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Towards real-time non-destructive lumber grading using x-ray images and modulus of elasticity signals Saravi, Albert 2004

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Towards Real-Time Non-Destructive Lumber Grading Using X-Ray Images and Modulus of Elasticity Signals By: A L B E R T S A R A V I B.Sc , Amir Kabir University of Technology, 1988 M . S c , Sharif University of Technology, 1991 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF THE REQUIREMENTS FOR THE D E G R E E OF DOCTOR OF PHILOSOPHY in THE F A C U L T Y OF G R A D U A T E STUDIES (Department of Electrical and Computer Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH C O L U M B I A November 2004 © Albert Saravi, 2004 11 Abstract The objective of this study was to examine relationships between the physical properties of a board and the strength of a board for use in an intelligent lumber grading system. Available research literature and commercial grading systems, that have taken steps in this direction, are described in this study. Some previous work on estimating the strength of lumber has been based on mathematical models derived from different approximation methods such as regression, function approximation or neural networks. The disadvantage of this approach is that a large training set of board scans are required to be representative of the various species, harvesting sites, and geometrical variations between boards (grain direction, knot location and size etc.) A major contribution of this thesis was to develop a mechanics based method of determining board strength that relies only upon simple in-mill measurement of X-Ray and Modulus of Elasticity (MOE) signals. In this thesis, an intelligent mechanics-based lumber grading system was developed to provide a better estimation of the strength of a board nondestructively. This system processed X-Ray-extracted geometric features (of 1080 boards that eventually underwent destructive strength testing) by using Finite Element Methods (FEM) to generate associated stress fields. In order to find a few significant mechanics-based features, the stress fields were then fed to a feature-extracting-processor which produced twenty six strength predicting features. The best strength predicting features were determined from the coefficient of determination (correlation r squared) between the features and actual strengths of the boards. The coefficient of determination of each feature (or combination of features) with the actual strength of the board were calculated and compared. A coefficient of determination of 0.42 was achieved by using a longitudinal (along the local grain angle) maximum stress concentration (MSC) feature to predict the estimated strength of lumber. In addition to the above feature, a Weibull based feature was defined and examined. Since it is based on the whole stress field; whereas, maximum stress concentration based feature is based on one point in stress field, we hoped to get a better correlation. By implementing a system using the Weibull based feature, a coefficient of determination of 0.47 was achieved which is slightly higher than the M S C based feature. i i i Next, a combination of X-Ray and Modulus of Elasticity (MOE) signals were used to estimate the strength of lumber. The MSC and the Weibull features were based on the X-Ray images. The M O E part of the system used the output of a Continuous Lumber Tester (CLT) machine which produced two M O E profiles to calculate an M O E based feature. The M O E based feature and MSC feature were combined in an algorithm and a coefficient of determination of 0.65 was achieved. In the final step, the F E M processor was replaced by a knowledge-based system comprised of a fast table lookup for all FEM-modeled knot locations and sizes. To the extent that this table is representative of the important strength-reducing factors in a board, it replaces the need for a large training set of actual boards with real knots of differing sizes and locations. This knowledge-based system will permit a real-time board strength estimation system to be developed. We were able to fine-tune the mechanics based algorithm and reduce the F E M calculation errors. By implementing the knowledge based system coefficients of determination of 0.44, 0.46 and 0.67 were achieved for the MSC, the Weibull and combined (MOE and X-Ray) algorithm-based features respectively. Finally, by implementing a system without the need for an F E M module, it opens up the opportunity to investigate the location of breakage by using the Maximum Stress Concentration theory of failure. First a system based on the X-Ray image was implemented to estimate the point of fracture location. If this point is located between start and end of each actual breakage, then we can say that we estimated the breakage point correctly and a measure named the hit factor was introduced. The hit factor for both the most critical knot and the second highest M S C knot reached 56% for a tolerance of [5 cm] from the actual breakage point. IV TABLE OF CONTENTS A B S T R A C T II T A B L E O F C O N T E N T S I V L I S T O F T A B L E S V I L I S T O F F I G U R E S V I I L I S T O F S Y M B O L S , N O M E N C L A T U R E A N D A B B R E V I A T I O N S X I I A C K N O W L E D G M E N T X V 1. I N T R O D U C T I O N A N D L I T E R A T U R E S U R V E Y 1 1.1 OBJECTIVE 1 1.2 W H A T IS L U M B E R GRADING? 1 1.3 COMMERCIAL G R A D I N G SYSTEMS 3 1.3.1 Visual Grading 3 1.3.2 Continuous Lumber Tester (CUT) 3 1.3.3 Intelligent Lumber Grading Machine 6 1.3.3.1 X-Ray Lumber Gauge 6 1.3.3.2 Linear Planermill Optimizer 8 1.3.3.3 Addvantage Chopsaw Optimizer 9 1.3.4 Quality Control of Strength Grading 9 1.4 L ITERATURE SURVEY 9 1.4.1 Previous UBC Research 10 1.4.1.1 Advanced Grading Technology for Structural Lumber 10 1.4.1.2 P R E C A R N Intelligent Lumber Grading System (JLGS) Project 12 1.4.2 Worldwide Efforts to Improve Lumber Grading System 15 1.4.3 Strength Estimation Method 16 1.5 A N E U R A L NETWORK A P P R O A C H 19 1.6 MOTIVATION FOR THIS THESIS 2 2 1.7 THESIS CONTRIBUTIONS 23 2. E S T I M A T I N G T H E S T R E N G T H O F L U M B E R U S I N G X - R A Y I M A G E S 25 2.1 GENERATION OF STRENGTH PREDICTING FEATURES 2 5 2.1.1 Geometric Feature Extraction Algorithm 26 2.1.2 FEM Processor 33 2.1.3 Feature Extracting Processor 43 2.2 PROCESS REFINEMENT 4 6 2.3 STRENGTH ESTIMATION 5 0 2.4 CONCLUSIONS 57 3. W E I B U L L - B A S E D F E A T U R E S F O R S T R E N G T H E S T I M A T I O N 58 3.1 EXPERIMENTAL METHODS 58 V 3.2 STRENGTH PREDICTING FEATURE E X T R A C T I N G PROCESSOR 5 9 3.3 RESULTS 61 3.4 CONCLUSION 63 4. M O E A N D X - R A Y F O R S T R E N G T H E S T I M A T I O N 64 4.1 EXPERIMENTAL M E T H O D S 65 4.2 RESULTS 7 0 4.3 CONCLUSION 75 5. R E A L T I M E S Y S T E M F O R E S T I M A T I N G T H E S T R E N G T H O F L U M B E R 76 5.1 K N O W L E D G E B A S E D L O O K U P T A B L E FOR R E P L A C I N G F E M 7 6 5.1.1 Look up Table Using "Mesh Size 5" and Breaking the Narrow Part at "0.5* radi" 76 5.1.2 Look up Table Using Mesh Size 1 and Breaking the Narrow Part at "0.2* radi" 78 5.1.3 Look up Table after Passing through Low Pass Filter 80 5.2 ESTIMATING THE STRENGTH U S I N G A K N O W L E D G E B A S E D PROCESSOR 82 5.2.1 X-Ray Based Knowledge Based Processor for Estimating the Strength 82 5.2.2 MOE, X-Ray and Knowledge Based Processor for Estimating the Strength 83 5.3 RESULTS 84 5.4 CONCLUSION 91 6. E S T I M A T I N G T H E L O C A T I O N O F T H E B R E A K A G E O F A B O A R D 92 6.1 EXPERIMENTAL M E T H O D S 9 2 6.1.1 X-Ray Based Fracture Location Estimator 92 6.1.2 MOE Based Fracture Location Estimator 95 6.2 RESULTS 9 9 6.3 CONCLUSION 101 7. A L G O R I T H M A L T E R N A T I V E S 102 7.1 M A X I M U M D IAMETER A L G O R I T H M 102 7.2 EROSION A L G O R I T H M 104 7.3 CUTTING 1 5 % F R O M B O T H E N D S 106 7.4 H U M A N A N D ENVIRONMENT FACTORS 107 7.5 RESULTS 108 7.6 CONCLUSION 117 8. C O N C L U S I O N S A N D F U T U R E S T U D Y 118 8.1 CONCLUSIONS 118 8.2 FUTURE DIRECTIONS 119 R E F E R E N C E S 120 A P P E N D I X I : C O R R E L A T I O N C O N F I D E N C E I N T E R V A L S 125 A P P E N D I X I I : Z T A B L E 127 A P P E N D I X I I I : C O R R E L A T I O N R T O F I S H E R S Z ' T A B L E 128 vi LIST OF TABLES T A B L E 1, D E T E C T E D KNOTS OF A SAMPLE BOARD 3 1 T A B L E 2 , E X T R A C T E D FEATURES ( 2 8 FEATURES) 4 3 T A B L E 3 , E X A M P L E : C O M P U T E D VALUES FOR FEATURES 1 TO 2 8 (MAXIMUM/MINIMUM V A L U E S IN RESPECTIVE STRAIN OR STRESS FIELD AS THEY A R E DEFINED IN T A B L E 2 ) FOR BOARD # 2 1 0 4 4 T A B L E 4 , THRESHOLD COMPUTATION FOR BOARD 2 1 0 4 7 T A B L E 5 , COEFFICIENT OF DETERMINATION FOR DIFFERENT FEATURES A N D THRESHOLD LEVELS 5 1 T A B L E 6, COEFFICIENT OF DETERMINATION FOR DIFFERENT ORDER OF REGRESSION 5 6 T A B L E 7 , COEFFICIENT OF DETERMINATION FOR W E I B U L L - B A S E D FEATURES FOR DIFFERENT N VALUES A N D THRESHOLD L E V E L OF 1 .25* M O D E 6 1 T A B L E 8, COEFFICIENT OF DETERMINATION A N D CORRELATION FOR DIFFERENT FEATURES USING M E S H SIZE 5 A N D BREAKING " 0 . 5 * RADL" 8 5 T A B L E 9 , COEFFICIENT OF DETERMINATION (WITH CONFIDENCE INTERVAL) A N D CORRELATION FOR DIFFERENT FEATURES USING MESH SIZE 1 A N D B R E A K I N G 0 . 2 USING M A X A L G O R I T H M 8 7 T A B L E 10 , RESULTS FOR " S U M M A T I O N A L G O R I T H M " SUITABLE FOR W E I B U L L BASED FEATURES FOR THE CASE MESH SIZE 1 A N D B R E A K I N G " 0 . 2 * RADL" TOGETHER WITH " M A X A L G O R I T H M " RESULTS FOR COMPARISON 9 0 T A B L E 1 1 , H I T FACTORS FOR THE X - R A Y A N D THE M O E BASED SIGNALS 1 0 0 T A B L E 1 2 , D E T E C T E D KNOTS FOR BOARD # 2 USING TWO DIFFERENT ALGORITHMS 1 0 4 T A B L E 1 3 , RESULTS FOR THE M A X I M U M D I A M E T E R A L G O R I T H M 1 0 8 T A B L E 14 , RESULTS FOR THE M A X I M U M D I A M E T E R A L G O R I T H M A N D EROSION A L G O R I T H M 1 0 9 T A B L E 15 , RESULTS FOR THE M A X I M U M DIAMETER A L G O R I T H M A N D 1 5 % CUT A L G O R I T H M 1 1 0 T A B L E 16 , RESULTS FOR THE M A X I M U M DIAMETER, EROSION A N D 1 5 % CUT ALGORITHM.. . . Ill T A B L E 17 , RESULTS FOR THE FIRST HALF , A N D THE SECOND H A L F COMPARED WITH ONES IN THE CHAPTERS 2 , 3 , A N D 4 (ALL SAMPLES) 1 1 2 T A B L E 18 , RESULTS FOR THE SECOND H A L F OF SAMPLES (MESH 1 A N D B R E A K I N G AT 0 . 2 A N D NO FILTERING) COMPARED TO THE ONE IN C H A P T E R 5 1 1 6 V l l LIST OF FIGURES F IGURE 1-1, STRENGTH VS PREDICTOR 2 F IGURE 1-2, B E A M DEFLECTION UNDER A CENTER L O A D , F 4 F IGURE 1-3, COMPUTERMATIC M A C H I N E 5 F IGURE 1-4, C L T - 1 M A C H I N E 6 F IGURE 1-5, X - R A Y BASED L O C A L DENSITY DETECTION MEASUREMENT SYSTEM 7 F IGURE 1-6, T H E X - R A Y I M A G E OF A K N O T 7 F IGURE 1-7, M E A S U R E M E N T OF THE GRAIN A N G L E (SLOPE OF GRAIN) USING A SINGLE LASER B E A M 8 F IGURE 1-8, REFLECTIONS OF P L A N E - P A R A L L E L L A S E R B E A M S 9 F IGURE 1-9, C A P A C I T A N C E - B A S E D S O G M E A S U R E M E N T 11 F IGURE 1-10, M I C R O W A V E - B A S E D SYSTEM FOR MEASURING WOOD GRAIN A N G L E 13 F IGURE 1-11, M E A S U R I N G W A V E VELOCITY U S I N G U L T R A S O U N D 14 F IGURE 1-12, S IMULATION OF A H U M A N V I S U A L G R A D I N G S Y S T E M 15 F IGURE 1-13, K N O W L E D G E - B A S E D V I S U A L G R A D I N G S Y S T E M FOR D E F E C T DETECTION 15 F IGURE 1-14, C R A M E R A N D G O O D M A N M O D E L FOR DETERMINING STRENGTH OF A B O A R D .. 17 F IGURE 1-15, LONGITUDINAL STRESS CONCENTRATION F A C T O R 17 F IGURE 1-16, T R A N S V E R S E STRESS CONCENTRATION F A C T O R 18 F IGURE 1-17, M A T H E M A T I C A L STRUCTURE OF A NEURON 19 F IGURE 1-18, N E U R A L NETWORK FOR L U M B E R GRADING SYSTEM 2 0 F IGURE 1-19, T H E PRINCIPLE OF NEURAL NETWORK 21 F IGURE 1-20, A CLASSIFICATION OF WOOD DEFECTS 21 F IGURE 1-21, B L O C K D I A G R A M FOR A N E U R A L - N E T W O R K - B A S E D PROCESS FOR DETERMINING STRENGTH OF L U M B E R 2 2 F IGURE 2 -1 , NON-DESTRUCT IVE TESTING S Y S T E M 25 F IGURE 2-2, B L O C K D I A G R A M OF A MECHANICS-BASED SYSTEM FOR PREDICTING THE STRENGTH OF L U M B E R 2 6 F IGURE 2 -3 , X - R A Y IMAGE OF A BOARD 27 F IGURE 2-4, COMPENSATING FOR THE EFFECT OF BOARD L A T E R A L M O V E M E N T D U E TO FEEDER 27 F IGURE 2-5 , X - R A Y IMAGE OF A BOARD AFTER INTERPOLATING THE I M A G E INTO THE R E A L SIZE OF THE BOARD (CONDENSED IN THE L O N G DIMENSION) 28 F IGURE 2-6, X - R A Y IMAGE OF A BOARD AFTER THRESHOLD WITH THE CONDENSED AXIS IN L O N G DIRECTION (VERTICAL AXIS). C O L O R E D A R E A REPRESENT KNOTS WITH A V A L U E GREATER THAN ZERO, THE REST IS WHITE COLOR WHICH REPRESENTS THE SOUND WOOD WITH A V A L U E OF ZERO. NOTE THAT KNOTS LOOK LIKE LINE BECAUSE THE VERTICAL AXIS IS CONDENSED 5 0 TIMES. EDGES OF THE BOARD ARE AT 1 [MM] A N D 9 0 [MM] 29 F IGURE 2-7, Z O O M E D IMAGE OF A PART OF X - R A Y IMAGE OF A BOARD AFTER THRESHOLD WITH THE S A M E SCALE IN HORIZONTAL (EDGES AT 1 A N D 9 0 [MM]) A N D VERTICAL AXES . THERE IS ONE KNOT PRESENT AT CENTER (20 [MM] , 1458 [MM] ) WITH RADIUS 10.7 [MM] 3 0 F IGURE 2-8, DETECTION OF THE CENTER OF E A C H KNOT (CENTER OF A R E A OF THE DETECTED SEGMENT), A N D DETECTION OF KNOT RADIUS ( M A X I M U M CIRCLE THAT C A N B E FITTED TOTALLY INSIDE THE SEGMENTED A R E A A N D CONTAINING THE CENTER OF THE SEGMENT) 31 V l l l F IGURE 2-9, GRAPHICAL REPRESENTATION OF DETECTED KNOTS FOR THE S A M E BOARD AS IN T A B L E 1 (CONDENSED IN THE L O N G DIMENSION) 32 F IGURE 2-10, GEOMETRIC FEATURE EXTRACTING PROCESSOR DETAILED ALGORITHM 33 F IGURE 2-11, M O D E L WITH A N EDGE KNOT (LEFT) ; MODIFIED M O D E L OF EDGE KNOT WITH A B R O K E N PART (RIGHT) 34 F IGURE 2-12, F L O W G R A I N A N A L O G Y M O D E L 35 F IGURE 2-13, T H E SIMULATED FLOW AROUND A L O N G CYLINDRICAL OBSTACLE (RADIUS A=5 [MM]) IS LOCATED (AT CENTER 10 [MM] , 8 [MM] ) IN A LAMTNAR INCOMPRESSIBLE FLOW. 36 F IGURE 2-14, PHYSICAL MODEL OF A BOARD (BECAUSE OF THE L O N G DIMENSION OF THE BOARD THE A B O V E PICTURE WAS CUT INTO 4 PIECES), CIRCLES A L TO A 8 ARE THE KNOTS A N D B A C K G R O U N D IS THE SOLID WOOD 37 F IGURE 2-15, F E M M O D E L OF A PART OF A BOARD (TOP), F E M M O D E L OF SLOPE OF G R A I N B Y ROTATING THE E L E M E N T COORDINATE SYSTEMS (BOTTOM). T H E G L O B A L AXES ARE DEPICTED AT TOP RIGHT OF THE PICTURE. A L L OTHER AXES A R E L O C A L A X E S WHICH LOOK LIKE L WHICH IS ROTATED ACCORDING THE SLOPE OF GRAIN AT CENTER OF CORRESPONDING ELEMENT 39 F IGURE 2-16, 2 -D IMAGE OF THE LONGITUDINAL (BOTTOM) STRESS FIELD A R O U N D TWO CLOSE KNOTS. T H E M A X I M U M STRESS CONCENTRATION FOR LONGITUDINAL (BOTTOM) STRESS HAPPENS AT THE S M A L L EDGE KNOT A N D IT'S V A L U E FOR THIS CASE IS 3.65 (IT WAS SHOWN B Y " M X " IN THE PICTURE). THIS IS ALSO FEATURE 5 AS WE DEFINED IT. T H E TOP PICTURE IS CLOSE U P OF THE S M A L L EDGE KNOT IN THE BOTTOM FIGURE 41 F IGURE 2-17, F E M (FINITE E L E M E N T M E T H O D ) PROCESSOR DETAILED AL GORI T H M 42 F IGURE 2-18, F E A T U R E E X T R A C T I N G PROCESSOR DETAILED ALGORITHM 45 F IGURE 2-19, A MECHANICS-BASED SYSTEM FOR ESTIMATING THE STRENGTH OF A BOARD 46 F IGURE 2-20, H ISTOGRAM FOR BOARD #210 47 F IGURE 2-21, THRESHOLD BASE AFTER PASSING THROUGH LOW PASS FILTER FOR THE BOARD #210 48 F IGURE 2-22, X - R A Y IMAGE A N D PIECEWISE THRESHOLD L E V E L OF 1.3 TIMES OF M O D E (SIDE VIEW) 49 F IGURE 2-23, G R A P H I C A L REPRESENTATION OF A C T U A L STRENGTH (VERTICAL AXIS) VS, P R E D I C T O R FEATURE 28, " M A X ( F E A T U R E 5 , 7 * F E A T U R E 7)" ( H O R I Z O N T A L A X I S ) , (R 2 = 0.42) 52 F IGURE 2-24, G R A P H I C A L REPRESENTATION OF A C T U A L STRENGTH (VERTICAL AXIS) VS, PREDICTOR FEATURE 7, M A X I M U M T RANSV ERSE STRESS CONCENTRATION ( L O C A L AXES + N O D A L SOLUTION) (HORIZONTAL AXIS) , (R 2 = 0.35) 53 F IGURE 2-25, GRAPHICAL REPRESENTATION OF A C T U A L STRENGTH (R 2 = 0.42) vs , PREDICTOR FEATURE 5, M A X I M U M LONGITUDINAL STRESS CONCENTRATION ( L O C A L A X E S + N O D A L SOLUTION), WITH REGRESSION LINE: A C T U A L - S T R E S S = 51.05-5.78 * F E A T U R E - 5 54 F IGURE 2-26, GRAPHICAL REPRESENTATION OF A C T U A L STRENGTH VS, PREDICTOR, USING TRANSLATION A N D REFLECTION (5.78* {8.83 -FEATURE-5}), (R 2 = 0.42) 55 F IGURE 2-27, GRAPHICAL REPRESENTATION OF A C T U A L STRENGTH VS FEATURE_5 (THRESHOLD 1.3) WITH REGRESSION LINES F R O M ORDER ONE TO EIGHT (WITH R 2=0.41 FOR LINEAR A N D SQUARE REGRESSIONS A N D R 2=0.42 FOR THE REST) 56 F IGURE 3-1, B L O C K D I A G R A M OF A MECHANICS-BASED SYSTEM USING W E I B U L L - B A S E D FEATURES FOR PREDICTING THE STRENGTH OF A BOARD 58 ix F IGURE 3-2, A N O M A L I E S IN A B O A R D 60 F IGURE 3-3, WEELBULL B A S E D FEATURE VS A C T U A L STRENGTH FOR 1080 BOARDS WITH R 2 OF 0.47 62 F IGURE 3-4, ESTIMATED STRENGTH OF 1080 BOARDS USING W E I B U L L B A S E D FEATURE 63 F IGURE 4-1, B L O C K D I A G R A M OF A SYSTEM USING M O E A N D X - R A Y FOR PREDICTING THE STRENGTH OF A BOARD 65 F IGURE 4-2, A V E R A G E M O E PROFILE OF C L T MACHINE FOR BOARD #210 AFTER DUMPING 15 F R O M BOTH ENDS 66 F IGURE 4-3,2-D IMAGE OF THE TRANSVERSE (TOP) A N D LONGITUDINAL (BOTTOM) STRESS FIELD AROUND ONE EDGE KNOT. T H E M A X I M U M STRESS CONCENTRATION FOR LONGITUDINAL (BOTTOM) STRESS HAPPEN AT 90 DEGREES A N G L E A N D IT'S V A L U E FOR THIS CASE IS 3.65. T H E M A X I M U M STRESS CONCENTRATION FOR TRANSVERSE (TOP) STRESS HAPPENS AT ALMOST 60(OR 120) DEGREES A N D IT'S V A L U E FOR THIS CASE IS 0.41. 67 F IGURE 4-4, C O M P A R I N G 1 / M S C A N D l/MSC+5.7 FUNCTIONS FOR THEIR SIMILARITY TO A LINEAR FUNCTION 69 F IGURE 4-5, AVERAGE-M IN IMUM -15% CUT BOTH END-FEATURE VS A C T U A L STRENGTH OF A D A T A B A S E OF MORE T H A N 1000 BOARDS WITH SECOND ORDER REGRESSION CURVE WITH A COEFFICIENT OF DETERMINATION OF 0.56 71 F IGURE 4-6, A V E R A G E - M I N I M U M - 15% CUT BOTH END-FEATURE A N D TRANSFORMED BASED ON II ORDER REGRESSION (ESTIMATED STRENGTH) VS A C T U A L STRENGTH OF A DATABASE OF MORE THAN 1000 BOARDS WITH A COEFFICIENT OF DETERMINATION OF 0.58 72 F IGURE 4-7, ESTIMATED STRENGTH USING A MIXED SIGNAL OF M O E A N D X - R A Y FOR ALGORITHM A l WITH A COEFFICIENT OF DETERMINATION OF 0.64.. 73 F IGURE 4-8, ESTIMATED STRENGTH USING A MIXED SIGNAL OF M O E A N D X - R A Y FOR ALGORITHM A2 WITH A COEFFICIENT OF DETERMINATION OF 0.65 73 F IGURE 4-9, ESTIMATED STRENGTH USING A MIXED SIGNAL OF M O E A N D X - R A Y FOR ALGORITHM A3 WITH A COEFFICIENT OF DETERMINATION OF 0.65 74 F IGURE 4-10, ESTIMATED STRENGTH USING A MIXED SIGNAL OF M O E A N D X - R A Y FOR ALGORITHM A4 WITH A COEFFICIENT OF DETERMINATION OF 0.65 74 F IGURE 5-1, M O D E L USED FOR CREATING THE " L O O K U P TABLE" 76 F IGURE 5-2, L O O K U P TABLE WITH M E S H SIZE 5 A N D B R E A K I N G AT "0.5 * RADI" 77 F IGURE 5-3, A N SLICE OF L O O K U P TABLE AT RADIUS OF 10 [MM] WITH M E S H SIZE 5 A N D B R E A K I N G AT "0.5 * RADl" 78 F IGURE 5-4, L O O K U P TABLE WITH M E S H SIZE 1 A N D B R E A K I N G AT "0.2 * RADI" 79 F IGURE 5-5, A N SLICE OF L O O K U P TABLE AT RADIUS OF 10 [MM] WITH M E S H SIZE 1 A N D B R E A K I N G THE NARROW PART AT "0.2 * RADI " 80 F IGURE 5-6, L O O K U P TABLE AFTER FILTERING 81 F IGURE 5-7, A N SLICE OF LOOK U P TABLE AT RADIUS OF 10 [MM] AFTER FILTERING 82 F IGURE 5-8, B L O C K D I A G R A M A KNOWLEDGE BASED SYSTEM TO ESTIMATE THE STRENGTH OF A BOARD 83 F IGURE 5-9, B L O C K D I A G R A M A KNOWLEDGE BASED SYSTEM TO ESTIMATE THE STRENGTH OF A BOARD USING SUMMATION OF FEATURES FOR A L L KNOTS IN THE BOARD 83 F IGURE 5-10, B L O C K D I A G R A M A M O E A N D X - R A Y BASED " K N O W L E D G E B A S E D PROCESSOR" FOR ESTIMATING THE STRENGTH 84 X F IGURE 5 - 1 1 , A C T U A L STRENGTH VERSUS ESTIMATED STRENGTH USING KNOWLEDGE B A S E SYSTEM WITH M E S H SIZE 5 A N D BREAKING OF NARROW PART AT " 0 . 5 * RADL" A N D M S C FEATURE 5 WITH COEFFICIENT OF DETERMINATION OF 0 . 2 0 8 5 F IGURE 5 - 1 2 , A C T U A L STRENGTH VERSUS ESTIMATED STRENGTH USING " K N O W L E D G E B A S E D SYSTEM" WITH M E S H SIZE 5 A N D BREAKING OF NARROW PART AT " 0 . 5 * R A D I " A N D A L G O R I T H M Al TO A 4 A N D M O E PROFILES WITH COEFFICIENT OF DETERMINATION OF 0 . 5 6 , 0 . 5 3 , 0 . 5 5 , A N D 0 . 5 5 RESPECTIVELY 8 6 F IGURE 5 - 1 3 , A C T U A L STRENGTH VERSUS ESTIMATED STRENGTH USING KNOWLEDGE B A S E SYSTEM WITH M E S H SIZE 1 A N D BREAKING OF NARROW PART AT " 0 . 2 * RADL" A N D M S C FEATURE 5 WITH COEFFICIENT OF DETERMINATION OF 0 . 4 4 8 7 F IGURE 5 - 1 4 , A C T U A L STRENGTH VERSUS ESTIMATED STRENGTH USING KNOWLEDGE B A S E D SYSTEM WITH M E S H SIZE 1 A N D BREAKING OF NARROW PART AT " 0 . 2 * RADL" A N D A L G O R I T H M Al TO A 4 A N D M O E PROFILES WITH COEFFICIENT OF DETERMINATION OF 0 . 6 6 , 0 . 6 7 , 0 . 6 7 , A N D 0 . 6 7 RESPECTIVELY 8 8 F IGURE 5 - 1 5 , A C T U A L STRENGTH VERSUS ESTIMATED STRENGTH USING FILTERED KNOWLEDGE BASE SYSTEM A N D M S C (IN X - R A Y B A S E D STRESS FIELD, FEATURE 5 ) WITH COEFFICIENT OF DETERMINATION OF 0 . 