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Time matching of separate cine camera views for three dimensional motion studies Lord, Bruce Allan 1985

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TIME MATCHING OF SEPARATE CINE CAMERA VIEWS FOR THREE DIMENSIONAL MOTION STUDIES by BRUCE ALLAN LORD B.Sc. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF THE FACULTY OF GRADUATE STUDIES S c h o o l of P h y s i c a l E d u c a t i o n and R e c r e a t i o n We a c c e p t t h i s t h e s i s as co n f o r m i n g t o th-e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA MASTER OF PHYSICAL EDUCATION i n June 1985 Bruce A l l a n L o r d , 1985 I n p r e s e n t i n g 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 r e q u i r e m e n t s f o r an advanced degree a t the 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 , I agree t h a t the l i b r 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 r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s un d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department of P h y s i c a l E d u c a t i o n , 1 Oct., 1985 The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T IW5 Date: Qc^JJl^' A b s t r a c t The purpose of t h i s s t u d y was t o e v a l u a t e the importance of ti m e - m a t c h i n g s e p a r a t e c i n e camera views i n t h r e e - d i m e n s i o n a l motion s t u d i e s and t o develop a n a l y t i c a l methods t o a c c o m p l i s h t h e time m a t c h i n g . An image space was c a l i b r a t e d u s i n g t w e n t y - f o u r c o n t r o l p o i n t s and motion p i c t u r e f i l m s a t about 60 frames per second were taken of a moving b a r , and of a s u b j e c t p u t t i n g a s h o t . Combinations of c o r r e c t l y and i n c o r r e c t l y matched views were compared f o r t h e i r a c c u r a c y i n d e t e r m i n i n g the p o s i t i o n s of s i x o b j e c t p o i n t s . An a l g o r i t h m was d e r i v e d which i n c l u d e d the t i m i n g v a r i a b l e i n the l e a s t squares s o l u t i o n f o r the X, Y, and Z c o o r d i n a t e s . These "best f i t " s o l u t i o n s f o r t h e t i m i n g and f o r the c o o r d i n a t e l o c a t i o n s were compared w i t h c r i t e r i o n v a l u e s . A l t e r a t i o n s i n the t i m i n g of views tended t o i n t r o d u c e a b i a s i n t o the c o o r d i n a t e l o c a t i o n s . The magnitude of the b i a s was a f u n c t i o n of the v e l o c i t y of the o b j e c t p o i n t s and of the camera p o s i t i o n i n g . To keep f i n a l c o o r d i n a t e e r r o r s below 5% r e q u i r e d the two views t o be matched t o w i t h i n 0.008 seconds. The time matching a l g o r i t h m was a b l e t o match the views t o w i t h i n 0.005 sec o n d s . The c o r r e s p o n d i n g c o o r d i n a t e s c o u l d v a r y by an average of 2.4% from the c o r r e c t ones. I t was c o n c l u d e d t h a t the a n a l y t i c a l time matching a l g o r i t h m c o u l d produce a c c e p t a b l e r e s u l t s i f extreme a c c u r a c y was not r e q u i r e d . i i i TABLE OF CONTENTS A b s t r a c t i i L i s t of F i g u r e s i v Acknowledgements v I n t r o d u c t i o n 1 Purpose 1 Methods 2 Theory 2 Proc e d u r e s 5 Data C o l l e c t i o n and A n a l y s i s 8 R e s u l t s and D i s c u s s i o n 9 N o i s e i n the Data 9 I n t e r p o l a t i o n E r r o r s , 9 Consequences of Mismatching t h e Timing 10 E f f e c t i v e n e s s of the Time Matching A l g o r i t h m 17 C o n c l u s i o n s 28 Recommendations 28 Re f e r e n c e s 29 APPENDIX 1 - DEFINITIONS 30 APPENDIX 2 - SYMBOLS USED IN THE PAPER 32 APPENDIX 3 - REVIEW OF LITERATURE 33 Time Matching Methods 33 E f f e c t s of N o i s e 38 BIBLIOGRAPHY 40 i v L i s t of F i g u r e s 1. The Time Ma t c h i n g Problem 4 2. Top View of Set-up f o r Test One 6 3. Top View of Set-up f o r Test Two 7 4. P o s i t i o n of the W r i s t i n the X D i r . f o r Test Two 11 5. P o s i t i o n of the Elbow i n the X D i r . f o r Test Two 12 6. P o s i t i o n of the W r i s t i n the Y D i r . f o r T e s t Two....13 7. P o s i t i o n of the Elbow i n the Y D i r . f o r Test Two 14 8. P o s i t i o n of the W r i s t i n the Z D i r . f o r Test Two....15 9. P o s i t i o n of the Elbow i n the Z D i r . f o r T e s t Two 16 10. Time D i f f e r e n c e a t Minimum Norm f o r Test One 18 11. Time D i f f e r e n c e a t Minimum Norm f o r Test Two 20 12. X C o o r d i n a t e V a r i a t i o n a t Best Match f o r the Wrist...21 13. X C o o r d i n a t e V a r i a t i o n a t Best Match f o r the Elbow...22 14. Y C o o r d i n a t e V a r i a t i o n a t Best Match f o r the Wrist...23 15. Y C o o r d i n a t e V a r i a t i o n a t Best Match f o r the Elbow...24 16. Z C o o r d i n a t e V a r i a t i o n a t Best Match f o r the W r i s t 25 17. Z C o o r d i n a t e V a r i a t i o n a t Best Match f o r the Elbow...26 18. Equipment Set-up f o r S p l i t Image Photogrammetry 34 V Acknowledgements I w i s h t o thank Dr. D. Gordon E. R o b e r t s o n f o r the g uidance and p a t i e n c e he has p r o v i d e d me i n the p r e p a r a t i o n of t h i s t h e s i s . Dr. R. W. B l a k e and Dr. I . M. Franks s u p p l i e d a s s i s t a n c e w i t h o u t which t h i s work c o u l d not have been co m p l e t e d . The work and c o u n s e l i n g of Dr. J . S. Walton was r e l i e d upon e x t e n s i v e l y . The t e c h n i c a l a s s i s t a n c e of Mr. K. Hsu i s a l s o g r a t e f u l l y acknowledged. F i n a l l y , I thank Joyce P r i c e f o r her h e l p i n p r o o f r e a d i n g , F l o r e n c e P r i c e f o r her a s s i s t a n c e w i t h t y p i n g , and Dave P r i c e f o r the use of h i s f a c i l i t i e s . 