r e f e r e n c e c o o r d i n a t e s y s t e m . I f 110 VIEWER F i g . 6 . 3 O p t i c o l geometry of a p r o j e c t i o n spo t on a r e f l e c t i v e s u r f a c e . the s u r f a c e i s an i d e a l d i f f u s e r e f l e c t o r , the i n c i d e n t l i g h t i s d i f f u s e d e q u a l l y i n a l l d i r e c t i o n s and the image i r r a d i a n c e o f the i n c i d e n t p r o j e c t i o n i s a f u n c t i o n o n l y o f the i n c i d e n c e a n g l e , 8 , b e t w e e n the i and n v e c t o r s . S u r f a c e s w i t h t h e i d e a l d i f f u s i v e r e f l e c t a n c e p r o p e r t y a r e known as L a m b e r t i a n s u r f a c e s . T h e r e f l e c t a n c e m o d e l o f L a m b e r t i a n s u r f a c e s i s g i v e n by [ H a l l and M c P h e r s o n 1 9 8 3 ] : I d i f f - kj cos 0 , - 9 O ° £ 0 £ 9 O ° o the r w i s e ( 6 . 1 ) w h e r e i s a c o n s t a n t d e t e r m i n e d by the c a m e r a l i g h t s e n s i t i v i t y and 0 i s the a n g l e o f i n c i d e n c e . I l l 6 . 2 . 2 SPECULAR REFLECTANCE MODEL I f the s u r f a c e r e f l e c t a n c e c h a r a c t e r i s t i c i s s p e c u l a r , then the maximum i r r a d i a n c e of the l i g h t source can be o b s e r v e d i n the d i r e c t i o n of the r e f l e c t i o n v e c t o r r, w h i c h i s l o c a t e d at an angle equal to the angle of i n c i d e n c e but on the o p p o s i t e s i d e of the s u r f a c e normal v e c t o r . The i r r a d i a n c e i n t e n s i t y decreases as the v i e w i n g v e c t o r v i s moved away f r o m r. The s p e c u l a r r e f l e c t a n c e model can be e x p r e s s e d as [ H a l l and McPherson 1983] : ^ p e c - k 2 c o s m * » - 9 0 ° ^ 9 0 ° , •= 0 , o therwise ( 6 . 2 ) where m r e p r e s e n t s ' t h e s h i n i n e s s of the s u r f a c e , and <j> i s the a n g l e between v and r. For an i d e a l m i r r o r - l i k e s u r f a c e , m i s i n f i n i t e l y l a r g e . For the s u r f a c e of a n i c k e l (5 cent p i e c e ) , m can be between 10 and 100. 6 . 2 . 3 COMBINED REFLECTANCE MODEL The r e f l e c t a n c e model of most s u r f a c e s c o n s i s t s of the d i f f u s i v e , s p e c u l a r and the ambient r e f l e c t i o n t e r m s . The s p e c u l a r component r e p r e s e n t s the r e f l e c t e d l i g h t i n t e n s i t y f r o m the c y l i n d r i c a l s u r f a c e . The d i f f u s i v e component' r e p r e s e n t s the l i g h t s c a t t e r i n g i n t e n s i t y on the s u r f a c e of the m a t e r i a l [Cook and T o r r a n c e 1981] . S i n c e the ambient t e r m i s s m a l l compared to the other two, i t i s 112 c o m b i n e d w i t h the d i f f u s i v e t e r m as a common t e r m , K , i n the r e f l e c t a n c e m o d e l . T h e c o m b i n e d m o d e l o f an r e f l e c t i v e s u r f a c e c a n t h e n be e x p r e s s e d a s : I - K + k 2 c o s m 0 , - 9 O ° £ 0 £ 9 O ° , K , o t h e r w i s e ( 6 . 3 ) Now we w o u l d l i k e to d e t e r m i n e the c o m b i n e d r e f l e c t a n c e m o d e l o f the c y l i n d r i c a l s u r f a c e u s i n g t h e image o f t h e r e f e r e n c e p r o j e c t i o n p a t t e r n a c q u i r e d i n the 3 - D m e a s u r e m e n t p r o c e s s d e s c r i b e d i n S e c t i o n 2 . 4 . T h e c a l i b r a t i o n p r o c e d u r e s o f t h e r e f l e c t a n c e m o d e l w i l l be p r e s e n t e d i n S e c t i o n 6 . 2 . 5 . 6 . 2 . 4 O P T I C A L GEOMETRY O F T H E P R O J E C T I O N S Y S T E M T h e o p t i c a l g e o m e t r i e s o f the c a m e r a and s l i d e p r o j e c t o r u s e d i n the s p o t p r o j e c t i o n s y s t e m w e r e o b t a i n e d by d e c o m p o s i n g t h e i r t r a n s f o r m a t i o n m a t r i c e s as e x p l a i n e d i n S e c t i o n 4 . 5 . F r o m t h e s e r e s u l t s , the a n g u l a r o f f s e t o f the c a m e r a a n d s l i d e p r o j e c t o r o p t i c a l axes f r o m the y - z p l a n e o f the r e f e r e n c e c o o r d i n a t e s y s t e m a r e o n l y 1 . 3 3 and 1 0 . 6 d e g r e e s , r e s p e c t i v e l y . S i n c e t h e s e o f f s e t s a r e q u i t e s m a l l , t h e y a r e i g n o r e d i n o r d e r t o r e d u c e the c o m p l e x i t y of the o p t i c a l c o n f i g u r a t i o n o f t h e i m a g i n g s y s t e m . T h e r e f o r e , f r o m t h i s p o i n t o n w a r d s , we w i l l assume the o p t i c a l a x e s of the c a m e r a and s l i d e p r o j e c t o r to be p a r a l l e l to the y - z p l a n e o f t h e c o o r d i n a t e s y s t e m . T h e s i m p l i f i e d o p t i c a l g e o m e t r y o f 113 x Fo r e f e r e n c e coord i note system y comero pr o j e c t o r F i g . 6 .4 O p t i c a l c o n f i g u r a t i o n of the camera and s l i d e p r o j e c t o r w i t h respect to the r e f e r e n c e c o o r d i n a t e system Fo. the i m a g i n g s y s t e m i s g i v e n i n F i g u r e 6 . 4 . T h e i n c l i n a t i o n i n c i d e n c e a n g l e 8 a r e d e f i n e d a s : the a n g l e b e t w e e n the p r o j e c t o r a x i s and t h e y - a x i s of the r e f e r e n c e s y s t e m ; the a n g l e b e t w e e n the c a m e r a and p r o j e c t o r a x e s ; the a n g l e b e t w e e n the c a m e r a a n d t h e y - a x i s ; and t h e a n g l e b e t w e e n the i n c i d e n t l i g h t v e c t o r and the s u r f a c e n o r m a l , r e s p e c t i v e l y . The c a m e r a a n g l e y c a n be o b t a i n e d by : a n g l e a , p r o j e c t i o n a n g l e /3, c a m e r a a n g l e y and the y = 180° - a - $ ( 6 . 4 ) F r o m t h e r e s u l t s g i v e n i n S e c t i o n a r e 7 4 ° a n d 3 9 ° , r e s p e c t i v e l y , e q u a l t o 6 7 ° . 4 . 6 , the v a l u e s o f a and $ F r o m E q u a t i o n ( 6 . 4 ) , y i s 1 1 4 The s i z e o f the r e f e r e n c e p r o j e c t i o n p a t t e r n on the s l i d e i s 8 . 8 9 mm s q u a r e , as shown i n F i g u r e 6 . 5 , and the f o c a l l e n g t h o f the p r o j e c t o r l e n s i s 90mm. T h e r e f o r e , the m a x i m u m d e v i a t i o n o f the p r o j e c t i o n l i g h t beams f r o m the o p t i c a l a x i s o f the s l i d e p r o j e c t o r i s 2 . 8 d e g r e e s . Due to t h i s 1 ow ma x i mum d e v i a t i o n , the i n c i d e n c e l i g h t v e c t o r s f r om the p r o j e c t o r l e n s c e n t e r to the i l l u m i n a t e d s p o t s on the c y l i n d r i c a l s u r f a c e a r e a p p r o x i m a t e d by the v e c t o r a l o n g the o p t i c a l a x i s of the p r o j e c t o r . S i m i l a r l y , the v i e w i n g v e c t o r s f r o m the p r o j e c t i o n s p o t s to the c a m e r a l e n s c e n t e r a r e a p p r o x i m a t e d by the v e c t o r a l o n g the o p t i c a l a x i s of the c ame r a . 8.89 mm 0.11mm • E B B B B a B B B B S B B a B B a B B e B B B S a B B B B a B ' • • B B B B B B B B B B B B B B B B B B a B B B B B B B B B B B I I I I I B I I I I I I B I I i l l l l l l l l B f i B I I I I I 0.28mm 8.89 mm I F i g . 6 .5 The d i m e n s i o n s of the r e f e r e n c e s p o t p r o j e c t i o n p a t t e r n . 115 The image i r r a d i a n c e i n t e n s i t y as a s u r f a c e o r i e n t a t i o n s i n d i r e c t i o n s a l o n g the y the x-z plan e of the r e f e r e n c e c o o r d i n a t e c o n s i d e r e d i n d i v i d u a l l y . y-z Plane Variations C o n s i d e r the normal v e c t o r s to the c y l i n d r i c a l s u r f a c e a l o n g the y-z plane of the r e f e r e n c e c o o r d i n a t e system. The c y l i n d r i c a l s u r f a c e i s o r i e n t e d such t h a t the r e c t a n g u l a r base of the s u r f a c e i s l o c a t e d on the x-y plane and the l o n g i t u d i n a l a x i s of the c y l i n d e r i s p a r a l l e l to the x - a x i s of the r e f e r e n c e system, as shown i n F i g u r e 6.6. The o o o o o o e o o e o x F i g . 6 .6 The c y l i n d r i c a l s u r f a c e w i t h l o n g i t u d i n a l a x i s p a r a l l e l to the x - a x i s and i s p l a c e d in f r o n t o f a background p l a n e , l o c a t e d on the x - y p l a n e of Fo . f u n c t i o n of -z piane and sy s t em are 116 c y l i n d r i c a l s u r f a c e f u n c t i o n i s g i v e n by ( a l l d i m e n s i o n s are g i v e n i n i n c h e s ) : z(x,y) - [ 12 2 - (y-8.625) 2 ] 1 / 2 - 9.20 , 0.0^12.0 and 0.875£y±16.375 = 0 , otherwise (6.5) The n o r m a l v e c t o r to the s u r f a c e e l e m e n t l o c a t e d at the p o i n t ( x , y , z ) i s g i v e n by: (y-8.625) n -= O.Oi + j + 1.0k .O-x-12.0 and / 12 2-(y-8.625) 2 .875^16.375 1.0 k. , otherwise (6.6) where i , j , and k a r e the u n i t v e c t o r s of the r e f e r e n c e c o o r d i n a t e s y s t e m . The a n g l e 8 b e t w e e n the s u r f a c e nornlal n v e c t o r and t h e y - a x i s o f the r e f e r e n c e s y s t e m i s g i v e n by : , / 12 2 - (y-8.625) 2 0.o£x£l2.0 and 8 = t a n " 1 — (y-fc.625) • 0.875^-16.375 90° , otherwise (6.7) C o n s i d e r the o p t i c a l g e o m e t r y o f the c y l i n d r i c a l s u r f a c e , as shown i n F i g u r e 6.7. The l i g h t i n c i d e n t v e c t o r i and the v i e w i n g v e c t o r v a r e a l o n g the o p t i c a l axes of the p r o j e c t o r and camera, r e s p e c t i v e l y . The a n g l e <f> between the r e f l e c t i o n v e c t o r r and the v i e w i n g v e c t o r v can be TV CAMERA 0 +ve Ffg. 6.7 Optical geometry of a projection spot on the cylindrical surface, with normal vectors In the direction along the y-z plane of F_ . 118 e x p r e s s e d i n t e r m s of 8 n , a , and £ a s : a> - 26 - /3 = 2( ( 1 8 0 ° - 8 n ) - a ) - $ ( 6 . 8 ) w h e r e 0 - 1 8 0 ° - 0 - a n S u b s t i t u t i n g f o r <f> i n E q u a t i o n ( 6 . 8 ) , t h e s u r f a c e r e f l e c t a n c e m o d e l b e c o m e s : I = K + k . c o s m ( 2 ( l 8 O ° - 0 - a ) - $ ) ( 6 . 9 ) w h e r e K , k ^ and m a r e the c o e f f i c i e n t o f the r e f l e c t a n c e mo d e l . F r o m t h e r e f l e c t a n c e m o d e l o f t h e c y l i n d r i c a l s u r f a c e , the r e l a t i o n s h i p b e t w e e n the s p o t image i n t e n s i t y and y - z p l a n e v a r i a t i o n s o f the s u r f a c e o r i e n t a t i o n c a n be e s t a b l i s h e d by u s i n g E q u a t i o n ( 6 . 9 ) . T h i s r e l a t i o n s h i p c a n t h e n be u s e d t o d e t e r m i n e the l i m i t i n g o r i e n t a t i o n s of the c y l i n d r i c a l s u r f a c e s u b j e c t e d to t h e o p t i c a l c o n f i g u r a t i o n o f t h e s p o t p r o j e c t i o n s y s t e m as s p e c i f i e d i n S e c t i o n 2 . 4 . x-z Plane Variations I n t h i s c a s e , the a x i s p a r a l l e l t o t h e y - a x i s o f t h e shown i n F i g u r e 6 . 8 . T h e o p t p r o j e c t o r r e m a i n s u n c h a n g e d , of the c y l i n d r i c a l s u r f a c e i s r e f e r e n c e c o o r d i n a t e s y s t e m as i c a l g e o m e t r y o f t h e c a m e r a and but the s u r f a c e n o r m a l v e c t o r s 119 x F i g . 6 . 8 The c y l i n d r i c a l s u r f a c e w i t h l o n g i t u d i n a l o x i s p a r a l l e l t o t h e y - a x i s ond i s p l a c e d in f r o n t o f o b a c k g r o u n d p l a n e , l o c a t e d on —•> the x - y p l a n e o f F o . a r e now a l o n g the x - z p l a n e . The s u r f a c e n o r m a l v e c t o r , the r e f l e c t i o n v e c t o r and t h e v i e w i n g v e c t o r a r e no l o n g e r c o p l a n a r as i n the p r e v i o u s c a s e . H e n c e , t h e o p t i c a l g e o m e t r y i s now m o r e c o m p l i c a t e d . A s shown i n F i g u r e 6 . 9 , t h e i n c i d e n c e l i g h t v e c t o r i and the v i e w i n g v e c t o r v a r e s t i l l o n the y - z p l a n e , b u t the s u r f a c e n o r m a l i s now a l o n g t h e x - z p l a n e and i s a t an a n g l e 8 f r o m the z - a x i s . T h u s , 6 m t h e i n c i d e n c e a n g l e , 8, b e t w e e n the i n c i d e n t l i g h t v e c t o r a n d t h e s u r f a c e norma 1 i s g i v e n a s : c o s 8 = cos8 sinoc ( 6 . 1 0 ) wh e r e a a x i s o f i s the a n g l e b e t w e e n the i n c i d e n t v e c t o r and the y -the c o o r d i n a t e s y s t e m . The c o s i n e v a l u e o f the a n g l e 120 A PROJECTION SPOT ON THE CYLINDRICAL SURFACE CAMERA F i g . 6.9 PROJECTOR O p t i c o l geometry of o p r o j e c t i o n s p o t on the c y l i n d r i c a l s u r f a c e w i t h normal v e c t o r s In the d i r e c t i o n s o l o n g the x - z p l a n e o f F o . 0 between the r e f l e c t i o n v e c t o r and the v i e w i n g v e c t o r i s g i v e n a s : cos 26 c o s <f> c o s ( l 8 0 - 2 a ) ( 2 c o s z 6 - 1 ) c o s ( l 8 0 - 2 a ) cos ( y - a) :os(y - a) (6.11) S u b s t i t u t i n g f o r cos8 i n E q u a t i o n (6.11) y i e l d s , co s <f> 2 2 ( 2 c o s 0 s i n a - 1 ) m cos( 1 8 0 - 2 a ) cos(}/ - a) (6.12) S u b s t i t u t i n g the e x p r e s s i o n of cos$ f r o m (6.10) i n t o the r e f l e c t a n c e mo d e l , as g i v e n i n Equat i o n Equa t i on 121 ( 6 . 3 ) , t h e image i r r a d i a n c e f r o m the p r o j e c t i o n s p o t s on the m e a s u r e m e n t s u r f a c e c a n be e x p r e s s e d a s : I = K + k . 2 2 ( 2 c o s 8 s i n < x - 1 ) m c o s ( l 8 0 - 2 o c ) c o s ( ^ - a ) ( 6 . 1 3 ) w h e r e K , k 2 arid rn a r e the c o e f f i c i e n t s o f the r e f l e c t a n c e mo d e l . U s i n g t h i s r e f l e c t a n c e m o d e l g i v e n i n E q u a t i o n ( 6 . 1 3 ) , the r e l a t i o n s h i p b e t w e e n t h e s p o t image i n t e n s i t y and t h e x - z p l a n e v a r i a t i o n s o f the c y l i n d r i c a l s u r f a c e o r i e n t a t i o n c a n be e s t a b l i s h e d . T h e r e l a t i o n s h i p c a n t h e n be u s e d t o d e t e r m i n e the l i m i t i n g o r i e n t a t i o n s of the c y l i n d r i c a l s u r f a c e s u b j e c t e d to t h e o p t i c a l c o n f i g u r a t i o n o f the s p o t p r o j e c t i o n s y s t e m as s p e c i f i e d i n S e c t i o n 2 . 4 . 6.2.5 R E F L E C T A N C E MODEL C A L I B R A T I O N R E S U L T S Now we w o u l d l i k e to d e t e r m i n e the r e f l e c t a n c e m o d e l u s i n g t h e image o f the r e f e r e n c e p r o j e c t i o n p a t t e r n a c q u i r e d i n the 3 - D m e a s u r e m e n t p r o c e s s o f the c y l i n d r i c a l s u r f a c e d e s c r i b e d i n S e c t i o n 2 . 4 . The g r e y - s c a l e image of the r e f e r e n c e p r o j e c t i o n p a t t e r n i s shown i n F i g u r e 6 . 1 0 . A p l o t o f t h e i r r a d i a n c e i n t e n s i t i e s o f the p i x e l s a l o n g row a d d r e s s 106 v e r s u s the c o l u m n a d d r e s s o f the p i x e l i s shown i n F i g u r e 6 . 1 1 . T h e p e a k s shown i n the p l o t c o r r e s p o n d to t h e image p o s i t i o n s o f the p r o j e c t i o n s p o t s . T h e s e p r o j e c t i o n s p o t s a r e u s e d to d e r i v e the r e f l e c t a n c e m o d e l o f 122 F i g u r e 6.10 G r e y - s c a l e image of the r e f e r e n c e p r o j e c t i o n pat t e r n the c y l i n d r i c a l t e s t s u r f a c e . The i n t e n s i t y l e v e l s and the image c o o r d i n a t e s of the spots were o b t a i n e d d i r e c t l y from the p l o t g i v e n i n F i g u r e 6 . 1 1 . The c o r r e s p o n d i n g 3 - D s p a t i a l c o o r d i n a t e s of these p r o j e c t i o n spots were o b t a i n e d from A p p e n d i x 1. The image i n t e n s i t y , image c o o r d i n a t e s and the s p a t i a l c o o r d i n a t e s of the p r o j e c t i o n s p o t s used i n the model c a l i b r a t i o n p r o c e s s are l i s t e d i n T a b l e 6 . 1 . Knowing the s p a t i a l c o o r d i n a t e s of the p r o j e c t i o n s p o t s , the s u r f a c e normal v e c t o r s at the spots can then be d e f i n e d by E q u a t i o n ( 6 . 6 ) . F i n a l l y , f r o m E q u a t i o n s ( 6 . 7 ) and ( 6 . 8 ) , the r e l a t i v e v i e w i n g angle ci between the r e f l e c t i o n v e c t o r r and the v i e w i n g v e c t o r v at each of these p r o j e c t i o n spots were o b t a i n e d and are l i s t e d 1 r B A C K G R O U N D ^ P L A N E I M A G E I N T E N S I T Y P R O F I L E 1 - l 1 r C Y L I N D R I C A L SURFACE f/nfjjjnsAIrT T 100 2 00 3 tfl IZI PIXEL R O W C O O R D [ N A T E S U00 1 B A C K G R O U N D P L A N E V 500 Image i n t e n s i t y p r o f i l e of the p r o j e c t i o n spots a c r o s s the c y l i n d r i c a l s u r f a c e 124 P i x e l S p a t i a l C o o r d i n a t e s I n t e n s i t y c o l u m n X Y Z a d d r e s s ( i n c h e s ) ( i n c h e s ) ( i n c h e s ) 133 1 0 . 