4 5 8 9 F IGURE 5 - 1 6 , A C T U A L STRENGTH VERSUS ESTIMATED STRENGTH USING "FILTERED KNOWLEDGE B A S E S Y S T E M " A N D A L G O R I T H M A L B A S E D M S C (IN X - R A Y BASED STRESS FIELD , FEATURE 5 ) A N D M O E PROFILES WITH COEFFICIENT OF DETERMINATION OF 0 . 6 6 9 0 F IGURE 5 - 1 7 , A C T U A L STRENGTH VERSUS W E I B U L L B A S E D FEATURE USING K N O W L E D G E B A S E SYSTEM WITH M E S H SIZE 1 A N D BREAKING OF NARROW PART AT " 0 . 2 * RADL" WITH COEFFICIENT OF DETERMINATION OF 0 . 4 8 9 1 F IGURE 6 - 1 , B L O C K D I A G R A M OF A SYSTEM FOR ESTIMATING THE LOCATION OF A FRACTURE BASED ON THE FIRST M A X I M U M M S C KNOT A N D USING A KNOWLEDGE BASED PROCESSOR INSTEAD OF F E M PROCESSOR 9 2 F IGURE 6 - 2 , T O P VIEW OF A PART OF THE B O A R D NEAR THE B R E A K I N G LINE 9 3 F IGURE 6 - 3 , F L O W CHART OF DETERMINING THE HIT FACTOR, T IS THE TOLERANCE 9 4 F IGURE 6 - 4 , F L O W CHART OF DETERMINING THE " J T I T _ F A C T O R - 1 & 2 " , T IS THE TOLERANCE . 9 5 F IGURE 6 - 5 , L O C A T I O N OF THE MIN IMUM V A L U E FOR THE M O E PROFILES P I , P 2 A N D P M AT 6 3 % , 7 3 % A N D 8 0 % OF LENGTH OF THE BOARD # 2 1 0 (POINTED B Y ARROWS) 9 6 F IGURE 6 - 6 , A N M O E B A S E D SYSTEM FOR ESTIMATING THE L O C A T I O N OF FRACTURE USING A V E R A G E M O E PROFILES " P M = ( P 1 + P 2 ) / 2 " 9 6 F IGURE 6 - 7 , H I T FACTOR FOR M O E PROFILE P M , T IS THE TOLERANCE 9 7 F IGURE 6 - 8 , H I T FACTOR FOR THE M O E PROFILES P M , P I OR P 2 , T IS THE TOLERANCE 9 8 F IGURE 6 - 9 H I T FACTOR FOR THE X - R A Y BASED "HIGHEST M S C KNOT" A N D THE "SECOND HIGHEST M S C KNOT" A N D BOTH THE FIRST A N D THE SECOND HIGHEST KNOTS 9 9 F IGURE 6 - 1 0 , H I T F A C T O R FOR M O E PROFILES 1 0 0 F IGURE 6 - 1 1 , H I T FACTOR FOR A L L THE CASES 1 0 1 F IGURE 7 - 1 , CENTER OF A R E A A L G O R I T H M 1 0 2 F IGURE 7 - 2 , A SCHEMATIC OF THROUGH (LEFT) A N D NON-THROUGH (RIGHT) KNOTS 1 0 2 F IGURE 7 - 3 , 3 D ANALYSIS USING SLICES 1 0 3 F IGURE 7 - 4 , M A X I M U M CIRCLE FIT INSIDE THE X - R A Y I M A G E 1 0 3 F IGURE 7 - 5 , FLAT -WISE X - R A Y IMAGE OF BOARD # 2 NEAR THE DETECTED KNOT AFTER THRESHOLDING A N D DIGITIZING 1 0 4 xi F IGURE 7-6, T O P VIEW OF U L T I M A T E STRENGTH TESTER IN UNIVERSITY OF BRITISH C O L U M B I A ( U B C ) W O O D PRODUCTS L A B O R A T O R Y 107 F IGURE 7-7, RESULTS FOR THE M A X I M U M D I A M E T E R A L G O R I T H M 109 F IGURE 7-8, RESULTS FOR THE M A X I M U M D I A M E T E R A L G O R I T H M A N D EROSION A L G O R I T H M 110 F IGURE 7-9, RESULTS FOR M A X I M U M DIAMETER A L G O R I T H M A N D 1 5 % CUT A L G O R I T H M ... 111 F IGURE 7-10, RESULTS FOR THE M A X I M U M D I A M E T E R A L G O R I T H M , EROSION A L G O R I T H M A N D 1 5 % CUT A L G O R I T H M 112 F IGURE 7 -11 , M S C FEATURE 5 FOR THE SECOND H A L F OF THE SAMPLES COMPARED TO THE ONE IN C H A P T E R 2, F IGURE 2 -25 (R 2 OF 0.49 VERSUS 0 .42) 113 F IGURE 7-12, W E I B U L L BASED FEATURE FOR THE SECOND H A L F OF SAMPLES COMPARED TO THE ONE IN C H A P T E R 3, F IGURE 3-3 (R 2 OF 0 .57 VERSUS 0 .47) 114 F IGURE 7-13, RESULTS FOR THE SECOND H A L F OF THE SAMPLES COMPARED TO THE ONE IN CHAPTER 4 , F IGURE 4-6 (R 2 OF 0 .59 VERSUS 0 .58) 114 F IGURE 7-14, RESULTS FOR SECOND H A L F OF THE SAMPLES WITH COEFFICIENT OF 0 . 6 9 , 0 . 7 1 , 0 .71 , A N D 0.71 FOR ALGORITHMS A 1 TO A 4 RESPECTIVELY 115 F IGURE 7-15, RESULTS FOR SECOND HALF OF THE SAMPLES (COMPARED TO RESULTS IN C H A P T E R 5 (MESH 1 A N D BREAKING AT "0 .2 * R A D I " A N D N O FILTERING)) 116 xii List of symbols, nomenclature and abbreviations 2D two dimensional 3D three dimensional 5 deflection [m] P the partial regression coefficients 0 angle p density [Kg. m 3 ] 6 angle e strain °~ stress [Pa] a ** the variance of the components of the vector x G y y the variance of the components of the vector y a t h e covariance of vector x and y . a constant A area [m2] A l , A2 , . . . Algorithm 1, Algorithm 2, ... AveMinCut 15_feature a feature based on minimum of averaged two M O E profiles and dumping 15% from both ends b constant C coefficient CFI , CF2,.. calibration Factors CLT Continuous Lumber Tester D, d diameter [m] E modulus of elasticity [Pa] edge distance of the center of each knot from the one side [m] EF End of Fracture [m] E X M O E in longitudinal direction [Pa] E Y M O E in transverse direction [Pa] EZ M O E in perpendicular direction to E X and E Y [Pa] X l l l F Force [N] F E M Finite Element Method G X Y shear rigidity in E X and E Y plan [Pa] i number I cross section second moment of inertia [Kg. m ] K constant, distance of the center of knot to edge of a board [m] L length [m] L F Location of Fracture [m] Max maximum M C moisture content M O E Modulus of Elasticity [Pa] M O R Modulus of Rupture [Pa] M S C Maximum Stress Concentration n number N N Neural Network N U _ X Y Poisson's ratio P1,P2 Profile 1, Profile 2 Pm average of profiles PI and P2 r radius [m] r correlation r coefficient of determination R knot location ratio radi knot radius [m] Regll second order regression S output S Stress [Pa] SF Start of Fracture [m] SOG Slope of Grain SPF Spruce, Pine, Fir T tensile strength [Pa] T tolerance [m] t time [s], variable U displacement [m] V velocity [m.s"1] v(t) function of t w weight W width [m] Weibull_B_F Weibull Based Feature X,x variable, input Y the dependent variable of tensile strength y(t) function of t XV Acknowledgment First, I want to thank my family for providing the means to reach to this point in my life to be able to finish my studies. M y supervisor professor Peter D. Lawrence, a gentleman and a scientist, shaped me as a better person by his guidance during 5 years of my study in U B C . I learned from him to be a better researcher, a better educator and above all to be a better human being. Thanks Peter. M y co-supervisor, Dr. Frank Lam, same as Peter, a gentleman and a scientist, advised me in many aspects of Wood Sciences. I learned a lot from him. Thanks Frank. I also want to thank NSERC to provide the funding for the research. I want to thank University of British Columbia to provide the means of my study. I want to thank my supervisory committee members, Professor Tim Salcudean, Dr. Robert Rohling, Professor Guy Dumont, and Professor Ian Cumming for their excellent comments during the study. I also want to thank other researchers and educators whom I worked during my study, Professor Jose R. Marti, Dr. C. Kevin Lyons, and Dr. Hossein Saboksayr. Finally, I want to thank my friends especially Pogy Kurniawan for their help and support. Chapter 1 1 1. Introduction and Literature survey One of the major uses of wood is for structural applications. The engineering design of timber structures under imposed loading requires knowledge of the mechanical properties of lumber. Commercially available "state-of-the-art" technologies to grade lumber for structural applications cannot accurately assess the strength of a board; therefore, the "characteristic" design values for lumber are very conservatively assigned to reduce the impact of inadequate grading accuracy. The only way of accurately determining the strength of lumber is by subjecting it to a certain amount of stress to destruction. The test results are only accurate for the destroyed boards; whereas, a statistical estimation of strength is available for the rest of the unbroken lumber population. To increase the reliability of grade estimation of lumber, an intelligent lumber grading system needs to be developed. Currently, state-of-the-art sensors for measuring various factors, which may affect the strength of wood, have been developed. Significant progress can further be made towards more precise grading of lumber for structural applications by establishing a sound theoretical relationship between the material properties of the various "weak" regions of each board and the mechanical behavior of the specimen when tested. The finite element method (FEM) offers a means to evaluate the stress field in a board if sufficient details of the physical structure of the wood are known. An F E M processor can then be used with a knowledge based system to classify and grade lumber for strength. 1.1 Objective The overall objective of this thesis was to develop a sound relationship between the material properties of the various regions of each board and the mechanical behavior of the specimen when tested. The remainder of this chapter focuses on commercial grading systems, and a study of previous work in the area of wood grading system. 1.2 What is Lumber Grading? According to the Canadian Lumber Grading Manual [1]: Chapter 1 2 " A lumber grade is a grouping of pieces, all slightly different within defined limits, with regard to the end use for which the grade is intended... The purpose of lumber grading is to sort lumber into different types of utility value, so that the same grade will represent the same value and can be used for the same purpose irrespective of the type of log or the mill from which it is produced." A l l grading systems are based on the use of predictors to estimate the strength properties [2]. In visual grading, for example, the size and type of visual defects are considered to classify the wood into grades with associated strength properties. In machine-stress grading, the modulus of elasticity (MOE) of the lumber is used as the prime predictor of strength properties. Visual grading and daily quality control procedures are also used to provide additional inputs to the grading process. The goal of strength grading is to have a tight relationship between the predictor and the strength (see the line in Figure 1-1). What is usually obtained in reality, is a grouping of data that looks more like an ellipse (see the ellipse in Figure 1-1), and it is highly desirable to obtain as narrow an ellipse as possible. Statistically, an exact match between the predicted and the actual strength means that the coefficient of determination (r2) equals 1, and if there is no relationship between the predicted and the actual strength, r 2 equals 0. Hence, in all grading systems, it is highly desirable to achieve r 2 as close to 1 as possible. To do so, it all depends on which predictors to use. Researchers are continually looking for better predictors or improving the r of existing predictors. Chapter 1 3 1.3 Commercial Grading Systems There are several lumber grading systems currently available. This section will describe the three major commercial stress grading systems, namely visual grading, the continuous lumber tester (CLT), and intelligent lumber grading machines. 1.3.1 Visual Grading Visual stress grading began almost 80 years ago because engineers wanted safe and economical working stresses. During World War I, the demand for timber was very high, resulting in the birth of visual-stress-grading rules by the United States Forest Products Laboratory in 1923 [3]. In Canada, the National Lumber Grades Authorities (NLGA) has been responsible for writing and maintaining Canadian Lumber Grading rules since 1971 [2]. In visual grading, a person called a grader will look at the board's four faces. Within seconds, the grader will look for visible defects, such as knots (size and location), slope of grain, wane, shakes, and splits, check the usage of the board, and finally decide which grade stamp (one of 5 grades) to be stamped on a board up to 5 meters in length [1],[2]. Certified graders should have been trained to follow sets of standard rules for determining the defects. The usage of lumber is divided into "general use" and "structural use." For "structural use" lumber, the important criteria are the knot, and grain angle of the lumber. 1.3.2 Continuous Lumber Tester (CLT) The CLT is based on the principle that the strength of lumber is positively correlated with modulus of elasticity (MOE). The machine measures M O E based on the fact that a beam under a center load will deflect [4],[5] ,[6]. Suppose there is a simple beam under a load F (see Figure 1-2). Chapter 1 4 Figure 1-2, Beam Deflection under a Center Load, F From mechanics [68], it is known that the deflection of a simple support beam with a center load is: s cen.er ^ £ j The equation can be rewritten for E as follows: where L is a simple fixed support span, I is the second moment of area of the lumber cross section,' which is a constant depending on the dimension of lumber, and E is the modulus of elasticity (MOE). This means, two variables, namely the force (F) and the deflection (Center), can be used to design a C L T machine. Two kinds of machine can be designed based on which variable (F or Center) is measured. The first kind of machine keeps a constant force and measures the deflection. The M O E is calculated (E = K7Center), and the strength is determined. A machine that uses this principle is called the computermatic [5] machine. In this kind of machine, boards are fed edgewise into the machine in a longitudinal direction. The individual piece is continually deflected in a narrow dimension by a given load. The amount of deflection caused by the load is measured on 15 [cm] (6-inches) intervals throughout the length of the piece. The deflection is then Chapter 1 5 converted to estimate of the M O E and strength and finally the grade stamp. In this machine,, the lowest M O E is the basis for grading the lumber in to the five classes. U L O A D " Deflection-Scnsoi A i l Cyl inder Computer To Stampers Figure 1-3, Computermatic Machine One concern about this machine is the amount of force used in the operation, where there is a possibility that a board with a low M O E may deflect more than an allowable amount by the machine. Hence, the machine should have control over the applied force which can be reduced in such a situation. The computermatic machine is no longer available in the market. The second type of machine solves the above problem by making the deflection constant and measuring the applied force. Assuming simple supports, the modulus of elasticity can be calculated as follows: F L 3 K F 4 8 l £ c e n t e r This principle is used by the CLT-1 machine [8] and 7200 HCLT machine [9], both manufactured by Metriguard Inc. Chapter 1 6 rrn'crn <W7 <SSS <t Force Sensorl Force Sensor 0 o r f n 7n dnnn //// //// ///. Figure 1-4, CLT-1 Machine In this type of machine, boards are fed into the machine flatwise in the longitudinal direction. Shown in Figure 1-4, the individual board is continually deflected into a double curvature at prescribed center point's deflections by rolls in both the up and down directions. The amount of force required to reach this certain deflection is converted into an M O E estimate of the piece. Basically, one half of the machine is the duplicate of the other half, except that the force sensors for each half are placed in the opposite sides (top and bottom) of the board. Hence, two sets of M O E will be produced and the values will be averaged. The M O E is recorded continuously along the length of each piece (every 4-foot span). The machine measures both the average E of the whole board and the lowest E of the piece and uses this information to predict the strength of each board. 1.3.3 Intelligent Lumber Grading Machine C A E Newnes Ltd. is one of the major sawmill equipment manufacturing companies that produces industrial intelligent lumber grading machines. This company produces three types of machines, namely: the X-Ray Lumber Gauge [10], Linear Planermill Optimizer [12] , and Addvantage Chopsaw Optimizer [13]. The basic principles of these machines will be described in the following sections. 1.3.3.1 X-Ray Lumber Gauge The basic principle of this machine is based on the detection of the presence of anomalies in wood, such as knots, wane, and resin canals; that can create high stress zones, and significantly Chapter 1 7 weaken the wood [10]. The variation of local density on wood usually shows the presence of these anomalies. X-Rays suffer attenuation as they pass through a dense media; as the density of the material increases, the attenuation also increases. Based on this fact, X-Ray systems are utilized quite extensively in various industries. For the measurement of local density of wood, an X-Ray can be produced and projected onto the board from one side of the board and the intensity of the ray that passes through the board is measured on the other side of the board (see Figure 1-5). A i i i— L w \ \ \ \ \ \ \ \ K-v-X-ray source X-ray Cross sectiai of the board Sensor array Figure 1-5, X-Ray based local density detection measurement system The intensity of the ray at each pixel on the sensor array is measured and transformed into the material density. This information is then transferred to a computer to generate an X-Ray image of the whole board. An X-ray image of a typical knot is shown in Figure 1-6. The density values are used to predict M O E and the strength of the lumber. It is also possible to extract the knot geometry from the X-Ray image [11]. Chapter 1 8 1.3.3.2 Linear Planermill Optimizer The linear planermill optimizer uses X-Ray and laser technology to predict the strength of lumber [12]. The X-Ray technology was described in the previous section. The laser technology is based on the conduction of light along the grain at the surface of the board [14]. A laser beam will be projected onto the surface of the board. Although the cross section of the projected beam is a circle, the reflected beam will resemble an ellipse, whose main axis is parallel to the grain direction of the board (see Figure 1-7). The slope of the grain can then be determined based on the slope of the main axis of the ellipse. The weakness of this technique is that it is affected by surface roughness, which commonly exists on the surface of a board. Instead of using a single laser beam, one can also use several plane-parallel beams where the projection of these beams will form a line. When these beams strike a board without any defect, the reflection will be a thickened line (like a rectangular strip). If a knot is present, the reflection will be a strip with a pinched section at the knot location (see Figure 1-8). This plane-parallel laser beam method is very useful for detecting the presence of knots on the lumber in a short time. While both the single beam and plane-parallel beam methods can be used to estimate the strength of lumber, the single beam method requires a longer processing time to achieve the same purpose. The single beam method has to scan an entire surface of the lumber point by point, whereas the plane-parallel beam method scans the entire surface line by line. Chapter 1 9 Laser beams can also detect other geometric defects of lumber. The machine uses both the X -Ray and plane-parallel laser data to evaluate the strength of lumber. 1.3.3.3 Addvantage Chopsaw Optimizer The Addvantage Chopsaw Optimizer uses X-Ray, laser and vision systems to detect the defects and predict the strength of lumber [13]. The vision system processes the image by an algorithm that detects the surface defects. This machine also cuts the lumber to get the optimum market value (i.e. cutting large lumber into smaller, clear pieces). To do so, the machine needs the vision system. 1.3.4 Quality Control of Strength Grading For quality control purposes, after a four-hour shift, ten pieces from each grade are randomly sampled and tested typically in edge bending (proof-loading). Then, by using statistical calculation (Qsum), the production process is monitored and controlled. This feedback system allows the machine to be adjusted such that the production properties are "on-grade". 1.4 Literature Survey Possibly because of the commercial interest in the development of the next generation of lumber strength grading, only a few papers have been published in this area. In the next sections, worldwide efforts in improving the non-destructive lumber grading system will be presented. Chapter 1 10 1.4.1 Previous U B C Research The lumber grading research previously carried out at the University of British Columbia (UBC) is summarized in this section. One study was done in the Department of Wood Science, and the other study was done by three different departments (Electrical Engineering, Mechanical Engineering and Wood Science) in collaboration with three companies (CAE, Landmark and PRECARN). 