1. I n t r o d u c t i o n The s t u d y of human motion i n t h r e e dimensions u s i n g c i n e f i l m i s not a new i d e a ( M i l l e r , 1970; Van Gheluwe, 1973). S e v e r a l t e c h n i q u e s a r e a v a i l a b l e ( W o l t r i n g , 1977; Walton, 1981; Dapena e t a l . , 1982) which c o n s i d e r a b l y s i m p l i f y the p r o c e d u r e s i n v o l v e d . One of the added c o m p l e x i t i e s of t h r e e - d i m e n s i o n a l t e c h n i q u e s over t w o - d i m e n s i o n a l ones i s t h e need t o time s y n c h r o n i z e s e p a r a t e v i e w s . A l t h o u g h s i n g l e view methods have been developed ( M i l l e r e t a l . , 1980) t h e y a r e of c o m p a r a t i v e l y low a c c u r a c y and unique s o l u t i o n s a r e not always a t t a i n a b l e ( W o l t r i n g , 1981). S e v e r a l time matching t e c h n i q u e s are i n use (as w i l l be o u t l i n e d l a t e r ) , but a l l l a c k g e n e r a l a p p l i c a b i l i t y t o movements such as t h o s e t h a t occur i n r e a l s p o r t i n g a c t i v i t i e s . Purpose The purpose of t h i s s t u d y was t o e v a l u a t e the importance of time matching- s e p a r a t e c i n e camera views i n t h r e e - d i m e n s i o n a l motion s t u d i e s and t o develop a n a l y t i c a l methods t o a c c o m p l i s h the time m a t c h i n g . 2. Methods Theory The s i m p l e s t photogrammetric p r o c e s s used t o l o c a t e an o b j e c t p o i n t i n t h r e e - d i m e n s i o n a l space u t i l i z e s a p a i r of t w o - d i m e n s i o n a l v i e w s . I n t u i t i v e l y , t h i s l e a d s t o t h e c o n c l u s i o n t h a t more i n f o r m a t i o n i s p r o v i d e d i n two views than i s needed t o s o l v e the t h r e e - d i m e n s i o n a l problem. T h i s i s c o n f i r m e d when the problem i s m a t h e m a t i c a l l y s o l v e d . A b d e l - A z i z and K a r a r a (1971) developed a m a t h e m a t i c a l model c a l l e d the D i r e c t L i n e a r T r a n s f o r m a t i o n (DLT) t o r e l a t e d i g i t i z e r c o o r d i n a t e s t o o b j e c t c o o r d i n a t e s i n r e a l s p a ce. Walton (1981) a p p l i e d a s i m i l a r t e c h n i q u e t o problems i n human movement. In Walton's method, s o l v i n g the f o l l o w i n g f o u r e q u a t i o n s y i e l d s v a l u e s f o r the t h r e e - d i m e n s i o n 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 . (1) ( 2 ) (3) (4) (Where u^, v-^, U2, v 2 a r e the measured d i g i t i z e r c o o r d i n a t e s f o r a p a r t i c u l a r o b j e c t p o i n t and c o e f f i c i e n t s 'a' t o '1' a r e known c a l i b r a t i o n v a l u e s ) . T h i s system of e q u a t i o n s i s o v e r d e t e r m i n e d , as t h e r e a re f o u r e q u a t i o n s and o n l y t h r e e unknowns (X, Y, Z ) . ( a l ~ e l u l ) X + ( D i _ f l u l ) Y + ( c 1 - g 1 u 1 ) Z = ( h l ~ e l v l ) X + ( ^ l ~ f l v l ) Y + ( k i ~ 9 i v i , z = ( a 2 ~ " e 2 U 2 * X + * b 2 - f 2 U 2 ) Y + ( c 2 _ g 2 U 2 ) Z = ( h , - e 2 v 2 ) X + ( j 9 - f 9 v , ) Y + ( k 9 - g 9 v 9 ) Z = 2 V 2 '2 V2 ( u 1 - d 1 ) ( u 2 - d 2 ) (v 2-d 2) 3. If the experiment was error free, any three of the four equations would y i e l d a correct solution. In r e a l i t y , a "best f i t " solution may be found by using a linear least squares technique. This solution, when substituted back into equations (1) to (4) results in residual errors. The square root of the sum of squares of the residuals for equations (1), (2), (3), and (4) i s ca l l e d the minimized euclidean norm of the residuals. This norm is an indicator of how well the four equations agree on the solution (Walton, 1981). To calculate accurate coordinates the two sets of data must be time matched. When this is not the case the residual errors should increase, although experimental variations might obscure this fact for a single point. I f , however, the norm is calculated for a l l of the points in a frame, an optimal time match should be revealed. An i t e r a t i v e procedure can be performed to determine which time match produces the minimum norm of the residuals. There is one problem which must be overcome before the time matching of separate d i g i t i z e r coordinates can be attempted. Each camera does not record events at the same instant of time, therefore a method for finding intermediate values for at least one camera must be found to enable matching between the views. Figure 1 i l l u s t r a t e s the problem. 4. THE TIME MATCHING PROBLEM time FIGURE 1. 5. I n F i g u r e 1, r e p r e s e n t s a d i g i t i z e r c o o r d i n a t e f o r camera one, w h i l e r e p r e s e n t s a d i g i t i z e r c o o r d i n a t e f o r camera two. The open c i r c l e s r e p r e s e n t the d e s i r e d v a l u e s from the camera two d a t a w h i l e r e c o r d e d data i s i n d i c a t e d by s o l i d d o t s . A l i n e a r i n t e r p o l a t i o n between the two a c t u a l v a l u e s w i l l produce a r e a s o n a b l e a p p r o x i m a t i o n t o the d e s i r e d i n t e r m e d i a t e v a l u e (Walton, 1981). P r o c e d u r e s 16 mm f i l m s were taken of two motions t o i n v e s t i g a t e the time matching problem. I n i t i a l l y a 50 cm l o n g wood bar was marked a t 10 cm i n t e r v a l s . 24 beads spaced a t 10 cm i n t e r v a l s were s t r u n g on c o r d t o s e r v e as c o n t r o l p o i n t s f o r c a l i b r a t i n g the camera. The c o n t r o l p o i n t s d e f i n e d a C a r t e s i a n C o o r d i n a t e System w i t h Z b e i n g v e r t i c a l . A Redlake I n d u s t r i e s Locam camera o p e r a t i n g a t 50 frames per second was used t o f i l m the bar a f t e r i t was put i n m o t i o n . F i g u r e 2 i s a diagram of the equipment s e t - u p . In t e s t two , an u n s k i l l e d s u b j e c t was f i l m e d w h i l e p e r f o r m i n g a shot p u t . Two Locam cameras were used, one o p e r a t i n g a t 60 frames per second and the o t h e r o p e r a t i n g a t 180 frames per second. The h i g h e r speed was used t o p e r m i t a c l o s e r e s t i m a t e d time match between the v i e w s . The s p a c i n g s between the 24 c o n t r o l p o i n t s were i n c r e a s e d t o 20 cm t o accommodate the l a r g e o b j e c t space r e q u i r e d f o r the s h o t put (110 x 108 x 100 cm). See f i g u r e 3 f o r a diagram of t h e s e t - u p . 