8 3 6 5 8 1 5 . 9 2 4 2 3 0 . 5 3 0 4 7 68 . 152 1 0 . 7 9 2 4 6 1 5 . 2 2 9 8 3 0 . 9 8 6 8 2 72 . 171 1 0 . 7 8 3 2 6 1 4 . 5 3 8 7 1 1 .44296 7 7 . 189 1 0 . 7 7 6 4 1 13 . 86571 1 . 8 2 2 8 6 79 . 205 10 . 77421 1 3 . 2 2 5 0 8 2 . 0 5 2 0 8 82 . 221 1 0 . 7 7 1 9 8 1 2 . 5 8 6 0 7 2 . 2 8 1 9 5 86 . 237 1 0 . 7 3 5 0 7 1 1 . 9 4 8 4 8 2 . 5 1 1 7 2 8 9 . 252 1 0 . 7 3 7 4 3 1 1 . 3 4 1 4 2 2 . 5 9 3 2 9 9 0 . 266 1 0 . 7 3 7 4 3 1 0 . 7 2 1 0 5 2 . 7 5 0 7 0 9 3 . 280 1 0 . 7 4 2 0 1 1 0 . 1 2 9 8 4 2 . 7 5 9 8 0 9 5 . 293 1 0 . 7 4 4 2 4 9 . 5 2 5 3 5 2 . 8 4 4 7 3 9 4 . 306 1 0 . 7 4 8 7 3 8 . 9 3 5 3 2 2 . 8 5 6 2 6 9 3 . 319 1 0 . 7 5 3 1 8 8 . 3 4 5 8 2 2 . 8 6 9 0 2 9 4 . 331 1 0 . 7 5 9 9 0 7 . 7 7 0 0 9 2 . 8 0 8 4 9 9 2 . 343 1 0 . 7 6 6 5 8 7 . 1 9 4 5 6 2 . 7 4 9 2 1 91 . 355 1 0 . 7 7 5 5 3 6 . 6 3 2 1 6 2 . 6 1 6 6 0 8 9 . 366 1 0 . 7 8 2 1 3 6 . 0 5 6 8 6 2 . 5 5 9 7 8 88 . 377 1 0 . 7 9 3 3 3 5 . 5 0 6 9 9 2 . 3 5 4 8 8 84 . 387 1 0 . 8 0 2 1 7 4 . 9 4 4 2 4 2 . 2 2 5 7 7 83 . 397 1 0 . 8 1 3 3 1 4 . 3 9 3 7 0 2 . 0 2 3 0 0 7 9 . 407 1 0 . 8 2 4 4 1 3 . 8 4 2 7 5 1 . 8 2 1 2 3 7 7 . 416 1 0 . 8 0 3 0 2 3 . 3 0 3 4 1 1 . 5 4 4 5 9 73 . 425 1 0 . 8 1 6 3 7 2 . 7 6 3 4 0 1 . 2 6 9 5 0 6 9 . 459 1 0 . 8 2 9 7 1 2 . 2 2 2 7 1 0 . 9 9 5 1 7 6 5 . 434 1 0 . 8 4 5 3 8 1 . 6 9 2 9 1 0 . 6 4 6 0 2 6 3 . 442 1 0 . 8 6 1 0 5 1 . 1 6 2 1 6 0 . 2 9 7 2 9 61 . T a b l e 6 . 1 Image and s p a t i a l c o o r d i n a t e s and the i n t e n s i t y o f p r o j e c t i o n s p o t s u s e d t o d e t e r m i n e the r e f l e c t a n c e m o d e l o f the c y l i n d r i c a l s u r f a c e . ( P i x e l r o w c o o r d i n a t e o f t h e s p o t s i s 106) t o g e t h e r w i t h t h e i r image i n t e n s i t y l e v e l s i n T a b l e 6 . 2 . I t i s e v i d e n t f r o m F i g u r e 6 . 1 1 t h a t t h e r e a r e two d i s t i n c t b i a s l e v e l s i n t h e image i r r a d i a n c e p l o t . T h e f i r s t one r e p r e s e n t s the a m b i e n t l i g h t r e f l e c t i o n a n d has an i n t e n s i t y l e v e l o f e i g h t . T h e s e c o n d one i s the c o m b i n e d a m b i e n t a n d d i f f u s i v e t e r m K and has an i n t e n s i t y l e v e l o f a b o u t 2 0 . G e o m e t r i c r e g r e s s i o n [ D o r n and M c C r a c k e n 1972] was u s e d to d e t e r m i n e the c o e f f i c i e n t s , m and k . , o f the 125 s p e c u l a r r e f l e c t a n c e c o m p o n e n t f r o m the image i r r a d i a n c e s and the r e l a t i v e v i e w i n g a n g l e s of t h e s e s p o t s as g i v e n i n T a b l e 6 . 2 . The v a l u e s o f m and k 2 w e r e d e t e r m i n e d to be .34 and 7 0 . 2 r e s p e c t i v e l y . H e n c e , the r e f l e c t a n c e m o d e l of the c y l i n d r i c a l s u r f a c e i s g i v e n b y : I = 20 + 7 0 . 2 cos 0 ( 6 . 1 4 ) w h e r e I i s the image i r r a d i a n c e of the s p o t f e a t u r e i n g r e y <t> C O S 0 MEAS . C A L . ERROR P E R C E N T A G E I N T E N - I N T E N - ERROR ( D E G ) ( D E G ) S I T Y S I T Y (%) 5 2 . 5 6 8 . 9 0 . 3 6 6 8 . 0 6 9 . 7 1 . 7 2 . 5 5 6 . 6 6 0 . 8 0 . 4 9 72 .0 7 5 . 1 3 . 1 4 . 3 6 0 . 5 5 3 . 1 0 . 6 0 7 7 . 0 79 . 1 2 . 1 2 . 8 64 . 1 4 5 . 8 0 . 7 0 79 .0 8 2 . 2 3 . 2 4 . 0 67 . 5 39 . 1 0 . 7 8 8 2 . 0 8 4 . 5 2 . 5 3 . 0 70 . 7 3 2 . 5 0 . 84 86 .0 86 . 3 0 . 3 0 . 3 73 . 9 26 . 2 0 . 9 0 8 9 . 0 87 . 7 - 1 . 3 - 1 . 5 76 .9 2 0 . 2 0 . 94 9 0 . 0 8 8 . 7 - 1 . 3 - 1 . 4 79 .9 1 4 . 1 0 . 9 7 9 3 . 0 8 9 . 5 - 3 . 5 - 3 . 8 82 . 8 8 . 4 0 . 9 9 9 5 . 0 8 9 . 9 - 5 . 1 - 5 . 3 8 5 . 7 2 . 6 1 .00 94 .0 9 0 . 2 - 3 . 8 -4 . 1 8 8 . 5 - 3 . 0 1 . 00 9 3 . 0 9 0 . 2 -2 . 8 - 3 . 0 91 . 3 - 8 . 7 0 . 9 9 94 .0 8 9 . 9 - 4 . 1 - 4 . 3 94 . 1 - 1 4 . 2 0 . 9 7 92 .0 8 9 . 5 - 2 . 5 -2 . 7 96 . 8 - 1 9 . 7 0 . 94 91 .0 88 . 8 -2 . 2 - 2 . 4 9 9 . 6 - 2 5 . 1 0 . 9 1 8 9 . 0 8 7 . 9 - 1 . 1 - 1 . 2 102 . 4 - 3 0 . 7 0 . 86 8 8 . 0 86 . 7 - 1 . 3 - 1 . 5 1 0 5 . 1 -36 . 1 0 .81 84 .0 85 . 3 1 . 3 1 . 6 107 . 9 -41 . 7 0 . 75 8 3 . 0 83 . 6 0 . 6 0 . 7 1 1 0 . 6 - 4 7 . 3 0 . 6 8 7 9 . 0 8 1 . 6 2 . 6 3 . 3 1 1 3 . 5 - 5 3 . 0 0 . 6 0 7 7 . 0 7 9 . 2 2 . 2 2 . 8 1 1 6 . 3 - 5 8 . 7 0 . 5 2 7 3 . 0 7 6 . 3 3 . 3 4 . 6 1 1 9 . 2 - 6 4 . 5 0 . 4 3 6 9 . 0 7 2 . 9 3 . 9 5 . 6 1 2 2 . 2 - 7 0 . 5 0 . 3 3 6 5 . 0 6 8 . 5 3 . 5 5 . 4 1 2 5 . 3 - 7 6 . 6 0 . 2 3 6 3 . 0 6 2 . 9 - 0 . 1 - 0 . 1 1 2 8 . 5 - 8 2 . 9 0 . 1 2 6 1 . 0 5 4 . 7 - 6 . 3 - 1 0 . 4 T a b l e 6 . 2 Image i n t e n s i t i e s and r e l a t i v e v i e w i n g a n g l e s <j> o f the p r o j e c t i o n s p o t s on t h e c y l i n d r i c a l s u r f a c e 126 s c a l e l e v e l s and <f> i s the a n g l e b e t w e e n the r and n v e c t o r s . The r e f l e c t a n c e m o d e l o f the c y l i n d r i c a l s u r f a c e i s shown i n F i g u r e 6 . 1 2 w i t h the m e a s u r e d image i n t e n s i t y o f the s p o t s u s e d i n t h e r e f l e c t a n c e mo d e l c a l i b r a t i o n p r o c e s s ma r k e d as "+" . T h e d e v i a t i o n s o f the m e a s u r e d i n t e n s i t y v a l u e s of t h e s e p r o j e c t i o n s p o t s f r o m the v a l u e s p r e d i c t e d by the r e f l e c t a n c e m o d e l and the p e r c e n t a g e e r r o r s a r e g i v e n i n T a b l e 6 . 2 . T h e s e r e s u l t s have d e m o n s t r a t e d t h a t the r e f l e c t a n c e m o d e l g i v e n i n E q u a t i o n ( 6 . 1 4 ) g i v e s a good r e p r e s e n t a t i o n o f the r e f l e c t a n c e c h a r a c t e r i s t i c s of the c y l i n d r i c a l t e s t s u r f a c e . 6 . 2 . 6 L I M I T I N G S U R F A C E O R I E N T A T I O N R E S U L T S B a s e d u p o n the r e f l e c t a n c e m o d e l o f the c y l i n d r i c a l s u r f a c e a n d the o p t i c a l g e o m e t r y o f the s p o t p r o j e c t i o n s y s t e m , we c a n d e t e r m i n e the r a n g e o f the s u r f a c e n o r m a l v e c t o r o r i e n t a t i o n s so t h a t the image i n t e n s i t i e s of the p r o j e c t i o n s p o t s a r e a b o v e a g i v e n t h r e s h o l d l e v e l . In the b i n a r y t h r e s h o l d i n g p r o c e s s p r e s e n t e d i n S e c t i o n 5 . 2 , a b i n a r y t h r e s h o l d l e v e l o f 45 was e m p i r i c a l l y s e l e c t e d to p r o d u c e an a c c e p t a b l e b i n a r y image o f the s p o t p a t t e r n . In o r d e r f o r a p r o j e c t i o n s p o t t o be i d e n t i f i e d by t h e b i n a r y t h r e s h o l d i n g p r o c e s s , the i n t e n s i t y l e v e l o f t h e s p o t m u s t e x c e e d 4 5 . F o r the n o r m a l v e c t o r s t o the c y l i n d r i c a l t e s t s u r f a c e o r i e n t e d a l o n g t h e y - z p l a n e o f the r e f e r e n c e c o o r d i n a t e s y s t e m , the r e l a t i o n s h i p o f the s p o t i n t e n s i t i e s R E F L E C T A N C E MODEL I I I I I I I I L_ -Be -ua) e ao ee R E L . VIEWING ANGLE <p (DEGREES) gure 6.12 R e f l e c t a n c e model of the c y l i n d r i c a l s u r f a c e 128 and the s u r f a c e n o r m a l a n g l e s 6 was d e t e r m i n e d by u s i n g n E q u a t i o n ( 6 . 9 ) . A p l o t o f the i n t e n s i t y f u n c t i o n v e r s u s the s u r f a c e n o r m a l a n g l e 8 n i s g i v e n i n F i g u r e 6 . 1 3 . U s i n g 45 as t h e m i n i m u m a c c e p t a b l e i n t e n s i t y l e v e l , t h e l i m i t i n g a n g l e s of the s u r f a c e n o r m a l v e c t o r s a l o n g the y - z p l a n e a r e f o u n d t o be 4 9 ° and 1 3 6 ° . S i m i l a r l y , f o r the n o r m a l v e c t o r s o r i e n t e d a l o n g the x - z p l a n e , the r e l a t i o n s h i p of the s p o t i n t e n s i t i e s and the s u r f a c e n o r m a l a n g l e s 8 was d e t e r m i n e d m by u s i n g E q u a t i o n ( 6 . 1 3 ) . A p l o t o f the i n t e n s i t y l e v e l v e r s u s t h e s u r f a c e n o r m a l a n g l e 8 m i s g i v e n i n F i g u r e 6 . 1 4 . T h e l i m i t i n g a n g l e s o f the s u r f a c e n o r m a l v e c t o r a l o n g the x - z p l a n e a r e f o u n d to be 4 8 ° and 1 3 1 ° . 6.3 IMAGE CENTROID SPACING MODEL We w o u l d l i k e to c o n s i d e r the s e p a r a t i o n b e t w e e n the image c e n t r o i d s of two a d j a c e n t p r o j e c t i o n s p o t s w i t h r e s p e c t t o t h e o r i e n t a t i o n of the m e a s u r e m e n t s u r f a c e . The s p o t p a t t e r n i s p r o j e c t e d o n t o a s e r i e s o f p l a n a r s u r f a c e s w i t h t h e n o r m a l v e c t o r s v a r y i n g o n l y a l o n g the y - z p l a n e o f t h e r e f e r e n c e p r o j e c t i o n s y s t e m , as shown i n F i g u r e 6 . 1 5 . By v a r y i n g the d i r e c t i o n o f the n o r m a l v e c t o r s o f t h e s e p l a n a r s u r f a c e , t h e r e l a t i o n s h i p b e t w e e n the c e n t r o i d s p a c i n g s w i t h r e s p e c t t o t h e s u r f a c e o r i e n t a t i o n a l o n g t h e y - z p l a n e c a n be o b t a i n e d . T h e i d e n t i c a l p r o c e d u r e c a n be r e p e a t e d f o r a n o t h e r s e r i e s o f p l a n a r s u r f a c e s wh o s e no rma 1 v e c t o r s v a r y o n l y a l o n g t h e x - z p l a n e , as shown i n F i g u r e 6 . 1 6 . F r o m t h e s e r e s u l t s , we c a n d e d u c e the l i m i t i n g o r i e n t a t i o n of the r-Z PLANE VARIATIONS T 1 1 1 1 1 1 n r F i g u r e 6.13 The image i n t e n s i t i e s v e r s u s the s u r f a c e normal a n g l e s w i t h r e s p e c t to the y - a x i s of F . X - Z PLANE VARIATIONS T cc o e, m 3 U R F A C E NORMAL ANGLE (DEGREES) F i g u r e 6.14 The image i n t e n s i t i e s v e r s u s the s u r f a c e normal a n g l e s w i t h r e s p e c t to the x - a x i s of F . 131 132 m e a s u r e m e n t s u r f a c e s u c h t h a t the m i n i m u m c e n t r o i d s p a c i n g i s n i n e p i x e l u n i t s . 6.3.1 A N A L Y T I C A L R E P R E S E N T A T I O N O F PLANAR S U R F A C E S y - z Plane V a r i a t i o n s We w o u l d f i r s t c o n s i d e r the a n a l y t i c a l s u r f a c e f u n c t i o n of a p l a n a r s u r f a c e w i t h the s u r f a c e n o r m a l o r i e n t e d a l o n g the y - z p l a n e o f the r e f e r e n c e c o o r d i n a t e s y s t e m , as shown i n F i g u r e 6.15 The s u r f a c e f u n c t i o n c a n be wr i 11 en a s : f ( x . y . z ) z + y t a n f ( 6 . 1 5 ) = o w h e r e f i s the i n c l i n a t i o n a n g l e b e t w e e n the p l a n a r s u r f a c e and the y - a x i s o f t h e c o o r d i n a t e s y s t e m . The s u r f a c e n o r m a l v e c t o r c a n be d e f i n e d a s : V f y z ( x , y , z ) 0.0 i + t a n f j + 1.0k (6.16) w h e r e i , j and k a r e t h e b a s i s v e c t o r s o f t h e r e f e r e n c e c o o r d i n a t e s y s t e m . T h e a n g l e b e t w e e n the s u r f a c e n o r m a l v e c t o r n and the y - a x i s i s y/-90°. The p l a n v i e w o f the s u r f a c e g e o m e t r y i s shown i n F i g u r e 6.17. I n o r d e r t o h a v e a m e a n i n g f u l p h y s i c a l r e p r e s e n t a t i o n of the p r o j e c t i o n p a t t e r n 133 F i g . 6.17 P lon view of the o p t i c a l geometry of a p r o j e c t i o n spot on the p lanar su r f ace shown in F igure 6.15. o n t o t h e s e a n a l y t i c a l s u r f a c e s , the r a n g e o f the s u r f a c e i n c l i n a t i o n a n g l e f m u s t be b e t w e e n the c a m e r a a n g l e , -y, to t h e p r o j e c t o r a n g l e a. I f f i s l e s s t h a n -y, the p r o j e c t i o n s p o t s on the s u r f a c e w i l l be o c c l u d e d f r o m the v i e w of the c a m e r a . On the o t h e r h a n d , the p r o j e c t i o n o f the s p o t p a t t e r n w i l l n o t be a b l e to r e a c h the s u r f a c e i f f i s g r e a t e r t h a n a. T h u s the a n a l y t i c a l a m p l i t u d e f u n c t i o n f y z o f a f l a t s u r f a c e c a n be e x p r e s s e d a s : f ( x . y . z ) z + y t a n f , f o r - y ^ f ^ o c ( 6 . 1 7 ) y z - 0 A s g i v e n i n S e c t i o n 6 . 2 , a and y a r e 7 4 ° and 6 7 ° , r e s p e c t i v e l y . T h e r e f o r e , f m u s t be b e t w e e n - 7 4 ° and 6 7 ° w i t h r e s p e c t to the y - a x i s o f the r e f e r e n c e c o o r d i n a t e s y s t e m . 134 x-z Plane Variations N o w , l e t u s c o n s i d e r t h e p l a n a r s u r f a c e s w h o s e n o r m a l v e c t o r s a r e i n t h e d i r e c t i o n a l o n g t h e x - z p l a n e o f t h e c o o r d i n a t e s y s t e m , a s s h o w n i n F i g u r e 6 . 1 6 . S i m i l a r t o t h e p r e v i o u s c a s e , t h e a n a l y t i c a l f u n c t i o n s o f t h e s e s u r f a c e s a r e g i v e n a s : f ( x , y , z ) = z + x t a n f , ( 6 . 1 8 ) w h e r e f i s t h e a n g l e b e t w e e n t h e p l a n a r s u r f a c e a n d t h e x -a x i s . T h e n o r m a l v e c t o r o f a s u r f a c e i s a t a n a n g l e ^ - 9 0 ° w i t h r e s p e c t t o t h e x - a x i s , w h e r e ^ i s t h e a n g l e b e t w e e n t h e p l a n e a n d t h e x - a x i s o f t h e c o o r d i n a t e s y s t e m . T h e s i d e v i e w o f t h e s u r f a c e g e o m e t r y i s s h o w n i n F i g u r e 6 . 1 8 . S i n c e t h e o p t i c a l a x e s o f t h e s l i d e p r o j e c t o r a n d c a m e r a a r e i n t h e y -z p l a n e , t h e r a n g e o f t h e s u r f a c e i n c l i n a t i o n a n g l e f o r p r o d u c i n g v a l i d s p o t p r o j e c t i o n g e o m e t r i e s i s f r o m - 9 0 ° t o 9 0 ° . T h e s u r f a c e n o r m a l v e c t o r c a n b e d e f i n e d a s : V f x z ( x , y , z ) t a n f i + O . O j + 1 . 0 k ( 6 . 1 9 ) 135 2 F i g . 6 . 1 8 S i d e v i ew of the o p t i c o l g e o m e t r y of o p r o j e c t i o i n spo t on the p l a n a r s u r f o c e shown in F i g u r e 6 . 1 6 i ond r o r e both on the y-2 p l a n e . 6 . 3 . 2 S P O T S P A C I N G C A L C U L A T I O N PROCEDURE A N D R E S U L T S L e t t h e s p a t i a l c o o r d i n a t e s o f a p r o j e c t i o n s p o t on t h e p l a n a r s u r f a c e be ( x . y . z ) and t h e a d d r e s s o f the s p o t be ( u , v ) . T h e h o m o g e n e o u s r e l a t i o n s h i p b e t w e e n the s p a t i a l c o o r d i n a t e s ( x , y , z ) and the s p o t p a t t e r n c o o r d i n a t e s ( u , v ) c a n be w r i t t e n a s : v t P l l P 1 2 P 1 3 P 1 4 X u t P 2 1 P 2 2 p 2 3 p 2 4 y ( 6 . t . P 3 1 p 3 2 P 3 3 P 3 4 . z 1 136 w h e r e t i s an a r b i t r a r y s c a l i n g f a c t o r , and {p^j} i s the t r a n s f o r m a t i o n o f the s l i d e p r o j e c t o r . y-z Plane Varlat ions F o r the p l a n a r s u r f a c e s w i t h n o r m a l v e c t o r s o r i e n t e d a l o n g t h e y - z p l a n e , the x , y , and z s p a t i a l c o o r d i n a t e s o f a p r o j e c t i o n s p o t on the c y l i n d r i c a l s u r f a c e c a n be s o l v e d by E q u a t i o n s ( 6 . 1 7 ) and ( 6 . 2 0 ) . The image c e n t r o i d c o o r d i n a t e s ( l , J ) o f the c o r r e s p o n d i n g s p o t f e a t u r e c a n t h e n be o b t a i n e d by a p p l y i n g the c a m e r a t r a n s f o r m a t i o n m a t r i x on the s p a t i a l c o o r d i n a t e s o f the p r o j e c t i o n s p o t . H e n c e , t h e image c e n t r o i d c o o r d i n a t e s a r e g i v e n a s : Is Js = s C l l C 1 2 C 1 3 C 1 4 X C 2 1 C 2 2 C 2 3 C 2 4 y . C 3 1 C 3 2 C 3 3 C 3 4 . z 1 a r b i t r a r y ! s e a l i n g f a c t o r a n d ( 6 . 2 1 ) i J t r a n s f o r m a t i o n m a t r i x o f the c a m e r a . T h e same p r o c e d u r e i s r e p e a t e d f o r t h e a d j a c e n t s p o t on the same r o w as the p r e v i o u s o n e . I f the image c o o r d i n a t e s o f the s e c o n d s p o t f e a t u r e i s d e n o t e d by ( M , N ) , t h e n t h e image s p a c i n g b e t w e e n the c e n t r o i d s o f t h e s e two p r o j e c t i o n s p o t s i s d e f i n e d b y : 137 AR - [ ( M - I ) 2 + ( N - J ) 2 ] 1 / 2 ( 6 . 2 2 ) T h e image s p a c i n g AR f o r the p l a n a r s u r f a c e s w i t h i n c l i n a t i o n a n g l e s , f , v a r y i n g f r o m 7 4 ° t o - 6 7 ° i n i n c r e m e n t s o f f i v e d e g r e e s , h a v e b e e n c a l c u l a t e d and a r e p l o t t e d i n F i g u r e 6 . 