1.4.1.1 Advanced Grading Technology for Structural Lumber This study was performed by the Department of Wood Science, U B C Faculty of Forestry, in four phases [15],[16]. The purpose of the study was to develop an advanced grading technology by improving the tensile strength prediction of lumber. More than a thousand (1080) pieces of 38-by-89 mm SPF (Spruce, Pine, Fir) lumber (4.9 m long) were evaluated. M O E data for the lumber were measured using a CLT machine, an SOG detector based on capacitance, and a laser-based SOG indicator. A l l of these machines were described in the previous sections, except for the capacitance-based SOG indicator, which will be explained as follows. The directional electrical property of wood allows grain angle measurements to be made on the basis of capacitance. The permittivity of a board parallel to the grain angle is much higher than the permittivity perpendicular to the grain angle [17]. A capacitance-based SOG system measures the impedance of a capacitor, whose dielectric is the board. As the grain of the board makes a larger angle with the electric field of the capacitor, the capacitance of the capacitor drops. Figure 1-9 shows a capacitor whose electric field is parallel to the grain angle. If the grain of wood were perpendicular to the surface of this page, the capacitance of the detector would drop. If the impedance of the capacitor for a few directions (at the same point on the piece of wood) is measured, the maximum capacitance (minimum impedance) direction is the one closest to the grain direction. Chapter 1 11 The capacitance is most affected by the grain on the surface of the board; hence, the interior grain angle of the board remains unmeasured. However, since the board is more likely to fail from its surface, this deficiency can be assumed to be a negligible effect. Compared to the laser-based grain angle measurement technique, this capacitance technique is less local; it results in an average grain angle over an area, but is less sensitive to the surface roughness of the board. Researchers are also working on other method to measuring grain angle and other surface of logs and lumber [18],[19]. After non-destructive test data were gathered, the boards were broken in the "tension testing machine" to evaluate the true tensile strength of the boards. Once all non-destructive and real data were collected, they were utilized in a multiple regression model consisting of 41 independent variables (Xjj, where j = 1, 2, 3, 41), expressed as follows: Y i = (30 + (3, X i ! + (32 X i 2 + ... + (341 X i 4 i i = 1, 2, 3 , n where: n = the number of specimens Yj = the dependent variable of tensile strength Pk = the partial regression coefficients (k = 0, 1, 2 , 4 1 ) . The regression analysis was applied and the partial regression coefficients evaluated. This model was then used to predict the strength of lumber. A similar study using multiple regression was done by researchers in the Institute for Wood Research, at the Technical University of Munich [20]. In this study, a visual grade, an X-Ray determined knot ratio, a bending result, a stress-wave result and an ultrasonic test result were correlated with board strength alone and in combination. Chapter 1 12 1.4.1.2 PRECARN Intelligent Lumber Grading System (ILGS) Project The P R E C A R N ILGS project was conducted by C A E , Landmark, P R E C A R N , and three U B C departments, namely Electrical Engineering, Mechanical Engineering, and Wood Science, to improve the state-of-the-art strength grading of lumber. The project was performed at both U B C and C A E . The research was divided into two areas, namely sensor development and methodology. In the sensor development, several sensors were tested and developed, including the microwave, X-Ray, laser, dynamic flatwise-bending CLT, and ultrasound sensors. The mechanisms of the X-Ray, laser and flatwise-bending CLT sensors were described in the earlier sections. The microwave and ultrasound will be explained as follows. Several wood properties can be estimated by transmitting electromagnetic waves through wood [21],[22]. The attenuation, phase shift and depolarization of a polarized 4.81-gigahertz wave (microwave) as it is transmitted through a wood specimen, can provide estimates for the moisture content (MC), density, and grain angle of a wood specimen. Because of the high frequency of microwaves, the measurements can be done very fast and do not require any contact with the wood being tested (the measurement can be done over a small window). The window reduces the resolution of the measurement, and since the wave carries through the board, the measurement would be an average through the board. The basis of this measurement technique is that the electric field would be attenuated and the phase would lag (shift) differently as its angle with the direction of the wood grain changes. This attenuation is maximum along the grain and minimum across the grain. Furthermore, the attenuation (in all directions) will be increased as the moisture content of the wood increases. If the attenuation of dry wood is known, this technique can be used to measure the moisture content of wood. The density of the wood can also be obtained from this technique, since the density of material between the transmitter and the receiver would cause a delay in the travel of electromagnetic wave. The microwave-based system is shown in Figure 1-10. Chapter 1 13 linearly polarized transmitting antenna (signal generator) wood being tested probe circularly polarized receiving antenna Figure 1-10, Microwave-based system for measuring wood grain angle Several years of work under the direction of Dr. Schajer at the U B C Department of Mechanical Engineering [22], led to a prototype microwave system. The P R E C A R N project evaluated this prototype system and found that it was not practical to use due to poor accuracy, high noise and instability. Therefore, the P R E C A R N project (including Dr. Schajer) developed an entirely different system called the Asynchronous Rotating Field Microwave System (ARFMS) for lumber grading purpose. The P R E C A R N project also studied the ultrasound technology to predict the modulus of elasticity of lumber, using the following physical relationship: M O E = V 2 p where V is the longitudinal wave velocity and p is the density of lumber. The density was obtained from the X-Ray results, but to measure the velocity, the P R E C A R N project designed a prototype ultra-sound system (see Figure 1-11). By measuring the time required by a pulse to travel from the transmitter to the receiver, the velocity can be calculated by: V = - t - ; M O E = p V 2 time Chapter 1 14 Transmitter Receiver Figure 1-11, Measuring Wave Velocity Using Ultrasound Despite the numerous attempts to achieve an acceptable result, the P R E C A R N project came to the conclusion that although it was theoretically possible to measure M O E by using the ultrasound technology, at the time of the project, the technology was not practical. However, other researchers are currently working on nondestructive evaluation of wood using the same principle of ultrasound (or acoustic) [23],[24],[25],[26]. The P R E C A R N team developed two analysis methods for the lumber grading system. The first method, developed by Dr. Lawrence and Dr. Grudic, was an estimator to predict the strength of lumber using data from MOE, X-ray and SOG. They utilized a method called SPORE employing functional approximation to achieve their results. The function approximation method uses a set of functions, which may predict the strength of a board. To do so, one needs to define the input parameters, namely the defects. Then by choosing trial functions, such as: S = ci]X] + CI2X2 + 03x3 + ... + bjX] 2 + b2X2 2 + b3X3 2 + ... where: xj, X2, X3,... are inputs (defect parameters); and aj, a.2, 03, bj, b2, b3, are constants (determined using the test data), the strength of a board may be estimated. In this method, the information from the theoretical approach may be used to choose the proper functions, such as logarithmic, cyclic, etc [27],[28],[29]. The second analysis method, developed by former U B C Ph.D. student, Dr. Hossein Saboksayr, derived the necessary transformation to determine the knot geometry from the X-Ray sensor data. This achievement was useful for geometrical modeling of a board [30],[31]. Chapter 1 15 1.4.2 Worldwide Efforts to Improve Lumber Grading System Researchers from the University of Oulu in Finland [32][33][34] attempted to simulate a human visual grader with an intelligent system such as the one depicted in Figure 1-12 . Detection Rules B o a r d / Camera Classification Rules Sound Wood Elimination Grading Rules Defect Recognition Quality Grading To the - Control of Grading Figure 1-12, Simulation of a Human Visual Grading System A camera obtained an image of the surface of a board. The image contained a large amount of data, which included the sound wood data. Since sound wood did not affect the visual grading, by eliminating it, the data from the image would reduce to only five to ten percent. Then by using a set of classification rules, different defects could be recognized. The recognized defects were utilized to classify the board into a certain grade based on a set of grading rules. The results from this machine were compared to the grading results by a human visual grader. The grading rules were then fine-tuned to achieve a reasonable result. A similar study was performed by some researchers in the Virginia Polytechnic Institute and State University (VPI&SU) [35]. A knowledge-based vision system for lumber grading was introduced (see Figure 1-13). Image Segmentation Extraction of Region Properties Defect Detection Derect Verification using Spatial Constraints Labels Figure 1-13, Knowledge-Based Visual Grading System for Defect Detection In this system, sound wood and background of the image were segmented and eliminated, and the properties of the remaining regions were extracted. By using the extracted properties, the kind of defects (knots, holes, splits, and wanes) would be detected. Whenever a defect was not Chapter 1 16 clearly identified, for example, a knot was detected as both a knot and a hole), then further defect verification had to be performed using spatial constraints. Based on the verified defects, the board would be assigned a label. Conners, et.al. [35] had tested the above system on 30 boards. Astrand and Astrom [36] designed a single chip sensor to detect wood anomalies based on the image of the surface of a board. 1.4.3 Strength Estimation Method No method is available in the literature today that is capable of accurately predicting the strength of a board from its stress field. Kunesh et.al., [37] performed a study on the effect of a knot on the tensile strength of 2-by-8-inch board. They chose 240 boards which had a round, tight and inter-grown knots with diameters of %", 1 Vi", 1 3/ 4", 2 Vi", 2 %", and 3 VS". Half of them had a knot in the center and the other half had a knot in the edge. Then they designed and performed a tension test to break the lumber to measure the tensile strength of the board. By using regression analysis, they developed a relationship between the diameter of a knot (as a predictor) and the tensile strength of a board. For a center knot, they noticed that linear regression is a good fit and they developed the following relationship: T = 8000 - 1992 D where D is the diameter of a knot in the center of the board in inches and T is the tensile strength of the board in psi. For an edge knot, they noticed that a third order polynomial regression is a better fit than a linear and the second order regression. They established the following formula for an edge knot: T = 10705 - 8720 D + 3401 D 2 - 495 D 3 where D is the diameter of an edge knot in inches, and T is the tensile strength of the board in psi. Cramer and Goodman had used the finite element method to determine the strength of lumber [38],[39],[40],[41]. In their work, they assumed there was one through knot in a board. The diameter of the knot was Vi of the width of the lumber (see Figure 1-14). Chapter 1 17 K Knot D = W/4 W R = K / W M S C — CfnaximurnV ^applied Figure 1-14, Cramer and Goodman Model for Determining Strength of a Board Depending on the distance from the center of knot to the edge of the board, a knot location ratio, R, can be determined. The maximum stress concentration, M S C , was determined at different knot location ratios by using a program called KMESH1. (see Figure 1-15 and Figure 1-16). By dividing the strength of clear lumber with the MSC of the desired board, the strength of the desired board can be determined. Similar work was also done by Zendbergs and Smith [42]. A comprehensive study to model the effect of knots on material properties in structural timber was done by Foley [43], [44]. Longitudinal Stress Concentration Factors 2 r | r I -. " 7 \ i - T r- i !• r i i 4 r i !~ 0 -I—!—:—: i ' '—,—1—'—:—••—!——!—:—:—!—:—'•—:—'—i—•—• i— 0 0.1 0.2 0.3 0.4 0.5 0.6 Knot Location Ratio Figure 1-15, Longitudinal Stress Concentration Factor Chapter 1 18 Transverse Stress Concentration Factors 0 0.1 0.2 0.3 0.4 0.5 0.6 Knot Location Factor Figure 1-16, Transverse Stress Concentration Factor Takeda and Hashizume [45],[46],[47] studied the effect of knot on the strength of a board. They found that the knot-area ratio was a better estimate for predicting the tensile strength of a board rather than the knot-diameter ratio. Lam et.al. [15],[16] performed non-destructive tests followed by tension tests on more than a hundred boards to investigate the cumulative percentage of distinct signal-feature occurrence in failure locations. They found out that for weak boards (tension strength between 1 and 10 MPa), the minimum grain angle, maximum grain angle and the minimum M O E are the prominent features in the failure of the boards. For intermediate boards (tension strength between 11 and 20 MPa), large edge knots, minimum grain angle, maximum grain angle and minimum M O E are the prominent features in failure of the boards. Finally, for strong boards, the small edge knot is the prominent features in the failure of the boards. Another strength estimation method is the use of correction factors (e.g. duration of load, size effects, moisture content, slope of grain) to estimate the tensile strength of a long board under tension [48],[49],[50],[51],[52]. This method suggests that the tensile strength of wood is closely related to the product (or function) of all the correction factors by the clear-board strength. Chapter 1 19 1.5 A N e u r a l N e t w o r k A p p r o a c h Neural networks are collections of elements that emulate some observed properties of biological nervous systems and draw on the analogies of adaptive biological learning [54],[55]. The key element of the neural networks is the information processing structure, which is composed of a large number of highly interconnected processing elements (analogous to neurons) and are tied together with weighted connections (analogous to synapses). As shown in Figure 1-17, a neuron output is determined by neuron inputs through a set of mathematical computation, which consists of two separate steps, namely: v(t) = x 0 w 0 + xiwi + x 2 w 2 + ... + x n w n , and y(t) = <|> (v). Neural Inputs y(t) Neural Output Figure 1-17, Mathematical structure of a neuron "Learning" in biological systems involves adjustments to the synaptic connections that exist between the neurons (wo, Wi, w 2 , ... , w n). Neural networks also learn in a similar way. There is a training algorithm in neural networks that iteratively adjusts the connection weights (synapses) based on a large set of input/output data. The field of neural networks has been around since the late 1950s, but only since the mid-1980s it has been useful for general applications due to the development of sophisticated algorithms. Since they are good pattern recognition engines and robust classifiers with the ability to make decisions about imprecise input data, neural networks have been applied in an Chapter 1 20 increasing number of complex real-world problems. The advantage of neural networks is their flexibility against distortions in the input data and their capability of learning. They are usually good at solving problems that do not have an algorithmic solution or for which an algorithmic solution is too complex to be found. Neural networks consist of several layers of neurons. There are always a layer for input, a layer for output, and several other layers in between depending on the nature of the problem. For a lumber grading system, input vectors consist of defects and geometric parameters depending on the nature of research. However, the output will always be the strength, hence, the output layer of the network will have only one neuron. Each neuron in the input layer has n inputs (see Figure 1-18). Input Input Middle Output Output vector layer layer layer vector Figure 1-18, Neural network for lumber grading system A description of how neural networks solve a complex problem based on a large set of input/output data is shown in Figure 1-19. A set of input and target output (generated from the original system) are sent to the neural network (NN). N N sends its own output (based on the input) to be compared with the target output. The difference (error) is then sent to the learning algorithm, which adjusts the N N accordingly. A new set of input and target output will now go to an "adjusted" N N , giving another output to be compared with the target output. This process will continue until the output from the "adjusted" N N is the same as the target output. Therefore, the more input and target output is given to the neural networks, the more similar the neural networks will be to the original system. Chapter 1 21 INPUT Figure 1-19, The principle of neural network When an N N is applied to an advanced lumber grading system, it is necessary to choose the proper input for the desired system. The output will be the strength of the board (e.g. the strength of a board containing defect(s)). Current studies use defects and some geometrical parameters as the input. The most important defects for lumber under tension can be^classified as in Figure 1-20. Wood Defect Knot Defects Local Density Grain Angle One More than Clear Resin Wane Knot one knots wood canal Figure 1-20, A classification of wood defects Using the different data acquisition techniques explained previously, one needs to use different algorithms to calculate the defects. For example, if the density map of lumber is available, by using an algorithm to look for the high density region, the center of a knot may be determined. Then by using another algorithm to find points whose densities are the average of the center point and the surrounding area, the boundary of the knot may be found. Consequently, a defect can be detected [56]. Chapter 1 22 By using the neural network tool in M A T L A B [54], one can implement a neural network to solve the above problem. The important point for any neural network is that one needs a diverse set of data to "train" the neural network. This means, the parameters of the board (non-destructively) has to be measured as the input of the neural network. Then by testing and breaking the lumber, the actual strength of that board (as the output of the neural network) will be determined. The procedure has to be repeated for many boards to get sufficient data, which represent the different variations in wood defects. This information can now be utilized to train the neural network. Subsequently, the trained neural network can be used to estimate the strength of lumber. The most current work in determining the strength of lumber has been performed by training a neural network using the database of various non-destructive and destructive tests [30],[31],[56]. The block diagram for the strength-determining process is as follows: Sensor Data Feature Extracting Processor Strength Predicting Features Neural Network Strength Estimate Figure 1-21, Block diagram for a Neural-Network-Based process for determining strength of lumber The effectiveness and most likely, the accuracy of the above process depend closely on the feature extraction to produce the most representative and the most important features as inputs for training the neural network. Extracting the most important features from a board can be done by using finite element method (FEM) and a mechanical approach instead of N N to reduce the strength predicting features. In the next chapter, a mechanics-based system for estimating of the strength of lumber is introduced and implemented. 1.6 Motivation for this Thesis Commercially available "state-of-the-art" technologies to grade lumber for structural applications cannot accurately assess the strength of a piece of lumber. Up-to-date-work for lumber grading is based on a correspondence between estimated strength and actual strength regardless the actual mechanical mechanism. Other works regarding mechanical mechanism Chapter 1 23 of breakage do not offer any algorithm for lumber grading. We decided to implement a lumber grading system based on the mechanics of breakage of lumber, to merge actual mechanical mechanisms and strength estimation. We also believe by doing this a better grade estimation of lumber will be achieved. 