6. TOP VIEW OF SET-UP FOR TEST ONE ] MIRROR 50 c m •O CONTROL POINTS 50 c m CAMERA FIGURE 2 . 7. 1.08 m TOP VIEW OF SET-UP FOR TEST TWO t r r (1.40 m HIGH) (1.25 m HIGH) 6.2m 3.7 0.3 m 2.8 m 0 (Z VERTICAL) o CONTROL POINTS 1.10 m FIGURE 3. 8. Data C o l l e c t i o n and A n a l y s i s F i l m s were p r o j e c t e d a t a p p r o x i m a t e l y 1/6 l i f e s i z e and d i g i t i z e d w i t h a Numonics G r a p h i c s C a l c u l a t o r i n t e r f a c e d w i t h a M i c r o n o v a MP 200 computer (Data G e n e r a l ) . T e s t one was i m p l i c i t y time matched because both images were r e c o r d e d on one frame a t the same i n s t a n t . To determine the most a c c u r a t e time match f o r t e s t two, the p o i n t of r e l e a s e of t h e s h o t f o r the s l o w e r camera (60 frames/s) was matched t o the f a s t e r camera's (180 f r a m e s / s ) r e l e a s e p o i n t . T h i s s h o u l d have been a c c u r a t e t o w i t h i n one t h i r d of one frame, or .006 s e c o n d s . The s i x marks on the bar were d i g i t i z e d i n the f i r s t t e s t . For the second t e s t markers a t the w r i s t , elbow, h i p , top of knee, bottom of knee, and a n k l e were d i g i t i z e d . As w e l l , two of the c o n t r o l p o i n t s were a l s o d i g i t i z e d i n each frame t o p r o v i d e an e s t i m a t e of the n o i s e i n the d a t a . Two computer programs (JSW3D and JSWFILT) from Walton (1981) were m o d i f i e d t o run on the Micronova computer. JSW3D was adapted t o i n t e r p o l a t e new U and V v a l u e s from one d i g i t i z e d view and produce X, Y, and Z c o o r d i n a t e s f o r a number of d i f f e r e n t time m a t c h i n g s . To o b t a i n the m i n i m a l time match an i n t e r v a l one frame wide on e i t h e r s i d e of the best e s t i m a t e d match was d e f i n e d . T h i s was d i v i d e d i n t o t h r e e segments and the segment c o n t a i n i n g the minimum was i s o l a t e d . T h i s segment was then f u r t h e r s u b d i v i d e d i n t o t h r e e p a r t s f o r the next i t e r a t i o n . The p r o c e d u r e stopped when a change of l e s s than 0.01 seconds was observed between i t e r a t i o n s . 9. R e s u l t s and D i s c u s s i o n N o i s e i n t h e Data The c o o r d i n a t e s of the c o n t r o l p o i n t s i n t e s t two s h o u l d have remained c o n s t a n t because t h e y were s t a t i o n a r y i n the o b j e c t s p ace. Any apparent movement c o u l d o n l y be due t o n o i s e from the d i g i t i z i n g or a n a l y s i s p r o c e d u r e s . The apparent movement due t o n o i s e , averaged over 30 f r a m e s , was 0.52 cm f o r X, 0.62 cm f o r Y, and 0.20 cm f o r Z. T h i s compares f a v o r a b l y w i t h v a l u e s W e l l s and Winter (1975) f o u n d . T h e i r data would p r e d i c t a v a l u e of 0.4 cm f o r a 1/6 l i f e s i z e image. To e x p r e s s t h e s e e r r o r s i n p e r c e n t a g e terms, the dimensions of the o b j e c t space were used as the maximum e x c u r s i o n s a p o i n t c o u l d t a k e . The o b j e c t space e n c l o s e d by the c o n t r o l p o i n t s had the f o l l o w i n g d i m e n s i o n s : X = 110 cm, Y = 110 cm, Z = 100 cm. C o r r e s p o n d i n g p e r c e n t a g e e r r o r s would be: X = 0.47%, Y = 0.57%, Z = 0.20%. I n t e r p o l a t i o n E r r o r s The U and V c o o r d i n a t e s used i n the time matching a l g o r i t h m were somewhat i n a c c u r a t e due t o i n t e r p o l a t i o n e r r o r . The magnitude of . t h i s e r r o r was e s t i m a t e d by c a l c u l a t i n g the d i f f e r e n c e between a c t u a l and i n t e r p o l a t e d v a l u e s f o r the 180 frames per second camera. 10. Using e v e r y t h i r d v a l u e t o i n t e r p o l a t e f o r the i n t e r m e d i a t e two r e s u l t e d i n an average d i f f e r e n c e of 0.5 cm, w i t h a maximum d i f f e r e n c e of 1.2 cm. S i n c e the a c t u a l i n t e r p o l a t i o n i n t e r v a l would be 1/3 as l a r g e , the i n t e r p o l a t i o n e r r o r s would a l s o be e x p e c t e d t o be 1/3 as l a r g e (0 .4 cm maximum and 0.17 cm a v e r a g e ) . T h i s i s o n l y about 0.15% e r r o r , thus i n t e r p o l a t i n g seems t o be a v a l i d p r o c e d u r e . Consequences of M i s m a t c h i n g t h e T i m i n g F i g u r e s 4 t o 9 i l l u s t r a t e the e f f e c t s of a l t e r i n g the t i m i n g f o r the w r i s t and elbow markers of the shot put t e s t . S i n c e the movement reached peak v e l o c i t y i n frames 17 t o 36 t h i s d a t a was s e l e c t e d f o r the g r a p h s . The l i n e s marked by an 'x' a r e the c o o r d i n a t e s f o r the best e s t i m a t e d time match. The l i n e s marked by an 'o' a r e f o r a t i m i n g mismatch of 0.015 seconds ( a p p r o x i m a t e l y one f r a m e ) . The l i n e s marked by a ' + ' i n d i c a t e a mismatch i n t i m i n g of 0.035 seconds (two f r a m e s ) . I t i s e v i d e n t t h a t the Y c o o r d i n a t e ( F i g s . 6 and 7) i s a f f e c t e d the most by the time o f f s e t , a l t h o u g h i t i s moving a t about 1/2 the speed. The maximum d i f f e r e n c e i n c o o r d i n a t e l o c a t i o n t h a t a mismatch of 0.015 seconds produced was about 12 cm (11% e r r o r ) . Note t h a t the s i t u a t i o n s i l l u s t r a t e d would be u n l i k e l y t o occur i n p r a c t i c e because a time mismatch of a f u l l frame (0.0167 seconds) would be e v i d e n t t o c a s u a l o b s e r v a t i o n . 11 POSIT ION OF THE WRIST IN THE X D IR. FOR TES 106 i 80 - 60 40 - 20 tf - x x BEST MATCH + + . 