1 9 . T h e l i m i t i n g i n c l i n a t i o n a n g l e f o f the s u r f a c e w h i c h c a u s e s t h e image s p a c i n g o f two a d j a c e n t s p o t t o be l e s s t h a n 9 p i x e l s a p a r t , i s f o u n d to + 2 8 ° w i t h r e s p e c t to the y - a x i s o f t h e r e f e r e n c e c o o r d i n a t e s y s t e m . T h e m i n i m u m i n c l u d e d a n g l e b e t w e e n the s u r f a c e n o r m a l v e c t o r s and t h e y -a x i s i s 62 . x-z Plane Varlat ions F o r t h e p l a n a r s u r f a c e s w i t h n o r m a l v e c t o r s o r i e n t e d a l o n g t h e x - z p l a n e , the x , y , and z s p a t i a l c o o r d i n a t e s o f a p r o j e c t i o n s p o t on the c y l i n d r i c a l s u r f a c e c a n be s o l v e d by E q u a t i o n s ( 6 . 1 8 ) and ( 6 . 2 0 ) . U s i n g the i d e n t i c a l p r o c e d u r e s as d e s c r i b e d a b o v e , the image s e p a r a t i o n s b e t w e e n two a d j a c e n t s p o t s on the same c o l u m n o f the p r o j e c t i o n p a t t e r n w e r e c a l c u l a t e d . A p l o t o f the image s e p a r a t i o n s v e r s u s t h e s u r f a c e i n c l i n a t i o n a n g l e f i s shown i n F i g u r e 6 . 2 0 . F r o m F i g u r e 6 . 2 0 , we c a n show t h a t any o f the p l a n a r s u r f a c e s w i t h the n o r m a l v e c t o r p a r a l l e l to the x - z p l a n e w i l l h a v e p r o d u c e d a m i n i m u m image s p a c i n g o f 15 p i x e l s CENTROID SPACING MODEL Y-Z 2 8 . 0 -(0 CD Z u a a. o QL \-Z LU U 2 0 . 0 1 2 . 0 -9 . 0 4 . 0 -u> C O INCLINATION ANGLES (DEGREE8) F i g u r e 6.19 C e n t r o i d s p a c i n g s v e r s u s s u r f a c e f o r s u r f a c e normal v e c t o r s a l o n g i n c l i n a t i o n a n g l e s the y - z p l a n e CENTROID SPACING MODEL X-Z (0 CO z u a CL to o oc \-z Ul u - 4 0 0 4 0 ^ INCLINATION ANGLES (DEQREE8) 8 0 Figure 6.20 Centroid spacings versus surface for surface normal vectors along inclination ang1es the x-z plane 140 b e t w e e n a n y two a d j a c e n t p r o j e c t i o n s p o t s . T h e r e f o r e , t h e s e s u r f a c e s w i l l n o t c a u s e any p r o b l e m s i n p r o v i d i n g the m i n i m u m image s p a c i n g b e t w e e n two a d j a c e n t s p o t s r e q u i r e d to g i v e a r e l i a b l e a d d r e s s d e c o d i n g r e s u l t . 6.4 C O M B I N E D R E S U L T S O F T H E R E F L E C T A N C E A N D C E N T R O I D S P A C I N G M O D E L S I n o r d e r to s a t i s f y the i n t e n s i t y l e v e l and c e n t r o i d s p a c i n g r e q u i r e m e n t s as s p e c i f i e d i n S e c t i o n 6 . 1 , the o p e r a t i n g r a n g e o f the s u r f a c e n o r m a l a n g l e a l o n g the y - z p l a n e o f t h e r e f e r e n c e c o o r d i n a t e s y s t e m F q i s b e t w e e n 4 9 ° and 1 3 6 ° f o r the i n t e n s i t y r e q u i r e m e n t , and a m i n i m u m o f 6 2 ° f o r the c e n t r o i d s p a c i n g r e q u i r e m e n t . H e n c e , the r e s u l t i n g r a n g e o f t h e c o m b i n e d s p e c i f i c a t i o n i s b e t w e e n 6 2 ° and 1 3 6 ° . I n o t h e r w o r d s , the i n c l u d e d a n g l e b e t w e e n the y - a x i s o f F q a n d t h e n o r m a l v e c t o r s c a n v a r y f r o m 6 2 ° to 1 3 6 ° . F o r t h e s u r f a c e n o r m a l v e c t o r s a l o n g t h e x - z p l a n e o f F q , t h e c e n t r o i d s p a c i n g p r o d u c e d by t h e s p o t p r o j e c t i o n s y s t e m i s a l w a y s be g r e a t e r t h a n the r e q u i r e d w i d t h o f 9 p i x e l u n i t s . H o w e v e r , the i n t e n s i t y r e q u i r e m e n t c a n o n l y be f u l f i l l e d i f t h e i n c l u d e d a n g l e b e t w e e n t h e s u r f a c e n o r m a l and t h e x - a x i s o f F i s b e t w e e n 4 9 ° and 1 3 1 ° . o 1 4 1 C H A P T E R 7 S U R F A C E R E C O N S T R U C T I O N 7.1 O V E R V I E W Range or d e p t h v a l u e s m e a s u r e d by a c t i v e or p a s s i v e s t e r e o p s i s t e c h n i q u e s a r e n o n - u n i f o r m l y s p a c e d s a m p l e s . I t i s d i f f i c u l t to g i v e an e x p l i c i t r e p r e s e n t a t i o n o f the s u r f a c e p r o p e r t i e s , s u c h as a m p l i t u d e and o r i e n t a t i o n , u s i n g t h i s s c a t t e r e d i n f o r m a t i o n . F u r t h e r m o r e , i n a mu 1 1 i - s e n s o r e n v i r o n m e n t , r a n g e i n f o r m a t i o n a v a i l a b l e f r o m v a r i o u s p r o c e s s e s n e e d s to be a s s i m i l a t e d to g i v e a c o m p l e t e s u r f a c e d e s c r i p t i o n o f t h e o b j e c t . T h e r e f o r e , a r e c o n s t r u c t i o n a l g o r i t h m i s r e q u i r e d to f i l l i n the gaps o f the u n s a m p l e d a r e a f o r t h e g e n e r a t i o n o f the f u l l s u r f a c e r e p r e s e n t a t i o n o v e r a r e g u l a r g r i d p a t t e r n f r o m a l l t h e s c a t t e r e d m e a s u r e m e n t s . R e c o n s t r u c t i o n i s d e f i n e d as a p r o c e s s o f f i t t i n g a s u r f a c e t o some s c a t t e r e d m e a s u r e m e n t p o i n t s w i t h a d d i t i o n a l c o n s t r a i n t s r e l a t e d t o some known f u n d a m e n t a l c h a r a c t e r i s t i c s o f the s u r f a c e , s u c h as s u r f a c e s m o o t h n e s s , or t h e b a n d l i m i t e d p r o p e r t y i n the s p e c t r u m o f t h e s u r f a c e f u n c t i o n . T h i s c h a p t e r d e a l s w i t h the b a s i c p r o b l e m s o f r e c o n s t r u c t i n g a v i s i b 1 e - s u r f a c e r e p r e s e n t a t i o n t o f i t the a m p l i t u d e m e a s u r e m e n t s a c q u i r e d by the s p o t p r o j e c t i o n i m a g i n g s y s t e m . T h e t h r e e - d i m e n s i o n a l s p a t i a l c o o r d i n a t e s ( x . , y . , z . ) o f e a c h s c a t t e r e d s a m p l e p o i n t a r e o b t a i n e d f r o m 142 the s p o t p r o j e c t i o n s y s t e m . S i n c e some o f the p r o j e c t i o n s p o t s ma y be o c c l u d e d f r om the v i ew o f the c ame r a , the number o f m e a s u r e m e n t p o i n t s g e n e r a t e d by the 32x32 s p o t p r o j e c t i o n p a t t e r n a r e l i m i t e d t o 1024 p o i n t s o r l e s s . The i r r a d i a n c e p a t t e r n o f the i l l u m i n a t e d s p o t s on the o b j e c t s u r f a c e i s o f t e n i r r e g u l a r and v a r i e s a c c o r d i n g to the shape o f t h e m e a s u r e m e n t s u r f a c e . We w o u l d l i k e to r e c o n s t r u c t a b i v a r i a t e s u r f a c e f u n c t i o n o v e r a s p e c i f i e d p l a n a r r e g i o n to f i t t h e s e i r r e g u l a r l y d i s t r i b u t e d s a m p l e s . T h e s p o t p r o j e c t i o n s y s t e m i s d e s i g n e d to c o v e r an a r e a o f 20x20 i n c h e s on the x - y p l a n e o f . t h e r e f e r e n c e c o o r d i n a t e s y s t e m . When an 80x80 l i n e s q u a r e s a m p l i n g l a t t i c e i s u s e d o v e r the w o r k a r e a o f t h e p r o j e c t i o n s y s t e m , e a c h e l e m e n t i n the t e s s e l a t i o n c o r r e s p o n d s t o a s q u a r e o f . 2 5 by . 2 5 i n c h e s . In o t h e r w o r d s , the g r i d p o i n t s i n the s a m p l i n g l a t t i c e h a v e a p l a n a r s p a t i a l r e s o l u t i o n o f one q u a r t e r o f an i n c h . T h r e e r e c o n s t r u c t i o n a l g o r i t h m s , n e a r e s t n e i g h b o u r , t h i n p l a t e m o d e l l i n g a n d w a r p i n g t r a n s f o r m a t i o n , w e r e i m p l e m e n t e d t o r e c o n s t r u c t the m e a s u r e m e n t s u r f a c e u s i n g the s c a t t e r e d s a m p l e s o b t a i n e d f r o m the p r o j e c t i o n s y s t e m [ T e r z o p o u l o s 1984 , C l a r k 1 9 8 5 ] . The c h a r a c t e r i s t i c s and c o n s t r a i n t s o f t h e s e t h r e e r e c o n s t r u c t i o n m e t h o d s a r e s u m m a r i z e d i n t h e n e x t t h r e e s e c t i o n s o f t h i s c h a p t e r . D e t a i l e d t h e o r e t i c a l d e r i v a t i o n s o f t h e r e c o n s t r u c t i o n a l g o r i t h m s i s b e y o n d t h e s c o p e o f t h i s t h e s i s , and c a n be f o u n d i n t h e c i t e d r e f e r e n c e s . A d i s c u s s i o n o f the r e s u l t s g e n e r a t e d by t h e s e s u r f a c e r e c o n s t r u c t i o n m e t h o d s , and the 143 a d a p t a b i l i t y o f t h e s e m e t h o d s f o r s h a p e - m e a s u r ement o f a i r c r a f t w i n g s u r f a c e s , a r e g i v e n i n the l a s t s e c t i o n . 7 . 1 . 1 S U R F A C E C O N S T R A I N T S T h e r e a r e some common c o n s t r a i n t s amongst the t h r e e r e c o n s t r u c t i o n a l g o r i t h m s . F i r s t , a s c a t t e r e d s a m p l i n g s e t i s f o r m e d f r o m the s p o t p r o j e c t i o n on the o b j e c t s u r f a c e . The s c a t t e r e d s a m p l i n g s e t i s a s e t o f g r i d p o i n t s on t h e 80x80 s a m p l i n g l a t t i c e , w h e r e t h e i l l u m i n a t e d s p o t s on t h e o b j e c t s u r f a c e a r e p r o j e c t e d o n t o the s a m p l i n g g r i d a n d a s s i g n e d to the c l o s e s t g r i d p o i n t as i l l u s t r a t e d i n F i g u r e 7 . 1 . T h u s , the s u r f a c e a m p l i t u d e v a l u e s o f t h e s e s a m p l e d g r i d p o i n t s a r e e q u a l to the z - c o o r d i n a t e s o f t h e F i g . 7 . 1 Sampled g r i d p o i n t s a s s i g n m e n t s . 144 c o r r e s p o n d i n g s p o t on the o b j e c t s u r f a c e . G r i d p o i n t s i n the l a t t i c e w i t h measurement samples a f f i x e d to them are d e f i n e d as the sampled g r i d p o i n t s . The o t h e r s a r e known as u n s a m p l e d g r i d p o i n t s . The s u r f a c e v a l u e s a t t h e s e sampled g r i d p o i n t s become the a m p l i t u d e c o n s t r a i n t s of the r e c o n s t r u c t e d s u r f a c e . S e c o n d , i t i s r e a s o n a b l e t o assume t h a t the measurement s u r f a c e i s p l a c e d i n s i d e the d e s i g n a t e d work s p a c e o f the s p o t p r o j e c t i o n s y s t e m . H e n c e , the s u r f a c e a m p l i t u d e v a l u e s o f the g r i d p o i n t s on the b o u n d a r y of the work s p a c e can be s e t to z e r o . T h e s e b o u n d a r y g r i d p o i n t s a r e a l s o added i n t o the s e t of s c a t t e r e d s a m p l i n g p o i n t s . The t h i r d c o n s t r a i n t i s o n l y a p p l i c a b l e to the t h i n p l a t e r e c o n s t r u c t i o n a l g o r i t h m . In t h i s method, the f i r s t p a r t i a l d e r i v a t i v e s o f the r e c o n s t r u c t e d s u r f a c e i s assumed to be c o n t i n u o u s . A c y l i n d r i c a l s u r f a c e i s p l a c e d on the x-y p l a n e of the r e f e r e n c e c o o r d i n a t e s y s t e m , s u c h t h a t the r e c t a n g u l a r base o f the s u r f a c e i s bounded by the l i n e s y=.875 i n c h , y= 16.375 i n c h e s , x= 0.0 and x=12.00 i n c h e s . The 3-D w i re - frame p l o t o f the c y l i n d r i c a l s u r f a c e g e n e r a t e d by the a n a l y t i c a l f u n c t i o n i s shown i n F i g u r e 7.2. A d i a g r a m o f the s c a t t e r e d sample p o i n t s i s shown i n F i g u r e 7.3. The v a l u e s of the u n s a m p l e d g r i d p o i n t s a r e r e p r e s e n t e d by a n e g a t i v e s u r f a c e v a l u e o f -0.5 i n c h . 7.2 NEAREST NEIGHBOUR RECONSTRUCTION ALGORITHM N e a r e s t n e i g h b o u r i n t e r p o l a t i o n i s the s i m p l e s t 145 F i g u r e 7.2 The o r i g i n a l c y l i n d r i c a l s u r f a c e . r e c o n s t r u c t i o n a l g o r i t h m . Under t h i s a l g o r i t h m , the r e c o n s t r u c t e d s u r f a c e v a l u e s of the sampled g r i d p o i n t s a r e u n c h a n g e d . The v a l u e s o f the unsampled g r i d p o i n t s a r e o b t a i n e d f r o m the s u r f a c e v a l u e of t h e i r n e a r e s t sampled g r i d p o i n t s . No a v e r a g i n g or s m o o t h i n g i s a t t e m p t e d on the r e c o n s t r u c t e d s u r f a c e v a l u e s . As a r e s u l t , the r e c o n s t r u c t e d s u r f a c e o b t a i n e d u s i n g t h i s a l g o r i t h m m i g h t n o t be smooth. I f any s u r f a c e d i s c o n t i n u i t i e s e x i s t on the o r i g i n a l s u r f a c e , the same d i s c o n t i n u i t i e s w i l l a l s o a p p e a r on the r e c o n s t r u c t e d s u r f a c e , a l t h o u g h the edge p o s i t i o n of the d i s c o n t i n u i t i e s m i g h t be a l t e r e d by the p r o c e s s . I f the 146 F i g u r e 7.3 T h e s c a t t e r e d s a m p l i n g p o i n t s s c a t t e r e d s a m p l e p o i n t s a r e t o o s p a r s e l y d i s t r i b u t e d , the u n s a m p l e d g r i d p o i n t may be q u i t e d i s t a n t f r o m i t s n e a r e s t s a m p l e p o i n t . C o n s e q u e n t l y , the s u r f a c e v a l u e o f t h i s s a m p l e p o i n t may be i n a p p r o p r i a t e to be u s e d i n t h e u n s a m p l e d g r i d p o i n t a n d may c a u s e l a r g e r e c o n s t r u c t i o n e r r o r . T h e n e a r e s t n e i g h b o u r m e t h o d w o r k s b e s t i f the s a m p l e p o i n t s a r e d e n s e and i s o t r o p i c a 1 l y d i s t r i b u t e d . 1 4 7 7 . 2 . 1 I M P L E M E N T A T I O N N e a r e s t n e i g h b o u r r e c o n s t r u c t i o n i s an i t e r a t i v e e x p a n s i o n a 1 g o r i t h m w h i c h r e p l a c e s t h e s u r f a c e v a l u e s of the u n s a m p l e d g r i d p o i n t s by the v a l u e s o f t h e i r n e a r e s t n e i g h b o u r i n g s a m p l e p o i n t s . I n i t i a l l y , t h e s u r f a c e v a l u e s of the s a m p l e d g r i d p o i n t s a r e d e f i n e d by the z - c o o r d i n a t e s of the c o r r e s p o n d i n g p r o j e c t i o n s p o t s o n t h e o b j e c t s u r f a c e . A 35 l a r g e i n v a l i d s u r f a c e v a l u e , -10 , i s a s s i g n e d to the r e m a i n i n g u n s a m p l e d g r i d p o i n t s . T h e g r i d p o i n t s w i t h 35 s u r f a c e v a l u e s e q u a l to -10 a r e known as " u n d e f i n e d p o i n t s " . D u r i n g e a c h i t e r a t i o n , a r a s t e r s c a n o p e r a t i o n i s p e r f o r m e d on e a c h p o i n t o f the g r i d l a t t i c e to c h e c k f o r i n v a l i d s u r f a c e v a l u e s . When an u n d e f i n e d p o i n t i s e n c o u n t e r e d , t h e a 1 g o r i t h m w i 1 1 s e a r c h i t s f o u r i m m e d i a t e n e i g h b o u r s f o r a v a l i d s u r f a c e v a l u e . T h e o r d e r of the s e a r c h s e q u e n c e a r o u n d the g r i d p o i n t s i s shown i n F i g u r e 7 . 4 . I f a v a l i d s u r f a c e v a l u e i s f o u n d , t h i s s u r f a c e v a l u e w i l l be u s e d t o r e p l a c e the v a l u e o f t h e u n d e f i n e d p o i n t . I f n o t , t h e s u r f a c e v a l u e o f t h i s p o i n t r e m a i n s i n v a l i d . The same r a s t e r s c a n o p e r a t i o n i s r e p e a t e d i n e a c h i t e r a t i o n . T h e t o t a l number o f i t e r a t i o n s t o be p e r f o r m e d i n the e x p a n s i o n a l g o r i t h m d e t e r m i n e s the s i z e o f t h e r e g i o n w h e r e t h e s u r f a c e v a l u e o f a s a m p l e d g r i d p o i n t w i l l be e x p a n d e d . I n t h e s p o t p r o j e c t i o n a p p l i c a t i o n , t h e number o f i t e r a t i o n s t o be p e r f o r m e d i n t h e r e c o n s t r u c t i o n a l g o r i t h m i s f i v e . The u n s a m p l e d g r i d p o i n t s w h i c h f a i l e d t o h a v e a d e f i n e d s u r f a c e 148 O O 1 2 g i ~3 l-COORDINATES OF THE SAMPLED LATTICE 1-1 I 1+1 J -1 J J+1 o ^  a ) -J Vi c V Vi V f • F i g . 7 . 4 The o r d e r o f s e a r c h sequence In the n e a r e s t n e i g h b o r a l g o r i t h m . value assigned, continue to have a large negative value 7.3 THIN PLATE MODEL RECONSTRUCTION It may not be p o s s i b l e to describe f u l l y the geometric p r o p e r t i e s of the measured surface w i t h only the 3-D s p a t i a l measurements of the sc a t t e r e d samples. I m p l i c i t assumptions or a p r i o r i knowledge of the surface c h a r a c t e r i s t i c s , such as the smoothness and boundary c o n d i t i o n s , can provide a d d i t i o n a l information f or the i n t e r p o l a t i o n process. These extra i n s i g h t s about the surface would improve the accuracy of the reconstructed s u r f a c e . A t h i n p l a t e model, a surface w i t h continuous f i r s t order p a r t i a l d e r i v a t i v e s over the region of i n t e r e s t , was 149 s u g g e s t e d by T e r z o p o u l o s [1984] to f i t the b e s t s u r f a c e t h r o u g h a s e t of s c a t t e r e d sample p o i n t s . 