1.7 Thesis Contributions The main contribution of this thesis is a means for computing the tensile strength of a board using only a few key mechanical parameters - for calibration, the author used the typical tensile strength of a knot free board of the species under test, and a typical value of the modulus of elasticity of knot-free boards of the species under test, and for strength estimation, an FEM-based estimate of the maximum stress concentration (MSC) produced by knot geometry as determined by X-Ray scanning of each board under test, and the measured modulus of elasticity profile of each board under test. The individual contributions are: • A deterministic M S C based system was first investigated using only X-Ray images to predict board strength. For this, the only information required for calibration was the typical tensile strength of a knot-free board (for this study a tabled value was used), and for strength estimation, an FEM-based estimate of the maximum stress concentration (MSC) produced by knot geometry derived from the X-Ray image. The present study shows that strength in any dataset board can be predicted from only knowledge of clear wood properties and information derived from the X-Ray scan of that board. A l l previous studies of which we are aware find experimental regression coefficients and correlations (or coefficients of determination) relating board strength and one or more measurements made on a single dataset of boards (e.g. [20] [59]). The difference then is that our study uses no information from the boards under test to determine regression coefficients. Nevertheless, the coefficient of determination of .42 was either better or comparable to the values determined by other researchers. The novel use of a Weibull-based estimate of strength was investigated. It was found to be a slightly better estimator of strength of lumber giving a coefficient of determination of .47 than the MSC-based feature. • A mixed system using the "Modulus of Elasticity" (MOE) and the X-Ray images was also evaluated to estimate the strength of a board, using for calibration, a typical value Chapter 1 24 of modulus of elasticity of knot-free boards of the species under test and the typical tensile strength of a knot free board of the species under test, and for estimation of tensile strength, the measured modulus of elasticity profile of each board under test, and an FEM-based estimate of the maximum stress concentration (MSC) produced by knot geometry as determined by X-Ray scanning of each board. The coefficient of determination for this was 0.72. This result is also comparable to results obtained by other researchers ([20] achieved .0.64) but this result has been obtained with only the parameters listed above and with no requirement for training data obtained from sample sets of boards with knots • A knowledge based system was implemented to estimate the strength of a board without using a real-time F E M processor and is capable of doing so in real-time with a coefficient of determination similar to the mixed system just described. This is novel because it allows the results of F E M computations to be utilized in real-time without actually carrying out the very extensive F E M computations on-line. • A system was developed to estimate the location of breakage initiation in tension from on-line X-Ray scans. Previous breakage initiation localization has been restricted to hand examination of individual samples [42]. Chapter 2 25 2. Estimating the Strength of Lumber Using X-Ray Images Previous works[31],[37],[38],[42] show that the tensile strength of a board is significantly affected by the presence of knots. Using this fact, a mechanics-based system for estimating the strength of lumber was implemented. This system extracts knot geometries from X-Ray images and uses them to generate a mechanical model of the corresponding board. The strength of the board was estimated from the mechanical model using FEM-generated stress field features. In this chapter, the overall objective is to find mechanics-based features from the X-Ray image of a board, to reduce the number of features needed and to enhance the estimation of the tensile strength of lumber as well as implementing a mechanic based system using only X-Ray image to estimate the strength of lumber. More than 1000 (1080), 38 [mm] by 89 [mm] (2"by 4") boards were sampled from interior British Columbia's SPF (Spruce, Pine, Fir). They were air dried to reach to the 12% moisture content. Then X-Ray images of them were scanned and measured and a database was established. Then all boards underwent destructive testing and their breaking strengths and break locations were measured [31],[56]. The terms "True Strength" or "Actual Strength" in this text refer to this measured breaking strength. 2.1 Generation of Strength Predicting Features The goal of a non-destructive testing system is to scan a board moving along a conveyor belt with several sensors, and compute an estimate of its "Strength" grade in real-time. The proper grade will be stamped on the board accordingly. The process is illustrated in Figure 2-1. Board Conveyor System Scanning System (Sensors) Estimate the Strength Grade Stamp Figure 2-1, Non-Destructive Testing System In order to non-destructively estimate the tensile strength of a board by using X-Ray sensor data, a mechanics-based system for estimating the strength of a board was implemented (see Chapter 2 26 Figure 2-2). Boards are passed through a system of conveyors to be scanned to develop X -Ray images of the boards. Then, by using only the X-Ray image, the geometrical features of the boards were extracted. The knot size and location were fed to an F E M (Finite Element Method) processor to generate an approximate physical model. The resultant physical model is comprised of two materials. One material represents all knots, and the other one represents sound wood. The physical model is segmented into "elements" by means of a "meshing" process (see Figure 2-15) to generate an F E M model. Each element is 4 sided and contain 8 nodes (one node in each corner {4 nodes} and one node in the middle of each side {4 nodes}). Two equations (in X and Y directions) will mathematically represent the displacement field of each node under load. The F E M model contains around a hundred of thousands of nodes. The F E M formulation results in a system of equations (twice the nodes) representing the differential equations that govern the mechanics of the continuum. By applying the proper loads, boundary conditions, and approximate grain direction (Figure 2-12), the system of equations is then solved numerically to obtain a resultant stress field (see Figure 2-16). Finally, a set of strength predicting features were extracted from F E M processor output, stress field. These features were ultimately used to estimate the board strength (see Section IV). Real Board X-Ray Geometric Feature Extracting Processor features Stress field Strength Predicting Features Estimated Strength (Section IV) FEM k Processor Feature Extracting Processor w Figure 2-2, Block Diagram of a mechanics-based system for predicting the strength of lumber Each block in Figure 2-2 is described in sequence in the following paragraphs except the last block "Estimated Strength" which will be explained in Section IV. 2.1.1 Geometric Feature Extraction Algorithm The X-Ray images were produced by using an array of sensors with a 2.5 [mm] distance between each cell. Boards were passed through the array of sensors and an image was produced. The useful part of the image contains around 35 pixels in width (depending on board vibration and bow) and 3000 pixels in length (depending on the feed rate). Chapter 2 27 Recorded X-Ray Signal of a Board 0 0 Figure 2-3, X-Ray image of a board The X - R a y image analysis involved extracting the useful parts of the image and compensating for the effect of vibration in the board conveyer system. X - R a y S igna l of a B o a r d after extract ing the useful part 0 5 0 0 i i 1 1 1 1 1000 1500 2000 2500 3000 35O0 X - R a y S i g n a l of a B o a r d after C o m p e n s a t i n g for the effect of vibrat ion 4 C 55 ' 3 0 " 2 5 " 2 0 " 15 " 15 ' 5 " 0 " MUBUar: r ' l » JS 1500 2 0 0 0 2 5 0 0 3 0 0 0 Figure 2-4, Compensating for the effect of board lateral movement due to feeder After that, the image was passed through a directional low-pass filter to reduce the image noise. Furthermore, the image was resized by interpolation in such a way that the size of the signal was the same as the real size of the board, which is 89*4900 [mm*mm] (see Figure Chapter 2 28 2-5). It means for 89*4900 [mm*mm] board, the resultant image w i l l have 89 cells in each row and 4900 cells in each column after resizing. X-Ray Signal of a Board after Interpolating the image into the real lumber size Condensed dimension [mm] Figure 2 - 5 , X-Ray image of a board after interpolating the image into the real size of the board (condensed in the long dimension) The image was then passed through a threshold filter to separate the knots based on the fact that the denser knots produce "high hil ls" in the X-Ray image, because of their density. The main strength reducing defect in a board is the presence of knots; hence, all areas of the image below the threshold (which represents sound wood) are assigned the value zero. The rest, representing knot wood contains a number greater than zero. Colored areas in Figure 2-6 represent knots with a value greater than zero; the rest is white color which represents the sound wood with a value of zero. Edges of the board are at 1 [mm] (shown by a dashed line in Figure 2-6 and Figure 2-7) and 90 [mm] in horizontal direction and 1 [mm] and 4901 [mm] in vertical direction. Chapter 2 29 X-Ray Image of the Board after Threshold S I c © 4500 4000 3500 a 3 0 0 0 V in a T3 C O U 2500 2000 1500 1 0 0 0 500 Figure 2-6, X-Ray image of a board after threshold with the condensed axis in long direction (vertical axis). Colored area represent knots with a value greater than zero, the rest is white color which represents the sound wood with a value of zero. Note that knots look like line because the vertical axis is condensed 50 times. Edges of the board are at 1 [mm] and 90 [mm]. A zoomed image of Figure 2-6 around the first knot from the bottom with the same scale in horizontal (edges at 1 and 90 [mm]) and vertical axes is shown in Figure 2-7. There is one Chapter 2 30 knot present at the center (20 [mm], 1458 [mm]) with radius 10.7 [mm]. Figure 2-7 is a contour plot and each contour represents the same value. The inner contours (red) have a higher value compared to the outer contours (blue). 1500 1490 1480 X-Ray Signal of a Board after Thresholed, Zoomed 1470 E E 1460 1450 1440 1430 Figure 2-7, Zoomed image of a part of X-Ray image of a board after threshold with the same scale in horizontal (edges at 1 and 90 [mm]) and vertical axes. There is one knot present at center (20 [mm], 1458 [mm]) with radius 10.7 [mm] Finally, information on all the knots such as geometry and location was detected from the threshold image by means of an algorithm comprised of three parts, segmentation, detection of the center of each knot, and detection of knot radius. At first a segment (knot) is detected by assigning zero to sound (knot-free) wood, finding the maximum value in the remaining part, and finding all the points that are connected to it. Secondly, the center of the area of the detected segment is calculated and assigned as the center of the detected knot. Finally, the maximum circle is calculated that can be fitted totally inside the segmented area and Chapter 2 31 containing the center of the segment. The radius of this circle is assigned as the radius of the corresponding knot (see Figure 2-8). Figure 2-8, Detection of the center of each knot (center of area of the detected segment), and detection of knot radius (maximum circle that can be fitted totally inside the segmented area and containing the center of the segment) A correction on the calculated radius was done for knots which inclined in the transverse direction by means of a geometrical relationship. After detecting the knot center and radius, all points belonging to this segment are set to zero, and the algorithm is repeated for detecting the next knot. Table 1, Detected knots of a sample board No. Y(mm) X(mm) Radius(mm) 1 440 18 9.9 2 729 86 18 3 746 36 11.9 4 1458 20 10.7 5 1716 23 10.5 6 2262 23 18 7 2614 86 3M 8 2960 14 8.3 9 3999 8 7.3 10 4578 30 6.3 11 4681 20 5.4 11 4760 52 48 Chapter 2 32 Knots which are smaller than 5 mm in radius are ignored except when all detected knots are less than 5 mm in radius. This reduction in knot number increases the speed of calculation without changing the final result. The result is illustrated in Table 1 and Figure 2-9 for one example board. The ignored knots in this example are knots number 2, 6, 7, and 12 with radius less than 5 [mm](underlined in Table 1). 4500 4000 E 3500 .1 3000 £ Z <D I 2500 XD § 2000 <x> •o o 1500 o 1000 500 Image of Detected Knots 10 20 30 40 50 [mm] 60 70 80 90 Figure 2-9, Graphical representation of detected knots for the same board as in Table 1 (condensed in the long dimension) The described algorithm up to this point is the "Geometric Feature Extracting Processor" box in Figure 2-2 with the X-Ray image as the input and geometrical features (radius and center of the detected knots) as the outputs. It is depicted in Figure 2-10 in detail. By comparing Figure 2-6 to Figure 2-9, a good match can be seen between detected knots and threshold X-Ray image which confirms the accuracy of Geometric Feature Extracting algorithm. Chapter 2 33 Start X-Ray Image Extracting the useful part Compensate vibration Low pass directional filter Threshold Filter and assignment • Detect a segment • Detect the center of the segment • Detect the biggest circle that can fit inside the segment and containing the center of the segment Figure 2-10, Geometric feature extracting processor detailed algorithm. 2.1.2 F E M Processor One proven software product to analyze the stress and strain field of solids is the ANSYS®, software package. In order to utilize ANSYS® effectively, an intensive ANSYS® training program in Everett, Washington, was attended in May 2000 [65]. ANSYS® can perform analysis on various problems, including heat transfer, fluid mechanics, structural analysis and more, but for this project, ANSYS® was specifically used to analyze the stress and strain fields of the lumber under axial tensile load. The results from ANSYS® also needed some post-processing using both ANSYS® and M A T L A B to estimate the strength of the lumber. Before explaining the F E M processor, two points (edge knot and grain angle which are used in F E M processor) will be explained first for continuity. Chapter 2 34 In the F E M processor stage, two practical points (edge knots and grain angle), were considered due to the results of a previous study [58][59] [60]. The first point is that the presence of a knot close to the edge of a board results in a very high longitudinal "Maximum Stress Concentration" (MSC is a normalized value comprised of the maximum stress in the mechanical field divided by average stress of the field) near the edge which represents localized failure near the edge but not the failure of the whole board (which needs to be determined). To accommodate the first point, the F E M model was modified by removing the narrow part of the board (see Figure 2-11). Figure 2-11, Model with an edge knot (Left); Modified model of edge knot with a broken part (Right) The second point is that a comparison of the "Maximum Stress Concentration" (MSC) calculated with and without local grain angle at points around a knot shows that the slope of grain is a significant feature in generating the stress field and cannot be ignored. We have measured these grain angles with a resolution of 1cm; but this is not of sufficient accuracy to be used in our model, and thus we developed the approximation described below. To accommodate the second point, an approximation of the slope of grain was generated by simulating the grain direction by analogy to fluid flow and reorienting the element coordinate system along the flow line direction [62][63] . Since our F E M model is only a 2D model, a 2D fluid flow approximation of grain angel will be sufficient. Previous studies show that fluid flow analogy provides a reasonable approximation of slope of grain [63]. Suppose a long cylindrical obstacle (radius a) is located in a laminar incompressible flow. Flow is in x direction with the speed of V 0 . w w Knot Chapter 2 35 Figure 2-12, Flow Grain Analogy Model Assuming Vo=l, the components of the local flow velocity vector V at point P at a distance r from center of the obstacle of radius a and angle 0 with respect to the global flow Vn direction axis X are [64]: n2 * Vn*Cn<; 0 n2*Cn<;Q \r=\Q*CosQ-- ° C O S =CosQ-a , y e = - V o * Sin 0 - * 2 * V ° ; ^ 6 = -Sin 6 -( 1) ( 2) y CY y „ \ Transforming the speed v r ' ' as a function of the global coordinates (XY) of point P , we can calculate the angle of flow at point P . The final results are: a = ATAN2(VX, Vy) where, Vx= 1 + 2*a2 *Y*X (Y2 + X2)2 ( 3) The results for a circular obstacle with a radius 5 [mm] at center (x=10 [mm], y=8 [mm]) are depicted in Figure 2-13. A practical point, which needs to be considered is that at the two locations (at 0 =0 and 180 degrees) on the surface of the obstacle, angle Q. is Chapter 2 36 indeterminate (because the veloci ty is zero) or the ve loc i ty di rect ion is not a g o o d approx ima t ion for the slope of the grain . B y us ing the center of area of each element to calculate the cor responding slope of g ra in w e can track the p r o b l e m (see F igure 2-15). Simulated flow around a long cylindrical obstacle 16 i 1 -i—•—>•—s*—=^ —i^ *—<= 0 I 1 1 1 1 0 5 10 15 20 X axis (parallel to flow direction) in [mm] Figure 2-13, The simulated flow around a long cylindrical obstacle (radius a=5 [mm]) is located (at center 10 [mm], 8 [mm]) in a laminar incompressible flow. In the presence of one knot, by calculating angle Q for each element in sound wood, the element's local coordinate axes were rotated by £2 degrees to simulate the grain angle. In the presence of more than one knot, the effect of all knots is summed to establish the resultant angle £2 . From the angle Q. equation, one can infer that for points far from the center of the knot, V x goes to one (V 0=l) and V y goes to zero; hence, angle Q, goes to zero, which is useful for reducing the processing time. Note that the local coordinate system for an element inside the knot will remain parallel to the global axes (XY). Chapter 2 37 The knot size and location were fed to an F E M processor to generate the physical model (see Figure 2-14) using ANSYS® [65]. Figure 2-14, Physical model of a board (because of the long dimension of the board the above picture was cut into 4 pieces), circles Al to A8 are the knots and background is the solid wood. Different material properties were assigned for knot and sound wood. For solid wood E X (MOE in the longitudinal direction), E Y (MOE in the transverse direction), E Z (MOE in the perpendicular direction to E X and EY) , N U _ X Y (Poisson's ratio, which is the ratio of transverse contraction strain to longitudinal extension strain in the direction of stretching force), and G X Y (shear rigidity in E X and E Y plane) were assigned 2141000 [psi] (14762 Chapter 2 38 [MPa]), 91200 [psi] ( 630 [MPa]), 142000 [psi] (980 [MPa]), 0.47 and 108000 [psi] ( 745 [MPa]) respectively [66]. For knot wood E X , E Y , EZ, N U _ X Y , and G X Y were assigned 50000 [psi] ( 345 [MPa]), 50000 [psi] (345 [MPa]), 2141000 [psi] ( 14773 [MPa]), 0.42, and 38000 [psi] ( 262 [MPa]) respectively [66]. In this analysis, the level of M O E will not affect the final Maximum Stress Concentration. This means that if we double all values for both solid and knot wood the final results will be the same. However, the ratio of knot and solid wood M O E will affect the results. If we only double the M O E a solid wood, it will change the results. The M O E ratio with the previously mentioned values comes to almost 43 which is a typical value for SPF. Then the first practical modifications were implemented due to the presence of edge knots (as previously described the F E M model was modified by removing the narrow part of the board (see Figure 2-11 )). After that, the physical model was divided in to pieces, which are called "elements" by means of a process called "meshing" to generate an F E M model (see Figure 2-15 (top part)). Then the second practical modifications were implemented due to the slope of grain (see Figure 2-12, Figure 2-13, and Figure 2-14). An approximation of the slope of grain was generated by simulating the grain direction by analogy to fluid flow and reorienting the element coordinate system along the flow line direction, (see Figure 2-15 (bottom part)). Solid element type "PLANE82" was chosen for the analysis. It is a 2D element with 4 sides and contains 8 nodes. Each element is mathematically represented by nodes. Two equations (in X and Y direction) represent each node mathematically. The resultant F E M model could contain millions of nodes (depends on the complexity of the geometry and precision {mesh size} used to calculate the results). The version of software which we have access to is capable of handling up to 200,000 equations. A variable named mesh size control the size of elements and consequently the precision of the results. ANSYS® has value 1 to 10 for mesh size. Mesh size equals one; represents the finest mesh and consequently the least calculation error. However, it will produce too many elements and nodes (more than the limit of our ANSYS® software license). A mesh size of 5 was chosen for this study to accommodate the equation limit. Depending on the number of knots, the resulting F E M model contains tens of thousands nodes for mesh size 5. Chapter 2 39 Figure 2-15, FEM Model of a part of a board (top), FEM Model of Slope of Grain By rotating the Element Coordinate Systems (bottom). The global axes are depicted at top right of the picture. All other axes are local axes which look like L which is rotated according the slope of grain at center of corresponding element. By accommodating all the mentioned points, a constant mechanical tension force was applied to both ends of the F E M structure to generate the stress fields (transverse, longitudinal, shear, and principal). Stress fields were calculated in global axes (XY) as opposed to, axes parallel and normal to the grain angle which are called "local axes". As it can be seen in Figure 2-15, some local axes near the knot (inside solid wood) are rotated almost 45 degrees compared to local axes inside the knot and far from the knot which are parallel to the global axes. Chapter 2 40 Considering the definition of MSC, which is the ratio of maximum stress to the average stress, by assigning the applied stress to one, the Maximum Stress Concentration (or minimum) can be obtained by finding the maximum (or minimum) value in the resulting stress field. The resulting stress field for a part of an example board is depicted in Figure 2-16. The bottom part of this figure is the longitudinal stress divided by the average applied stress (due to the force applied to the ends of the board), and the top picture is close up of the small edge knot in the bottom figure. Different colors and shading represents different levels of stress. Chapter 2 41 NODAL SOLUTION STEP=1 (AVG) SUN --1.391 SMNB=-1.'