0 3 3 s MISMATCH .-+ •20 J — i 1 1 { \ 1 r . 0 2 . 0 5 .08 .12 .15 .18 .2 TIME •'£> F I G U R E 4. POSITION OF THE ELBOW IN THE X DIR. FOR TEST 188 >'8 68 4© - 28 - 8 —x BEST MATCH —+ .033s MISMATCH 2 0 r ~! i ! \ r ! 1 1 82 .85 .88 .12 .15 .18 .22 .25 .28 .3 TIME <.&> FIGURE 5. 13. POSITION OF THE WRIST IN THE V DIR. FOR TEST 2 ?0 60 -+ - + o-o. P 0 1 S 50 - i ' " v x I T 1 40 -\ 0 N 30 •+ .•+ .+ • + " • + + - + u.. ro 10 o. o. / ,o, + '+ /+' • +f .o .,, .,, BEST MATCH o o .0167s MISMATCH + + .033s MISMATCH i 1 r 1 j 1 , , .02 .05 .08 . 12 . 15 .18 .22 .25 .23 .32 TIME <s> FIGURE 6 14 POSITION OF THE ELBOW IN THE V DIR. FOR TES 3 6 i 70 - —x BEST MATCH -—o .0167s MISMATCH + .033s MISMATCH +••• + o - o 60 50 40 30 H 20 o - o + • .r.r . .+ .+...+...+...+...+ . / a. 'o vo \ , o 'o .ol ~ i 1 1 \ \ i 1 r 1 — .02 .05 .08 .12 .15 .18 .22 .25 .28 TIME <s: FIGURE 7 15. POSITION OF THE WRIST IN THE Z DIR. FOR TEST 2 1 8 6 167 P 0 1 5 4 T 141 I 0 N 129 m 116-1 / . r v ,x>+ ,-?:\ rt /,o- • ..o /..o..+ 163 x x BEST MATCH o o .0167s MISMATCH + + .033s MISMATCH + 9 y |~ f ~1 1 1 ! 1 1 1 1 G2 .85 .08 .12 .15 .18 .22 .25 .28 .32 TIME <s> FIGURE 8. 16 P O S I T I O H OF T H E E L B O W IN T H E Z D I R . F O R T E S T 2 178 i 1 5 6 - P 0 C; I T I 0 N 141 -) i ~i' r 113 - n 99 - 8 4 -J » * : '5' .•+ x x B E S T M A T C H o c- .8167s M I S M A T C H + + .833s M I S M A T C H ?Q 1 f . ! 1 1 p .82 .85 .88 .12 .15 .1 1 1 T I M E <s> F I G U R E 9 . 17. There i s , u n f o r t u n a t e l y , no s i m p l e e x p l a n a t i o n f o r the v a r i a n c e i n e f f e c t s . The Y c o o r d i n a t e might p o s s i b l y have been most a f f e c t e d because both camera axes were c l o s e s t t o b e i n g i n the Y p l a n e . I t i s obvi o u s t h a t the d i g i t i z e r c o o r d i n a t e s of f a s t e r moving p o i n t s w i l l be more i n e r r o r f o r a g i v e n t i m i n g mismatch, but due t o t h e n a t u r e of s i m u l t a n e o u s e q u a t i o n s , t h i s e r r o r c o u l d appear i n any of the X, Y, or Z o b j e c t c o o r d i n a t e s . The camera p o s i t i o n i n g has a s i g n i f i c a n t e f f e c t upon t i m i n g e r r o r s because i t determ i n e s the c o e f f i c i e n t s of the e q u a t i o n s . I t i s a l s o n o t a b l e t h a t the d e v i a t i o n i n l o c a t i o n i s p r o p o r t i o n a l t o the amount of time o f f s e t . Thus, i f the maximum e r r o r s due t o t i m i n g i n t h i s experiment were t o be kept below 1% (about 1 cm) the two views would have t o be matched t o w i t h i n 0.0015 s (.014/11). I f 5% e r r o r s were t o l e r a b l e , matching would have t o be a c c u r a t e t o w i t h i n .008 seconds. E f f e c t i v e n e s s of t h e Time M a t c h i n g A l g o r i t h m F i g u r e 10 i l l u s t r a t e s the c o n s i s t e n c y w i t h which the minimum norm c o r r e s p o n d e d t o the c o r r e c t time match f o r the f i r s t t e s t . S i n c e a m i r r o r was used t h e r e i s a b s o l u t e c e r t a i n t y of c o r r e c t time m a t c h i n g . Over the f i r s t 35 frames (0.6 s) the correspondence was q u i t e p o o r , w i t h maximum d e v i a t i o n s of up t o 0.06 seconds (3 f r a m e s ) . 18. TIME DIFFERENCE AT MINIMUM NORM FOR TRIAL 1 06 "i T I M E D I F F .04 A 02 H 0 . 0 x X X X X X -.02 -j X X 04 -j .... '< > x y x x>" x x x x x x x x x >rA x-v x • w x x x XX X " X X X 06 — , —j — r —I : 1 — i i 1 r .20 .40 .60 .80 1.0 1.2 1.4 1.6 1 8 TIME <s> FIGURE 10. 19. The match from frames 35 t o 95 was s u b s t a n t i a l l y b e t t e r . O b s e r v a t i o n of the f i l m r e v e a l e d t h a t the bar r e v e r s e d d i r e c t i o n a t frame 18, and t h u s , moved v e r y s l o w l y i n t h a t time p e r i o d . T h i s made d i f f e r e n t i a t i o n between frames d i f f i c u l t . The mean time match over 95 frames was 0.005 seconds d i f f e r e n t from the e x a c t v a l u e . T h i s i s a p p r o x i m a t e l y one q u a r t e r of one frame a t 50 frames per second. The time match a t a minimum norm f o r the f i l m of a shot put i s i l l u s t r a t e d i n f i g u r e 11. A l t h o u g h the mean (-0.007 s) approaches the e s t i m a t e d e x a c t v a l u e , i n d i v i d u a l times v a r y by up t o 0.04 seconds, as i n d i c a t e d by the l a r g e s t a n d a r d d e v i a t i o n (0.016 s ) . There i s a l s o a low f r e q u e n c y v a r i a t i o n i n the v a l u e s o b t a i n e d , which does not seem t o be due t o random n o i s e . F i g u r e s 12 t o 17 show the e f f e c t s t h e s e time v a r i a t i o n s have on the c a l c u l a t e d c o o r d i n a t e s . The best e s t i m a t e s of the c o o r d i n a t e l o c a t i o n s are the same as those i n f i g u r e s 4 t o 9 ("x" p o i n t s ) , w h i l e the c o o r d i n a t e s a t a minimum norm are shown by "+" p o i n t s . The X and Y c o o r d i n a t e s d e v i a t e c o n s i d e r a b l y from t h e c l o s e s t e s t i m a t e d p a t h . A l t h o u g h t h e b e s t e s t i m a t e d p a t h may not be e x a c t , i t i s c l e a r l y more a c c u r a t e than the one d e r i v e d from the time matching a l g o r i t h m . The mean d e v i a t i o n f o r the X c o o r d i n a t e was 2.4 cm. w i t h a maximum of 8 cm. The mean d e v i a t i o n f o r the Y c o o r d i n a t e was 2.9 cm. w i t h a maximum of 12 cm. 26. TIME DIFFERENCE AT MINIMUM NORM FOR TRIAL .06 T. I M E .84 i 0 2 D I 8.0 F F . - 02 - . 0 4 A 86 • X X X .•XXX X "I "1 I I l — ~ T 1 — — r — — I 1 —i .89 .18 .27 36 .46 .55 .64 .72 .81 .96 1.8 TIME <s> FIGURE 11. 21 X COORDINATE VARIATION AT BEST MATCH FOR THE WRIST 16C1 -, 80 60 A 40 >@ 1 0 ' / X x — x BEST ESTIMATE o o MINIMUM NORM ,/x /.o .o • O-' . ' / X /O -o ; D , ' X . '>X- •o o - o-- - >0 —T 1 i — i 1——i 1 i — r — \ . 02 . 05 . 08 . 12 .15 .18 . 22 . 2 5 . 28 . 