7.3.1 PHYSICAL MODEL The i n p u t to the t h i n p l a t e r e c o n s t r u c t i o n a l g o r i t h m i s the s c a t t e r e d s a m p l i n g s e t d e f i n e d o v e r the 80x80 g r i d l a t t i c e i n the work space of the s p o t p r o j e c t i o n s y s t e m . The i n p u t c o n s t r a i n t s can be v i s u a l i z e d as v e r t i c a l p i n s a t t a c h e d t o the sampled g r i d p o i n t s w i t h i n the r e g i o n . The h e i g h t o f t h e s e p i n s a r e e q u a l to the a m p l i t u d e v a l u e s of the a s s o c i a t e d p r o j e c t i o n s p o t on the o b j e c t s u r f a c e . A t h i n p l a t e i s t h e n u s e d to f i t o n t o the t i p s of t h e s e v e r t i c a l p i n s . S i n c e measurement e r r o r e x i s t s f o r e ach s u r f a c e s a m p l e , the t h i n p l a t e c a n n o t be f i t t e d d i r e c t l y onto the t i p o f t h e p i n s . I n s t e a d , i t r e s t s on some i d e a l s p r i n g s a t t a c h e d t o the end o f the v e r t i c a l p i n s . I f t h e s e i d e a l s p r i n g s a l l have z e r o n a t u r a l l e n g t h and c o n t r o l l a b l e s t i f f n e s s , t h e n the d i s p l a c e m e n t o f the s p r i n g s f r o m t h e i r n a t u r a l p o s i t i o n s can be u s e d t o s i m u l a t e the measurement e r r o r o f t h e s u r f a c e a m p l i t u d e s o f the sample p o i n t s . The b e s t t h i n p l a t e s u r f a c e t o f i t t h i s s e t o f s c a t t e r e d samp 1e p o i n t s i s the one w h i c h e x p e r i e n c e s the minimum amount of d e f o r m a t i o n p o t e n t i a l e n e r g y on the p l a t e s u r f a c e and the a t t a c h e d s p r i n g s . The g r a p h i c a l r e p r e s e n t a t i o n of the s u r f a c e and the p o s i t i o n c o n s t r a i n t s o v e r the s a m p l i n g l a t t i c e i s shown i n F i g u r e 7.5. 150 7.3.2 SURFACE CONSTRAINTS In a d d i t i o n to the p o s i t i o n and p o t e n t i a l e n e r g y c o n s t r a i n t s , a s u r f a c e smoothness c o n s t r a i n t i s a l s o imposed on the r e c o n s t r u c t e d s u r f a c e . I k e u c h i and H o r n [1981] have shown i n t h e i r p a r a l l e l c y l i n d e r e x p e r i m e n t t h a t d i s c o n t i n u i t i e s i n s u r f a c e d e p t h and f i r s t p a r t i a l d e r i v a t i v e s a r e r e a d i l y d e t e c t e d by human p e r c e p t i o n , but c h a n g e s i n the s e c o n d p a r t i a l d e r i v a t i v e s of a v i s i b l e s u r f a c e a r e m o s t l y u n n o t i c a b l e . T hese r e s u l t s i m p l y t h a t s u r f a c e m o d e l s w i t h c o n t i n u o u s s e c o n d or h i g h e r o r d e r p a r t i a l d e r i v a t i v e s a r e e x c e s s i v e l y smooth. A t h i n p l a t e s u r f a c e w h i c h has c o n t i n u o u s f i r s t p a r t i a l d e r i v a t i v e s , 151 w o u l d p r o v i d e the a p p r o p r i a t e s m o o t h n e s s c h a r a c t e r i s t i c f o r o u r a p p l i ca t i o n . A s m e n t i o n e d e a r l i e r , the m e a s u r e m e n t s u r f a c e i s a l w a y s p l a c e d i n s i d e t h e d e s i g n a t e d w o r k s p a c e o f the s p o t p r o j e c t i o n s y s t e m . I t i s r e a s o n a b l e to a s s i g n z e r o a m p l i t u d e v a l u e s to t h e g r i d p o i n t a l o n g the b o u n d a r y o f t h e r e g i o n . F u r t h e r m o r e , the r e c o n s t r u c t e d s u r f a c e w i l l a l s o h a v e z e r o p a r t i a l d e r i v a t i v e w i t h r e s p e c t to the o u t w a r d n o r m a l v e c t o r a l o n g t h e s u r f a c e b o u n d a r y . T h i s m o d e l i s t h e n known as c1 amp e d p l a t e mo d e l . 7 . 3 . 3 I M P L E M E N T A T I O N I t h a s b e e n shown t h a t , the d i s c r e t e p o t e n t i a l e n e r g y f u n c t i o n a l o f a r e c o n s t r u c t e d s u r f a c e v o b t a i n e d by t h e t h i n p l a t e m o d e l c a n be w r i t t e n as [ T e r z o p o u l o s 1 9 8 4 , p 8 6 ] : E (v ) 79 79 Z Z [v. , . -2v . .+v. , .] 2h i=2 j -1 i + l , j i , j i - l , j 79 79 + £ £ [ v i + i , j + r v i , j + r v i + i , j + v i , j r 80 79 + I Z [v. . . - 2 v . .+v. . J ' i = l j - l ^ J " 1 + - Z /B (v - d )" - r i XS XS XS 2 {xs} (7.1) w h e r e v . . i s t h e r e c o n s t r u c t e d s u r f a c e v a l u e o f a n y g r i d 152 p o i n t ( i , j ) , v i s the r e c o n s t r u c t e d s u r f a c e v a l u e at a X s s a m p l e d g r i d p o i n t , d ^ g i s the m e a s u r e d s u r f a c e v a l u e at a s a m p l e d g r i d p o i n t , h i s the s i z e o f a g r i d l a t t i c e c e l l , a n d /fi i s the s u r f a c e s m o o t h n e s s p a r a m e t e r s at t h e s a m p l e d X s g r i d p o i n t s . I f t h e s m o o t h n e s s o f t h e s u r f a c e i s e v e n , the v a l u e s o f & o f a l l t h e g r i d p o i n t s a r e e q u a l to a X s c o n s t a n t , $ . S i n c e the 20x20 w o r k i n g a r e a o f the s p o t p r o j e c t i o n s y s t e m i s p a r t i t i o n e d i n t o an 80x80 g r i d l a t t i c e , t h e s i z e o f the l a t t i c e c e l l s i s a p p r o x i m a t e l y e q u a l to a q u a r t e r o f an i n c h . B y f i r s t t a k i n g t h e p a r t i a l d e r i v a t i v e s of the p o t e n t i a l e n e r g y f u n c t i o n a l w i t h r e s p e c t t o t h e r e c o n s t r u c t e d s u r f a c e v a l u e o f e a c h g r i d p o i n t and t h e n e q u a t i n g t h e r e s u l t s t o z e r o , the m i n i m i z e d s u r f a c e a m p l i t u d e v a l u e o f a g r i d p o i n t ( i , j ) i s g i v e n b y : 2 (20+h $ . . )v . . - 8(v. , .+v. , .+v. . „ + v . . „ ) L J L J l - l . J i+l, J i . j - 1 1 .3+1 " 2 ( v i - l . j - l + v i + l . j - l + v i - l , j + l + v i + l , j + l ) " ( v i - 2 , j + V i + 2 , j + V i , j - 2 + V i , j + 2 ) - h 2/fid. (7.2) 1 > J whe r e /fl . = /fl , ( i , j ) i s a s a m p l e d g r i d p o i n t 1 » 3 = 0 , o the r w i s e . F r o m t h e s u r f a c e a m p l i t u d e c o n s t r a i n t s o f the g r i d p o i n t s as s p e c i f i e d i n S e c t i o n 7 . 1 . 1 , the m e a s u r e d s u r f a c e v a l u e s d . . i , 3 at the g r i d p o i n t s a r e g i v e n as : 153 i . J z . i . J 0 ( i , j ) i s a s a m p l e d g r i d p o i n t o t he rwi s e . ( 7 . 3 ) w h e r e z^ ^ i s t h e s u r f a c e a m p l i t u d e o f the p r o j e c t i o n s p o t a s s o c i a t e d w i t h ( i , j ) . T h e m i n i m i z a t i o n p r o c e s s o f the p o t e n t i a l e n e r g y f u n c t i o n a l i s c a r r i e d out on e v e r y g r i d p o i n t o f the s a m p l i n g l a t t i c e o v e r the w o r k s p a c e o f the p r o j e c t i o n s y s t e m . T h u s , a s y s t e m o f 6400 l i n e a r e q u a t i o n s i s f o r m e d f r o m t h i s p r o c e s s . T h e s y s t e m of e q u a t i o n s c a n be e x p r e s s e d i n ma t r i x f o r m a s : A v ( 7 . 4 ) w h e r e A i s t h e H e s s i a n m a t r i x o f the p o t e n t i a l e n e r g y f u n c t i o n , a n d t h e e n t r i e s o f A a r e g i v e n by 3 2 E ( v ) 3 V i , j 8 V k , l 1 6 i , j & 80 1 = k , 1 k 80 ( 7 . 5 ) d i s a 6400x1 c o l u m n v e c t o r c o n s i s t i n g o f t h e m e a s u r e d s u r f a c e a m p l i t u d e s o f t h e s a m p l e p o i n t s , a n d v i s the s o l u t i o n v e c t o r o f t h e m i n i m i z e d s u r f a c e . T h e s o l u t i o n v e c t o r v c a n be s o l v e d by u s i n g the G a u s s - S e i d e l r e l a x a t i o n m e t h o d [ D o r n a n d M c C r a c k e n 1 9 7 2 ] . Due t o t h e s i z e o f m a t r i x A , i t i s d i f f i c u l t to w r i t e o u t t h e e n t i r e m a t r i x e x p l i c i t l y . I t c a n be r e a l i z e d f r o m 154 E q u a t i o n s ( 7 . 2 ) o r ( 7 . 3 ) t h a t the s u r f a c e v a l u e o f a g r i d p o i n t i s a l i n e a r c o m b i n a t i o n o f the r e c o n s t r u c t e d s u r f a c e v a l u e s o f i t s 12 n e i g h b o u r i n g g r i d p o i n t s . T h i s l i n e a r c omb i n a t i o n r e l a t i o n s h i p o f the s u r f a c e v a l u e s c a n be b e t t e r d e m o n s t r a t e d by the c o m p u t a t i o n a l m o l e c u l e s , as shown i n F i g u r e 7 . 6 [ T e r z o p o u 1 o s , 1 9 8 4 ] . The number g i v e n i n e a c h m o l e c u l e r e p r e s e n t s the s c a l a r m u l t i p l i c a t i o n f a c t o r a s s o c i a t e d w i t h the g r i d p o i n t . F o r the g r i d p o i n t s on or n e a r the b o u n d a r y o f t h e s a m p l i n g l a t t i c e , some o f the 12 n e i g h b o u r i n g g r i d v a l u e s may n o t e x i s t . T h e r e f o r e , s p e c i a l m o l e c u l e s a r e a r r a n g e d to h a n d l e t h e s e s i t u a t i o n s . T h e c o m p u t a t i o n a l m o l e c u l e s f o r the g r i d p o i n t s on or n e a r t h e l e f t c o r n e r a n d the l o w e r edge o f the s a m p l i n g l a t t i c e i s d e p i c t e d i n F i g u r e 7 . 7 . R o t a t e d v e r s i o n s o f t h e g i v e n = g r i d p o i n t ( i , j ) F i g . 7.6 C o m p u t o t l o n o I m o l e c u l e s f o r the I n t e r i o r g r i d p o i n t s ( T e r z o p o u l o s , 1984. ) 155 F i g u r e 7.7 C o m p u t a t i o n m o l e c u l e s ( T e r z o p o u l o s , 1984) at b o u n d a r y n o d e s . 156 m o l e c u l e s can be u s e d f o r the o t h e r c o r n e r s and b o u n d a r y edges o f the l a t t i c e . A s e t o f i n i t i a l v a l u e s i s r e q u i r e d by the G a u s s -S e i d e l method as the f i r s t a p p r o x i m a t i o n to the s o l u t i o n o f the s y s t e m of l i n e a r e q u a t i o n s . The s u r f a c e v a l u e s o b t a i n e d f r o m the n e a r e s t n e i g h b o u r method ar e u s e d as the i n i t i a l a p p r o x i m a t e d s o l u t i o n of the r e l a x a t i o n p r o c e s s . T h i s s e t of i n i t i a l s u r f a c e v a l u e s i s s u b s t i t u t e d i n t o E q u a t i o n ( 7 . 3 ) to f o r m a new s e t o f s u r f a c e v a l u e s f o r the n e x t i t e r a t i o n . The c a l c u l a t i o n s i n v o l v e d f o r the g r i d p o i n t i n e a c h i t e r a t i o n a r e r e p r e s e n t e d by the c o m p u t a t i o n m o l e c u l e s g i v e n i n F i g u r e 7.6 or F i g u r e 7.7. The s e t of s u r f a c e v a l u e s o b t a i n e d by the i t e r a t i v e r e l a x a t i o n p r o c e s s c o n v e r g e s to f o r m the s o l u t i o n v e c t o r v o f the minimum p o t e n t i a l e n e r g y f unc t i ona 1 . A u t o m a t i c s t o p p i n g c r i t e r i a f o r the G a u s s - S e i d e l r e l a x a t i o n p r o c e s s w h i c h c h e c k the d i f f e r e n t i a l c hanges between s u c c e s s i v e i t e r a t i o n s can be implemented to t e s t f o r the c o n v e r g e n c e c o n d i t i o n o f the s o l u t i o n . However, the s o l u t i o n o f the r e c o n s t r u c t e d s u r f a c e g e n e r a t e d by the r e l a x a t i o n method al w a y s a p p e a r s to be s t a b l e a f t e r a h u n d r e d i t e r a t i o n s . T h u s , the number o f i t e r a t i o n s p e r f o r m e d by the G a u s s - S e i d e l method t o compute the s o l u t i o n of the r e c o n s t r u c t e d s u r f a c e i s i m p i r i c a l l y s e l e c t e d to be 100 i t e r a t i o n s . T e r z o p o u l o s [1984] s u g g e s t e d t h a t the s u r f a c e s m o o t h n e s s p a r a m e t e r i s g i v e n by <S=yh , where y i s a c o n s t a n t and k i s the d e g r e e of the s u r f a c e a n a l y t i c a l 157 f u n c t i o n . In the c y l i n d r i c a l s u r f a c e m e a s u rement, k-2 and h « . 2 5 i n c h e s . The v a l u e s of y g i v e n i n T e r z o p o u l o s ' r e c o n s t r u c t i o n examples were between 0.2 to 2.0. From E q u a t i o n ( 7 . 1 ) , fi i s f o u n d to be a s s o c i a t e d o n l y w i t h the measurement e r r o r t e r m (v -d ) . S i n c e the a v e r a g e xs xs 6 measurement e r r o r o f the s u r f a c e a m p l i t u d e was l e s s t h a n one t e n t h o f an i n c h i n the c y l i n d r i c a l s u r f a c e m e a s u rement, the e f f e c t s o f fi on the r e c o n s t r u c t e d s u r f a c e v a l u e s w o u l d be m i n i m a l . The s u r f a c e r e c o n s t r u c t i o n . r e s u l t s o b t a i n e d by u s i n g y-0.8 and 2.5 were f o u n d to be v i r t u a l l y i d e n t i c a l . 7 . 4 WARPING TRANSFORMATION RECONSTRUCTION The w a r p i n g t r a n s f o r m a t i o n a l g o r i t h m i s a s u r f a c e r e c o n s t r u c t i o n t e c h n i q u e b a s e d on s e r i e s e x p a n s i o n s of the s u r f a c e f u n c t i o n r a t h e r t h a n on the m i n i m i z a t i o n of a v a r i a t i o n a l p r i n c i p l e as i n the c a s e o f the t h i n p l a t e s p l i n e m e t h o d . S i m i l a r to the t h i n p l a t e m e t h o d , the s t a r t i n g p o i n t of t h i s a l g o r i t h m i s the s c a t t e r e d s a m p l i n g s e t c o l l e c t e d o v e r an 80x80 s a m p l i n g l a t t i c e l o c a t e d i n the w o r k s p a c e o f the s p o t p r o j e c t i o n i m a g i n g s y s t e m . The sample p o i n t s a r e the 32x32 p r o j e c t i o n s p o t s on the o b j e c t s u r f a c e . The p r o j e c t i o n s p o t s a r e u s u a l l y n o n - u n i f o r m l y s p a c e d but h o m o g e n e o u s l y d i s t r i b u t e d o v e r the o b j e c t s u r f a c e . T h i s h o m o g e n e i t y p r o p e r t y i s u s e d l a t e r to f o r m the w a r p i n g t r a n s f o r m a t i o n i n the s u r f a c e r e c o n s t r u c t i o n p r o c e s s . The s e t o f g r i d p o i n t s on the 80x80 s a m p l i n g l a t t i c e i s f o r m a l l y d e f i n e d by: 158 {i| x - mvj+nvj, l £ m , n ± 8 0 , V j - O . O ) and v 2 - ( 0 , l ) } (7.6) w h e r e and v 2 a r e the b a s i s v e c t o r s o f the s q u a r e s a m p l i n g l a t t i c e . T h e s c a t t e r e d s a m p l i n g s e t Z g o b t a i n e d by the s p o t p r o j e c t i o n s y s t e m i s d e f i n e d b y : Z c - { z l z - i . v . + y . v , i=l N , and O^x. ,y .^80 } S S S 1 1 1 . & 1 1 (7.7) w h e r e N i s the number o f s c a t t e r e d s a m p l e p o i n t s . I t i s o b v i o u s t h a t Z i s a s u b s e t o f X . T h e o b j e c t i v e of t h e s s u r f a c e r e c o n s t r u c t i o n p r o c e s s i s t o e v a l u a t e t h e v a l u e s o f t h e b i v a r i a t e s u r f a c e f u n c t i o n f ( x ) , f o r e v e r y x e X , f r o m t h e g i v e n s c a t t e r e d s a m p l i n g s e t Z . T h e g r a p h i c a l s r e p r e s e n t a t i o n o f the p o i n t s i n X a n d Z g i s i l l u s t r a t e d i n F i g u r e 7 . 8 . 80 x 80 sampled l o t t i c e ft ^ f \ f N f s f N f s f \ f \ f >> ? ' s k / / \ s / ' \ \ ) ' \ k ) f \ k } f \ f s f \ k ) f \ K 7 f \ \ f ? V. ?f \ \ ? ' fi k / h fi \ ? >\ fi \ ) >\ ^ \ ? f \ \ ) f \ < s s / f \ \ ? f \ f ) \ \ } f \ \ \l f \ y ^f s f s >» ? f \ k / f \ s ? f \ \ ) f \ \ ? f \ \ > f \ f / S v. )f \ \ ? f \ k / f \ \ ? f s \ ) f \ \ } f \ S } f \ k ? f \ \ ? f \ k ) f s f / s f s \ / f \ f s \ / r \ \ / f \ \ ) f \ s ? f \ f fi h ^ k ? f N f ) \ \ / f \ \ / f fi >\ \ \ / f \ s ) c s \ ) f N \ ) f \ ^ K f \ f \ f \ f ) \ f \ < V. ( \ \ / r \ f N s ) f fi s /*\ \ f s f \ f \ f ) s \ ? f \ k ) f \ \ ) f \ \ ) f \ \ } f S < V. f \ 9 ? f \ k ? f \ \ ? ' s \ ) f s f ) s \ } f \ k } f \ k / f \ f s f s \ } f \ \ ? f \ \ / / \ \ } f s f s f / \ k / f \ \ } ( fi W \ ' s f \ s, ) f > C s s. /" fi n fi \ } w ^  f / \ ? k ) 9 ^ 9 'X «= g r i d p o i n t s ® •= Sampled g r i d p o i n t s F i g . 7.8 Graphica l r ep re sen ta t i on of the sompled g r i d po ints and the sampling l a t t i c e . 159 7 . 4 . 