!84 SMX -3.6S4 '3.90S AN HAY 7 2003 19:54:06 Figure 2-16, 2-D image of the Longitudinal (bottom) stress field around two close knots. The maximum stress concentration for longitudinal (bottom) stress happens at the small edge knot and it's value for this case is 3.65 (it was shown by "MX" in the picture). This is also Feature 5 as we defined it. The top picture is close up of the small edge knot in the bottom figure. Chapter 2 42 The algorithm described up to this point is in the " F E M " box in Figure 2-2 and is depicted in Figure 2-17 in detail. • Start Z Geometrical Features Radius, Center of Knots Physical model Generator Physical Model Breaking the narrow edge parts r Modified Physical Meshing i F E M Model r Flow Grain Analogy Slope of Grain Embedded r F E M Model Applying Loads and Boundary conditions Set of linear equations Solver , Displacement Field Post processing Different Stress (or Strain) fields _ " 1 End Figure 2-17, FEM (Finite Element Method) processor detailed algorithm By generating the resulting stress (strain or displacement) fields, 26 different features were extracted from them, which will be explained in the next section. Chapter 2 43 2.1.3 Feature Extract ing Processor By generating the resulted stress (strain or displacement) fields, 26 different features were extracted from them which are summarized in Table 2. A l l features are extremes (minimum or maximum) points in the 2-D fields. Two combined features were also investigated. Features 27 and 28 are based on a linear, and a non-linear combination of features 5 and 7 respectively. Table 2, Extracted features (28 features) 1. Maximum Longitudinal Stress Concentration (Global axes +Nodal solution) 2. Minimum Longitudinal Stress Concentration(Global axes + Nodal solution) 3. Maximum Transverse Stress Concentration(Global axes + Nodal solution) 4. Minimum Transverse Stress Concentration(Global axes + Nodal solution) 5. Maximum Longitudinal Stress Concentration (Local axes + Nodal solution) 6. Minimum Longitudinal Stress Concentration Local axes + Nodal solution) 7. Maximum Transverse Stress Concentration(Local axes + Nodal solution) 8. Minimum Transverse Stress Concentration Local axes + Nodal solution) 9 to 16 Repeat 1 to 8 by replacing Nodal with Elemental solution. 17. Maximum Shear Stress Concentration (Global axes + Nodal solution) 18. Minimum Shear Stress Concentration(Global axes + Nodal solution) 19. Maximum Second Principal Stress Concentration(Global axes + Nodal solution) 20. Minimum Second Principal Stress Concentration(Global axes + Nodal solution) 21. Maximum Longitudinal Strain (Global axes + Nodal solution) 22. Minimum Longitudinal Strain(Global axes + Nodal solution) 23. Maximum Transverse Strain(Global axes + Nodal solution) 24. Minimum Transverse Strain Concentration Global axes + Nodal solution) 25. Total Longitudinal elongation 26. Maximum Transverse Deflection 27. Feature 5+10*Feature 7 28. max(Feature 5,7*Feature 7) Note: Local = Axes Parallel and Normal to Grain angle Nodal and Elemental are two different ways that A N S Y S software uses to calculate strain and stress. In an Elemental solution, element contours are discontinuous across element boundaries for the selected elements. In a Nodal solution, contours are determined by linear interpolation within each element from the nodal values, which are averaged at a node whenever two or more elements connect to the same node. Chapter 2 44 For the board #210 (see Figure 2-16), the locations of the maximum stress (in this case, feature 5) are labeled as "Mx" . Its numerical value is represented in Table 3. The numerical value for all other computed features for the sample board #210 is also represented in Table 3. Positive values in the table show tension and negative values show compression in the extracted point. The Poisson effects together with the geometry generate compression in some points of the board even though the ends of the boards are in tension. Table 3, Example: Computed values for features 1 to 28 (maximum/minimum values in respective strain or stress field as they are defined in Table 2) for board #210 (1) 3.65 (2) -.365 (3).416 (4)-1.66 (5)3.65 (6)-1.81 (7).394 (8)-.393 (9)3.68 (10)-.848 (11).439 (12)-1.76 (13)3.68 (14J-1-91 (15).394 (16)-.635 (17).87 (18)-.848 (19).248 (20)-.322 (21).443E-5 (22)-0.486E-5 (23).232E5 (24)-.473E-5 (25)609E-3 (26)2.41 E -3 (27) 7.59 (28)3.65 The described algorithm up to this point is in the "Feature Extracting Processor" box in Figure 2-2, which is depicted in Figure 2-18 in detail. After generating the mentioned features, the highest correlated feature, compared to real strength (boards were subjected to destructive testing to measure their strength), is used to predict the estimated strength (described in Section 2.3). Chapter 2 45 Figure 2-18, Feature Extracting Processor detailed algorithm Chapter 2 46 2 . 2 Process refinement At this stage, a feedback process is used to improve the geometric feature extraction algorithm by using a database of more than a thousand boards. Adjusting the strength predicting rules will be explained in future chapters. Sensor Data Geometric Feature Extractor Geometric Feature FEM Stress Field Model Board Feature Extracting Processor * — Revising feature extracting processor algorithm Improving geometric feature extractor algorithm Strength Predicting Features Strength Estimating Processor Adjusting the Strength Predicting Rules Training Algorithm o Strength Estimate Real Strength Figure 2-19, A mechanics-based system for estimating the strength of a board The geometric feature extraction algorithm was improved by a piecewise threshold filtering procedure based on the statistical mode (maximum number of image pixels which have the same absorption value) of the X-Ray image. Knowing that 90 to 99 percent of a board is solid wood then the calculated statistical mode of the X-Ray image represents solid wood. Mean of the X-Ray image can also represent solid wood; however, for boards with either too many knots or too many holes, the mean cannot represent solid wood value properly. This is why the mode is a better representation of solid wood. In the first study a simple threshold for a whole board, based on the 60 % of the mean and 40% of the maximum value was used to calculate the threshold in an algorithm named 1TF. The presumption for the algorithm 1TF is that a board has at least one knot and there are not too many holes and knots on the board as well. The result from algorithm 1TF was not satisfactory. Then another algorithm (2TF) was employed. At this stage 1.3 times the mode of X-Ray image was used as threshold for the whole board. The results were better than the Algorithm 1TF. Chapter 2 47 Histogram of a X-Ray Signal 14000 i 1 1 r 2000 X-Ray Absorption Intensity Figure 2-20, Histogram for board #210 The histogram of X-Ray absorption intensity of board #210 is shown in Figure 2-20. For this specimen, the mode value of 1266 occurred at the maximum frequency 13605 pixels out of 436100 (89*4900) pixels of whole X-Ray image. Since the piece contained primarily sound wood, this mode value of X-Ray absorption intensity indicated the occurrence of sound wood. As we can compare with the mean and median value as a base, the mode is a better representation of the presence of sound wood. By choosing 1.3 times the mode as a threshold for the occurrence of knot wood (1645.8 for this case) pixels with values greater than the threshold indicate the presence of knots. Knot wood usually comprises less than 2% of the whole board (for this case it is less than 1 percent) for our database. Table 4, Threshold Computation for board 210 Histogram Mean =1221.7, Max= 2055. Mode= 1266.0, Median=1236.2 Algorithm 1TF: Threshold = .6*Mean + 4*Max = 1555.2 Algorithm 2TF: Threshold = 1.3*Mode = 1645.8 Chapter 2 48 The results were further improved by changing the algorithm 2TF to piecewise threshold filtering algorithm (algorithm 3TF). In this algorithm, the X-ray image of a board was subdivided into 50 slices along the length of the board, and for each slice a threshold, based on the mode within the slice, was established. The calculated value was assigned to the whole slice. In other words, 50 different thresholds were calculated for a given board and in each part (one out of 50 parts), which is 89mm by 98mm (4900/50) all cells have the same threshold value. Then threshold values for all pieces were assembled and passed through a low pass filter, to establish the threshold value for the whole board (see Figure 2-21). Piecewise Threshold Base and X-Ray Image L J Condensed dimension [mm] Figure 2-21, Threshold base after passing through low pass filter for the board #210 Then the X-Ray image was subtracted from the calculated piecewise threshold. A l l positive values were considered as knot wood and the rest considered as sound wood (see Figure 2-22). Chapter 2 49 Piecewise Threshold Base 2500 1000 2000 3000 Condensed dimension [mm] 4000 5000 Figure 2-22, X-Ray image and piecewise threshold level of 1.3 times of mode (side view) A preliminary visual inspection was performed to compare the three 1TF, 2TF, and 3TF algorithms for the threshold level. A set of 10 randomly chosen boards were visually inspected. For a knotless board algorithm 1TF produced many knots which did not exist. 2TF and 3TF gave us similar results except for a case of a board with too many defects (rotten part, wane, shakes, and splits) in one side of it. Algorithm 2TF misclassified some sound wood as knot wood which resulted in some bigger detected knots. Then we came to the conclusion that algorithm 3TF is a better choice for calculating the threshold level. In this study this algorithm (3TF) was repeated for different multiplicands of the mode value (1.2, 1.25, 1.275, and 1.30) for calculating the threshold filter. Chapter 2 50 2.3 Strength Estimation The coefficient of determination (correlation square r ) was used to assess the effectiveness of the predictors to estimate the strength of a board. The definition of the coefficient of Where: 6X X is the variance of the components of the vector x which contains the true strength of the boards. 6 y y is the variance of the components of the vector y which contains the corresponding feature value (or predicted strength ) of the boards. 6 x y is the covariance of vectors x and y . The determination of which feature is a better predictor for estimating the strength of a board was done by calculating the coefficient of determination ( r 2 ) for 1080 boards ( Table 5). The combined feature 28 gives the highest r 2 of 0.42 as well as feature 5; however, feature 5 has a lower computational requirement as well as a strong physical meaning leading to a mechanics based approach to strength estimation. From these observations, feature 5, "Longitudinal Maximum Stress Concentration" (Local axes + Nodal solution), is a good feature for predicting the actual strength of a board. By analyzing the results in Table 5, it can be concluded that a threshold value between 1.25 to 1.3 times the modes are better than 1.2 times the mode for a threshold filter. The probable reason is that at 1.2 times of the mode for threshold, part of the solid wood may be misclassified as knot wood. 2 determination r is: Chapter 2 51 Table 5, Coefficient of determination for different features and threshold levels Decisive Feature Threshold Coefficient of Determination (correlation square r2) Threshold 1.2Mode ' 1.25Mode 1.275Mode 1.3Mode 1 0.37 0.41 0.40 0.41 2 0.29 0.31 0.30 0.31 3 0.30 0.35 0.35 0:36 4 0.10 0.17 0.21 0.19 5 0.37 0.42 0.40 0.41 6 0.25 0.27 0.28 0.28 7 0.32 0.35 0.36 0.37 8 0.17 0.21 0.23 0.24 9 0.37 0.41 0.40 0.41 10 0.27 0.29 0.29 0.28 11 0.27 0.34 0.34 0.35 12 0.08 0.14 0.17 0.16 13 0.37 0.41 0.40 0.41 14 0.22 0.25 0.26 0.25 15 0.28 0.35 0.35 0.34 16 0.14 0.20 0.21 0.24 17 0.30 0.38 0.38 0.38 18 0.33 0.37 0.36 0.36 19 0.32 0.35 0.35 0.37 20 0.07 0.13 0.14 0.15 21 0.31 0.35 0.36 0.39 22 0.26 0.28 0.31 0.29 23 0.32 0.35 0.35 0.36 24 0.23 0.27 0.27 0.28 25 0.29 0.28 0.27 0.29 26 0.35 0.34 0.34 0.33 27 0.37 0.40 0.40 0.41 28 0.37 0.42 0.40 0.41 Maximum 0.37 0.42 0.40 0.41 Graphical representations of predictors 5, 7, and 28 versus actual strength of lumber are depicted in Figure 2-23, Figure 2-24 and Figure 2-25. We chose feature 5 as our selected feature to predict the actual strength of a board for the rest of this work. Chapter 2 52 Feature 28 vs Actual Strength with Regression Line Feature 28 (combined features 5 and 7) Figure 2-23, Graphical representation of actual strength (vertical axis) vs, predictor feature 28, "max (feature 5,7*Feature 7)" (horizontal axis), (r2 = 0.42) As we can see in Figure 2-23 a set of points with the value one (feature 28=1) belongs to boards without any detected knots. It is the same for feature 5 in Figure 2-25. For feature 7, knotless boards have value of zero as in Figure 2-24. Chapter 2 53 As we can find in Table 5, stress driven feature 5 is a better predictor of strength of a board compared to other features that are driven from the strain (features 21 to 24) and displacement field (features 25 and 26). Chapter 2 54 Feature 5 vs Actual Strength with Regression Line 70 i 1 1 1 1 1 1 1 0 1 2 3 4 5 6 7 8 9 Feature 5 Figure 2-25, Graphical representation of actual strength (r2 = 0.42) vs, predictor feature 5, Maximum Longitudinal Stress Concentration (Local axes + Nodal solution), with regression line: Actual-Stress = 51.05 - 5.78 * Feature-5 By using the maximum stress theory for the failure of a board, the strength of the board is calculated as follows: T sound wood M S C where T is the tensile strength of a board; T S 0 Unci wood is the tensile strength of a knot free board, and M S C is the maximum stress concentration. If we follow any of the maximum stress (or strain) theory for failure (maximum principle stress theory, maximum shear stress theory or maximum strain theory), we need to divide sound wood strength by the maximum stress (or strain) concentration to reach the predicted strength of the lumber [66]. However, this is a nonlinear transformation and we lose a part of the correlation. The way that was used here to estimate the strength of a board was by using feature 5 directly by means of a linear transformation. Thus the strength of a board can be estimated by a linear translation of feature 5 to the left by 8.83 units (=51.05/ 5.78 calculated Chapter 2 55 from regression analyses) then reflect the whole graph around the vertical axis, and then multiply the results by a second constant (5.78 calculated from regression analyses): predicted _ strength = 5.78 * (8.83 - Feature _ 5) ( 5 ) As we can expect, the resulting coefficient of determination stays the same because both translation and reflection are linear transformations (see Figure 2-25 and Figure 2-26). This approach uses only feature 5 alone to estimate the strength. Estimated Strength vs Actual Strength with Regression Line 70 i 1 1 1 1 1 Estimated Strength calculated from Feature 5 [MPa] Figure 2-26, Graphical representation of actual strength vs, predictor, using translation and reflection (5.78*{8.83-Feature-5», (r5 = 0.42) The results illustrate that the maximum stress concentration (MSC) plays an important role in the strength of a board and can be used to predict the strength of a board under mechanical tension. The results also show that the MSC parallel and normal to the grain angle are well correlated with actual strength of the boards. The relationship of strength predicting feature_5 (with threshold 1.3) and true strength is investigated by using regression analyses. We wanted to investigate if the relationship of Chapter 2 56 feature_5 and actual strength is linear or square (or higher order). The results are summarized in Table 6 and depicted in Figure 2-27. Table 6, Coefficient of determination for different order of regression Order (r2) Co CT c 2 c 3 c 4 c 5 c 6 c 7 c 8 Linear 0.41 47.25 5.30 Square 0.41 48.11 -5.80 0.06 Cubic 0.42 43.21 -0.47 -1.47 0.127 4 0.42 44.58 -2.53 -0.56 -0.03 0.009 5 0.42 58.23 -27.3 14.07 -3.84 0.459 -.02 6 0.42 53.7 -17.8 7.09 -1.37 0.011 0.02 -.001 7 0.42 57.71 -27.2 15.26 -4.98 0.899 -0.1 0.0076 -.0003 8 0.42 153.4 -273 263.7 -137 42 -7.8 0.876 -.0536 0.001 Feature 5 vs Actual Strength with Regression Curves of order one to eight Feature 5 Figure 2-27, Graphical representation of actual strength vs feature_5 (threshold 1.3) with regression lines from order one to eight (with r2=0.41 for linear and square regressions and r2=0.42 for the rest) Chapter 2 57 The results show that there is not any significant increase in the correlation of feature_5 and actual strength for higher order regressions. Therefore the relationship feature_5 and actual strength was assumed to be linear. 2.4 Conc lus ions It is well-known that "Maximum Stress Concentration" (MSC) is an important factor that affects the failure of a board sample under tension. This study has shown that when this feature is computed for each board in a sample size of 1080 randomly-selected commercial boards, it was found that using only X-Ray derived longitudinal maximum stress concentration itself provides a feature that is well correlated with the experimentally-measured board strength, and could be used to provide a measure of board strength through a simple linear transformation. The slope of grain and the modified model for knots close to the edge of a board must be considered for this type of study to reach a realistic result. This study has also shown that stress-based features, in particular feature 5 - the longitudinal Maximum Stress Concentration, is better correlated with board strength than strain and displacement features. It was also shown that the relationship between feature_5 and actual strength is linear. Previous work at U B C was achieved r 2 = 0.40 using 16 geometrical features (extracted from X-Ray images) as inputs to a trained neural network (NN) to estimate the strength of aboard [31]. The current study [61] has reached slightly better results (r2 = 0.42) with a single feature as the predictor compared to previous U B C work (r = 0.40) and 16 input features as the predictors. Chapter 3 58 3. Weibull-Based Features for Strength Estimation Because of the complexity of lumber, its failure mechanisms are still under investigation. "Maximum Stress Theory" is a failure analysis theory that is currently employed by researchers, and "Maximum Stress Concentration" (MSC) is used to predict the strength of lumber, i.e. where T is the tensile strength of a board; T S 0 U nd wood is the tensile strength of a knot free board, and M S C is the "Maximum Stress Concentration". This method only uses one point of the stress field (namely the maximum stress) to predict the strength of a board. In the previous, chapter, mechanics-based features, such as the "Maximum Stress Concentration" factor, were obtained from the X-ray image of a board using the finite element method [58]. A coefficient of determination of r2=0.42 was achieved by using a "Maximum Longitudinal Stress Concentration Feature" [63]. That study used a feature that was based on one point in the stress field. Hence, we hoped, by introducing a Weibull-based feature which is based on the whole stress field, we would obtain a better correlation. The current chapter describes the creation of Weibull-based features from the associated stress field of a board (from the X-Ray image and the finite element method), to enhance the estimation of the tensile strength of lumber. 