3! TIME Xs) FIGURE 12. COORDINATE VARIATION AT BEST MATCH FOR THE ELBOW 100 -1 8 8 68 H 48 i 58 A 8 -4 —x BEST ESTIMATE /° .'.-'X o — o MINIMUM NORM , x - ° /.o o -o •x 1 /' ,Q-' .C'!< u • .-x •o 0-0-6"0'° -28 | I | 1 - r — — t ] - i - j——i .82 .85 .68 .12 .15 .18 .22 .25 . 2 3 .32 TIME •:•> FIGURE 13. Y COORDINATE VARIATION AT BEST MATCH FOR THE WRI 76 60 x — x BEST ESTIMATE o o MINIMUM NORM . . » > 50 -i*'>x. 40 30 - 10 "O.-O "X U - i - i % s o - o o - o ''O-O : D 1 | | J 1 l i t 1 1 . 02 .05 .68 .12 .15 .18 .22 .25 . 2 8 .32 TIME <s> F IGURE 14. 24. Y COORDINATE VARIATION AT BEST MATCH FOR THE ELBOW 80 7 70 - P 0 S I T I 0 N c ro x x BEST ESTIMATE o o MINIMUM NORM .o 60 -x-*o 50 - 40 30 - •O V o - o \ \ -o-o • \ > .--x—v T ,x ,o '• ,o / ;. ' o i i r i i : i i i i i .02 .05 .08 .12 .15 .18 .22 .25 .28 .32 TIME Xs) FIGURE 15. 25. COORDINATE VARIATION AT BEST MATCH FOR THE WRIST c ?88 ISO -j P 0. S 160 -j I •T I 0 N 120 -I 148 A x x BEST ESTIMATE & MINIMUM NORM / X yy- .-'X 188 i " " X 8 8 — r - 1 1 — — r - r | | T — — T 1 .82 .85 .88 .12 . 15 .18 .22 .25 .28 .32 TIME C s > FIGURE 16. 26 COORDINATE VARIATION AT BEST MATCH FOR THE ELBOW 180 -i 160 P 0 Cr I T I 0 N c ro 140 120 A 100 x x BEST ESTIMATE AND MINIMUM NORM 30 A e y - i T " — ~ T r~ 1 i——i 1 r- 1 . 02 .05 .08 .12 .15 .18 .22 .25 . 2 8 .32 TIME <s> F IGURE II There was no d i s c e r n a b l e d i f f e r e n c e between the two c u r v e s f o r the Z c o o r d i n a t e . These e r r o r s e x p r e s s e d as a p e r c e n t a g e of the o b j e c t space dimensions are X = 2.2% e r r o r and Y = 2.7% e r r o r . I f m i s t a k e s of l o g i c i n the m a t h e m a t i c a l t h e o r y a r e d i s c o u n t e d , the o n l y f a c t o r which can account f o r the o b s e r v e d v a r i a b i l i t y i s e x p e r i m e n t a l e r r o r . The g r e a t e s t e r r o r s are most l i k e l y found i n the p r o j e c t i o n and d i g i t i z a t i o n p r o c e d u r e s . The e r r o r i n r e c o n s t r u c t i n g the measured l o c a t i o n s of the c o n t r o l p o i n t s averaged 0.5 cm i n each p l a n e . N o i s e i n t r o d u c e d i n t o the r e s u l t s by the d a t a c o l l e c t i o n and r e d u c t i o n p r o c e s s would l i m i t the e f f e c t i v e n e s s of the time matching methods. 28. C o n c l u s i o n s The d a t a p r e s e n t e d above s u p p o r t the f o l l o w i n g c o n c l u s i o n s : 1. Time matching methods used i n t h r e e - d i m e n s i o n a l cinematography s h o u l d be a c c u r a t e t o w i t h i n p l u s or minus e i g h t m i l l i s e c o n d s t o l i m i t c o o r d i n a t e e r r o r s t o l e s s than 5%. 2. The time matching a l g o r i t h m p r e s e n t e d was a b l e t o time match two views t o w i t h i n 5 m i l l i s e c o n d s of the c o r r e c t v a l u e . Recommendations Based on the u n d e r s t a n d i n g g a i n e d as a r e s u l t of t h i s s t u d y , t h e f o l l o w i n g recommendations f o r the i n v e s t i g a t i o n of t i m i n g i n t h r e e - d i m e n s i o n a l cinematography a r e s u g g e s t e d . 1. A movement s h o u l d be f i l m e d from s e v e r a l d i f f e r e n t a n g l e s t o e v a l u a t e the e f f e c t s of camera p o s i t i o n i n g on time m i s m a t c h i n g . 2. The n o i s e which appears i n the raw data must be reduced t o a lower l e v e l , perhaps by d i g i t a l f i l t e r i n g , or w i t h s u p e r i o r equipment. REFERENCES 1. A b d e l - A z i z , Y. I . and K a r a r a , H. M. (1971). " D i r e c t L i n e a r T r a n s f o r m a t i o n from Comparator C o o r d i n a t e s i n t o O b j e c t Space C o o r d i n a t e s i n Close-Range Photogrammetry". P r o c e e d i n g s of t h e Symposium on Close-Range Photogrammetry, J a n . 26-29, 1971. F a l l s Church Va.: American S o c i e t y of Photogrammetry. 2. Dapena, J . , Harman, E. A., and M i l l e r , J . A. (1982). "Three D i m e n s i o n a l Cinematography With C o n t r o l O b j e c t of Unknown Shape". J . of Biomechanics 15, 11-19. 3. M i l l e r , D. I . (1970). "A Computer S i m u l a t i o n Model of the A i r b o r n e Phase of D i v i n g " . Ph.D. T h e s i s , Penn. S t a t e U n i v . 4. M i l l e r , N. R., S h a p i r o , R. and M c L a u g h l i n , T. M. (1980). "A Technique f o r O b t a i n i n g S p a t i a l K i n e m a t i c Parameters of B i o m e c h a n i c a l Systems from C i n e m a t o g r a p h i c Data". J . of Biomechanics 13, 535-547. 5. Van Gheluwe, B. (1973). "A New Three D i m e n s i o n a l F i l m i n g Technique I n v o l v i n g S i m p l i f i e d A l i g n m e n t and Measurement P r o c e d u r e s " . I n : N e l s o n , R. C. and Morehouse, C. A. (Eds.) Biomechanics IV. B a l t i m o r e : U n i v e r s i t y Park P r e s s . 6. Walton, J . S. (1981). "Close Range Cine-Photogrammetry: A G e n e r a l i z e d Technique f o r Q u a n t i f y i n g Gross Human M o t i o n " . Ph.D. T h e s i s , U n i v . of Penn.. 7. W o l t r i n g , H. J . (1977). Measurement and C o n t r o l of Human Movement. Nijmegan: P e t e r s and Haarsma. 8. W o l t r i n g , H. J . (1981). "Comment on the Paper 'A Technique f o r O b t a i n i n g S p a t i a l K i n e m a t i c Parameters of Segments of B i o m e c h a n i c a l Systems From C i n e m a t o g r a p h i c D a t a 1 " . J . of Biomechanics 14, 277. 