1 S A M P L I N G THEOREMS F o r the s a k e o f c o m p l e t e n e s s , we w i l l b e g i n w i t h a b r i e f r e v i e w o f t h e f u n d a m e n t a l 1 - D and 2 - D u n i f o r m s a m p l i n g t h e o r e m s , f o l l o w e d by a d i s c u s s i o n o f the e x t e n s i o n s o f t h e s e t h e o r e m s f o r s u r f a c e r e c o n s t r u c t i o n f r o m n o n - u n i f o r m l y s p a c e d s a m p l e s . R e c o n s t r u c t i o n o f a b a n d l i m i t e d 1 - D f u n c t i o n f r o m a s e t o f u n i f o r m l y s p a c e d s a m p l e s has b e e n a w e l l s t u d i e d s u b j e c t s i n c e t h e d a y s o f E . T . W h i t t a k e r [ 1 9 1 5 ] and C . E . S h a n n o n [ 1 9 4 9 ] . T h e u n i f o r m 1 - D s a m p l i n g t h e o r e m i s s t a t e d as f o i l o w s : Theorem 7.1 [Papoulis 1966] If a bandlimited signal / ( f ) with Fourier transform co -F(co) = S f ( t ) e ' j < j i t d t , for \u\<urn/T (7.8) -co »= 0 , otherwise is sampled at the points tn=nT, n=0,±i,±2,..., then, f ( t ) can be reconstructed exactly from its samples f(nT) as foilows: oo sin[oJt-nT)] fit) = Z f(nT) ± (7.9) n=-co luAt-nT)] A s an e x t e n s i o n o f t h e 1 - D s a m p l i n g t h e o r e m , a m u l t i - d i m e n s i o n a 1 u n i f o r m s a m p l i n g t h e o r e m was l a t e r d e v e l o p e d by P e t e r s e n a n d M i d d 1 e t o n , [ 1 9 6 6 ] f o r s i g n a l s t h a t 160 h a v e b e e n s a m p l e d on a m u l t i - d i m e n s i o n a 1 s a m p l i n g l a t t i c e . In the 2 - D c a s e , we c o n s i d e r a b i v a r i a t e f u n c t i o n hCf, ) d e f i n e d o v e r a 2 - D E u c l i d e a n s p a c e H , s a m p l e d at a s e t o f r e g u l a r s a m p l i n g l a t t i c e p o i n t s E , s u c h t h a t , s E s " {£SI Ss " Vl+^V ^ 2 m 0 ' ± U ± 2 a n d ^ s £ Z } ( 7 ' 1 0 ) w h e r e a n d v 2 a r e the b a s i s v e c t o r s o f the s a m p l i n g l a t t i c e . T h e F o u r i e r t r a n s f o r m o f t h e s a m p l e d f u n c t i o n , H (CJ) , c a n t h e n be e x p r e s s e d a s : s H ( u ) = Z H ( u + u ) , f o r any u efl ( 7 . 1 1 ) s ^ s s s s w h e r e H g ( w ) i s the s p e c t r u m o f the o r i g i n a l f u n c t i o n h ( £ ) and f}g i s t h e s e t o f p e r i o d i c l a t t i c e p o i n t s i n t h e 2 - D 0 -d o m a i n . T h e e l e m e n t s i n t h i s p e r i o d i c l a t t i c e a r e d e f i n e d b y : fJ s - { u J tos - l 3 u 1 + l 4 u 2 , 1 3 , 1 4 - 0 , ± 1 , ± 2 . . . . } (7.12) w h e r e u ^ . u ^ a r e the b a s i s v e c t o r o f t h e p e r i o d i c l a t t i c e i n t h e f r e q u e n c y d o m a i n . T h e b a s i s v e c t o r s i n t h e s p a t i a l and f r e q u e n c y d o m a i n s a r e r e l a t e d b y : U J . V J - u2*2 " 2 J T ' A N D U 1 ' V 2 ™ U 2 ' V 1 " ° ( 7 . 1 3 ) I f E 8 i s o n a h e x a g o n a l s a m p l i n g l a t t i c e w h e r e V j « ( 2 / V 3 , 0 ) a n d v 2 - ( l / 7 3 , l ) , t h e n t h e b a s i s v e c t o r s o f t h e f r e q u e n c y l a t t i c e become u 1 » ( n / v ' 3 , n / 3 ) and u 2 « ( 0 , 2 r r / 3 ) . T h e h e x a g o n a l 161 s a m p l i n g l a t t i c e s i n the s p a t i a l and f r e q u e n c y domain a r e d e p i c t e d i n F i g u r e 7.9. The u n i f o r m 2-D s a m p l i n g t h e o r e m c a n t h e n be s t a t e d as f o i l ows : Theorem 7.2 [Petersen and MiddIeton,1966] If there exists a 2-D function M£) and i t s Fourier t rans f orm H(w) , such that, 1) H(u) " 0 for any u e f l 0 , for some region ft in the frequency domain; 2) the spectrum r e p e t i t i o n s in the function H (u>) do s not overlap; then, for any a r b i t r a r y £ e E , the b i v a r l a t e function h(C,) can be exactly reconstructed from i t s uniform samples as follows: M O - ? *(Zs)g(Z<s) • for any £sels (7.14) ~s where is the inverse Fourier transform of a lowpass f i l t e r function G(u) defined by: G(u) - Q - a r b i t r a r y = 0 where Q i s a constant equal to the area of the sampling l a t t i c e c e l l in the Z -domain. s The u n i f o r m s a m p l i n g theorems s t a t e d above a r e v a l i d , for any dieQQ , for any wefj ,«-&) eft o s o , for any u - u eft ; *> o 162 A" (o) Spotiol dotnoln (b) Frequency domain F i g . 7.9 Horizontal sampling l o t t l c e ' s In spa t i a l and frequency domain's 163 o n l y w h e n t h e s a m p l e s o f the f u n c t i o n s a r e t a k e n f r o m a r e g u l a r s a m p l e l a t t i c e . T h e y do n o t h a v e a n y p r o v i s i o n s to h a n d l e s c a t t e r e d s a m p l e s f r o m an i r r e g u l a r l y s p a c e d s a m p l i n g l a t t i c e , s u c h as the s a m p l e s o b t a i n e d by t h e s p o t p r o j e c t i o n s y s t e m . P a p o u l i s [ 1 9 6 6 ] i n v e s t i g a t e d the e r r o r o f a 1 - D f u n c t i o n , f ( t ) , r e c o n s t r u c t e d f r o m s a m p l e s t a k e n at v a r i a b l e s a m p l i n g i n t e r v a l s . He t r e a t e d e a c h s a m p l i n g p e r i o d as a r e g u l a r i n t e r v a l w i t h an a d d i t i o n a l r a n d o m t i m e d e l a y t e r m 0. T h e s a m p l i n g t i m e t o f the n * * 1 s a m p l e i n the 1 - D t ime s e r i e s i s t - n T - 0 ( n T ) ( 7 . 1 5 ) n w h e r e T i s t h e r e g u l a r s a m p l i n g p e r i o d . S u b s t i t u t i n g f o r T «=nT y i e l d s , t «= T - 0 ( f ) . P a p o u l i s s t a t e d t h a t i f t h e r e n n n n e x i s t s : 1) a n o n - l i n e a r t r a n s f o r m a t i o n y: t->f, s u c h t h a t , T - y ( t ) , f o r any a r b i t r a r y t ( 7 . 1 6 ) 2) a new b a n d l i m i t e d f u n c t i o n h ( f ) d e f i n e d o v e r the u n i f o r m l y s p a c e d i n t e r v a l , s u c h t h a t , h ( T ) = f [ T - 0 ( f ) ] , f o r any a r b i t r a r y t ( 7 . 1 7 ) a n d h ( T ) - f [ n T - 0 ( n T ) ] = f ( t ) ( 7 . 1 8 ) n n t h e n , t h e o r i g i n a l f u n c t i o n f ( t ) c a n be e x a c t l y r e c o n s t r u c t e d f r o m t h e r e g u l a r l y s p a c e d s a m p l e s o f h ( T Q ) . B a s e d o n P a p o u l i s ' r e s u l t s f o r the 1 - D s i g n a l , C l a r k [1985] 164 u t i l i z e d a n o n - l i n e a r or f o r m u l a t e the n o n - u n i f o r m 2 - D i s s t a t e d as f o i l ows : w a r p i n g , t r a n s f o r m a t i o n to s a m p l i n g t h e o r e m . The t h e o r e m Theorem 7.3 [Clark,1985] Consider a bivariate function f{x) over the 2-D Euclidean space X, If this function ts sampled at a set of scattered points Xg, and there exists 1) a one-to-one continuous and Invertible transformation, y: X-*l, such that , £ - y(') . for any arbitrary xeX, and £eE (7.19) and Ks - y(*s) . for any xjEX CX, and t ^ e ^ C l ; (7.20) 2) a new function / i ( £ ) Is defined over the Z-space , such that, a) MCJ - fiy'Uv) - fix), for any arbitrary xtX; (7.21) b) the Fourier t r ansf orm H(u) of h(Et) satisfies conditions 1) and 2) of Theorem 7.2; then, for any arbitrary £ e l , the function M £ ) c a n be reconstructed exactly from the regularly spaced samples of M £ ) follows: s M£) - 5 . for any arbitrary EJEI, £ ^ (7.22) ~s where g(^) is the reconstruction function whose frequency spectrum is given in Equation (7.14). Finally, for any xe.X, 165 the reconstructed bivariate function f ( x ) , can be derived from Equation (7.22). He nee, f i x ) -« £ f ( x s ) g [ y ( x ) - y ( x s ) ] , for xseXs (7.23) s 7 . 4.2 IMPLEMENTATION Now, w i t h the s a m p l i n g theorems a l l i n p l a c e , we w i l l r e c o n s t r u c t the s u r f a c e f u n c t i o n to f i t a s e t of s c a t t e r e d samples o b t a i n e d f r o m the sp o t p r o j e c t i o n i m a g i n g s y s t e m . More s p e c i f i c a l l y , we w o u l d l i k e to r e c o n s t r u c t the v a l u e o f the s u r f a c e f u n c t i o n at any g i v e n g r i d p o i n t x on the 80x80 l i n e s s amp l i n g l a t t i c e , f r om the s c a t t e r e d s amp l i n g s e t Z s d e f i n e d i n E q u a t i o n ( 7 . 7 ) . F o r e x a c t s u r f a c e r e c o n s t r u c t i o n , an i n f i n i t e number o f sample p o i n t s a r e r e q u i r e d . However, i n p r a c t i c a l a p p l i c a t i o n s , o n l y s amp l e s t a k e n f r om a f i n i t e r e g i o n a b o u t the g i v e n g r i d p o i n t w i l l be u s e d . C o n s e q u e n t l y , the r e c o n s t r u c t e d s u r f a c e i s s u b j e c t to t r u n c a t i o n and a l i a s i n g e r r o r s . I f o n l y N samples a r e s e l e c t e d f r om t h e s c a t t e r e d s amp l i n g s e t t o g e n e r a t e the r e c o n s t r u c t e d s u r f a c e v a l u e , i t has been shown [ C l a r k 1985] t h a t , t h e r e c o n s t r u c t i o n e r r o r s c a n be m i n i m i z e d by c h o o s i n g the s e t o f N c l o s e s t s a m p l i n g p o i n t s {z } , ab o u t x. In our s s o a p p l i c a t i o n , N g i s e q u a l to s e v e n . I t was m e n t i o n e d e a r l i e r t h a t the d i s t r i b u t i o n of the 166 s c a t t e r e d s a m p l e p o i n t s a r e n o n - u n i f o r m b u t h o m o g e n e o u s . By t a k i n g a d v a n t a g e o f t h i s p r o p e r t y , C l a r k d e r i v e d a h e u r i s t i c a l g o r i t h m t o d e t e r m i n e a o n e - t o - o n e w a r p i n g f u n c t i o n y , s u c h t h a t t h e p o i n t s i n ^ z s ^ 0 a r e m a p p e d o n t o t h e v e r t i c e s and t h e c e n t e r o f a h e x a g o n i n t h e u n i f o r m h e x a g o n a l l a t t i c e i n t h e £ - s p a c e , as i l l u s t r a t e d i n F i g u r e 7 . 1 0 . U n d e r t h i s w a r p i n g f u n c t i o n , the image p o i n t £ = y ( * ) o f any a r b i t r a r y x i n s i d e t h e t r i a n g u l a r r e g i o n shown i n t h i s d i a g r a m c a n be e s t i m a t e d u s i n g t h e t r i l i n e a r i n t e r p o l a t i o n f u n c t i o n , f r o m t h e v e r t i c e s o f the e n c l o s i n g t r i a n g l e s b o t h i n the x - s p a c e and t h e £ - s p a c e [ C l a r k 1 9 8 5 ] , I f U j . x ^ X j } and { ^ . f ^ , ^ } a r e t h e v e r t e x s e t s o f t h e e n c l o s i n g t r i a n g l e s i n the x -s p a c e a n d £ - s p a c e , r e s p e c t i v e l y , t h e n the image p o i n t £ i s F i g . 7.10 Warp ing t r a n s f o r m a t i o n f rom X - s p o c e to £ - s p a c e 167 g i v e n by: £ - >(x) - £ 1 I ( x , x 1 , x 2 , x ; } ) + l ^ 2 ( x , x 2 , x 3 , x 1 ) + £ 3l ( x , x 3 , x 1 , x 2 ) (7.24) where I ( x , x j , x 2 , X j ) i s the t r i l i n e a r i n t e r p o l a t i o n f u n c t i o n d e f i n e d by: I ( x , x 1 > x 2 > x 3 ) = [ x 1 ( x 2 2 - x 2 3 ) + x 2 ( x 1 3 - x 1 2 ) + ( x 2 3 x 1 2 - x 2 2 x 1 3 ) ] / A, (7.25) A f 1 2 ^ < - 1 2 N , 1 2 X J , 1 2>> and X j - (x 1 > x j ) , x 2 •= (x 2 > x 2 ) , » 3 •= (x 3 > x 3 ) and x = (x ,x ) A summary of the pro c e d u r e s used to c a l c u l a t e the r e c o n s t r u c t e d s u r f a c e v a l u e of a g r i d p o i n t x i s g i v e n as f o i l ows : 1) F i n d the c l o s e s t s c a t t e r e d sample p o i n t , z^, about x; 2) F i n d the set of the c l o s e s t sample p o i n t s { z . z,} of 1 o Z q i n each of the s i x 60° s e c t o r r e g i o n s , a s shown i n F i g u r e 7.10; 3) Determine the image set {£ } of the elements i n ^ s o 4) Determine the v e r t e x s et of the t r i a n g l e c o n t a i n i n g x; 5) Determine the c o r r e s p o n d i n g v e r t e x set of the e q u i l a t e r a l t r i a n g l e c o n t a i n i n g £, the image of x; 6) Use t r i l i n e a r i n t e r p o l a t i o n to determine the p o s i t i o n of £ i n the £-space; 7) Compute the s u r f a c e v a l u e f ( x ) from {z } and {£ } u s i n g s o ^s o the r e c o n s t r u c t i o n e q u a t i o n [ C l a r k 1985,pl63] : 168 f ( x ) - h ( > ( x ) - Z h ( C = ) g ( C - C - ) . f o r > ( 7 . 2 5 ) 0 whe r e - 2 . 1 6 | £ - £ . | 2 g ( C - C i ) " — ~ 1 . f o r ^ i c { C 8 > 0 ( 7 . 2 6 ) Z e " 2 - 1 6 l ^ i | 2 <Vo 7 . 5 S U R F A C E R E C O N S T R U C T I O N R E S U L T S T h e r e c o n s t r u c t i o n o f the c y l i n d r i c a l s u r f a c e f r o m t h e s e t o f known s c a t t e r e d s a m p l e s u s i n g the n e a r e s t n e i g h b o u r a l g o r i t h m i s shown i n F i g u r e 7 .11 and t h e e r r o r map o f t h e r e c o n s t r u c t e d s u r f a c e i s g i v e n i n F i g u r e 7 . 1 2 . T h e 3 - D w i r e - f r a m e p l o t o f t h e t h i n p l a t e m o d e l r e c o n s t r u c t e d s u r f a c e and i t s e r r o r map g e n e r a t e d a f t e r 100 i t e r a t i o n s a r e g i v e n i n F i g u r e 7 . 1 3 and 7 . 1 4 , r e s p e c t i v e l y . T h e r e c o n s t r u c t i o n o f t h e c y l i n d r i c a l s u r f a c e by means o f t h e w a r p i n g t r a n s f o r m a t i o n a l g o r i t h m i s shown i n F i g u r e 7 . 1 5 a n d t h e e r r o r map o f t h i s r e c o n s t r u c t e d s u r f a c e i n g i v e n i n F i g u r e 7 . 1 6 . T h r e e q u a n t i t a t i v e e r r o r t e r m s , maximum a b s o l u t e e r r o r , m e a n e r r o r and r o o t - m e a n - s q u a r e e r r o r , a r e u s e d to d e s c r i b e t h e d e v i a t i o n o f the r e c o n s t r u c t e d s u r f a c e f r om t h e o r i g i n a l c y l i n d r i c a l s u r f a c e shown i n F i g u r e 7 . 2 . The e r r o r t e r m s o f t h e s u r f a c e s g e n e r a t e d by t h e t h r e e r e c o n s t r u c t i o n a l g o r i t h m s a r e l i s t e d i n T a b l e 7 . 1 . A s shown i n t h e e r r o r 169 F i g u r e 7 . 1 1 N e a r e s t n e i g h b o u r r e c o n s t r u c t i o n s u r f a c e maps o f t h e r e c o n s t r u c t e d s u r f a c e s , i t i s e v i d e n t t h a t t h e m a j o r c o n t r i b u t i o n s t o t h e s e e r r o r t e r m s a r e f r o m t h e g r i d p o i n t s i n t h e a r e a a r o u n d the e d g e s o f the c y l i n d r i c a l s u r f a c e . I f s u r f a c e d i s c o n t i n u i t i e s a r e i g n o r e d on t h e r e c o n s t r u c t e d s u r f a c e a n d a t t e n t i o n i s f o c u s e d on i t s s m o o t h i n t e r i o r r e g i o n , t h e e r r o r t e r m s o f t h e r e c o n s t r u c t e d s u r f a c e s i n s i d e t h e s m o o t h r e g i o n , as l i s t e d i n T a b l e 7 . 2 , 170 F i g u r e 7 . 1 2 E r r o r map o f the n e a r e s t n e i g h b o u r r e c o n s t r u c t i o n s u r f a c e . R e c o n s t r u c t i o n Me t h o d s N e a r e s t Ne i g h b o u r T h i n P l a t e W a r p i n g T r a n s f o r m a t i o n M a x . A b s o l u t e E r r o r ( i n c h e s ) 2 . 8 7 2 . 3 4 2 . 4 0 Me an E r r o r s ( i n c h e s ) . 0 7 . 0 9 .08 RMS E r r o r s ( i n c h e s ) .31 .29 .34 T a b l e 7 . 1 T h e e r r o r t e r m s o f t h e r e c o n s t r u c t i o n s u r f a c e s 171 Recons t r u c t i o n Me thods Ne a r c s t Ne i ghbour T h i n P l a t e Max. Ab s o l u t e E r r o r ( i n c h e s ) .54 .25 Mean E r r o r s ( i n c h e s) .04 .05 RMS E r r o r s ( i n c h e s ) 12 09 W a r p i n g T r a n s f o r m a t i o n 35 ,05 10 T a b l e 7.2 The e r r o r terms i n the i n t e r i o r r e g i o n of the r e c o n s t r u c t e d c y l i n d r i c a l s u r f a c e s 172 F i g u r e 7 . 1 4 E r r o r map o f t h i n p l a t e r e c o n s t r u c t i o n s u r f a c e . a r e s i g n i f i c a n t l y 1 owe r t h a n i n the p r e v i o u s c a s e wh en the e n t i r e s u r f a c e was c o n s i d e r e d . T h e s e r e s u l t s g i v e a more t y p i c a l r e p r e s e n t a t i o n o f t h e a c c u r a c y o f t h e m e a s u r e m e n t s y s t e m f o r o b j e c t s w i t h c o n t i n u o u s s u r f a c e a m p l i t u d e , s u c h as a h e m i s p h e r i c a l s u r f a c e . T h e c o m p u t a t i o n t i m e s on a V A X - 1 1 / 7 5 0 r e q u i r e d f o r the t h r e e s u r f a c e r e c o n s t r u c t i o n a l g o r i t h m s f o r an 80x80 g r i d s u r f a c e a r e l i s t e d i n T a b l e 7 . 3 . B e c a u s e of the 113 gut t T a r , s f o t m a t i o n t e c o n s t i u c t i o x i f a Cse< t i on A l g ° T * R e c o n s t r u c t N e ighbour t e Mo d e l NeaT es T l a n s f o r m a t i o n W a I p i n g 1 .42 IOI.O 142-0 „ r r u c t i o n 174 s i m p l i c i t y of the n e a r e s t n e i g h b o u r a l g o r i t h m , i t s c o m p u t a t i o n time i s much l e s s t h a n the o t h e r two m ethods. F i g u r e 7.16 E r r o r map of the w a r p i n g r e c o n s t r u c t i o n s u r f a c e t r a n s f o rma t i o n 175 7.6 D I S C U S S I O N I t has b e e n d e m o n s t r a t e d f r o m the e r r o r maps t h a t the r e c o n s t r u c t i o n e r r o r s o c c u r m a i n l y n e a r the edge of the c y l i n d r i c a l s u r f a c e . T h i s i s c a u s e d by the a l i a s i n g e r r o r a s s o c i a t e d w i t h 32x32 d i s c r e t e s a m p l i n g p a t t e r n u t i l i z e d by the v i s i o n s y s t e m . I n i t s c u r r e n t c o n f i g u r a t i o n , the s a m p l e s o b t a i n e d f r o m the s p o t p r o j e c t i o n s y s t e m a r e n o t c a p a b l e to d e s c r i b e t h e s h a p e o f the s u r f a c e n e a r the d i s c o n t i n u i t i e s . A s w i t h a n y s u r f a c e r e c o n s t r u c t i o n a l g o r i t h m , t h e y w i l l n o t be p o s s i b l e t o r e c o n s t r u c t new s u r f a c e f e a t u r e s t h a t a r e n o t d e s c r i b e d by t h e s a m p l e d d a t a . I f a s u p p l e m e n t a r y i m a g i n g s y s t e m i s a v a i l a b l e to l o c a t e the a m p l i t u d e d i s c o n t i n u i t i e s , t h e n a p i e c e - w i s e s u r f a c e r e c o n s t r u c t i o n a l g o r i t h m c a n be a p p l i e d s e p a r a t e l y t o e a c h g r o u p o f s a m p l e p o i n t s o b t a i n e d f r o m t h e d i f f e r e n t c o n t i n u o u s l o c a l r e g i o n s . A b a c k l i g h t i n g i m a g i n g s y s t e m as i l l u s t r a t e d i n F i g u r e 7 . 