3.1 Experimental Methods In this study a mechanics-based system using Weibull-based features for estimating the strength of a board was implemented (see Figure 3-1). sound wood M S C Geometric Feature Extracting Processor Geometric features Stress field Strength Predicting Feature Weibull -Based Strength Predicting Features Real Board X-Ray Strength Estimate * FEM Extracting Processor Figure 3-1, Block Diagram of a mechanics-based system using Weibull-based features for predicting the strength of a board Chapter 3 59 Boards were passed through a conveyor system and scanned by an X-Ray image system. From the X-Ray image, geometrical features (knot location and diameter) were extracted then by applying the geometrical features to an F E M processor, stress fields were calculated. These steps are the same as in the previous chapter. Then a feature based on the Weibull approach was extracted from the stress field and was used to estimate the strength. The new module "Strength Predicting Feature Extracting Processor" in Figure 3-1 will be described in detail in next section. 3.2 Strength Predicting Feature Extracting Processor In a previous study [58], 26 different features were extracted from the stress field based on "maximum stress theory" for failure of a board. In this chapter Weibull-based features were extracted from the stress field to estimate the strength of a board. This process was repeated for 1080 boards to check the robustness of the approach. A possible representation of failure of lumber is based on the "maximum stress theory" of failure. We need to divide sound wood strength by the "maximum stress concentration" to reach the predicted strength of lumber [66], [67], [68]. This approach is based on one point in the stress field which has either a maximum or a minimum value. One of the oldest studies for improving the strength estimation of a board is the Weibull distribution to estimate the tensile strength of a long board under tension [69],[70]. This method suggests that the tensile strength of a board is closely related to the weakest link in the series of infinitesimal links [71]. Supposed there is a long board under tension. This board is divided into several sections, each containing some defects (anomalies). It is also assumed that the strength of each section is not related to (independent from) the other sections; this assumption is not absolutely correct, but it is applicable in most cases. Therefore, the strength of the board will be determined by the weakest link. By finding the weakest link and developing a model for the strength estimation of this specific link, one can determine the strength of the board and consequently grade it. The strength estimation of the weakest link could be established by finite element methods, due to the complex geometry of wood. Chapter 3 60 windows related to each anomaly anomalies Figure 3-2, Anomalies in a Board Another possible strength estimation method is the use of "brittle-fracture theory", shear strength, and the Weibull model to estimate the probability of failure of a long board under tension [72],[73]. This method suggests that the probability of failure of a certain volume of lumber when the stresses are known is given by: F = 1 - exp - 1 V t-f m J dV ( 6) where; m, n and t* are material constants, V * is a reference volume, t is the stress, and F is the probability of failure [72]. The Weibull weakest link theory [70] can be used to relate the strength distributions of different stressed volumes of wood product [69],[72],[73], [74]. The "brittle-fracture theory" suggests that the probability of failure of a certain volume of lumber when the stresses are known can be calculated. In this study we did not intend to use the probability of failure directly, because of the need to have an estimate of failure not the probability of failure in an industrial application; however, we intended to introduce features based on the Weibull formulation to directly estimate the strength of lumber. As can be seen from the formulation, the Weibull theory for failure is based on the entire stress field; whereas, the "maximum stress theory" for failure is based on one point in the stress field. This is why we believe a Weibull-based feature will result in a better estimation of strength of lumber. From a previous study [58] we chose the "Maximum Longitudinal Stress" for calculating Weibull-based feature as follows: Chapter 3 61 Weibull Based Feature = ( 7) where S is the stress inside the small area AA (area of the corresponding element in which the average stress is S). Sa is the average stress over the total area A , when there is no abnormality (knot) present. The calculations were repeated for different values of n (1, 2, 3, 3.5, 4, 5, 6, and 7). 3.3 Resul ts More than 1000 (1080), 38 [mm] by 89 [mm] (2"by 4") boards were sampled from interior British Columbia's SPF (Spruce, Pine, Fir). They were air dried to reach to the 12% moisture content. Then X-Ray images of them were measured and a database was established. Then all boards underwent destructive testing and their breaking strengths were measured. Table 7, Coefficient of determination for Weibull-Based Features for different n values and threshold level of 1.25* Mode n Correlation r Coefficient of . . 2 Determination r 1 -0.6142 0.34 2 -0.6659 0.44 3 -0.6861 0.47 3.5 -0.6879 0.47 4 -0.6864 0.47 5 -0.6801 0.46 6 -0.6737 0.45 7 -0.6686 0.45 The determination of the effectiveness of the Weibull-based feature predictor for estimating the strength of a board was done by calculating the coefficient of determination factor r2 for the 1080 boards. The results for different n values and threshold level of 1.25* Mode (see Table 5 in Chapter 2) are shown in the Table 7. For n=3.5 the Weibull-based feature gives the highest r of 0.47. The corresponding "maximum stress concentration" (MSC) based feature resulted in r = 0.42. This is a significant increase of 13.8% which we already expected. The results for n=3.5 is also Chapter 3 62 depicted in Figure 3-3. As we can see in Figure 3-3 a set of points with the value zero (Weibull-based feature) belongs to boards without any detected knots. Weibull Based Feature vs Actual Strength 0.2 0.3 0.4 0.5 Weibull Based Feature 0.7 Figure 3-3, Weilbull Based Feature vs Actual strength for 1080 boards with r of 0.47 Then the lumber strength was estimated by using linear regression as follows: predicted^ strength = 43.81- (65.7 * Weibull_Based_ Feature ) ( 8 } As it can be expected, the estimated strength has the same coefficient of determination as the Weibull-based feature (r2 =0.47). The results are also depicted in Figure 3-4 for 1080 boards. Another way to reach the predicted strength of the lumber is to use a linear translation to the Weibull-based feature to the left by C2=0.67 unit (the higher estimated bound for the Weibull-based feature) then reflect the whole graph around the vertical axis, then multiply the results to a proper number ( CF=65.7) for calibration. If we rewrite the Eq. 8: predicte(Lstrength = CF^{C2-Weibull_Based_Featur^ ( 9 ) where; CF =65.7 and C2=43.81/65.7=0.67. CF is a calibration factor and it can be calculated by using few samples to calibrate the measurement system. C2 is the value of the Chapter 3 63 Weibull-based feature for a very weak board (the higher estimated bound for the Weibull-based feature). It can be calculated numerically by calculating the value for a board with the largest knot possible. Estimated Strength vs Actual Strength with Regression Line 70 i 1 1 1 1 1 Estimated Strength calculated from Weibull Based Feature [MPa] Figure 3-4, Estimated Strength of 1080 boards using Weibull Based Feature 3.4 C o n c l u s i o n The results illustrate that the Weibull-based feature is a better estimator of strength of lumber compared to the "maximum stress concentration" (MSC) based feature. The application of Weibull-based feature by including all points in the entire stress field, improves the estimation of the strength of lumber. The Weibull-based features [77] improve the estimation by more than 13.8% and 18.1% compared to the strength estimation with MSC-based features [58],[59] and neural network method [31],[56] respectively. Chapter 4 64 4. M O E and X-Ray for Strength Estimation Based on the well-known principle that the strength of lumber is positively correlated with the modulus of elasticity (MOE), the Continuous Lumber Tester (CLT) was developed as a lumber grading machine. The machine measures the force required to bend a member under center point loading to a prescribed deflection. The M O E of the piece is estimated and a grade is assigned accordingly. In this type of machine, boards are fed into the machine flat-wise in the longitudinal direction. The individual board is continually deflected to a certain amount by rolls in both the up and down directions. The amount of force required to reach this certain deflection is converted to an M O E estimate of the piece. Basically, one half of the machine is the duplicate of the other half, except that the force sensors for each half are placed in the opposite sides (upper and bottom) of the lumber (see Figure 1-4). Hence, two sets of M O E will be produced and the values will be averaged. The M O E is recorded continuously along the length of each piece (every 4-foot span). The machine measures both the average E of the whole board and the lowest E of the piece and uses this information to assess the strength of each board. Researchers found different coefficients of determination between M O E and M O R using a similar machine. Green and Rosales reported coefficients of determination of 0.21 and 0.26 for Ramon and Danto [75]. They also reported 0.15 for mixture of Danto and Ramon. Green and MacDonald reported coefficients of determination 0.31, 0.46, 0.25, 0.30, 0.52, 0.56, and 0.53 for Red Maple, Northern Red Oak, Yellow-Poplar, Aspen-Cottonwood, Southern Pine, Douglas Fir-Larch, and Hem-Fir, respectively [76]. Saboksayr used a neural network based system to estimate the strength of lumber. In this method, 16 statistical features were extracted from M O E profiles and by using a learning system, a neural network was trained to predict the strength of the lumber. A coefficient of determination of 0.56 was achieved in this study. He also used same approach to estimate the strength of lumber using 16 statistical features extracted from X-Ray images. He achieved a coefficient of determination of 0.40 using neural network and ten fold cross correlation. He also trained another neural network using both M O E and X-Ray statistical features and reached a coefficient of determination of 0.61 [31],[56]. Chapter 4 65 The overall objective is to find mechanics-based features from the X-ray image and M O E profiles of a board, to enhance the estimation of the tensile strength of the lumber and reducing the number of strength predicting features needed to predict the strength. 4.1 Experimental Methods A mixed system of M O E and X-Ray used to estimate the strength of a board, using only one combined feature extracted from M O E profiles and X-Ray image. Real Board X-Ray Projector X-Ray Image Geometric Feature Extracting Processor CLT Machine Geometric Features MOE Profiles Feature Extracting Processor Feature Extracting Processor Stress Field FEM Processor Strength Predicting Feature Amalgamator Strength Estimate Figure 4-1, Block diagram of a system using MOE and X-Ray for predicting the strength of a board The M O E part of the system uses the output of the CLT machine which contains two profiles (top and bottom). Due to lumber curvature, one profile may have higher value than the other one. By averaging the two profiles this effect will be compensated. Based on past results, we postulate that a board will break in the area where the minimum M O E of the averaged profile Chapter 4 66 is located. However, by considering the grip length required for tension testing, 15% of the beginning part and the end part of each profile separated and dumped. The remaining part (from 15% to 85%) is the base for calculating the minimum value. One example is depicted in Figure 4-2. We named this feature AveMinCutl5_feature, and this method AveMinCutl5 Algorithm. x 1 0 Average MOE profile of CLT machine for board # 2 1 0 20 70 80 30 40 50 60 board length in percentage Figure 4-2, Average MOE profile of CLT machine for board #210 after dumping 15 from both ends The X-Ray part of the system uses a mechanics-based system for estimating the strength of a board. It is the same as the ones described in Chapters 2 and 3. Boards were passed through an X-Ray scanning system and geometrical features were extracted from the X-Ray images. The geometrical features were fed to the F E M processor. Then proper loads were applied to the two ends and boundary conditions were applied to the F E M model. The F E M formulation results in a system of equations representing the differential equations that govern the mechanics of the continuum. By applying the proper loads and boundary conditions, the Chapter 4 system of equations was then solved numerically to obtain a resultant stress field. The stress field was used to calculate the strength predicting features. Figure 4-3,2-D image of the transverse (top) and Longitudinal (bottom) stress field around one edge knot. The maximum stress concentration for longitudinal (bottom) stress happen at 90 degrees angle and it's value for this case is 3.65. The maximum stress concentration for transverse (top) stress happens at almost 60(or 120) degrees and it's value for this case is 0.41. The stress (strain and displacement) fields yielded different features for strength prediction. Two features namely "Maximum Longitudinal Stress Concentration" (Eq. 10 ) and Weibull based feature (Eq. 11) which give good correlations to actual strength of board were extracted to be used in an amalgamation processor [58] [59]. MSC Feature - Maximum _ Longitudinal _ Stress Average_Stress ^ Chapter 4 68 S Sa Weibull _ Based _ Feature • A ( 1 1 ) where S is the stress inside the small area A A (area of the corresponding element in which the average stress is S). Sa is the average stress over the total area A , when there is no abnormality (knot) present. The calculations were repeated for different values of n (1, 2, 3, 3.5, 4, 5, 6, and 7); however, n=3.5 was used in this study because of better correlation. Then four different algorithms were used to estimate the strength of the lumber. The first two algorithms were based on the maximum stress theory for the failure of a board; the strength of the board is calculated as follows: T„, T = sound wood M S C where T is the tensile strength of a board; T s o u n d W 0 O d is the tensile strength of a knot free board, and M S C is the maximum stress concentration. By replacing " T s o u n d W O O d " with "CFl*AveMinCutl5_feature" and " M S C " to "MSC_Feature + 5.7" we arrived at Eq. 12. For the sake of easier referencing we named Eq. 12 Algorithm 1 (or A l ) . „ . , _ . ^ AveMinCutf5_feature Estimated_ Strength_ Al = CFI * MSC _Feature+5.1 ( 1 2 ) where CFI is a calibration factor which can be calculated by tension testing to destruction a few samples (for this study CF1= 0.00015). Then CFI is corrected such that the error ( |Ttru e -Testimated |) becomes minimum. The MSC_Feature for a value less than 2 is considered to be 2. This correction is based on the assumption that a knot-free board can be approximated by a board with a very small knot to avoid a singularity in our algorithm. The value 5.7 in the second equation is to linearize the transformation. From Chapter 2 we know the experimental relationship between M S C and true strength is close to linear. Then, we can say a close to linear A l is better than highly non-linear A l . Considering the MSC_Feature may vary between 2 to 9, the 1/ MSC_Feature graph is very far from a straight line (top curve at Figure 4-4). By changing to l/MSC+5.7, it looks more like a straight line. The 5.7 is almost the middle of the M S C range (5.5=[9+2]/2). The resulting estimate of strength was not highly Chapter 4 69 dependent on a particular value of the linearization parameter which could be selected in the range for 4 to 8. Comparing 1/{MSCFeature5) and 1/(MSCFeature+5.7) ° , 0-5 co 0.4 ™ 0.3 + o to 0.2 0.1 n r n r i i 1/MSCFeature5 1/(MSCFeature5+5.7) 0 1 2 3 4 5 6 7 8 9 10 MSC Feature 5 Figure 4-4, Comparing 1/MSC and l/MSC+5.7 functions for their similarity to a linear function In the same way we arrive at Eq. 13 by replacing "T s o u n d wood" to "CF2*AveMinCutl5_feature" and " M S C " to "Weibull_Based_Feature + 0.45". For the sake of easier referencing we name Eq. 13 Algorithm 2 (or A2). ^ . , n i >^ ^T-^± AveMinCutl5_feature Estimated _ Strength _ Al = C r 2 * Weibull _ Based _ Feature + 0.45 ^ ^ where; CFI and CF2 are calibration factors (for this study CF1= 0.00015 and CF2=0.00001). In algorithm 1(A1), the MSC_Feature for a value less than 2 is considered to be 2, in the same way in A2, the Weibull_Based_Feature for a value less than 0.075 is considered to be 0.075. This correction is based on the assumption that a knot-free board can be approximated Chapter 4 70 by a board with a very small knot to avoid the singularity in our algorithms. The values 5.7 and 0.45 in eq. 12 and 13 are used to linearize the transformation. By using linear and second order regression two features (Weibull_Based_feature and AveMinCutl5_feature) were amalgamated in algorithm 3 and algorithm 4 respectively. The similar investigation was done in Section 2.3 for the MSC_Based_feature_5 and the results for linear and second order regression (and also higher orders) were summarized in Table 6. As was concluded there was not any significant increase in higher order regression for the MSC_Based_feature_5. This is why we only examined the Weibull_Based_feature at this stage by algorithm 3 (linear regression) and algorithm 4 (second order regression). Estimated Strength _ A3 = ( 1 4 ) AveMinCutl 5_F*(C1+C2 * Weibull _B_F) v ' This approach needs linear regression; however, if we rewrite the equation as follows: Estimated Strength A3 = ~ ( 1 5 ) CF3 * AveMinCutl5_F * (F3Max - Weibull_ B_F) where CF3 is a calibration factor and F 3 m a x is the maximum value of the feature. A board with this maximum value has a very low strength (weak board). It can be calculated numerically by calculating the value for a board with the largest knot possible. This will reduce calibration time (because you only need to calibrate one factor compared to two factors) when it is implemented in a real machine. By using second order regression, algorithm A4 was implemented for estimating the strength of lumber as follows: Estimated _ Strength _ AA - ( 16 ) AveMinCutl5_F * (C, + C 2 * Weibull _B_F + C 3 * Weibull _B_F2) Coefficients C i , C2, and C 3 were calculated using second order regression. 4.2 Resu l ts More than 1000, 38 [mm] by 89 [mm] (2"by 4") boards were sampled from interior British Columbia's SPF (Spruce, Pine, Fir). They were air dried to reach to the 12% moisture Chapter 4 71 content. Then X-Ray images and M O E profiles of them were measured and a database was established. Then all boards underwent destructive testing and their breaking strengths were measured. By applying the AveMinCutl5 Algorithm to a database of more than 1000 boards, a coefficient of determination of 0.56 was achieved. The results with a second order regression curve are depicted in Figure 4-5. AveMinCut15-feature vs Actual Strength with II order Regression curve 0 I i i i i i i I 0 0.5 1 1.5 2 2.5 3 3.5 AveMinCut15-feature[MPa] Q 6 Figure 4-5, Average-Minimum-15 % cut both end-feature vs actual strength of a database of more than 1000 boards with second order regression curve with a coefficient of determination of 0.56. By using a linear regression we achieved the same coefficient of determination of 0.56 for strength estimation as AveMinCutl5 feature. By using a second order regression we achieved a coefficient of determination of 0.58. Using higher order regression 3, 4, 5 and 10 gives coefficients of determination of 0.58, 0.58, 0.58, and 0.58 respectively. The results support the second order regression predictor is a reasonable estimator for M O E profile. By using a transformation as follows based on second order regression: AveMinCutRegll = 5.3839 *10"12AveMinCutl52 -1.0078 * 10"6AveMinCutl5 + 6.8444 ( 17) The results are depicted in Figure 4-6. Chapter 4 72 70 60 50 ro £ 40 CD (_> < 20 10 AveMinCutl5Regll-feature vs Actual Strength 10 20 30 40 AveMinCutl5Regll-feature [MPa] 50 60 Figure 4-6, Average-Minimum-15% cut both end-feature and transformed based on II order regression (estimated strength) vs actual strength of a database of more than 1000 boards with a coefficient of determination of 0.58. For the next step, the AveMinCutl5 feature is used in combination with X-Ray image extracted features to predict the strength of the boards. By applying the A1,A2, A3, andA4 algorithms to a data-base of more than 1000 boards, using both X-Ray and M O E extracted features, coefficients of determination of 0.64, 0.65, 0.65, and 0.65 were achieved for A l , A2, A3, and A4 respectively. The results for algorithm A l to A4 are depicted in Figure 4-7 to Figure 4-10. Chapter 4 7 3 70 60 h 50 CD CL £ 40 CO 20 10 Mixed Signal of M O E and X-Ray using algorithm A1 10 20 30 40 50 Estimated Strength [MPa] 60 70 Figure 4-7, Estimated Strength using a mixed signal of MOE and X-Ray for algorithm Al with a coefficient of determination of 0.64. 