30. APPENDIX 1 - DEFINITIONS OF TERMS (from Walton,1981) ACCURACY: The amount any s t a t e d v a l u e d i f f e r s from i t ' s t r u e v a l u e i s termed i t ' s a c c u r a c y . CONTROL POINT: An o b j e c t p o i n t f o r which the o b j e c t - c o o r d i n a t e s a r e known a - p r i o r i i s r e f e r r e d t o as a c o n t r o l p o i n t . DIGITIZER COORDINATES: An o r d e r e d p a i r of numbers, e x p r e s s e d i n d i g i t i z e r u n i t s , and used t o d e s c r i b e the l o c a t i o n of a p o i n t i n a r e a l or v i r t u a l secondary image r e l a t i v e t o the d i g i t i z e r - r e f e r e n c e - f r a m e , a r e r e f e r r e d t o as d i g i t i z e r c o o r d i n a t e s . EXPOSURE TIME: The p e r i o d d u r i n g which a p h o t o g r a p h i c e m u l s i o n i s exposed t o l i g h t i s known as the exposure t i m e . I n cinematography, the exposure t i m e i s d e t e r m i n e d by the e q u a t i o n : exposure time = (Frame Rate) x ( S h u t t e r F a c t o r ) FIELD OF VIEW: The f i e l d of view of a camera ( w i t h a p a r t i c u l a r o b j e c t i v e ) i s t h a t p o r t i o n of the o b j e c t space which can be r e c o r d e d by the camera. FRAME RATE: The f r e q u e n c y w i t h which a c i n e camera r e c o r d s d i s t i n c t images i s known as the frame r a t e of the camera. 3 1 . INTERMITTENT CAMERA: An i n t e r m i t t e n t camera i s one i n which the f i l m i s h e l d s t i l l w h i l e each exposure i s made . OBJECT COORDINATES: An o r d e r e d p a i r or t r i p l e t 'of numbers which a r e e x p r e s s e d i n o b j e c t u n i t s and used t o d e s c r i b e the l o c a t i o n of an o b j e c t p o i n t w i t h r e s p e c t t o the o b j e c t r e f e r e n c e frame are r e f e r r e d t o as o b j e c t c o o r d i n a t e s . OBJECT POINT: Any p o i n t i n the o b j e c t space which i s s u b j e c t e d t o p h o t o g r a p h i c e x a m i n a t i o n i s r e f e r r e d t o as an o b j e c t p o i n t . PHOTOGRAMMETRY: Photogrammetry i s d e f i n e d as the s c i e n c e or a r t of o b t a i n i n g r e l i a b l e measurements by means of p h o t o g r a p h y . SHUTTER FACTOR: I n cinematography, the r a t i o of the t o t a l t i m e between s u c c e s s i v e frames t o the exposure time per frame i s known as the s h u t t e r f a c t o r . TIMING MARKS: I n some c i n e cameras t h e r e i s a p r o v i s i o n f o r marking the border of the f i l m by e x p o s i n g i t t o a s m a l l , p u l s e d l i g h t s o u r c e . These marks are c a l l e d t i m i n g marks. 32. APPENDIX 2 - SYMBOLS USED IN THIS PAPER a , b , c , d , e , f , g , h , j , k , 1 - The e l e v e n c a l i b r a t i o n c o e f f i c i e n t s needed t o t r a n s f o r m d i g i t i z e r c o o r d i n a t e s t o o b j e c t c o o r d i n a t e s . u - The h o r i z o n t a l c o o r d i n a t e measured from a f i l m ( i n d i g i t i z e r u n i t s ) , v - The v e r t i c a l c o o r d i n a t e measured from a f i l m ( i n d i g i t i z e r u n i t s ) . X,Y,Z - The t h r e e c o o r d i n a t e s of a C a r t e s i a n system ( i n o b j e c t u n i t s w i t h Z v e r t i c a l ) . 33. Appendix 3 - REVIEW OF LITERATURE There are many t e c h n i q u e s t h a t have been used i n the f i e l d of b i o k i n e m a t i c data a c q u i s i t i o n . W o l t r i n g , (1984) su r v e y s e l e c t r o g o n i o m e t r y , u l t r a s o u n d , s t o b o s c o p i c photography, o p t o e l e c t r o n i c s , a c c e l e r o m e t r y , and cinematography. T h i s d i s c u s s i o n w i l l d e a l o n l y w i t h c i n e m a t o g r a p h i c t e c h n i q u e s , or t h o s e concerned w i t h a n a l y s i n g data r e c o v e r e d from c i n e f i l m of an a t h l e t i c p e rformance. Time M a t c h i n g Methods The time matching problem i s not one which has r e c i e v e d e x t e n s i v e a n a l y s i s i n the l i t e r a t u r e . Only one s t u d y d e a l s d i r e c t l y w i t h the time matching problem (Garnov and Dubovick, 1965). These a u t h o r s developed e x p r e s s i o n s f o r f i n d i n g the number of matched p a i r s of photos t o be e x p e c t e d from two cameras r u n n i n g i n d e p e n d e n t l y . I n v e s t i g a t o r s who have completed t h r e e - d i m e n s i o n a l s t u d i e s of human movement have used t h r e e methods t o time match t h e i r d a t a . The " s p l i t image" t e c h n i q u e ( P i e r r y n o w s k i , 1981; Van Wi j k and Ziemann, 1976) i n v o l v e s r e c o r d i n g two s e p a r a t e views on one f i l m w i t h t h e use of m i r r o r s or p r i s m s . F i g u r e 18 i s an i l l u s t r a t i o n of the set-up used by P i e r r y n o w s k i (1981). 34. EQUIPMENT SET-UP FOR SPLIT IMAGE PHOTOGRAMMETRY (as used by Pierrynowski, 1981) MIRROR IMAGE VOLUME TO VIRTUAL CAMERA CAMERA FIGURE 18. The l i m i t a t i o n s of t h i s method i n c l u d e : (1) . The l a r g e m i r r o r s needed are e x p e n s i v e , cumbersome, and a r e l i k e l y t o i n t r o d u c e d i s t o r t i o n s . (2) . The image s i z e i n the f i l m frame i s h a l v e d , r e d u c i n g the s i g n a l - t o - n o i s e r a t i o of the d a t a . (3) . The method i s s i m p l y i m p r a c t i c a l i n some s i t u a t i o n s such as l a r g e r s c a l e movements i n s p o r t i n g a c t i v i t i e s . The second method i n v o l v e s placement of an i d e n t i f y i n g mark on the f i l m so s y n c h r o n i z a t i o n can be completed l a t e r . There ar e s e v e r a l v a r i a t i o n s of t h i s t e c h n i q u e , r a n g i n g from an e x t e r n a l l i g h t v i s i b l e i n both cameras, t o i n t e r n a l t i m i n g l i g h t s b u i l t i n t o the cameras. The most b a s i c method uses an event s i g n a l , o f t e n a l i g h t , which i s v i s i b l e t o s e p a r a t e cameras. F u k a s h i r o e t a l . (1981) used t h i s method t o c o o r d i n a t e e i g h t cameras i n t h e i r t w o - d i m e n s i o n a l s t u d y of the t r i p l e jump. Hobart and P r o v e n z a , (1983) and B e n - S i r a e t a l . (1971) a v o i d e d the p r oblem of v i s i b i l i t y by u s i n g a b r i g h t f l a s h t o overexpose the f i l m i n s e p a r a t e cameras. The f l a s h , however, would o n l y be v i s i b l e i n low l i g h t s i t u a t i o n s . The problem w i t h t h e s e methods i s t h a t they can o n l y be a c c u r a t e t o the n e a r e s t h a l f frame. The f i r s t frame on each f i l m where the l i g h t i s v i s i b l e can be