1 7 , c a n be u s e d to d e t e r m i n e the b o r d e r of t h e c y l i n d r i c a l s u r f a c e on the x - y p l a n e o f the r e f e r e n c e c o o r d i n a t e s y s t e m . F r o m the image o u t l i n e and the p e r s p e c t i v e t r a n s f o r m a t i o n m a t r i x o f the c a m e r a , t h e s p a t i a l c o o r d i n a t e s o f t h e b o u n d a r y p o i n t s on the x - y p l a n e c a n be c a l c u l a t e d . T h i s b o u n d a r y i n f o r m a t i o n c a n t h e n be u s e d to d e f i n e t h e b o u n d a r y g r i d p o i n t s to be u s e d i n t h e p i e c e - w i s e r e c o n s t r u c t i o n a l g o r i t h m . The b a c k l i g h t i n g s y s t e m i s n o t i m p l e m e n t e d and i t i s l e f t as a n o t h e r f u t u r e i m p r o v e m e n t f o r t h e s p o t p r o j e c t i o n i m a g i n g s y s t e m . 176 Fig. 7.17, A suggested bocklighting projection system. T h e s u r f a c e r e c o n s t r u c t i o n r e s u l t s f r o m the n e a r e s t n e i g h b o u r m e t h o d a p p e a r s to be a b l e to r e t a i n the s h a p e o f some s i m p l e d i s c o n t i n u i t y f e a t u r e s , s u c h as the s t r a i g h t e d g e s o f t h e c y l i n d r i c a l s u r f a c e . B e c a u s e of the s u r f a c e s m o o t h n e s s c o n s t r a i n t i m p o s e d i n t h e t h i n p l a t e m o d e l , and t h e l o w p a s s f i l t e r c h a r a c t e r i s t i c s o f t h e r e c o n s t r u c t i o n e q u a t i o n i n the w a r p i n g t r a n s f o r m a t i o n a l g o r i t h m , t h e d i s c o n t i n u i t y f e a t u r e s on the o r i g i n a l s u r f a c e c a n n o t be r e p r o d u c e d . H o w e v e r , i f the b o u n d a r y o f the s u r f a c e d i s c o n t i n u i t y i s l o c a t e d by the s u p p l e m e n t a r y m e a s u r e m e n t s y s t e m , a p i e c e - w i s e m e t h o d c a n be i m p l e m e n t e d t o a v o i d the a s s i g n m e n t o f s u r f a c e v a l u e s f r o m a d i f f e r e n t r e g i o n a c r o s s t h e b o u n d a r y o f the d i s c o n t i n u i t y . O t h e r t h a n t h e r o u g h t e x t u r a l a p p e a r a n c e of the 177 r e c o n s t r u c t i o n s u r f a c e g e n e r a t e d by the w a r p i n g t r a n s f o r m a t i o n m e t h o d , t h e mean and rms e r r o r s o f the s u r f a c e , as g i v e n i n T a b l e 7 . 2 , a r e a l m o s t i d e n t i c a l t o t h a t o f the s m o o t h t h i n p l a t e r e c o n s t r u c t i o n s u r f a c e . C u r r e n t l y , o n l y s e v e n s c a t t e r e d s a m p l e p o i n t s i n the n e i g h b o u r h o o d of an u n s a m p l e d g r i d p o i n t a r e u s e d i n the r e c o n s t r u c t i o n p r o c e s s . I f a d d i t i o n a l s a m p l e p o i n t s a r e u s e d , the t r u n c a t i o n e r r o r o f the r e c o n s t r u c t e d s u r f a c e c a n t h e n be r e d u c e d . T h e r e f o r e , t h e w a r p i n g t r a n s f o r m a t i o n h a s the p o t e n t i a l o f p r o v i d i n g a m o r e a c c u r a t e r e c o n s t r u c t e d s u r f a c e f o r the e x p e n s e o f i n c r e a s e d c o m p u t a t i o n t i m e . I t m u s t be e m p h a s i z e d t h a t the w a r p i n g t r a n s f o r m a t i o n a l g o r i t h m d o e s n o t i m p o s e a n y s u r f a c e s m o o t h n e s s c o n s t r a i n t s on t h e r e c o n s t r u c t i o n p r o c e s s . T h e r e f o r e , m e a s u r e m e n t s u r f a c e s w i t h shape c h a r a c t e r i s t i c s w h i c h do n o t s a t i s f y the c o n t i n u o u s f i r s t p a r t i a l d e r i v a t i v e s c o n d i t i o n c a n be b e t t e r r e c o n s t r u c t e d by the w a r p i n g t r a n s f o r m a t i o n t e c h n i q u e . T h i s was C l a r k ' s p r i m a r y i n t e n t i o n t o d e v e l o p the w a r p r e c o n s t r u c t i o n t e c h n i q u e . 7 . 7 R E C O N S T R U C T I O N A L G O R I T H M S E L E C T I O N FOR A I R C R A F T - W I N G D A T A T h e shape o f t h e a i r c r a f t w i n g s u r f a c e i s v e r y s i m i l a r t o t h a t o f t h e c y l i n d r i c a l s u r f a c e u s e d i n t h e s p a c e - e n c o d e d s p o t p r o j e c t i o n e x p e r i m e n t . T h e y b o t h h a v e s m o o t h i n t e r i o r c u r v e s u r f a c e s and a r e b o u n d e d by e d g e s w i t h s h a r p c u r v a t u r e s . T h e r e f o r e , t h e c o n c e p t o f t h e s p o t 178 p r o j e c t i o n i m a g i n g s y s t e m u s e d i n the s h a p e m e a s u r e m e n t of t h e c y l i n d r i c a l s u r f a c e s h o u l d be d i r e c t l y a p p l i c a b l e f o r a i r c r a f t - w i ng s u r f a c e s m e a s u r e m e n t . B a s e d on t h e r e s u l t s o f the c y l i n d r i c a l s u r f a c e , one w o u l d s e l e c t the m o s t a p p r o p r i a t e r e c o n s t r u c t i o n m e t h o d i n f i t t i n g a s u r f a c e to the s e t of s c a t t e r e d s a m p l e p o i n t s o b t a i n e d by t h e s p o t p r o j e c t i o n s y s t e m . F o u r c r i t e r i a a r e c o n s i d e r e d i n t h e s e l e c t i o n p r o c e s s : 1) A c c u r a c y o f the r e c o n s t r u c t i o n m e t h o d . 2) S m o o t h n e s s o f the r e c o n s t r u c t e d s u r f a c e . 3) A d a p t a b i l i t y t o p i e c e - w i s e r e c o n s t r u c t i o n . 4 ) C o m p u t a t i o n t i m e r e q u i r e m e n t . 7.7.1 A C C U R A C Y A N D SMOOTHNESS T h e t h r e e r e c o n s t r u c t i o n m e t h o d s t e s t e d w i t h the c y l i n d r i c a l s u r f a c e m e a s u r e m e n t s , a l l s t a r t e d w i t h a common s e t o f s c a t t e r e d s u r f a c e s a m p l e p o i n t s . T h e t h r e e q u a n t i t a t i v e e r r o r s o f the r e c o n s t r u c t e d s u r f a c e g e n e r a t e d by the t h i n p l a t e m o d e l a r e c o n s i s t e n t l y l o w e r t h a n the e r r o r s o f t h e s u r f a c e s p r o d u c e d by the o t h e r two m e t h o d s . F u r t h e r m o r e , t h e t h i n p l a t e m o d e l r e c o n s t r u c t e d s u r f a c e i s s m o o t h a n d has c o n t i n u o u s f i r s t p a r t i a l d e r i v a t i v e s , w h e r e a s the o t h e r t y p e s o f r e c o n s t r u c t e d s u r f a c e s a r e n o t . The a i T c r a f t - w i n g s u r f a c e i s c o n s i d e r e d to be s m o o t h w i t h the e x c e p t i o n o f a f e w f a s t e n i n g d e v i c e s on the w i n g . In t h i s r e s p e c t , t h e t h i n p l a t e r e c o n s t r u c t i o n s u r f a c e s h o u l d p r o v i d e a c l o s e r r e s e m b l a n c e to the p h y s i c a l s h a p e of the 179 w i n g s u r f a c e . 7.7.2 P I E C E - W I S E R E C O N S T R U C T I O N I f the e d g e s o f the w i n g a r e l o c a t e d by a s u p p l e m e n t a r y i m a g i n g s y s t e m , the l o c a t i o n and s u r f a c e v a l u e s at the d i s c o n t i n u i t i e s c a n t h e n be d e f i n e d p r i o r to the r e c o n s t r u c t i o n p r o c e s s . S e p a r a t e s r e g i o n o v e r the s a m p l i n g l a t t i c e c a n be e s t a b l i s h e d f r o m the d i s c o n t i n u i t y b o u n d a r i e s . In t h e w a r p i n g t r a n s f o r m a t i o n r e c o n s t r u c t i o n , the s p i r a l s e a r c h o p e r a t i o n i s u s e d to l o c a t e the s e v e n n e a r e s t s a m p l e g r i d p o i n t s a b o u t the r e c o n s t r u c t i o n p o i n t . The d i s c o n t i n u i t y b o u n d a r y m u s t be u s e d to p r e v e n t the s e a r c h o p e r a t i o n s f r om s e l e c t i n g s amp l e p o i n t s f r om a d i f f e r e n t r e c o n s t r u c t i o n r e g i o n . C o m p l e x a l g o r i t h m s a r e r e q u i r e d to d e t e r m i n e the l o c a t i o n o f the s e a r c h p o i n t s a t t h e s e i r r e g u l a r l y s h a p e d b o u n d a r i e s . I n n e a r e s t n e i g h b o u r r e c o n s t r u c t i o n , t h e p i e c e - w i s e r e c o n s t r u c t i o n a l g o r i t h m s i m p l y i n v o l v e s a t e s t o f the f o u r i m m e d i a t e n e i g h b o u r i n g g r i d p o i n t s f o r d i s c o n t i n u i t y c o n d i t i o n s . T h e r e c o n s t r u c t e d s u r f a c e v a l u e i s o b t a i n e d o n l y f r o m the n e i g h b o u r i n g s a m p l e d g r i d p o i n t t h a t i s n o t on the d i s c o n t i n u i t y b o u n d a r i e s . A s r e c a l l e d f r om t h e t h i n p l a t e r e c o n s t r u c t i o n , t h e s u r f a c e v a l u e o f a g r i d p o i n t i s a l i n e a r c o m b i n a t i o n o f t h e s u r f a c e v a l u e s o f i t s 12 n e i g h b o u r i n g g r i d p o i n t s as d e f i n e d i n the c o m p u t a t i o n a l m o l e c u l e s d e p i c t e d i n F i g u r e 7.8. T o i n c o r p o r a t e p i e c e - w i s e r e c o n s t r u c t i o n i n t h e t h i n p l a t e mo d e l , e a c h of t h e s e 12 g r i d p o i n t s mu s t be t e s t e d f o r 180 d i s c o n t i n u i t y c o n d i t i o n . I f any d i s c o n t i n u i t y p o i n t s a r e f o u n d among t h e m , the s u r f a c e v a l u e s o f t h e s e p o i n t s c a n n o t be u s e d i n the r e c o n s t r u c t i o n e q u a t i o n . T h e a p p r o p r i a t e c o m p u t a t i o n a l m o l e c u l e s m u s t be s e t - u p to t a k e c a r e o f t h e s e d i s c o n t i n u i t y p o i n t s . A n e x a m p l e o f t h e c o m p u t a t i o n a l m o l e c u l e s f o r a c o u p l e o f s u r f a c e d i s c o n t i n u i t y c o n f i g u r a t i o n s a r e g i v e n i n F i g u r e 7 . 1 7 . T h e p i e c e - w i se r e c o n s t r u c t i o n a l g o r i t h m has n o t b e e n i n c o r p o r a t e d i n t o the a b o v e s u r f a c e r e c o n s t r u c t i o n m e t h o d s . I t i s l e f t as a f u t u r e i m p r o v e m e n t f o r t h e i m a g i n g s y s t e m . 7.7.3 COMPUTATION TIME B e c a u s e o f the s i m p l i s t i c a p p r o a c h t a k e n by t h e n e a r e s t n e i g h b o u r a l g o r i t h m , the c o m p u t a t i o n t i m e r e q u i r e d to p e r f o r m an 80x80 g r i d s u r f a c e i s c o n s i d e r a b l y f a s t e r t h a n the o t h e r two m e t h o d s . In r e a l - t i m e a p p l i c a t i o n s w i t h s t r i n g e n t t i m i n g r e q u i r e m e n t s , the n e a r e s t n e i g h b o u r a l g o r i t h m o f f e r s a n a d v a n t a g e i n c o m p u t a t i o n t i m e . A l t h o u g h the t i m e s f o r t h e o t h e r two m e t h o d s a r e c o n s i d e r a b l y g r e a t e r t h a n t h e n e a r e s t n e i g h b o u r m e t h o d , t h e c o m p u t a t i o n t i m e s a r e s t i l l l e s s t h a n t h r e e m i n u t e s . S i n c e the c o m p u t a t i o n t i m e of the r e c o n s t r u c t i o n i s p r o p o r t i o n a l to the s q u a r e o f the g r i d s i z e , t h e t i m e r e q u i r e d by the t h i n p l a t e m o d e l and the w a r p i n g t r a n s f o r m a t i o n m e t h o d s c a n i n c r e a s e q u i t e r a p i d l y as the g r i d s i z e o f t h e r e c o n s t r u c t e d s u r f a c e i n c r e a s e s . 181 7.7.4 DISCUSSIONS I n t h e f i n a l a n a l y s i s , the c h o i c e o f the a p p r o p r i a t e r e c o n s t r u c t i o n a l g o r i t h m f o r a i r c r a f t w i n g s u r f a c e shape m e a s u r e m e n t i s a t r a d e - o f f b e t w e e n the a c c u r a c y and s m o o t h n e s s o f the r e c o n s t r u c t e d s u r f a c e v e r s u s the c o m p u t a t i o n t i m e r e q u i r e m e n t . I n an a p p l i c a t i o n w h e r e a more a c c u r a t e s h a p e o f the s u r f a c e i s r e q u i r e d , the t h i n p l a t e r e c o n s t r u c t i o n t e c h n i q u e i s t h e b e s t m e t h o d . F o r o t h e r a p p l i c a t i o n s w h e r e r e s p o n s e t i m e i s c r i t i c a l , the n e a r e s t n e i g h b o u r a l g o r i t h m m i g h t be a m o r e s u i t a b l e s e l e c t i o n . 182 C H A P T E R 8 CONCLUSIONS AND F U T U R E RECOMMENDATIONS 8 . 1 SUMMARY A N D CONCLUSIONS I n t h i s t h e s i s , the a l g o r i t h m of a 3 - D v i s i o n s y s t e m b a s e d u p o n the s p a c e - e n c o d i n g s p o t p r o j e c t i o n t e c h n i q u e was d e v e l o p e d . The v i s i o n s y s t e m was i m p l e m e n t e d u s i n g o n l y o p t i c a l and image p r o c e s s i n g e q u i p m e n t w h i c h was r e a d i l y a v a i l a b l e i n the E l e c t r i c a l E n g i n e e r i n g D e p a r t m e n t . A c y l i n d r i c a l s u r f a c e was made to the g i v e n s h a p e s p e c i f i c a t i o n s and was u s e d as a c a l i b r a t i o n s u r f a c e f o r the v i s i o n s y s t e m . B e s i d e s t h i s c y l i n d r i c a l s u r f a c e , s e v e r a l o t h e r t e s t o b j e c t s i n c l u d i n g a l a n d i n g g e a r c o v e r p l a t e f o r the C F - 1 8 a i r c r a f t w e r e u s e d f o r t e s t i n g the v i s i o n s y s t e m . The m e a s u r e m e n t e r r o r s o f the s a m p l e p o i n t s on the c y l i n d r i c a l s u r f a c e w e r e o b t a i n e d f r o m the d i f f e r e n c e b e t w e e n the m e a s u r e d a m p l i t u d e s and the c a l c u l a t e d v a l u e s f r o m the s h a p e s p e c i f i c a t i o n s . The maximum and a v e r a g e m e a s u r e m e n t e r r o r s of the s a m p l e p o i n t s i n the s m o o t h i n t e r i o r r e g i o n o f the c y l i n d r i c a l s u r f a c e was f o u n d to be l e s s t h a n a q u a r t e r of an i n c h and a t e n t h of an i n c h , r e s p e c t i v e l y . S i m i l a r e n c o u r a g i n g m e a s u r e m e n t r e s u l t s on t h e s e o t h e r t e s t o b j e c t s we re a l s o o b t a i n e d . S i n c e the p e r f o r m a n c e o f the v i s i o n s y s t e m h a d b e e n w e l l i l l u s t r a t e d by the m e a s u r e m e n t r e s u l t s of the c y l i n d r i c a l s u r f a c e , the 183 r e s u l t s o f the o t h e r s u r f a c e s w e r e n o t i n c l u d e d i n the t h e s i s . P r o c e d u r e s f o r c a l i b r a t i n g the t r a n s f o r m a t i o n m a t r i c e s of the c a m e r a and s l i d e p r o j e c t o r u s e d i n the v i s i o n s y s t e m w e r e d e v e l o p e d . The RANSAC a l g o r i t h m was u t i l i z e d to e l i m i n a t e the g r o s s e r r o r p o i n t s i n the c a l i b r a t i o n s a m p l e p o p u l a t i o n s o f t h e s e d e v i c e s , so t h a t o n l y good d a t a p o i n t s w e r e u s e d f o r the d e r i v a t i o n of the t r a n s f o r m a t i o n m a t r i c e s . The a v e r a g e e r r o r b e t w e e n the c a l c u l a t e d s p a t i a l c o o r d i n a t e s o f the c a l i b r a t i o n p o i n t s by u s i n g t h e s e m a t r i c e s and t h e i r m e a s u r e d c o o r d i n a t e v a l u e s w e r e o n l y 0 . 0 6 i n c h i n the c a s e o f the c a m e r a , and 0 . 