70 60 50 CD CL & 40 30 20 10 Mixed Signal of MOE and X-Ray using algorithm A2 10 20 30 40 50 Estimated Strength [MPa] 60 70 Figure 4-8, Estimated Strength using a mixed signal of MOE and X-Ray for algorithm A 2 with a coefficient of determination of 0.65. Chapter 4 74 Mixed Signal of M O E and X-Ray using algorithm A3 20 30 40 Estimated Strength [MPa] 60 Figure 4-9, Estimated Strength using a mixed signal of MOE and X-Ray for algorithm A3 with a coefficient of determination of 0.65. Mixed Signal of M O E and X-Ray using algorithm A4 20 30 40 Estimated Strength [MPa] 60 Figure 4-10, Estimated Strength using a mixed signal of MOE and X-Ray for algorithm A4 with a coefficient of determination of 0.65. Chapter 4 75 By using a linear (algorithm 3) and second order (algorithm 4) regression we achieved the same coefficient of determination of 0.65. 4.3 C o n c l u s i o n Using the described algorithms [78] [79] to estimate the strength of lumber compare to works done before has two major benefit. First by using only one mixed feature (compare to 32 in Neural network based system) to estimate the strength, it will reduce the cost of fine tuning and make it more practical for use in industry. Secondly, it provides a better estimation (a higher coefficient of determination 0.65 compared to 0.61). The other benefit of this algorithm is that straightforward and easy to implement in industry compared to neural network based system, if we replace the F E M processor. The next chapter deals with replacing the F E M processor with a knowledge-based system for this reason. Chapter 5 76 5. Real Time System for Estimating the Strength of Lumber For implementing a real time system to predict the strength of lumber it is necessary to replace the F E M module in the previous strength predicting system, which was introduced in previous chapters. By replacing the F E M processor by a knowledge based processor, based on a look up table or other approximation system, it was expected that one could estimate the strength in a reasonably short time. 5.1 K n o w l e d g e B a s e d L o o k up T a b l e fo r R e p l a c i n g F E M The first step in replacing the F E M processor is creating a "look up table" for different knot sizes and different distances from the edge of a board. Since the biggest knot size in our database is 35 [mm], a range of 1 [mm] to 43 [mm] with increments of l[mm] is used for the knot radius( named radi). The boards we used, have widths of 89 [mm] then we vary the distance of the center of each knot from the one side (named edge) from 0 [mm] to 89[mm]. Because of symmetry, edge=0,l,2..,43, 44 is the same as edge=89,88,87,...,46,45 (e.g. edge=SS is the same as edge=l) , we changed edge from 0 to 44.5 (center of the board), and calculate the MSC for each edge values. The values for 45 to 89 edge value are equal to values from 44 to 0. ____ Knot I I I O^Radius Figure 5-1, Model used for creating the "look up table" 5.1.1 Look up Table Using "Mesh Size 5" and Breaking the Narrow Part at "0.5* radi" Mesh size 5 and breaking the narrow part at "0.5 * radi" are the same condition we used in previous algorithms (see Figure 2-11) to estimate the strength as in Chapter 2. The results for W=89 [mm] Chapter 5 77 the whole range of knot radius and distance from edge are depicted in Figure 5-2. The results for half range is depicted the other half is symmetrical. Look up table using Mesh size 5, and breaking the narrow part at 0.5* radi Radius of Knot 0 0 Distance from the Edge Figure 5-2, Look up table with mesh size 5 and breaking at "0.5 * radi" By inspecting the results, it can be seen that the look up table contains a significant amount of noise. The reason for the noise is that the mesh size is not very fine. A slice of the results for a 10 [mm] radius knot in different distances from edge is depicted in Figure 5-3. Chapter 5 78 Look up table for knot radius of 10 o CO 4.5 = 4 CD O o O 0 ) (f) a i CO 3.5 X 2.5 0 5 10 15 20 25 30 35 40 45 Distance from the Edge Figure 5-3, An slice of look up table at radius of 10 [mm] with mesh size 5 and breaking at "0.5 * radi" As it can be seen from the Figure 5-3 , there is error due to mesh size and numerical error (the curve should have a peak between 10 [mm] and 10 [mm]*1.5=15 [mm]). We still can see the edge knots (distance from the edgelO to 20 [mm]) have higher M S C than center knots (distance from the edge 35 to 45 [mm]). However, the desired range for the high M S C is a narrower range such as 10 to 15 [mm] for edge knots. For accomplishing this we need to break the narrow part as early as we can such as 0.2 radi. Note that a very small amount such as 0.05 radi will produce very high MSC which is not desirable and can not describe the real board model. 5.1.2 Look up Table Using Mesh Size 1 and Breaking the Narrow Part at "0.2* radi" In the next step, we used the finest possible mesh size to reduce the error. We also change the breaking algorithm, by breaking the narrow part at 0.1, 0.2, 0.3, and 0.4 radi only for 10[mm] Chapter 5 79 radius knot to see which is better representing a board model. From the results we chose 0.2 radi. The results for the whole range of knot radius and distance from edge (for 0.2 radi) are depicted in Figure 5-4. By inspecting the results, it can be seen that the look up table, with mesh size 1 and breaking at "0.2 * radi", contain a lower amount of noise compared to look up table with mesh size 5 and breaking at "0.5 * radi". We can conclude that the reason for the noise in the look up table, with mesh size 5 and breaking at "0.5 * radi", was the mesh size itself. A slice of results for a 10 [mm] radius knot in different distance from edge is depicted in Figure 5-5. As it was mentioned in previous section (see Figure 2-11), if we do not break the narrow part a high (infinite) MSC will occur at 10 [mm] (where edge is equal to radi). As it can be seen in the slice, the results have a high MSC in 10 to 15 [mm] range which is desirable and describe the real board more accurately. Chapter 5 80 Look up table for knot radius of 10 15 20 25 30 35 Distance from the Edge 45 Figure 5-5, An slice of look up table at radius of 10 [mm] with mesh size 1 and breaking the narrow part at "0.2 * radi" As we can see, there is still a small amount of noise which will be reduced more in the next section. 5.1.3 Look up Table after Passing through Low Pass Filter We also decreased the noise by passing the look up table through a low pass filter (moving average by the length of 5 cells). The filter was passed through each slice with a constant knot diameter. We also tried a 5 by 5 low pass mask (and 2D filtering) but the results were not desirable. The resulting look up table is depicted in Figure 5-6. Chapter 5 81 A slice of results for a 10 [mm] radius knot in different distance from edge is depicted in Figure 5-7. As it can be seen the results are very desirable and most of the noise is filtered out. Chapter 5 82 Look up table for knot radius of 10, after filtering 35 40 15 20 25 30 Distance from the Edge Figure 5-7, An slice of look up table at radius of 10 [mm] after filtering 45 In the next section we use the resulting "look up table" to estimate the strength of boards. 5.2 Estimating the Strength Using a Knowledge Based Processor In this section we implement and examine two systems containing a knowledge based processor in the form of a "look up table". It also means that these systems do not contain an F E M processor. It also means that this system can be realized as a real-time system to estimate the strength of boards in lumber mill. 5.2.1 X-Ray Based Knowledge Based Processor for Estimating the Strength A system containing the described look up table was implemented to estimate the strength of boards. X-Ray images of boards were passed through a geometric feature extractor and the calculated geometric features were then fed to the knowledge based processor. The output of the knowledge based processor is the maximum "maximum stress concentration" (MSC). It Chapter 5 83 can be transverse, longitudinal, local, or global as described in previous chapters. Then the MSC data were fed to the strength estimator to calculate the strength of the board. The block diagram of the system is depicted in Figure 5-8. Real Board X-Ray Geometric Feature Extracting Processor iGeometricj features Knowledge based Processor Max MSC Strength Estimator Estimated Strength Figure 5-8, Block diagram a knowledge based system to estimate the strength of a board. We also examined another possibility for using the Weibull based features. We used the summation of all values calculated from the look up table for each knot in the board. We named this method the "Summation Algorithm" and the previous one the "Max Algorithm". Geometric Feature Extracting Processor Geometric features Knowledge based Processor Summation of features E Real Board X-Ray —• Strength 9 • Estimator Figure 5-9, Block diagram a knowledge based system to estimate the strength of a board using summation of features for all knots in the board. The system was improved by adding M O E profiles to it. The next section will describe the combined system of M O E and X-Ray and knowledge Based Processor for Estimating the strength of boards. 5.2.2 M O E , X-Ray and Knowledge Based Processor for Estimating the Strength A system containing the described "look up table" was implemented to estimate the strength of boards using both X-Ray images and M O E profiles. X-Ray images of boards were passed through a geometric feature extractor and the calculated geometric features were then fed to the knowledge based processor (look up table). Output of the knowledge based processor is the estimate of the "maximum stress concentration" (MSC). It can be transverse, longitudinal, local, or global as described in previous chapters. At the same time as the boards were passed, a C L T machine and their M O E profile were measured and fed to the feature extractor to calculate the "Average-Minimum-15% cut both ends" feature which is described in Chapter 4. Chapter 5 84 Real Board X-Ray Projector X-Ray Image Geometric Feature Extracting Processor CLT Machine Geometric! Features MOE Profiles Feature Extracting Processor Average-Minimum-15% cut Feature MSC Knowledge based Processor Strength Predicting Feature Amalgamator Strength Estimate Figure 5-10, Block diagram a MOE and X-Ray based "knowledge Based Processor" for estimating the strength Then the M S C together with M O E feature were fed to strength predicting feature amalgamator to calculate the strength of the board which X-Ray and M O E profiles were fed in the first place. The block diagram of the system is depicted in Figure 5-10. 5.3 Resul ts The results for the threshold of 1.25 for three algorithms were examined. For the sake of briefness we chose feature 5 from Chapter 2, Weibull feature with n=3.5 from Chapter 3 and features correspond to Algorithms A l , to A4 from Chapter 4 (all 4 in one graph).The first try was based on a look up table with mesh size 5 and breaking at "0.5 * radi" for the narrow part. The second try was based on a look up table with mesh size 1 and breaking at "0.2 * radi" for narrow part. The last try is filtered database of second try. Chapter 5 85 The results for the first part "look up table" with mesh size 5 and breaking at "0.5 * radi" for narrow part are as follows: Table 8, Coefficient of determination and correlation for different features using mesh size 5 and breaking "0.5 * radi" Correlation r Coefficient of determination r 2 MSC-Feature 5 -0.4514 0.20 Weibull feature with n=3.5 corresponding to feature 5 -0.4886 0.24 A l 0.7491 0.56 A2 0.7283 0.53 A3 0.7443 0.55 A4 0.7445 0.55 The result for feature 5 based strength estimator is depicted in Figure 5-11. Estimated Strength vs Actual Strength with Regression Line Estimated Strength calculated from Feature 5 [MPa] Figure 5-11, Actual Strength versus Estimated Strength using knowledge base system with mesh size 5 and breaking of narrow part at "0.5 * radi" and MSC feature 5 with coefficient of determination of 0.20. Chapter 5 86 The results for algorithms A l to A4 are depicted in Figure 5-12. Note that algorithm A l which is based on feature 5 has a good mechanics based justification. However; A2, A3, and A4 algorithms are based on Weibull features (using the "Max Algorithm" for the knowledge based system), and the value for only the most significant knot calculated and other knots are ignored. Note that in "Summation Algorithm" all knots were considered. 20 40 Estimated Sj^ngth [MPa] 60 20 40 Estimated Strength [MPa] 60 20 40 60 Estimated Strength [MPa] 20 40 60 Estimated Strength [MPa] Figure 5-12, Actual Strength versus Estimated Strength using "knowledge based system" with mesh size 5 and breaking of narrow part at "0.5 * radi" and Algorithm Al to A4 and MOE profiles with coefficient of determination of 0.56,0.53, 0.55, and 0.55 respectively . The results for the second try, using a look up table with mesh size 1 and breaking at "0.2 * radi" for narrow part are listed in Table 9. There are significant increases in feature 5 and Weibull feature correlations compared to previous try. We calculated the confidence interval for the coefficient of determination r 2 for MSC, Weibull, A l , A2, A3, and A4 features and algorithms (see Table 9) [83],[84]. We also calculated the "Standard Error Estimate" for the "estimated strength mean value" using the correlation r. The "estimated strength mean value" (using algorithm A2) was 27.03 with the "Standard Error Estimate" (using correlation r) of ±6.13. For the same set, the Chapter 5 87 "measured strength mean value" was 24.84 with the "Standard Error Estimate" of ±6.83 [85],[86]. Table 9, Coefficient of determination (with confidence interval) and correlation for different features using mesh size 1 and breaking 0.2 using Max Algorithm Correlation r 7 Coefficient of determination r (with confidence interval) MSC-Feature 5 -0.6662 0.44 ± 0.06 Weibul l feature with n=3.5 corresponding to feature 5 -0.6780 0.46 ± 0.06 A l 0.8116 0.66 ± 0.05 A 2 0.8199 0.67 ± 0.05 A 3 0.8157 0.67 ± 0.05 A 4 0.8192 0.67 ± 0.05 The results for feature_5 based strength estimator are depicted in Figure 5-13. Estimated Strength vs Actual Strength with Regression Line 70 I 1 1 1 1 1 1 1 1 r Estimated Strength calculated from Feature 5 [MPa] Figure 5-13, Actual Strength versus Estimated Strength using knowledge base system with mesh size 1 and breaking of narrow part at "0.2 * radi" and MSC feature 5 with coefficient of determination of 0.44. Chapter 5 88 The results for algorithms A l to A4 are depicted in Figure 5-14. Note that the coefficient of determination and correlation are increased significantly. 0 20 40 Estimated Sjrjmgth [MPa] 0 20 40 60 Estimated Strength [MPa] 60 CL 40 ZD 20 0 20 40 60 Estimated Strength [MPa] 0 0 20 40 60 Estimated Strength [MPa] Figure 5-14, Actual Strength versus Estimated Strength using knowledge based system with mesh size 1 and breaking of narrow part at "0.2 * radi" and Algorithm Al to A4 and MOE profiles with coefficient of determination of 0.66, 0.67, 0.67, and 0.67 respectively . The next try was based on the filtered look up table. The result for feature_5 based strength estimator is depicted in Figure 5-15. The coefficient of determination increased slightly from 0.44 to 0.45. This increase is not significant. The same way for A l algorithm, coefficient of determination stayed the same (0.66). Other filters also examined but the results were not better. Since A2 is mechanics based using the Weibull-based feature (A3 and A4 are regression analyses which are done previously by other researchers, we use them for comparison with our methods) the slope of regression line for A2 algorithm was also calculated. For a confidence of 99%, a value 1.01± 0.02 was obtained for A2 algorithm. This interval contains the desired slope of 1 between estimated and actual strength. Chapter 5 89 Estimated Strength vs Actual Strength with Regression Line Estimated Strength calculated from Feature 5 [MPa] Figure 5-15, Actual Strength versus Estimated Strength using filtered knowledge base system and MSC (in X-Ray based stress field, feature 5) with coefficient of determination of 0.45 The results for A l algorithm using filtered look up table is depicted in Figure 5-16. The last try was based on the "Summation Algorithm". In the summation method, all knots MSC and Weibull values calculated and add up to generate the output value for M S C feature_5 and Weibull feature. For example a board with 2 knots with the MSC_feature_5 value 3 and 5 respectively will output 8 using the "Summation Algorithm" (output will be 5 for "Max Algorithm"). This method were implemented to investigate the Weibull based feature and strength predicting algorithm A2, A3, and A4 which are based on Weibull feature. We wanted to see how much summation method will change the strength estimation, and if it is significant. Chapter 5 90 70 60 50 CO •_ J C 40 cu I "55 u < 30 20 10 Mixed Signal of MOE and X-Ray using algorithm A1 1 1 1 1 t • 10 20 30 40 50 Estimated Strength [MPa] 60 70 Figure 5-16, Actual Strength versus Estimated Strength using "filtered knowledge base system" and Algorithm Al based MSC (in X-Ray based stress field , feature 5) and MOE profiles with coefficient of determination of 0.66 The results for summation algorithm are summarized in Table 10. For comparison with the "Max Algorithm" the other condition is the same as in Table 9. Table 10, Results for "Summation Algorithm" suitable for Weibull based features for the case mesh size 1 and breaking "0.2 * radi" together with "Max Algorithm" results for comparison Coefficient of determination r 2 Summation Agorithm Max Algorithm MSC-Feature 5 0.23 0.44 Weibull feature with n=3.5 corresponding to feature 5 0.48 0.46 A l 0.43 0.66 A2 0.64 0.67 A3 0.66 0.67 A4 0.66 0.67 Chapter 5 91 A s it can be seen only coefficient of determination for Weibul l feature slightly increased. We expect this because Weibul l feature is based on the whole board. However, this increase does not occur for features based on the algorithm A 2 , A 3 , and A 4 which use the Weibul l feature. Weibull Based Feature vs Actual Strength 70 I 1 1 1 1 1 1 6Tji-0 1 ' ' ' 1 1 ' ' 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Weibull Based Feature Figure 5-17, Actual Strength versus Weibull based feature using knowledge base system with mesh size 1 and breaking of narrow part at "0.2 * radi" with coefficient of determination of 0.48 5.4 Conclusion From results of this chapter we can conclude that using a knowledge based system to replace the F E M processor is practical without losing accuracy. It also provides us a tool to implement a real time system to estimate the strength of lumber [80]. The results raised the intriguing a question that by considering only the highest M S C knot we reach the same as considering all knots. Is only highest M S C knot is the only reason for breakage in boards? How many percentage of breakage is due to the highest M S C knot? To answer these questions we implement a system to estimate the location of breakage which is described in the next chapter. Chapter 6 92 6. Estimating the Location of the Breakage of a Board By implementing a real time system without the need for an F E M module, it opens an opportunity to investigate the location of breakage by using the "maximum stress concentration" theory of failure. 6.1 Experimental Methods 1080 boards underwent to different scanning instruments and their MOE, and X-Ray signals were recorded as described in the previous chapters. Then all boards went through tensile strength testing until complete breakage. The strength of each tested board was recorded (This information was used in the previous chapters). Two more measurements were also performed; the location of beginning and the end of each breakage. For example, in board #0001, fracture starts from 4180 [mm] and ends at 4330[mm]. 