matched, but i f the i l l u m i n a t i o n o c c u r s when one s h u t t e r i s c l o s e d , t h e r e would be some e r r o r . B e n - S i r a e t ajL. (1971) acknowledge t h i s when th e y s t a t e "... the p r o c e d u r e i s not s u f f i c i e n t l y a c c u r a t e f o r c r i t i c a l t h r e e - d i m e n s i o n a l a n a l y s i s . " A v a r i a n t of t h i s method used by M i l l e r e t a l . (1980 ) i s s u b j e c t t o the same i n a c c u r a c y . M i l l e r used two Locam cameras o p e r a t i n g a t 100 Hz., and s i m u l t a n e o u s l y f i r e d t h e i r i n t e r n a l event marking l i g h t s . He noted t h a t the i n t e r m i t t e n t motion of the f i l m i n the gate of the camera i n t r o d u c e d an u n c e r t a i n t y of p l u s or minus one frame i n the m a t c h i n g . P i e r r y n o w s k i (1982) att e m p t e d t o reduce the u n c e r t a i n t y by matching the c l o s e s t frame of one Locam r u n n i n g a t 160 Hz. w i t h another Locam r u n n i n g a t 40 Hz.. T h i s method may have en a b l e d him t o d e c rease h i s e r r o r t o w i t h i n p l u s or minus 1/4 frame at 40 Hz., which i s e q u i v a l e n t t o a time p e r i o d of 0.006 sec o n d s . T h i s method p l a c e s l i m i t s on the speed ranges a l l o w e d and i n c r e a s e s c o s t s . An improvement upon- s i m p l y r e c o r d i n g an event mark i s the i n c l u s i o n of an a c t u a l time r e c o r d on the f i l m . T h i s i s p o s s i b l e v i a an i n t e r n a l or e x t e r n a l t i m e r . B l i e v e r n i c h t (1967) p r o v i d e s p l a n s f o r an e x t e r n a l m e c h a n i c a l t i m e r but i t has a low r e s o l u t i o n of 0.01 seconds. Walton, (1970) d e s c r i b e s an e l e c t r o n i c t i m i n g u n i t of 0.001 seconds r e s o l u t i o n . T h i s d e v i c e i n c l u d e s a master u n i t and s e v e r a l s l a v e u n i t s f o r s e p a r a t e cameras. Walton (1981) used t h i s t i m e r t o r e c o r d a time f o r each f i l m frame d u r i n g d i g i t i z a t i o n . C o o r d i n a t e s from s e p a r a t e f i l m s were then matched u s i n g l i n e a r i n t e r p o l a t i o n . Walton f u r t h e r smoothed the t i m i n g r e c o r d u s i n g the assumption t h a t h i s cameras were r u n n i n g a t a c o n s t a n t speed. He found the assumption was v a l i d f o r AC powered cameras, but DC powered cameras d i d not have s t a b l e frame r a t e s . Borms e t a l . (1973) a l s o used a t i m e r v i s i b l e t o both cameras i n h i s s t u d y of the t w i s t i n g s o m e r s a u l t . These methods a c h i e v e d an a c c u r a t e time match q u i t e s u c c e s s f u l l y . The o n l y d i s a d v a n t a g e was i n e n s u r i n g t h a t a t i m e r was v i s i b l e t o each camera. A s l i g h t l y d i f f e r e n t approach was used by Newton e t a l (1977 ) which i n v o l v e d the use of a s p l i t l e n s t o s i m u l t a n e o u s l y f i l m a s t o p w a t c h . H is r e s o l u t i o n was o n l y 0.001 seconds and d i f f i c u l t i e s i n i n c l u d i n g the same stopwatch i n s e v e r a l camera views a re a n t i c i p a t e d . 38. The t h i r d and most e l e g a n t s o l u t i o n t o the time matching problem i s t o use e l e c t r o n i c a l l y s y n c h r o n i z e d cameras. Walton (1981) d e s c r i b e s two such systems t h a t are a v a i l a b l e . The l a r g e c a p i t a l c o s t s and c a b l e c o n n e c t i o n s between cameras a r e t h e o n l y drawbacks t o t h i s s o l u t i o n . To summarize, t h e s e t h r e e time matching s o l u t i o n s seem t o be t h e o n l y ones i n use. T h e i r l i m i t a t i o n s are r e c o g n i z e d , which c o n t r i b u t e s s i g n i f i c a n t l y t o the l a c k of w i d e s p r e a d a c c e p t a n c e of t h r e e - d i m e n s i o n a l a n a l y s i s . E f f e c t s of N o i s e The a b i l i t y t o a c h i e v e an a c c u r a t e time match by s e a r c h i n g f o r the minimum norm of r e s i d u a l s s h o u l d be dependent o n l y upon the e x p e r i m e n t a l a c c u r a c y . S e v e r a l a u t h o r s have a n a l y z e d the c h a r a c t e r i s t i c s of the n o i s e t o be e x p e c t e d i n a c i n e m a t o g r a p h i c a n a l y s i s of human movement. Capozzo e t a l . (1975) a s s e r t e d t h a t the e r r o r i n t r o d u c e d by m u l t i p l e and independent causes can be assumed random w i t h normal d i s t r i b u t i o n . W e l l s and Winter (1980) a s s e s s e d the e x p e c t e d n o i s e l e v e l s under f i l m i n g c o n d i t i o n s v a r y i n g from a c o n t r o l l e d l a b o r a t o r y s e t t i n g t o an outdoor s p o r t s event.. They found n o i s e i n a 1/3 l i f e s i z e image t o be 2 mm, w h i l e t h a t i n 1/14 s i z e image was 1 cm. Atwater (1981) a t t r i b u t e d most of the s y s t e m a t i c e r r o r s which might appear t o the d i g i t i z i n g p r o c e d u r e s . She recommended m o n i t o r i n g the s k i l l and p r e c i s i o n of the d i g i t i z i n g p e r s o n n e l . Lanshammar (1982) determined t h a t measurement n o i s e would be w h i t e i f i t changed by s e v e r a l q u a n t i z a t i o n s t e p s between samples. 