0 7 i n c h i n the c a s e o f the s l i d e p r o j e c t o r . The o p t i c a l p a r a m e t e r s of the c a m e r a and s l i d e p r o j e c t o r w e r e r e c o v e r e d f r o m t h e s e m a t r i c e s u s i n g the d e c o m p o s i t i o n p r o c e s s s u g g e s t e d by G a n a p a t h y [ 1 9 8 4 ] . The m e a s u r e d d i s p l a c e m e n t v e c t o r s of the c a m e r a and s l i d e p r o j e c t o r w i t h r e s p e c t to the r e f e r e n c e c o o r d i n a t e s y s t e m w e r e f o u n d to be c o n s i s t e n t w i t h the c a l c u l a t e d v a l u e s . T h i s c o n f i r m e d the v a l i d i t y o f the t r a n s f o r m a t i o n m a t r i c e s o b t a i n e d by the c a l i b r a t i o n p r o c e s s . A n image p r o c e s s i n g a l g o r i t h m was d e v e l o p e d to r e c o g n i z e t h e s p o t f e a t u r e s i n the images o f the s p a c e -e n c o d e d p r o j e c t i o n p a t t e r n s . B i n a r y t h r e s h o l d i n g t e c h n i q u e was u s e d to d e t e c t the s p o t f e a t u r e s i n t h e s e i m a g e s . The edge f e a t u r e d e t e c t i o n r e s u l t s o b t a i n e d by u s i n g the b i n a r y t h r e s h o l d i n g m e t h o d i n a c o n t r o l l e d l i g h t i n g e n v i r o n m e n t w e r e c o m p a r a b l e to the r e s u l t s o b t a i n e d by o t h e r more s o p h i s i c a t e d m e t h o d s . The c o l u m n a d d r e s s e s o f the p r o j e c t i o n 184 s p o t s w e r e d e c o d e d by the p r e s e n c e / a b s e n c e c o n d i t i o n s of the s p o t s i n the images o f the s p a c e - e n c o d e d p a t t e r n s . T h e . s p a t i a l c o o r d i n a t e s of the s p o t s w e r e c o m p u t e d u s i n g the d e c o d e d c o l u m n a d d r e s s e s and the t r a n s f o r m a t i o n m a t r i c e s of the c a m e r a and s l i d e p r o j e c t o r . We h a v e i l l u s t r a t e d t h a t s u r f a c e d e p t h d i s c o n t i n u i t y w o u l d a f f e c t the r e l i a b i l i t y of the a d d r e s s d e c o d i n g p r o c e s s f o r the s p o t s l o c a t e d i n the i m m e d i a t e v i c i n i t y to the d i s c o n t i n u i t y . T h i s was f o u n d to be an i n h e r e n t p r o b l e m o f the s p o t p r o j e c t i o n s t r u c t u r e d -l i g h t m e t h o d . T h e o n l y s o l u t i o n to t h i s p r o b l e m was to i m p l e m e n t an e r r o r d e t e c t i o n s p a c e - e n c o d i n g scheme f o r the p r o j e c t i o n p a t t e r n s so the s p o t s w i t h an i n c o r r e c t l y d e c o d e d a d d r e s s w o u l d be d e l e t e d f r o m the s a m p l e p o p u l a t i o n . I t was f o u n d f r o m the image i n t e n s i t i e s of the p r o j e c t i o n s p o t s on the c y l i n d r i c a l s u r f a c e t h a t the i r r a d i a n c e i n t e n s i t i e s and the s e p a r a t i o n s o f the p r o j e c t i o n s p o t s w e r e a f u n c t i o n o f the o r i e n t a t i o n of the m e a s u r e m e n t s u r f a c e . T h e d e t e c t a b i 1 i t y o f the s p o t f e a t u r e s and the r e l i a b i l i t y o f the s p o t a d d r e s s d e c o d i n g p r o c e s s w e r e d e t e r m i n e d by t h e s e two v a r i a b l e s . A r e f l e c t a n c e m o d e l f o r the c y l i n d r i c a l s u r f a c e was d e r i v e d f r o m t h e image i n t e n s i t i e s and the s u r f a c e n o r m a l v e c t o r s at the p r o j e c t i o n s p o t s on the s u r f a c e . T h e r e l a t i o n s h i p b e t w e e n the s e p a r a t i o n o f the image c e n t r o i d s o f the p r o j e c t i o n s p o t s and t h e o r i e n t a t i o n o f the s u r f a c e was a l s o e s t a b l i s h e d . F r o m t h i s r e f l e c t a n c e and c e n t r o i d s e p a r a t i o n m o d e l s , the l i m i t i n g o r i e n t a t i o n o f a m e a s u r e m e n t s u r f a c e w i t h a s i m i l a r r e f l e c t a n c e c h a r a c t e r i s t i c s as the c y l i n d r i c a l t e s t s u r f a c e 185 c o u l d be f o u n d . T h r e e r e c o n s t r u c t i o n a l g o r i t h m s , n e a r e s t n e i g h b o u r , t h i n p l a t e m o d e l , and w a r p i n g t r a n s f o r m a t i o n , were i n v e s t i g a t e d f o r f i t t i n g a s u r f a c e to the s c a t t e r e d measurement samples o b t a i n e d f r o m the s p o t p r o j e c t i o n s y s t e m . The n e a r e s t n e i g h b o u r a l g o r i t h m r e q u i r e d the l e a s t amount of c o m p u t a t i o n t i m e , but the e r r o r of the r e c o n s t r u c t e d s u r f a c e was the h i g h e s t . The a v e r a g e and rms r e c o n s t r u c t i o n e r r o r s of the o t h e r two methods were q u i t e c o m p a r a b l e . However, the t h i n p l a t e r e c o n s t r u c t i o n s u r f a c e o f f e r e d a smooth v i s i b l e - s u r f a c e r e p r e s e n t a t i o n and a lower maximum e r r o r . A l l t h r e e r e c o n s t r u c t i o n s u r f a c e s e x p e r i e n c e d l a r g e e r r o r s i n the a r e a s n e a r the edges of the c y l i n d r i c a l s u r f a c e . T h e s e r e s u l t s i n d i c a t e d t h a t the t h r e e implemented a l g o r i t h m s had a l l f a i l e d to p r o v i d e a good r e p r e s e n t a t i o n of the s u r f a c e d i s c o n t i n u i t i e s . By i g n o r i n g the r e c o n s t r u c t i o n e r r o r s near the edges of the c y l i n d r i c a l s u r f a c e , the e r r o r s of the p o i n t s i n s i d e the smooth i n t e r i o r r e g i o n were s u b s t a n t i a l l y l o w e r . These r e s u l t s were a more t y p i c a l r e p r e s e n t a t i o n of the a c c u r a c y of the s y s t e m f o r smooth c u r v e d s u r f a c e s . The m a i n c o n t r i b u t i o n of the t h e s i s i s the d e v e l o p m e n t of a low c o s t 3-D v i s i o n s y s t e m f o r m e a s u r i n g the shape of smooth f e a t u r e l e s s c u r v e d s u r f a c e s . T h r e e s u r f a c e r e c o n s t r u c t i o n a l g o r i t h m s a r e implemented i n the s y s t e m f o r g e n e r a t i n g a v i s i b l e - s u r f a c e r e p r e s e n t a t i o n f r o m the s c a t t e r e d samples p r o v i d e d by the s p o t p r o j e c t i o n s y s t e m . The t h e s i s a l s o p r o v i d e s a s o l i d b a s i s f o r the 3-D 186 v i s i o n s y s t e m to be u s e d f o r the f u t u r e a i r c r a f t w i n g m e a s u r e m e n t a p p l i c a t i o n . 8 .2 RECOMMENDATIONS FOR FUTURE WORK A number of p r o b l e m s a s s o c i a t e d w i t h the v i s i o n s y s t e m w e r e d i s c u s s e d i n t h i s t h e s i s . S o l u t i o n s to some of t h e s e p r o b l e m s w e r e i n v e s t i g a t e d and s u g g e s t e d as f u t u r e i m p r o v e m e n t s f o r the s y s t e m . The m i s a l i g n m e n t p r o b l e m o f the p r o j e c t i o n s p o t p a t t e r n s c a n be e l i m i n a t e d by u s i n g a l i q u i d c r y s t a l d i s p l a y ( L C D ) l i g h t s h u t t e r d e v i c e to p r o d u c e the s p o t p a t t e r n s . T h i s e l e c t r o - o p t i c a l d e v i c e made of n e m a t i c l i q u i d c r y s t a l d i s p l a y o p e r a t i n g i n the t r a n s m i s s i v e v i e w i n g mode i s an i d e a l s o l u t i o n f o r t h i s p r o b l e m . The l i q u i d c r y s t a l d i s p l a y c a n be c o n f i g u r e d to o p e r a t e w i t h l i g h t s e g m e n t s on a b l a c k b a c k g r o u n d . I t c a n be p a c k a g e d i n t o a t h i n s l a b , the s i z e of a s t a n d a r d two i n c h by two i n c h c a r d b o a r d h o l d e r f o r the 35 mm s l i d e s . S i m i l a r to l i q u i d c r y s t a l d i s p l a y s i n w a t c h e s and i n s t r u m e n t p a n e l s , the n e m a t i c l i q u i d c r y s t a l i n the l i g h t s h u t t e r i s s a n d w i c h e d b e t w e e n two p l a n e s o f t h i n g l a s s . The n o m i n a l s p a c i n g b e t w e e n the f r o n t and b a c k g l a s s p l a t e s i s 10 to 20 um. A s e a l i n g m a t e r i a l i s u s e d to h e r m e t i c a l l y i s o l a t e the l i q u i d c r y s t a l f r o m m o i s t u r e and o x y g e n i n the a t m o s p h e r e . S t r i p s o f t r a n s p a r e n t c o n d u c t i v e c o a t i n g m a t e r i a l c o n s i s t i n g o f a m i x t u r e o f i n d i u m and t i n o x i d e a r e e t c h e d o n t o the i n n e r s i d e s o f the g l a s s p l a n e s i n a c r i s s -187 c r o s s f a s h i o n . The o u t e r s i d e s o f the g l a s s p l a n e s a r e l a m i n a t e d w i t h p o l a r i z e r s o r i e n t e d i n p e r p e n d i c u l a r d i r e c t i o n s . When the e l e c t r i c f i e l d i s a p p l i e d to the a p p r o p r i a t e e l e c t r o d e s , the l i q u i d c r y s t a l m o l e c u l e s i n the a r e a u n d e r the i n f l u e n c e of the f i e l d , r e v e r s e o r i e n t a t i o n and a l l o w l i g h t e n e r g y to p a s s t h r o u g h . The u n e n e r g i z e d p o r t i o n o f the l i q u i d c r y s t a l r e m a i n s u n c h a n g e d and b l o c k s a l l the i n c i d e n t l i g h t f r o m p a s s i n g t h r o u g h . H e n c e , a s p o t p a t t e r n i s p r o d u c e d by p r o j e c t i n g l i g h t t h r o u g h the l i q u i d c r y s t a l s h u t t e r . One of the l i q u i d c r y s t a l s h u t t e r m a n u f a c t u r e r s c l a i m s the i n t e n s i t y c o n t r a s t r a t i o b e t w e e n the t r a n s p a r e n t and opaque c o n d i t i o n to be 2 0 : 1 [ E x c e l I n d u s t r i e s 1 9 8 5 ] . T h e r e l i a b i l i t y of the s p o t a d d r e s s d e c o d i n g p r o c e s s c a n be i m p r o v e d by e n c o d i n g the s p o t p a t t e r n s w i t h an e r r o r d e t e c t i o n s c h e m e , s u c h as the G r a y - c o d e or the Hamming c o d e , i n p l a c e o f the b i n a r y scheme w h i c h i s c u r r e n t l y u s e d . A s a r e s u l t , the p r o j e c t i o n s p o t s w i t h i n c o r r e c t l y d e c o d e d c o l u m n a d d r e s s e s c a n be r e c o g n i z e d and r e m o v e d f r o m the s u r f a c e s a m p l e p o p u l a t i o n . I f s u r f a c e d e p t h d i s c o n t i n u i t i e s e x i s t on the m e a s u r e m e n t s u r f a c e , the l o c a t i o n o f the d i s c o n t i n u i t i e s s h o u l d be d e t e c t e d b e f o r e s u r f a c e r e c o n s t r u c t i o n i s p e r f o r m e d . W i t h t h i s i n f o r m a t i o n , p i e c e - w i s e r e c o n s t r u c t i o n a l g o r i t h m s c a n be i m p l e m e n t e d s e p a r a t e l y f o r the s a m p l e p o i n t s o b t a i n e d f r o m e a c h c o n t i n u o u s l o c a l r e g i o n . F o r e x a m p l e , a b a c k l i g h t i n g v i s i o n s y s t e m may be u s e d to d e t e r m i n e t h e b o r d e r o u t l i n e s o f the c y l i n d r i c a l s u r f a c e and 188 the o u t l i n e s c a n t h e n be u s e d to d e f i n e l o c a l p i e c e - w i s e r e c o n s t r u c t i o n r e g i o n s . the b o u n d a r y of the 189 R E F E R E N C E S A g i n G . and B i n f o r d T . 1983 , " C o m p u t e r d e s c r i p t i o n of c u r v e d o b j e c t s : P r o c . I n t e r n a t i o n a l J o i n t C o n f . on A r t i f i c i a l I n t e l l i g e n c e , A u g . 1983 , p p . 6 2 9 - 6 4 0 . A l t s c h u l e r M . et a l . 1979 , " M e a s u r i n g s u r f a c e s s p a c e - e n c o d e d by a l a s e r -p r o j e c t e d d o t ma t r i x " S P I E V o l . 1 8 2 , P r o c . o f I m a g i n g A p p l i c a t i o n s f o r A u t o m a t e d I n d u s t r i a l I n s p e c t i o n and A s s e m b l y , p p . 1 8 7 - 1 9 7 . 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TERM ' «• # X Y Z Z co l (PIXELS) (INCHES) (INCHES) (INCHES) (INCHES) (INCHES) 87 4 9 14 42 17 88 181 4 1 14 42 17 26 114 4 14 43 16 63 126 9 3 14 48 16 87 13» 3 4 14 48 13 46 132 3 14 41 1 4 86 169 8 , 8 1 4 38 1 4 26 176 6 7 14 38 13 66 191 8 14 39 13 86 284 7 9 14 36 12 46 217 7 19 14 36 11 86 238 7 11 14 37 11 26 243 8 1 2 14 34 16 66 2SS 8 13 14 33 19 87 268 8 1 4 14 39 9 47 288 9 1 3 14 32 8 89 293 9 1 6 14 33 8 29 386 9 1 7 14 33 7 79 318 9 1 6 14 34 7 1 1 331 9 1 9 14 34 6 32 344 9 28 14 33 3 92 357 18 21 14 32 3 33 378 18 22 14 32 4 73 382 18 23 14 33 4 13 399 18 24 14 33 3 36 488 18 23 14 34 2 97 421 18 26 14 34 2 37 434 11 27 14 31 1 78 447 11 28 14 31 1 19 468 12 29 14 28 8 68 474 12 38 14 28 6 88 486 12 31 1 4 29 -8 38 87 28 13 86 17 87 188 28 1 13 86 17 27 113 28 13 87 16 67 126 21 3 13 84 16 86 139 22 4 13 81 13 46 132 22 13 61 14 86 163 22 0 13 82 14 23 178 23 • 7 13 79 13 63 191 23 13 79 13 83 284 23 9 13 88 12 43 217 24 18 13 77 1 1 83 238 24 11 13 77 1 1 29 243 24 12 13 77 18 63 233 23 13 13 73 18 87 268 23 14 13 73 9 47 288 23 19 13 76 8 69 293 23 16 13 76 8 29 369 26 17 13 74 7 71 318 26 18 13 74 7 11 331 26 19 13 73 6 92 344 26 29 13 73 9 92 397 26 21 13 73 3 33 378 26 22 13 76 4 73 382 26 23 13 76 4 19 393 27 24 13 73 3 96 488 27 29 13 74 2 97 421 27 26 13 74 2 37 434 27 27 13 74 1 78 447 27 26 13 73 1 19 468 28 29 13 72 8 68 473 28 39 13 72 8 91 486 28 31 13 72 -8 97 87 37 9 13 26 17 87 188 37 1 13 26 17 26 113 37 13 27 16 66 126 38 3 13 24 16 86 139 36 4 13 24 13 49 132 39 9 13 21 14 89 6 86 e 88 9 96 6 14 9 98 9 14 8 14 8 99 9 14 9 86 9 98 9 96 8 86 9 88 9 96 8 86 9 68 8 96 8 86 9 88 9 96 8 87 8 88 8 87 8 87 8 88 9 97 8 88 8 88 8 88 8 89 8 88 9 99 8 18 8 89 8 18 6 11 8 88 9 11 6 84 6 99 8 64 6 86 8 88 9 66 8 88 8 99 8 68 8 81 9 89 9 81 6 83 9 98 8 63 -8 83 9 88 9 83 -8 81 9 88 9 81 6 81 9 88 8 81 8 83 9 98 9 83 6 86 9 88 8 86 6 88 9 88 8 88 8 83 9 88 9 83 8 86 9 88 8 96 8 89 8 88 9 89 8 12 9 88 8 12 8 13 8 98 9 13 8 18 9 88 8 18 8 29 8 88 9 29 8 24 8 88 8 24 8 93 8 86 9 93 8 93 8 88 8 93 8 93 8 88 9 93 8 83 8 88 8 93 8 93 8 88 9 93 6 83 8 89 8 93 8 83 8 99 9 93 8 86 8 99 9 96 8 86 8 99 9 96 8 87 8 98 9 87 8 68 9 99 8 98 8 89 9 89 9 89 8 18 9 98 8 19 8 83 9 99 8 83 8 89 9 99 8 93 -8 82 8 99 8 82 8 88 9 99 8 99 -8 86 9 99 6 96 -6 84 8 99 8 94 -8 82 8 99 8 82 9 88 8 99 9 99 8 82 8 99 8 82 8 84 8 99 9 94 -8 81 8 99 8 81 8 82 9 99 8 82. 8 89 8 99 8 63 8 87 9 99 8 87 8 18 8 99 9 18 8 13 9 88 8 13 8 16 8 98 9 16 8 29 9 88 8 28 8 23 8 88 9 23 6 84 9 88 9 94 8 84 8 88 9 84 e 84 8 88 9 84 8 64 8 88 9 84 8 84 8 88 9 84 8 84 8 88 9 84 < 0 0 5 ^ 0 > C * > - 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8 82 0 00 0 82 23 - 8 82 6 00 8 82 61 8 85 8 08 0 85 53 1 42 1 25 0 17 86 1 88 1 61 8 1 9 28 2 18 1 98 0 28 56 2 33 2 1 4 0 1 9 94 2 49 2 34 0 15 32 2 64 2 50 0 1 4 72 2 73 2 62 0 18 1 1 2 81 2 71 0 1 0 52 2 82 2 77 0 05 93 2 83 2 88 0 03 34 2 84 2 80 0 04 77 2 78 2 78 8 01 19 2 72 2 72 8 88 62 2 67 2 64 6 03 85 2 53 2 53 8 61 91 0 51 8 34 0 17 20 1 84 8 84 8 20 51 2 33 2 39 8 07 94 2 28 2 23 8 83 39 2 88 2 84 8 04 84 1 88 1 81 0 82 38 1 52 1 56 0 04 76 1 24 1 28 0 03 22 8 97 8 96 8 01 69 8 62 8 68 6 82 1 8 8 28 8 21 0 02 62 8 88 8 06 0 00 83 8 03 8 00 e 03 55 8 86 8 60 8 06 81 8 05 8 80 8 05 21 0 04 8 00 8 04 60 8 84 8 00 6 04 51 1 48 1 27 0 22 85 1 79 1 61 0 18 197 286 158 7 9 . 86 13. .26 2. ,09 1 . , 90 8 . , 19 222 158 8 9. , 88 12, . 56 2 . 32 2. . 1 4 6 . 18 237 158 9 9 01 1 1 , . 94 2 . 48 2 . , 34 0 . , 1 4 252 158 10 9. 61 1 1 . .32 2. .63 2 . , 50 0 . 13 266 158 1 1 9. .01 16. .71 2 . ,71 2 . 62 6 . ,89 288 158 12 9 . 01 10. . 1 1 2 . 88 2 . , 72 0 . 68 293 1 58 1 3 9. . 02 9 . 52 2 . ,81 2 . , 77 0 . 83 386 1 58 1 4 9 . , 02 8 . 93 2 . 82 2 . 88 0 . 82 319 1 58 15 9 . , 03 8. .34 2 . ,83 2 . 80 0 . 03 331 158 1 6 9 . 03 7 . 76 2 . 77 2 . , 78 0 . 01 3*3 1 58 1 7 9 . 04 7 . 1 9 2 . ,71 2 . , 72 8 . 01 355 1 58 18 9 . 04 6 . 62 2 . 65 2 . , 64 0 . 01 366 1 58 19 9 . 05 6 . 85 2 . , 52 2 . 53 0 . 01 377 1 58 20 9 , . 86 5 , . 49 2 . 39 2 . , 39 6 . 00 387 1 58 21 9 , .07 4 , .94 2 . , 1 9 2 . , 23 0. . 04 133 158 3 9 . 02 15. .91 0. . 58 0 . , 34 0 . 15 153 158 4 9 . 02 15. . 28 1 . , 03 0 . , 85 0 . 18 397 1 58 22 9 . 08 4 . 39 1 . .98 2 , .84 0 . 05 487 158 23 9 . 89 3 .84 1 , . 78 1 , .81 0 . 03 416 158 24 9 . 18 3 . 30 1 . .51 1 , . 56 0 . 06 425 158 25 9 . 1 1 2 . 76 1 , .23 1 , . 28 0 . 05 434 1 58 26 9 . 12 2 .22 0 . 96 0 . 96 0 . 00 459 159 29 9 . 12 0 .62 - 0 , .81 0. . 88 0 .01 472 1 59 30 9 .13 0 . 03 0 . 82 0 . 88 0 . 02 442 1 59 27 9 . 10 1 . 69 0 .61 0. . 68 0 . 00 449 159 28 9 . 1 1 1 . 18 0. . 18 0. .21 0 . 03 485 159 31 9 . 1 3 - 0 . 55 0 . 85 0 . 88 0 .05 331 174 16 8 . 48 7 . 76 2 , . 76 2 . 78 0. .02 88 1 75 0 8 . 43 17 . 88 0, . 84 0 . 88 8 . 64 181 175 1 8 . 43 17 . 20 0. . 83 8 . 88 6 . 63 1 1 4 175 2 8 . 44 16 . 60 0. . 83 8 . 00 0 .03 153 175 4 8 . 42 1 5 . 1 9 1 , . 82 0 . 85 0 . 1 7 1 72 175 5 8 . 42 14 .51 1 , .47 1 . 27 0 .21 189 175 6 8 . 42 13 . 85 1 .78 1 .61 0 . 17 286 175 7 8 . 42 13 . 19 2 .88 1 .98 0 . 18 222 175 8 8 . 42 12 . 56 2, .31 2, . 1 4 0. . 16 237 175 9 6 . 42 1 1 . 93 2, . 46 2 . 34 0. . 12 252 1 74 10 8 . 45 1 1 .31 2 . 62 2 , . 50 0 . 12 266 174 1 1 8 . 46 1 0 .71 2 . 70 2, . 62 0 . 88 288 174 12 8 . 46 1 0 . 1 1 2 . 79 2 . 72 0. . 87 294 174 1 3 8 . 46 9 .58 2 . 87 2 . 77 0 . 09 387 1 75 1 4 8. . 43 8 .91 2 , . 88 2 . 80 0. . 08 319 1 74 15 8 . 47 8 . 34 2 , . 82 2 , . 80 0. .81 343 1 74 1 7 8 . 49 7 . 19 2 , . 78 2 , . 72 0 . 82 355 1 74 18 8 . 49 6 .61 2 , .64 2 . 64 0 0 . . 00 366 1 74 19 8 . 58 6 .05 2 , .51 2 , . 53 . 02 377 174 28 8, .51 5 . 49 2 , . 38 2 , . 39 0 . 01 387 174 21 8 .  52 4 .94 2 , . 1 7 2 , . 23 0 . 05 397 174 22 8 . 52 4 . 39 1 , .97 2 , .04 0 . 06 487 175 23 8 . 58 3. . 84 1 . .77 1 , 81 0 . 04 416 175 24 8 .51 3 . 38 1 . . 49 1 , , 56 0 . 07 134 175 3 8 . 43 15, . 89 0 . 56 0 . 36 0 . 20 425 175 25 8 . 52 2 , . 76 1 . , 22 1 . , 28 0 . 06 434 175 26 8 .53 2 , . 22 0. , 94 0. . 96 0 . 01 442 1 75 27 8. , 54 1 , . 69 0 . , 59 0 . 60 0 . 01 458 1 76 28 8. .52 1 , . 17 0. , 24 0. ,21 8 . 04 459 1 75 29 8. . 56 8 . 62 - 0 . . 83 0. , 00 0 . 03 472 1 75 38 8 . 57 8 . 04 8. .01 0. , 00 0 . 01 4B5 175 31 8 . , 57 - 8 . .55 0 . 04 0. , 00 0 . 04 252 191 18 7 . 87 1 1 . .31 2. ,61 2. , 30 0 . 1 1 267 191 1 1 7 . 87 1 8 . .69 2 . , 76 2 . 63 0 . 1 4 281 191 12 7 . , 87 18. .09 2 . , 85 2. , 72 8 . 13 294 191 13 7 . 88 9 . 50 2 . 86 2 . , 78 0. .68 387 191 1 4 7 . 88 8 . 91 2. , 87 2. , 88 6 . 66 319 191 15 7 . 89 8 . 34 2 . 81 2. , 88 0 . 00 332 191 1 6 7 . 89 7 , .75 2 . ,82 2. , 77 6 . 64 343 191 1 7 7 . 98 7. . 1 9 2. , 68 2. . 72 6 . 64 355 191 1 8 7 . 91 6 . 61 2 . , 63 2. , 64 6 . 61 366 191 1 9 7 . 91 6 . 85 2 . 49 2. , 53 6 . 63 377 191 28 7 . 92 5 . 49 2 . 36 2. . 39 6 . 63 387 191 21 7 . 93 4 . 94 2 . , 16 2. , 23 8 . 07 398 191 22 7 . 93 4 . 38 2 . 03 2. . 83 6 . 06 88 191 8 7 . , 87 17. . 88 0 . ,03 0 . , 88 8. . 63 181 192 1 7 . 84 1 7 . . 1 9 8 . ,82 0 . 60 8. . 02 1 1 4 192 2 7 . ,84 1 6 . .39 8 . 02 0 . , 00 8. . 02 134 192 3 7 . 84 1 5 . . 88 0 . ,55 0 . , 36 0 . 1 9 154 1 92 4 7 . ,83 1 5 . , 1 7 1 . .08 0 . 86 0 . , 22 172 191 5 7 . , 87 1 4 . . 58 1 . , 46 1 . , 27 0. . 1 9 1 98 191 6 7 . ,86 1 3 . , 83 1 . .84 1 . , 62 0 . , 22 286 191 7 7 . 86 1 3 . . 1 9 2 . ,07 1 . , 98 0. . 1 6 222 191 8 7 . 86 12. .55 2 . ,36 2. , 15 0 . , 15 238 191 9 7 . 87 1 1 . .92 2 . ,53 2. , 35 0. .18 407 191 23 7 . , 94 3. .84 1 . , 76 1 . .81 0 . 66 417 191 24 7 . 95 3. .29 1 . ,55 1 . , 56 0 . 66 426 191 25 7 . 96 2. . 75 1 . 28 1 . , 27 0 . 01 434 191 26 7 . 97 2. .22 0 . 93 0 . , 96 0 . 63 442 191 27 7 . 99 1 . .76 6. ,58 0. , 68 8. .82 459 191 29 8. .81 8. .62 - 0 . ,64 0. . 88 6. .64 472 192 38 7 . 98 8. .84 - 8 . ,61 0. .88 6 . 61 450 191 28 8 . 88 1 . . 1 7 0 . ,23 0. ,21 8 . 03 485 192 31 7 . 98 - 8 . .55 6 . , 83 0. .86 8. .03 223 288 8 7 . 28 12. .53 2 . , 36 2 , 15 6 . 21 238 288 9 7 . 28 1 1 . .91 2 . 51 2. . 35 6 . , 1 7 253 288 18 7 . ,28 1 1 . .29 2. , 67 2. ,51 6. . 16 267 288 1 1 7 . , 28 18. . 69 2 . , 75 2. .63 6 . 12 281 288 12 7 . ,29 18. .09 2. 83 2. .72 6 . 12 294 288 13 7 . 29 9 . 50 2. , 84 2. . 78 6 . 07 387 208 1 4 7 . , 38 8 . 91 2. , 85 2 , .88 8 . 05 326 288 15 7 . 38 8. . 32 2 . , 87 2 . 86 6 . 66 332 208 16 7 . 31 7 , . 75 2. .81 2. . 77 6 . 03 344 287 17 7 . 35 7. .17 2. .75 2 , .72 0 . 03 355 207 1 8 7 . 35 6 . 61 2. 61 2 , .64 0 . 02 366 208 19 7. .33 6 . 05 2. . 48 2 . 53 0 . 05 377 207 28 7 . 37 5, . 49 2. .35 2 , . 39 0. . 04 387 287 21 7 , . 38 4 . 94 2 , . 15 2. . 23 0. . 08 398 207 22 7 , .38 4 . 38 2 .02 2 . 83 0 .81 487 207 23 7 . 39 3 .84 1 , . 74 1 .81 0 .07 417 208 24 7 , .36 3 .29 1 , . 54 1 .56 6 .01 426 207 25 7. .41 2 . 75 1 , . 27 1 .27 6 . 00 88 289 8 7 . 24 1 7 . 79 8. . 02 0 .88 6 .02 198 101 2 8 9 1 7 . 2 5 1 7 . 1 9 8 . 01 0 . 8 8 0 . 01 1 1 4 2 8 9 2 7 . 2 5 1 6 . 5 9 8 . 01 0 . 0 0 0 . 81 1 3 4 2 8 9 3 7 . 2 5 1 5 . 8 8 8 . . 5 4 0 . 3 7 8 . 1 7 1 5 4 2 8 8 4 7 . 2 8 1 5 . . 1 7 1 . , 0 7 0 . 8 7 0 . 21 1 7 2 2 8 8 5 7 . 2 8 1 4 . 5 8 1 . , 4 5 1 . 2 7 0 . 1 8 1 9 6 2 8 8 6 7 . 2 8 1 3 . . 8 3 1 . , 8 3 1 . 6 2 0 . 21 2 0 7 2 8 8 7 7 . 2 8 1 3 . 1 7 2 . 13 1 . 91 0 . 2 2 4 3 4 2 8 8 2 6 7 . 3 8 2 . 2 2 8 . . 9 2 0 . 9 6 0 . 8 4 4 4 2 2 8 7 2 7 7 . 4 3 1 . 78 0 . 5 7 0 . 6 0 0 . 8 4 4 5 8 2 8 8 2 8 7 . 4 8 1 . 17 8 . 2 2 0 . 21 0 . 81 4 5 9 2 8 8 2 9 7 . 41 8 . 6 2 - 0 . . 0 5 0 . 0 0 0 . 8 5 4 7 2 2 8 8 3 8 7 . 4 2 8 . 8 4 - 8 . 0 2 0 . 00 0 . 8 2 4 8 5 2 8 8 31 7 . 4 2 - 8 . 5 5 0 . 01 0 . 0 0 0 . 01 2 9 4 2 2 4 13 6 . 7 4 9 . 4 9 2 . 8 3 2 . 78 0 . 0 6 3 9 7 2 2 4 14 6 . 7 5 8 . 91 2 . 8 4 2 . 80 0 . 8 4 3 2 8 2 2 4 15 6 . 7 5 8 . 3 2 2 . 8 5 2 . 80 0 . 0 5 3 3 2 2 2 4 16 6 . 7 6 7 . 74 2 . . 79 2 . 7 7 8 . 8 2 3 4 4 2 2 4 1 7 6 . 7 6 7 . 1 7 2 . . 7 3 2 . 7 2 8 . 0 2 3 5 5 2 2 4 18 6 . 7 7 6 . 61 2 . 6 0 2 . 6 4 8 . 8 4 3 6 6 2 2 4 19 6 . 7 7 6 . 8 5 2 . . 4 7 2 . 5 3 8 . 06 3 7 7 2 2 4 2 8 6 . 7 8 5 . 49 2 . . 3 4 2 . 39 8 . 8 5 3 8 8 2 2 4 21 6 . 7 9 4 . 9 3 2 . , 21 2 . 2 2 8 . 81 3 9 8 2 2 4 2 2 6 . 8 8 4 . 3 8 2 . . 0 1 2 . 8 3 8 . 8 2 2 8 7 2 2 5 7 6 . 6 9 1 3 . 17 2 . , 12 1 . 91 8 . 21 2 2 3 2 2 5 8 6 . 6 9 1 2 . 5 3 2 . 3 5 2 . 15 8 . 1 9 2 3 8 2 2 5 9 6 . 6 9 1 1 . 91 2 . 5 0 2 . 3 5 8 . 1 5 2 5 3 2 2 5 18 6 . 7 8 1 1 . 2 9 2 . 6 6 2 . 51 8 . 15 2 6 7 2 2 5 1 1 6 . 7 8 1 8 . 6 9 2 . . 7 4 2 . 6 3 8 . 1 1 2 8 1 2 2 4 12 6 . 7 4 1 8 . 8 6 2 . 8 2 2 . 7 2 8 . 18 4 8 8 2 2 4 2 3 6 . 8 8 3 . 8 3 1 . 8 0 1 . 81 8 . 88 4 1 7 2 2 4 2 4 6 . 81 3 . 2 9 1 . 5 3 1 . 56 8 . 8 3 4 2 6 2 2 4 2 5 6 . 8 2 2 . 7 5 1 . 2 5 1 . 2 7 8 . 8 2 4 3 4 2 2 4 2 6 6 . 8 3 2 . 2 2 0 . 90 0 . 96 8 . 8 5 4 4 2 2 2 4 2 7 6 . 8 4 1 . . 78 8 . . 5 6 8 . 68 8 . 8 5 4 5 8 2 2 4 2 8 6 . 8 5 1 . . 17 8 . , 2 1 8 . 21 8 . 8 8 4 6 8 2 2 5 2 9 6 . 8 2 8 . , 6 1 8 . , 8 1 8 . 8 8 8 . 81 8 8 2 2 6 8 6 . 6 5 1 7 . , 78 8 . , 8 1 8 . 8 8 8 . . 01 181 2 2 6 1 6 . , 6 6 1 7 . . 18 8 . , 8 8 8 . 8 8 8 . 0 0 1 1 4 2 2 6 2 6 . . 6 6 1 6 . , 58 8 . . 8 8 8 . 8 8 8 . 0 0 1 3 4 2 2 5 3 6 . 6 9 1 5 . , 8 7 8 . . 5 3 8 . . 3 7 0 . . 16 1 5 4 2 2 5 4 6 . . 6 9 1 5 , , 16 1 , . 8 6 0 . . 8 7 8 . , 19 1 7 2 2 2 5 5 6 . 6 9 1 4 . , 4 9 1 . . 44 1 . . 2 8 0 . 16 1 9 8 2 2 5 6 6 . 6 9 1 3 . , 8 2 1 , . 8 2 1 . , 6 2 0 . . 19 4 7 3 2 2 5 3 8 6 . , 8 2 8 . . 8 3 8 , . 8 4 0 . . 8 8 0 . . 0 4 4 8 5 2 2 5 31 6 . , 8 3 - 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0 . 8 1 8 . 8 8 8 . 8 1 4 7 3 2 5 7 3 8 5 . 7 1 8 . 8 3 0 . 8 2 8 . 8 8 8 . 8 2 4 8 6 2 5 7 31 5 . 7 1 - 8 . 5 6 0 . 0 5 8 . 8 8 8 . 8 5 1 5 4 2 5 9 4 5 . 5 1 15 . 1 6 1 . 0 4 8 . 8 7 8 . 1 7 1 7 3 2 5 9 5 5 . 5 2 1 4 . 4 7 1 . 49 1 . 2 9 0 . 2 8 1 9 8 2 5 8 6 5 . 5 5 13 . 8 1 1 . 8 0 1 . 6 3 8 . 17 2 8 7 2 5 8 7 5 . 5 5 13 . 16 2 . 10 1 . 9 2 8 . 18 8 8 2 6 8 8 5 . 4 7 17 . 7 7 - 0 . 0 1 8 . 8 8 8 . 8 1 1 8 2 2 5 9 1 5 . 5 1 1 7 . 1 6 0 . 06 8 . 8 8 8 . 8 6 1 15 2 5 9 2 5 . 5 1 16 . 5 6 0 . 0 6 8 . 8 8 8 . 8 6 1 3 4 2 5 9 3 5 . 5 1 15 . 8 6 0 . 5 1 8 . 3 8 8 . 1 3 3 3 3 2 7 3 16 5 . 8 7 7 . 7 3 2 . 8 3 2 . 7 7 8 . 8 6 199 344 273 1 7 5 .68 7 . 17 2 . 76 2 . 72 6 .62 356 273 18 5 . 86 6 . 59 2 .64 2 .63 6 .68 367 273 19 5 . 89 6 . 83 2 .51 2 .52 6 . 82 378 273 28 5 . 89 5 . 47 2 . 37 2, . 39 8. .81 254 274 18 5 .81 1 1 .27 2 . 76 2 .51 6. . 18 268 274 1 1 5 .82 10 . 66 2 . 78 2, . 63 8 . 1 4 281 274 12 5 .82 18 . 88 2 . 79 2 . 72 6 . 87 295 274 13 5 . 83 9 . 47 2 . 87 2 . 78 8 . 89 388 273 1 4 5 .86 8 . 89 2 .88 2, .88 8 . 68 328 273 1 5 5 .07 8 .31 2 . 82 2 . 86 8 .61 388 273 21 5 . 16 4 . 92 2 . 1 7 2 .22 8 . 85 398 273 22 5 . 1 1 4 . 38 1 . 97 2 . 03 8 . 86 488 273 23 5 . 1 1 3 . 83 1 . 77 1 .81 8 .84 417 273 24 5 . 12 3 . 29 1 . 49 1 . 56 8 .87 426 273 25 5 . 12 2 . 75 1 . 22 1 . 27 8 .86 434 273 26 5 .13 2 . 23 8 . 87 0 . 96 8 . 89 442 273 27 5 . 1 4 1 . 78 8 . 52 0 .61 8 .09 458 273 28 5 . 1 4 1 . 1 7 8 . 1 7 0 .21 8 . 04 468 273 29 5 . 1 5 8 . 62 - 8 . 63 0. . 00 8 .03 473 273 38 5 . 1 5 0 . 03 8 . 88 0 . 00 8 . 00 486 273 31 5 . 16 - 0 . 55 8 . 84 0 . 00 8 .04 191 275 6 4 .97 1 3 . 79 1 .86 1 .64 6 .22 287 275 7 4 .97 13 . 1 6 2 . 89 1 . 92 6 . 1 7 223 275 8 4 .97 12 . 52 2 .31 2 . 1 6 6 . 1 6 239 274 9 5 .81 1 1 . 89 2 . 54 2 . 36 8 . 1 9 115 276 2 4 . 92 1 6 . 55 8 . 65 0 . 00 0 .05 154 275 4 4 . 96 15 . 15 1 .63 0 .88 8 . 1 5 1 73 275 5 4 .96 1 4 . 46 1 .48 1 .29 6 . 19 88 277 8 4 . 88 17 . 77 - 8 . 82 8 .80 0 . 02 182 276 1 4 . 92 17 . 15 8 . 85 6 .00 6 . 05 1 35 276 3 4 . 93 15 . 84 8 .58 8 . 39 0 . 1 8 282 298 12 4 . 47 10 . 86 2 .85 2 .72 0 . 13 . 88 295 298 13 4 . 48 9 . 47 2 .86 2 . 78 0 388 298 1 4 4 . 48 8 . 88 2 .87 2 . 80 0 . 06 328 298 15 4 . 49 8 .31 2 .88 2 . 88 0 . 00 333 298 16 4 . 49 7 .72 2 . 82 2 . 77 0 . 04 344 298 17 4 . 58 7 . 16 2 .68 2 . 72 0 . 03 356 289 18 4 . 54 6 .59 2 .62 2 . 63 0 .01 367 298 19 4 .51 6 .63 2 . 49 2 . 52 0 . 03 378 289 28 4 . 55 5 . 47 2 . 36 2 . 39 0 . 02 388 289 21 4 .55 4 .92 2 . 16 2 . 22 0 . 06 398 289 22 4 . 56 4 .38 1 .96 2 .83 0 . 07 488 289 23 4 .56 3 .83 1 . 75 1 .81 8 . 05 417 289 24 4 . 57 3 .29 1 . 48 1 . 56 0 . 08 426 289 25 4 .57 2 . 75 1 . 28 1 . 27 8 . 87 468 298 29 4 . 56 0 .62 - 8 . 84 8 . 00 0 . 84 224 291 8 4 . 42 12 .58 2 . 38 2 . 16 8 .21 239 291 9 4 . 43 1 1 .88 2 .53 2 . 36 0 . 1 7 254 291 18 4 . 43 1 1 .26 2 .68 2 .51 0 . 1 7 268 291 1 1 4 . 44 10 . 66 2 . 76 2 . 63 0 . 1 3 434 298 26 4 . 54 2 .23 6 . 85 8 . 96 0 . 1 1 442 298 27 4 .55 1 .70 6 .58 8 .61 0 . 1 0 438 298 28 4 .56 1 . 1 7 6 . 1 6 0 .21 0 . 05 473 298 38 4 .56 0 .63 - 8 .81 8 . 00 0 .01 486 298 31 4 . 57 - 0 . 55 8 . 82 0 . 00 0 0 . 02 191 292 6 4 . 38 13 .79 1 .85 1 . 64 .21 288 291 7 4 .42 1 3 . 1 4 2 . 15 1 . 93 0 . 22 182 293 1 4 . 33 1 7 . 1 4 0 . 04 0 . 00 0 . 04 115 293 2 4 . 34 1 6 .55 0 .64 0 . 00 0 . 04 135 293 3 4 . 34 15 .84 0 .57 0 . 40 0 . 1 7 155 292 4 4 . 38 15 .13 1 .69 6 . 89 0 . 20 1 73 292 5 4 . 38 1 4 . 46 1 .47 1 .29 0 . 1 8 88 294 8 4 . 29 1 7 .76 - 0 .63 6 . 08 0 . 83 333 386 16 3 . 94 7 .72 2 .86 2 . 77 0 . 03 345 386 17 3 . 95 7 . 15 2 .74 2 .72 0 . 03 356 386 18 3 .95 6 . 59 2 .61 2 . 63 6 .62 367 386 19 3 .96 6 . 63 2 . 48 2 .52 0 .64 378 386 28 3 .96 5 . 47 2 . 35 2 .39 0 .64 388 386 21 3. .97 4 .92 2 . 1 4 2 . 22 0 . 68 254 387 18 3, .88 1 1 .26 2 . 67 2 .31 0 . 1 6 268 387 1 1 3 : 89 10 .66 2 . 75 2 . 63 0 . 1 2 282 387 12 3 . 89 10. . 86 2 .84 2 , . 72 0 , . 1 1 295 387 13 3 . 98 9. . 47 2 .84 2. . 78 0 , . 87 388 387 14 3 . 98 8 .  88 2 . 85 2 . 86 6 . 65 321 387 15 3. .91 8 . 38 2 . 87 2. .86 0 . 06 398 306 22 3. . 97 4 . 38 1 . . 94 2. . 83 8 . 09 488 366 23 3 . 98 3 , .83 1 . 74 1 . .81 0 . 07 417 386 24 3. . 98 3. .29 1 . . 46 1 . ,56 8 . 09 426 386 25 3. . 99 2. .75 1 . . 19 1 . 27 0 . , 08 435 386 26 3 . 99 2. .21 0 . 92 8 . , 95 8 . 84 443 386 27 4 . 88 1 . . 69 0. .57 6 . ,68 0 . , 83 458 386 28 4. 60 1 . , 1 7 0 . , 1 4 6 . ,21 0 . ,07 468 386 29 4 . 61 6 . 62 - 0 . . 85 8. 88 0 . ,05 473 306 38 4 . 01 6 . 83 - 0 . . 82 6 . 88 0 . 02 486 367 31 3. 98 - 0 . .55 0 . ,81 8 . 68 0 . 01 288 308 7 3. 84 13. 13 2. . 1 4 1 . 93 0 . 21 224 388 8 3. 84 12. .58 2 . ,36 2. 16 0. 20 239 388 9 3 . 85 1 1 . 88 2. 52 2. 36 0 . 1 6 174 389 5 3 . 79 1 4 . 44 1 . 53 1 . 30 0 . 23 191 388 6 3. 83 13. 79 1 . 84 1 . 64 0 . 20 1 82 318 1 3. 74 1 7 . 1 4 0. 83 8. 00 6 . 83 115 318 2 3. 75 16. 54 0. 82 6 . 00 0. 02 135 389 3 3. 79 15. 83 6. 56 8. 40 8. 1 5 155 389 4 3. 79 15. 13 1 . 88 8. 89 0. 19 88 311 8 3. 78 17. 75 - 8 . 64 8. 00 8 . 04 296 323 13 3. 35 9. 45 2. 91 2. 78 . 8. 1 3 388 323 1 4 3 . 35 8. 88 2 . 84 2. 88 8 . 04 321 323 15 3 . 36 8 . 29 2 . 85 2. 88 8 . 05 333 323 16 3 . 36 7 . 72 2 . 79 2. 77 8 . 02 345 323 17 3 . 37 7 . 15 2. 73 2. 72 8 . 02 356 323 18 3 . 37 6 . 59 2 . 66 2 . 63 8 . 03 367 323 1 9 3. 38 6. 83 2 . 47 2 . 52 6 . 06 378 323 26 3. 38 5. 47 2 . 34 2. 38 8 . 05 389 322 21 3 . 42 4 . 91 2 . 21 2. 22 6 . 01 399 322 22 3. 42 4 . 36 2. 00 2. 82 8 . 02 488 323 23 3. 39 3. 83 1 . 73 1 . 81 8 . 08 418 322 24 3. 43 3. 28 1 . 53 1 . 55 8 . 62 427 323 25 3. 48 2 . 74 1 . 25 1 . 27 6 . 61 451 322 28 3. 45 1 . 16 8 . 21 8. 26 8 . 68 461 323 29 3. 42 8. 61 6. 01 8 . 88 6 . 81 200 2 3 9 3 2 4 9 3 . 3 8 1 1 2 5 4 3 2 4 18 3 . 38 1 1 2 6 8 3 2 4 1 1 3 . 3 1 10 2 8 2 3 2 3 12 3 . 3 3 18 4 3 5 3 2 3 2 6 3 . 48 2 4 4 3 3 2 3 2 7 3 . 4 1 1 4 7 3 3 2 3 38 3 . 4 2 8 4 8 6 3 2 3 31 3 . 4 3 - 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8 . . 8 4 6 . 8 8 8 . 8 4 5 5 8 , . 8 8 8 . 8 8 8 . 8 8 13 2 . 13 1 . 9 3 8 . 2 8 49 2 . 3 5 2 . . 1 7 8 . . 1 9 4 3 1 . . 5 2 1 , . 3 1 8 . . 2 2 78 1 . . 8 2 1 . . 6 4 8 . 18 13 8 . . 8 2 8 . 8 8 8 . 8 2 5 3 8 . . 8 1 8 . 8 6 8 . 8 1 8 3 8 . . 5 4 8 . 41 8 . 1 4 12 1 . . 8 7 8 . 9 8 8 . 1 8 7 5 - 8 . . 8 5 8 . . 8 8 8 . 8 5 8 5 2 . 8 1 2 . 7 2 8 . 8 9 46 2 . 8 2 2 . 7 8 8 . 8 4 88 2 . 8 3 2 . 8 6 8 . 6 2 29 2 . 8 4 2 . 8 6 8 . 8 4 7 2 2 . 7 8 2 . 7 7 8 . 6 1 15 2 . . 7 2 2 . 7 2 8 . 6 6 5 9 2 . 5 9 2 . 6 3 8 . 6 5 8 2 2 . 5 3 2 . 5 2 8 . 6 1 4 7 2 . 3 2 2 . 38 6 . 6 6 91 2 . 19 2 . 2 2 8 . 8 2 3 6 1 . 9 9 2 . 6 2 6 . 6 3 81 1 . 7 9 1 . 88 8 . 6 1 2 8 1 . 5 1 1 . 5 5 8 . 8 4 74 1 . 2 4 1 . 2 7 6 . 6 3 2 2 8 . 8 9 8 . 9 5 8 . 8 6 6 9 8 . 5 4 8 . 66 6 . 6 6 16 8 . 19 6 . 26 6 . 8 1 61 6 . 6 8 8 . 88 8 . 9 8 6 2 8 . 6 3 6 . 88 6 . 8 3 49 2 . 3 4 2 . 17 8 . 17 8 7 2 . 49 2 . 36 8 . 13 2 5 2 . 6 5 2 . 5 2 8 . 13 6 5 2 . 7 3 2 . 6 3 8 . 89 5 5 - 6 . 6 2 6 . 86 6 . 8 2 76 1 . 8 9 1 . 6 5 6 . 2 4 13 2 . 1 1 1 . 9 3 6 . 1 8 12 1 . 8 6 6 . 9 6 6 . 1 6 4 3 1 . 5 1 1 . 3 1 6 . 2 8 1 4 - 6 . 8 7 6 . 6 8 6 . 8 7 5 3 8 . 8 8 8 . 8 8 6 . 8 8 8 2 8 . 5 3 6 . 4 1 6 . 12 74 - 8 . . 8 6 8 . 8 8 8 . 86 7 2 2 . 7 7 2 . . 7 7 8 . . 8 1 15 2 , . 7 1 2 . . 7 2 8 . . 8 1 5 7 2 . 6 5 2 . . 6 3 6 . . 8 2 81 2 , . 5 1 2 . 5 2 8 . 86 4 6 2 , . 3 8 2 . . 3 8 6 . . 6 6 91 2 , , 18 2 , . 2 2 8 . . 6 4 3 6 1 . . 9 8 2 , . 8 2 6 . . 8 5 81 1 . , 7 8 1 . . 8 8 8 . . 8 3 2 5 2 . . 6 4 2 . , 5 2 6 . . 12 6 4 2 . , 7 9 2 . . 6 4 8 . . 1 5 8 5 2 . . 8 8 2 . . 7 2 6 . . 8 8 4 5 2 . . 88 2 . , 7 8 8 . . 1 6 8 6 2 . . 8 9 2 . . 8 8 8 . . 8 9 2 9 2 . 8 3 2 . . 8 8 8 . . 8 3 2 8 1 . . 5 8 1 . . 5 5 8 . . 8 5 7 4 1 . 2 3 1 . 2 7 8 . , 8 4 2 8 8 . 9 5 6 . . 9 4 6 . . 8 1 6 9 8 . 5 3 6 . 6 6 8 . . 8 7 1 6 8 . 1 8 6 . 2 6 6 . 8 2 61 - 8 . 8 2 6 . 6 8 8 . 8 2 6 3 8 . 81 6 . 8 8 6 . . 61 5 6 8 . 8 5 6 . 8 8 6 . . 6 5 49 2 . 3 3 2 . . 1 7 8 . . 16 8 5 2 . 5 6 2 . 3 6 6 . . 19 76 1 . 8 8 1 . 6 5 8 . . 2 2 12 2 . 18 1 . 9 3 6 . . 1 7 16 1 . 13 8 . 91 8 . 21 4 3 1 . 58 1 . 31 6 . . 1 9 1 4 - 8 . 8 8 8 . 8 6 8 . . 6 8 5 2 - 8 . 81 8 . 8 8 6 . . 81 8 2 8 . 5 2 8 . 41 8 . 1 1 7 4 - 6 . 8 7 8 . 8 6 8 . 8 7 4 5 2 . 8 7 2 . 7 8 8 . 8 9 8 6 2 . 8 8 2 . 8 8 8 . 8 7 2 7 2 . 8 9 2 . 8 8 8 . 8 9 7 8 2 . 8 3 2 . 7 7 8 . 8 6 1 3 2 . 7 7 2 . 71 8 . 8 5 5 7 2 . 6 3 2 . 6 3 8 . 8 8 81 2 . 5 8 2 . 5 2 8 . 8 2 4 5 2 . 3 7 2 . 3 8 8 . 81 91 2 . 1 7 2 . 2 2 8 . 8 5 36 1 . 9 6 2 . 8 2 8 . 86 81 1 . 7 6 1 . 8 8 6 . 8 4 2 8 1 . 4 9 1 . 5 5 6 . 8 6 7 4 1 . 21 1 . 2 7 8 . 8 5 2 6 8 . 9 4 8 . 9 4 8 . 81 61 - 8 . 8 3 8 . 8 8 8 . 8 3 6 3 8 . 8 8 8 . 8 6 8 . 8 8 8 5 2 . 5 4 2 . 3 7 8 . 18 2 3 2 . 7 8 2 . 5 2 8 . 1 8 6 3 2 . 7 8 2 . 6 4 8 . 1 4 8 3 2 . 8 6 2 . 7 2 8 . 1 4 6 8 8 . 5 9 6 . 59 8 . 8 6 16 8 . 17 8 . 2 8 8 . . 6 4 5 6 8 . 8 3 6 . 8 8 8 . . 6 3 18 2 . 17 1 . 9 4 8 . 2 3 4 7 2 . 3 9 2 . 17 8 . 2 2 41 1 . 5 7 1 . 3 2 8 . 2 4 7 5 1 . 8 7 1 . 6 6 8 . 21 8 9 1 . 12 6 . 9 2 8 . 2 8 7 3 - 8 . 6 8 0 . 8 6 8 . 8 8 201 1 02 378 1 1 . 48 17 113 378 2 . 48 1 6 1 36 377 3 1 . 44 1 5 357 389 18 1 . 12 6 368 389 19 1 . 12 6 379 389 28 1 . 12 5 269 391 1 1 .82 1 8 283 398 12 1 .86 18 296 398 1 3 1 87 9 389 398 1 4 1 87 8 322 398 15 87 8 33* 398 1 6 1 88 7 346 389 1 7 12 7 398 389 21 13 4 488 389 22 1 1 3 4 489 389 23 1 13 3 419 389 24 13 3 428 389 25 13 2 436 389 26 1 13 2 444 389 27 1 13 1 451 389 28 1 13 1 462 389 29 1 1 4 8 474 389 38 1 1 4 8 487 398 31 1 1 -8 248 391 9 81 1 1 255 391 18 1 82 1 1 289 392 7 e 96 13 225 392 8 e 97 12 1 74 393 5 8 91 1 4 192 392 6 8 96 13 1 56 393 4 8 91 15 181 395 1 8 81 17 115 395 2 8 82 16 1 36 394 3 8 86 1 5 88 396 8 8 77 1 7 296 487 13 8 49 9 389 486 1 4 8 53 8 322 486 15 8 53 8 334 486 1 6 8 53 7 346 486 17 8 54 7 358 486 18 8 54 6 369 486 19 8 54 6 379 486 28 8 54 5 398 486 21 8 55 4 488 486 22 8 55 4 418 486 23 8 55 3 419 486 24 8 55 3 428 485 25 8 58 2 436 486 26 8 55 2 444 486 27 8 55 1 462 486 29 8 55 8 475 486 38 8 56 8 255 487 18 8 47 1 1 269 487 1 1 8 48 18 283 487 1 2 8 48 18 452 486 28 8 55 1 488 486 31 8 56 -8 225 488 8 8 43 1 2 248 488 9 8 43 1 1 1 92 489 6 8 38 13 289 488 7 8 42 13 175 418 5 8 33 1 4 1 1 5 41 1 2 8 26 1 6 136 411 3 8 28 15 156 418 4 8 32 15 88 41 1 8 8 26 17 181 412 1 8 23 1 7 12 -8 81 0 88 8 81 52 -8 82 0 00 8 82 88 8 59 0 43 8 1 6 57 2 62 2 63 8 01 81 2 49 2 52 8 03 45 2 36 2 38 8 02 63 2 77 2 64 8 13 83 2 35 2 72 8 12 44 2 86 2 78 8 08 86 2 87 2 88 8 06 27 2 88 2 88 0 07 78 2 81 2 77 8 04 13 2 75 2 71 0 04 98 2 23 2 21 8 02 35 2 83 2 82 0 01 81 1 75 1 88 8 05 27 1 55 1 54 0 00 73 1 27 1 26 8 81 28 8 93 8 95 8 02 68 8 58 8 59 8 81 16 8 15 8 28 8 05 68 8 03 8 88 8 83 83 -8 01 0 88 8 01 56 8 02 8 88 8 82 85 2 53 2 37 8 1 7 23 2 69 2 52 8 1 7 18 2 15 1 94 8 21 47 2 38 2 18 8 28 42 1 48 1 32 8 16 75 1 85 1 66 8 28 89 1 10 8 92 8 19 13 -8 10 8 88 8 18 51 -8 03 8 88 8 83 79 8 58 8 43 8 1 4 72 -8 09 8 88 8 89 44 2 84 2 78 8 86 86 2 85 2 88 8 85 27 2 86 2 88 8 86 78 2 80 2 77 8 83 1 3 2 74 2 71 8 83 56 2 68 2 63 8 85 88 2 55 2 52 8 83 45 2 35 2 38 8 84 89 2 22 2 21 8 88 35 2 01 2 82 8 81 88 1 81 1 79 8 82 27 1 54 1 54 0 81 73 .1 26 1 26 8 88 21 8 91 8 95 0 83 68 8 56 8 59 8 03 68 8 62 8 88 0 02 82 8 05 8 88 8 05 23 2 67 2 52 0 15 63 2 75 2 64 8 12 83 2 83 2 72 8 1 1 15 8 22 8 28 8 02 56 8 08 0 88 8 88 46 2 37 2 18 8 19 84 2 52 2 37 8 15 73 1 84 1 66 8 1 8 18 2 1 4 1 94 8 28 48 1 54 1 33 8 22 51 -8 04 0 88 8 04 79 8 57 0 44 8 1 3 88 1 09 0 92 8 1 7 72 -8 10 0 88 8 1 0 12 -8 1 1 0 88 8 1 1 

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