39. These a u t h o r s seem t o agree t h a t most of the e x p e c t e d n o i s e s h o u l d be random. I f t h i s i s t r u e , the a v e r a g i n g p r o c e d u r e s proposed would a l l o w the time matching a l g o r i t h m t o produce an a c c u r a t e r e s u l t . S e v e r a l a u t h o r s have e v a l u a t e d methods t o reduce the amount of n o i s e i n b i o m e c h a n i c a l d a t a . Pezzack e t a l . (1977) c o n c l u d e d t h a t d i g i t a l f i l t e r i n g c o u l d e f f e c t i v e l y remove most h i g h f r e q u e n c y n o i s e . He c o n s i d e r e d p o l y n o m i a l c u r v e f i t t i n g t o be i n s u f f i c i e n t l y a c c u r a t e . Soudan e t a l . (1979) ad v o c a t e d the use of s p l i n e f u n c t i o n s f o l l o w e d by d e t e r m i n a t i o n of the F o u r i e r c o e f f i c i e n t s . The main advantage of s p l i n e f u n c t i o n s over d i g i t a l f i l t e r i n g i s the accommodation of gaps and d i s c o n t i n u i t i e s i n the d a t a . Hatze (1981) p r o v i d e s an e x t e n s i v e r e v i e w of the use of F o u r i e r s e r i e s f o r e l i m i n a t i n g n o i s e . He notes t h e i r advantages as c o m p u t a t i o n a l e f f i c i e n c y , and a u t o m a t i c f i l t e r i n g parameter s e l e c t i o n . D i s a dvantages are l i s t e d as the e q u i d i s t a n t data s p a c i n g r e q u i r e m e n t , and no data may be m i s s i n g . M i l l e r e t a l . (1980) e v a l u a t e d s e v e r a l smoothing t e c h n i q u e s and c o n c l u d e d t h a t d i g i t a l f i l t e r i n g , F o u r i e r a n a l y s i s , and c u b i c s p l i n e s a r e each best s u i t e d t o d i s t i n c t s i t u a t i o n s . 40. BIBLIOGRAPHY A t w a t e r , A. E. (1981). " K i n e m a t i c A n a l y s i s P r o c e d u r e s i n B i o m e c h a n i c a l Cinematography" I n : Terauds, J . ( E d . ) , Second I n t e r n a t i o n a l Symposium of B i o m e c h a n i c s , Cinematography, and High Speed Photography. P r o c e e d i n g s of t h e S.P.I.E., v o l . 291. B e n - S i r a , D., S t o n e r , L. J . and Lu e d t k e , D. (1978). "A Sim p l e P r o c e d u r e f o r Event M a r k i n g When F i l m i n g With One or Two Cameras". Research Q u a r t e r l y 49(3), 381-384. B l i e v e r n i c h t , D. L. (1967). "A M u l t i d i m e n s i o n a l Timing Device f o r Cinematography". Research Q u a r t e r l y 38, 146-148. Borms, J . , Duquet, W. and H e b b e l i n c k , M. (1973). " B i o m e c h a n i c a l A n a l y s i s of the F u l l T w i s t i n g Back S o m e r s a u l t " . I n : Biomechanics I I I , K a r g e r , B a s e l . pp.429-433. Cappozzo, A., Leo, T. and P e d o t t i , A. (1975). "A G e n e r a l Computing Method f o r the A n a l y s i s of Human Locomotion". J . of B i o m e c h a n i c s . (8), 307-320. F u k a s h i r o , S., J i m o t o , Y., K o b y a s h i , H. and M i y a s h i t a , M. (1981). "A B i o m e c h a n i c a l Study of the T r i p l e Jump". M e d i c i n e and S c i e n c e i n S p o r t s and E x e r c i s e . 13(4), 233-237. Garnov, V. V. and Dubovic, A. S. (1965). " S t e r e o s c o p i c F i l m i n g of R a p i d P r o c e s s e s by Two In d e p e n d a n t l y O p e r a t i n g Moving P i c t u r e Cameras". NASA T e c h n i c a l T r a n s l a t i o n F-337, c a t a l o g u e 16728, Oct. 1965, non d e p o s i t o r y . Washington, D.C.: NASA. Ha t z e , H. (1981). "The Use of O p t i m a l l y R e g u l a r i z e d F o u r i e r S e r i e s f o r E s t i m a t i n g H i g h er-Order D e r i v a t i v e s of N o i s y B i o m e c h a n i c a l Data". J . of B i o m e c h a n i c s . H o b a r t , D. J . and P r o v e n z a , D. V. (1973). " P r a c t i c a l S o l u t i o n s t o Problems i n C i n e m a t o g r a p h i c A n a l y s i s " . JOHPER, Jan.,44, p. 99. 41. Lanshammer, H. (1982). "On P r e c i s i o n L i m i t s f o r D e r i v a t i v e s N u m e r i c a l l y C a l c u l a t e d From N o i s y Data". J . of B i o m e c h a n i c s . ( 1 5 ) , 459-470. M i l l e r , N. R., S h a p i r o , R. and M c L a u g h l i n , T. M. (1980). "A Technique f o r O b t a i n i n g S p a t i a l K i n e m a t i c Parameters of Segments of B i o m e c h a n i c a l Systems from C i n e m a t o g r a p h i c Data". J . of B i o m e c h a n i c s . ( 1 3 ) , 535-547. Newton, J . , P r o v a n c h e r , J . , Abramson, D. and L e w i s , S. (19 7 7 ) . " I n e x p e n s i v e Timing Method f o r Cinematography". Re s e a r c h Q u a r t e r l y . May, 4 8 ( 2 ) , 484-488. P e z z a c k , J . C , Norman, R. W. and W i n t e r , D. A. (1977). "An Assesment of D e r i v a t i v e D e t e r m i n i n g Techniques Used f o r M o t i o n A n a l y s i s " . J . of B i o m e c h a n i c s . ( 1 0 ) , 377-382. P i e r r y n o w s k i , M. R. (1981). "Three D i m e n s i o n a l F i l m i n g U s i n g the D i r e c t L i n e a r T r a n s f o r m a t i o n " . U n p u b l i s h e d Paper . P i e r r y n o w s k i , M. R. (1982). "A P h y s i o l o g i c a l Model f o r the S o l u t i o n of I n d i v i d u a l Muscle F o r c e s D u r i n g Normal Human W a l k i n g " . Ph.D. T h e s i s , Simon F r a s e r U n i v e r s i t y , Vancouver, Canada. Soudan, K. and D i e r c k x , P. (1979) " C a l c u l a t i o n of D e r i v a t i v e s and F o u r i e r C o e f f i c i e n t s of Human Motion Data, W h i l e U s i n g S p l i n e F u n c t i o n s " . J . of B i o m e c h a n i c s . ( 1 2 ) , 21-26. Van W i j k , M. C. and Ziemann, H. (1976). "The Use of Non-M e t r i c Cameras i n M o n i t o r i n g High Speed P r o c e s s e s " . Photogrammetric E n g i n e e r i n g and Remote S e n s i n g . ( 4 2 ) , 91-102. Walto n , J . S. (1970 ). "A High Speed Timing U n i t f o r Cinematography". Research Q u a r t e r l y . 41(2),213-216. Walton, J . S. (1981). "Close Range Cine-Photogrammetry: A G e n e r a l i z e d Technique f o r Q u a n t i f y i n g Gross Human M o t i o n " . Ph.D. T h e s i s , U n i v . of Penn.. W e l l s , R. P. and W i n t e r , D. A. (1980) "Assessment of S i g n a l and N o i s e i n the K i n e m a t i c s of Normal, P a t h a l o g i c a l and S p o r t i n g G a i t s " . I n : Human Locomotion I , P r o c e e d i n g s of th e S o c i e t y f o r B i o m e c h a n i c s . Oct. 27-29, pp.92-93. 42. W o l t r i n g , H. J . (1984). I n : Human M o t o r A c t i o n s - B e r n s t e i n R e a s s e s s e d . H . T . A . W h i t i n g ( E d . ) . E l s e v i e r S c i e n c e P u b l . B . V . ( N o r t h H o l l a n d ) .

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