6.1.1 X-Ray Based Fracture Location Estimator First a system based on X-Ray images of a board was implemented to estimate the location of the point that the fracture occurs. If this point is located between the start and the end of each breakage then we can say that we estimate the breakage point correctly. Real Board X-Ray Geometric Feature Extracting Processor Geometric] features Knowledge Based System MSC location Based on The first Maximum Knot Location of Fracture Figure 6-1, Block diagram of a system for estimating the location of a fracture based on the first maximum MSC knot and using a knowledge based processor instead of FEM processor Boards were passed through the X-Ray sensing machine and their geometric features were extracted from their images as described in Chapter 2. Then the geometrical features were fed to the "knowledge based processor" to predict the location of the center of the knot that contains the maximum M S C . This module was also described in previous chapters for calculating the value of M S C whereas in this chapter the location of the corresponding knots (with highest MSC) is calculated. This procedure was repeated for a database of more than Chapter 6 93 thousand boards and the location of the "maximum M S C knot" for each board was calculated. Then a measure named the "hit factor" was introduced as depicted in Figure 6-3. The start and end of each fracture were measured and stored from a destructive test named as "Start Fracture" (SF) and "End Fracture" (EF). Because of the slippage in the scanning device and the measurement accuracy, a tolerance (named T) was introduced (see Figure 6-2). Tolerance (T) •^7 V Breaking Line x ^ & "7 4 -'"~c^c^ e Tolerance Figure 6-2, Top view of a part of the Board near the breaking line The location of the fracture was calculated from the location of the center of the knot causing the maximum M S C and named the "Location of Fracture" (LF). Then if this L F was located between the SF-T and EF+T the value for hit factor is 1 and 0 elsewhere (see Figure 6-3). Chapter 6 94 Figure 6-3, Flow chart of determining the hit factor, T is the tolerance Then this procedure was repeated for the location of the "second highest M S C knot". This procedure is the same as Figure 6-1 and Figure 6-3 except that the "second highest knot" location is replaced with the "highest knot" location. We named the location of the "second highest knot" as LF2 and the corresponding hit factor as "Hit_Factor_LF2". In addition to "Hit_Factor_LF" and "Hit_Factor_LF2" another hit factor defined as depicted in Figure 6-4 . We named this as "Hit_Factor_l&2". This means if the real fracture located either in the "highest MSC knot" or "second highest M S C knot". The logic for calculating this factor is the possibility of breakage of a board either in the first or the second maximum stress concentration due to the different wood properties in the longitude of the board. Chapter 6 95 'Location of Fracture based on maximum M S C End of fracture Location of Fracture based on the second maximum M S C (LF2) Start of fracture Figure 6-4, Flow chart of determining the "Hit_Factor-l&2", T is the tolerance 6.1.2 M O E Based Fracture Location Estimator Secondly, a system based on M O E was implemented to estimate the location of the point that the fracture happened. M O E profiles contain profile PI and P2. Due to bow (curvature) in boards usually PI and P2 are different. By averaging these two profiles PI and P2, profile "Pm=(Pl+P2)/2" is calculated. Chapter 6 96 X IQ6 Average M O E profile of CLT machine for board #210 30 40 50 60 70 board length in percentage 90 Figure 6-5, Location of the minimum value for the MOE profiles PI, P2 and Pm at 63%, 73% and 80% of length of the board #210 (pointed by arrows) Then 15% from both ends of the profile Pm were cut and the rest is used for the calculation of location of the breakage. Then the location of minimum value in Pm was calculated as the location of the fracture and named the "LFPm" (see Figure 6-6), MOE Profiles Real (P1,P2) Pm=(P1+P2)/2 Board w Cutting from 15% to 85% Location of Minimum Value of Pm Location of Fracture (LFPm) • Figure 6-6, An MOE based system for estimating the Location of fracture using average MOE profiles "Pm=(Pl+P2)/2" Then the Location of Fracture "LFPm" is used for calculating the "Hit_Factor_Pm" which is depicted in Figure 6-7. Chapter 6 97 Figure 6-7, Hit factor for MOE profile Pm, T is the tolerance This procedure was also repeated for the PI and the P2 alone. The procedure is exactly the same as depicted in Figure 6-6 and Figure 6-7 except changing Pm to PI for calculating LFP1 and "Hit_Factor-Pl" and the same way for P2 , LFP2 and "Hit_Factor-P2". It is also calculated the hit factor based on either PI or P2 and named "Hit_Factor_PlP2". The procedure is similar as shown in Figure 6-4 except changing L F to LFP1 and changing LF2 to LFP2. Finally a "hit factor" is calculated based on either PI or P2 or Pm and named "Hit_Factor_PlP2Pm". The procedure is shown in Figure 6-8 . Chapter 6 98 Start of fracture Location of Fracture based on minimum M O E profile Pm=(Pl+P2)/2 End of fracture Location of Fracture based on minimum M O E profile PI Location of Fracture based on minimum M O E profile P2 Figure 6-8, Hit factor for the MOE profiles Pm, PI or P2, T is the tolerance Chapter 6 99 6.2 Results For the "X-Ray image based hit factor", the "maximum MSC knot" algorithm produced 46% of the correct hits (considering 5 [cm] tolerances). For the "second highest MSC knot" it went to 31%. The both first and the second "highest MSC knot" algorithm produced 56% (see Table 11 and Figure 6-11). Hit Factors for X-Ray detected Knots 1.2 -1 " 2 0.8 £ 0.6 | 0.4 0.2 0 i 0 100 200 300 400 500 Tolerance [cm] Figure 6-9 Hit factor for the X-Ray based "highest M S C knot" and the "second highest M S C knot" and both the first and the second highest knots For the " M O E based hit factor", the minimum Pm algorithm produced 20% of correct hits (considering 5 [cm] tolerance). For the same tolerance, the minimum point for PI and P2 algorithms gave almost the same results of 18% (and 19%). "Either PI or P2 algorithm" gave 29%. "Either P1,P2 or Pm algorithm" gave 33% (Table 11 and Figure 6-10). Chapter 6 100 Hit Factor for MOE profiles 0 1 1 1 1 1 0 100 200 300 400 500 Tolerance [cm] Figure 6-10, Hit Factor for MOE profiles The results for all cases are depicted in the Figure 6-11 and written in Table 11. Table 11, Hit Factors for the X-Ray and the MOE based signals Tolerance First knot Second Knot Both knot Pm= (P1+P2)/2 P1 P2 P1 or P2 P1 or P2 or Pm 0 0.36 0.27 0.44 0.16 0.14 0.14 0.22 0.26 1 0.40 0.28 0.48 0.17 0.14 0.15 0.23 0.28 2 0.43 0.30 0.53 0.18 0.15 0.16 0.25 0.29 5 0.46 0.31 0.56 0.20 0.18 0.19 0.29 0.34 10 0.47 0.33 0.57 0.27 0.22 0.23 0.37 0.42 20 0.49 0.36 0.60 0.38 0.32 0.31 0.48 0.55 50 0.56 0.46 0.72 0.50 0.45 0.45 0.63 0.69 100 0.69 0.59 0.83 0.62 0.61 0.61 0.77 0.83 200 0.88 0.85 0.95 0.83 0.85 0.83 0.93 0.94 300 0.97 0.98 0.99 0.96 0.97 0.96 0.99 0.99 400 1 1 1 1 1 1 1 1 490 1 1 1 1 1 1 1 1 Chapter 6 101 A s can be seen, X - R a y based hit factors are better than the M O E based hit factors for predicting the location of breakage of lumber. Hit Factor 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 .—• *"* i i , I 1 1 1 i c irst knot Second knot -1 T i r c r T 1 — — — - - - - - - . + Both Knots Pm=(P1+P2)/2 — P1 P2 P1 orP2 - - P1 or P2 or Pm L _ X if 1 UF 1 1 1 1 _ 1 1 1 | r r T 1 1 1 1 1 1 I 100 200 300 400 Tolerance [ c m ] Figure 6-11, Hit factor for all the cases 500 6.3 C o n c l u s i o n From the results we can conclude that the " X - R a y based hit actors" ( 5 6 % for both knots) can predict the location of fractures better than " M O E based hit factors" (34% for P I , P2 or Pm). The results also show that the presence of knots play a significant role in the generation of fractures. It also says that other factors besides knots play a role in the fracture mechanism. Chapter 7 102 7. Algorithm Alternatives Looking at the different algorithms, there are some possibilities to improve the strength estimation in general. This is the reason for this chapter. 7.1 Maximum Diameter Algorithm In Chapter 2, a 2D analysis was carried out that considered the center of area of.knot part of the X-Ray image as the center of the knot and an average (the maximum circle is calculated that can be fitted totally inside the segmented area and containing the center of the segment). For easy comparison we named the previous algorithm in Chapter 2 used for detecting the knot's geometry as "Center of Area Algorithm". Figure 7-1, Center of Area Algorithm The question we wanted to answer in this section is what if we somehow analyze the problem in 3D. Is there anyway to do this in 2D space with some acceptable accuracy? For answering to this question, we first investigate the most likely geometry of real knots. For our analysis we look at knots as through and non-through knots. Figure 7-2, A schematic of through (left) and non-through (right) knots Chapter 7 103 Then we divided the board to many horizontal sections. Then if the section was thin enough we can analyze each section and calculate the stress field. Figure 7-3,3D analysis using slices If we only need the maximum stress then we do not need to analyze all slices. From previous experience, we know that the slice with the largest knot portion will create the maximum stress for the knot. Since the flat-wise X-Ray image will contain this information we can find the maximum stress slice by finding the maximum circle that fits inside the X-Ray image. Figure 7-4, Maximum circle fit inside the X-Ray image We named this algorithm, the "Maximum Diameter Algorithm". An example using both algorithms ("Maximum Diameter Algorithm" and "Center of Area Algorithm") is illustrated in Table 12. Chapter 7 104 Table 12, Detected knots for board #2 using two different algorithms X coordinate of knot center Y coordinate of knot center Radius of knot Maximum Diameter Algorithm 2859.0 60.0 7.6 Center of Area Algorithm 2860.0 59.0 7.3 Note that knots which are smaller than 5 [mm] in diameters are ignored. The thresholded and digitized X-Ray image of board#2 near the detected knot is depicted in Figure 7-5. 2900 | | | | | | | | | | | | | | | l | | | [ | [ | | | i m 2890 j i j j j j j j j j j j j i j i j j j j j j j j j j j j j j i j j 2880 i j j I j j i j j j j j i j j j j j i j j j j j i J E E E E i ;:EEE:i:EiE;iEE:E:i:E:! 2860 ::j::::jj::::pF"i\ME!MMMMMMMME!MMiMME!M1 EE!EEEEE: 2850 r^ffl fTTITnTtTl^^ff M; M M M M M M M M H M M ^  M M M M M; M M=; M M E 1: H M j 2840 liiHiiHiMiiMMiiiiiiiiM 2 83 0 l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l[ l l l l l l^ 90 80 70 60 50 40 30 20 10 0 Figure 7-5, Flat-wise X-Ray image of board #2 near the detected knot after thresholding and digitizing 7.2 Erosion Algorithm The question we wanted to answer in this section is what if we use a different method to extract the geometry. Do we get a better result? Erosion is one of the fundamental morphological operations (the other one is dilation). Erosion removes pixels on object boundaries. The number of pixels removed from the objects in an image depends on the size and shape of the "structuring element" used to Chapter 7 105 process the image. The effect of erosion in an image is that the areas with knot pixels shrink in size, and holes within those areas become larger. Since there is no hole in the knot X-Ray image, the only effect of erosion will be the shrinking of the knot image. Since the X-Ray image is two-dimensional (2D), the structuring element used is 2D or flat [81], [82] . Flat or 2D, a structuring element consists of a matrix of 0's and I's, typically much smaller than the image being processed. The center pixel of the structuring element, called the "origin", identifies the pixel of interest. The pixels in the structuring element containing I's define the "neighborhood" of the structuring element. Many different forms of structuring elements are used in image processing such as; Diamond; 0 1 0 1 1 1 0 1 0 Square; 1 1 1 1 1 1 1 1 1 The best structuring element is circle [81]. However, it is difficult to obtain a circular shape. We notice that if we use a radius 5 structuring element then we can have almost a circular structuring element. 0 0 0 0 0 ] L 0 0 0 0 0 0 0 1 1 1 1 L 1 1 1 0 0 0 1 1 1 1 1 L 1 1 1 1 0 0 1 1 1 1 ] I 1 1 1 1 0 0 1 1 1 1 ] L 1 1 1 1 0 1 1 1 1 1 ] I 1 1 1 1 1 0 1 1 1 1 ] I 1 1 1 1 0 0 . 1 1 1 1 ] I 1 1 1 1 0 0 1 1 1 1 ] L 1 1 1 1 0 0 0 1 1 1 ] L 1 1 1 0 0 0 0 0 0 0 ] L 0 0 0 0 0 Chapter 7 106 Since 52=32+42, a radius 5 structuring element will have equal distance in 12 directions (bold face l's) out of 28 possible directions. It looks more like a circle than any smaller radius structuring element. Then for each pixel of the X-Ray image the origin of a structuring element is located on it and the minimum of the neighborhood elements (all the board pixels covered by the structuring element) will be the new value for the board pixel corresponding to the origin. This procedure is repeated for all pixels in the X-Ray image. The resulting new X-Ray image will be an eroded image after one time erosion. The X-Ray thresholded image has many zero values; then for the pixels with zero value the new value will be zero and there is no need to check the entire board. It will increase the speed of the process. The erosion process is repeated until the whole X-Ray image becomes zero. After every erosion, the knot center and radius are calculated as in the algorithm in Chapter 2. Because the first erosion will decrease the diameter of knots 5 [mm] then 5 [mm] was added to the detected knots. In the same way for knots detected in second and third ... erosion 10[mm] and 15 [mm] ... were added to detected knots. It will be obvious that for the knots detected in the last erosion, the same knot will be detected in previous erosions. We looked at the knots detected in and around the same location and we chose the one that was detected with the higher number of erosions. 7.3 Cutting 15% from Both Ends The "Ultimate Strength Tester" in the University of British Columbia (UBC) Wood Products Laboratory was used to obtain the actual tensile strength of each board. This machine has two grips which hold both ends of a board. One of the grips is fixed and the other grip is moved by a pneumatic system until the board breaks completely. Chapter 7 107 73 {cm] Grip 65 [cm] Grip B o a r d G r i p Grip /77 Figure 7-6, Top view of Ultimate Strength Tester in University of British Columbia (UBC) Wood Products Laboratory The measured pressure of the pneumatic system multiplied by the effective cross section gives us the applied tensile force and stress (by dividing the force by cross section area of the board). The maximum measured tensile stress was recorded as the actual tensile strength of each board. Due to the grip length of the machine which is 65 [cm] and 8 [cm] from the end of each board, 73 [cm] of both ends of each board did not undergo the applied stress. By calculating (73/490=0.15) the percentage, 15% of the both ends of each board do not affect the breaking strength. Hence, we examined this effect by making a physical model with knots from 15% to 85% of the length of the lumber. The rest of the algorithm is the same as the previous chapters. 7.4 H u m a n a n d E n v i r o n m e n t Fac to rs There is a possibility that human and environment factors affected the results. I decided to divide the sample in two parts. The first half, around 500 boards is the ones which were tested from the beginning to the middle of the tests. The second half is the rest, from the middle to the end of the tests. I assumed that the people who do the test become more experienced and that the second half will give better results. The most important environmental factor can affect the results is the boards' humidity. The boards were air dried to 12% moisture content. However, I expected the second half of the boards have lower moisture content than the first half. I expected that the lower moisture content boards gives better results, due to the fact that knots are more effective in dry wood compared to wet ones. Chapter 7 108 7.5 Results First, the results for the threshold of 1.25 for four different algorithms were examined and compared with the one in previous chapters. The first try was based on the "Maximum Diameter Algorithm" described in Section 7.1. The second try based on the "Maximum Diameter Algorithm" and "Erosion Algorithm" which described in Section 7.2. The third try based on the "Maximum Diameter Algorithm" and "15% cut from both ends Algorithm" as it described in Section 7.3. The fourth try based on the combination of "Maximum Diameter Algorithm", "Erosion Algorithm" and "15% cut of both ends Algorithm". For the sake of brevity we chose feature-5 from Chapter 2, the Weibull feature with n=3.5 from Chapter 3 and features corresponding to Algorithms A l , to A4 from Chapter 4 (all 4 in one graph). The results for the "Maximum Diameter Algorithm" are depicted in Table 13, and Figure 7-7. Table 13, Results for the Maximum Diameter Algorithm Coefficient of Coefficient of determination r determination r From previous results MSC-Feature 5 0.42 0.39 Weibull feature with n=3.5 from feature 5 0.47 0.49 A l 0.64 0.65 A2 0.65 0.66 A3 0.65 0.66 A4 0.65 0.66 Chapter 7 109 20 40 60 Estimated S g n g t h [MPa] 20 40 60 Estimated Sj^ngth [MPa] 20 40 Estimated Strength [MPa] 60 20 40 60 Estimated Strength [MPa] Figure 7-7, Results for the Maximum Diameter Algorithm The results for the "Maximum Diameter Algorithm" and "Erosion Algorithm" are depicted in Table 14 and Figure 7-8. Table 14, Results for the Maximum Diameter Algorithm and Erosion Algorithm Coefficient of Coefficient of determination r 2 determination r From previous results MSC-Feature 5 0.42 0.39 Weibull feature with n=3.5 from feature 5 0.47 0.47 A l 0.64 0.64 A2 0.65 0.66 A3 0.65 0.66 A4 0.65 0.66 Chapter 7 no Figure 7-8, Results for the Maximum Diameter Algorithm and Erosion Algorithm The results for the Maximum Diameter Algorithm and 15% cut Algorithm are depicted in Table 15 and Figure 7-9. Table 15, Results for the Maximum diameter Algorithm and 15 % cut Algorithm Coefficient of determination r 2 From previous results Coefficient of determination r 2 MSC-Feature 5 0.42 0.34 Weibull feature with n=3.5 corresponding to feature 5 0.47 0.47 A l 0.64 0.63 A2 0.65 0.66 A3 0.65 0.66 A4 0.65 0.66 Chapter 7 111 A1 A2 Estimated Strength [MPa] Estimated Strength [MPa] Figure 7-9, Results for Maximum diameter Algorithm and 15% cut Algorithm The results for the "Maximum Diameter Algorithm", "Erosion Algorithm" and "15% cut Algorithm" are depicted in Table 16 and Figure 7-10. Table 16, Results for the Maximum diameter, Erosion and 15% cut algorithm Coefficient of determination r From previous results Coefficient of determination r 2 MSC-Feature 5 0.42 0.39 Weibull feature with n=3.5 corresponding to feature 5 0.47 0.47 A l 0.64 0.64 A2 0.65 0.65 A3 0.65 0.65 A4 0.65 0.66 Chapter 7 112 A1 0 20 40 60 Estimated Strength [MPa] A2 0 20 40 60 Estimated Strength [MPa] Figure 7-10, Results for the Maximum Diameter Algorithm, Erosion Algorithm and 15% cut Algorithm Table 17, Results for the first half, and the second half compared with ones in the Chapters 2,3, and 4 (all samples) _— Coefficient of determination r First Half Second Half A l l Samples MSC-Feature 5 0.32 0.49 0.42 Weibull feature with n=3.5 corresponding to feature 5 0.34 0.57 0.47 M O E Based Feature AveMinCutl 5RegII 0.56 0.59 0.58 A l 0.56 0.69 0.64 A2 0.59 0.71 0.65 A3 0.58 0.71 0.65 A4 0.58 0.71 0.65 Considering the coefficient of determination 0.41 for Feature-5 from Chapter 2 , 0.47 for Weibull-based feature from Chapter 3, and 0.63 to 0.65 for A l to A4 algorithms from Chapter 7 113 Chapter 4, there is no significant change in results in using any above algorithms. Note that these results produced using the same F E M processor. Finally, the results for the "human and environment factors" are depicted in Table 17, Figure 7-11, Figure 7-12, Figure 7-13, and Figure 7-14. 70, Feature 5 vs Actual Strength 60 50 40 f 30 J-20 10 5 Feature 5 Figure 7-11, MSC feature 5 for the second half of the samples compared to the one in Chapter 2, Figure 2-25 (r2 of 0.49 versus 0.42) Chapter 7 114 Weibull Based Feature vs Actual Strength (0 CL CO 0.2 0.3 0.4 0.5 Weibull Based Feature 0.6 0.7 Figure 7-12, Weibull based feature for the second half of samples compared to the one in Chapter 3, Figure 3-3 (r2 of 0.57 versus 0.47) AveMinCut15Regll-feature vs Actual Strength 70 60 50 h 40 30 20 V 10 10 20 30 40 AveMinCut15Regll-feature [MPa] 50 60 Figure 7-13, Results for the second half of the samples compared to the one in Chapter 4, Figure 4-6 (r2 of 0.59 versus 0.58) Chapter 7 115 As can be seen there is a very significant increase in r from 0.42 to 0.49 for the MSC feature and 0.47 to 0.57 for the Weibull based feature (10% absolute equal to 20% relative) in the second half of sample results. Other features also had a significant increase. There are three main sources for this change. First there is possibility the boards belong to two different regions or species. The second reason may lie in the change of moisture content of boards during the storing period. Dry lumber is more knot-sensitive. Finally, people who record the data become more experienced over time and the second half is less prone to human errors. 0 20 40 60 Estimated Sjijength [MPa] 0 20 40 60 Estimated Strength [MPa] Estimated Strength [MPa] 0 20 40 60 Estimated Strength [MPa] Figure 7-14, Results for second half of the samples with coefficient of 0.69,0.71,0.71, and 0.71 for algorithms Al to A4 respectively The results for the second half of the samples corresponding to Chapter 5 are depicted in Table 18, and Figure 7-15. As it can be seen a coefficient of 0.73 has been reached using A2 algorithm. The corresponding Figure 7-15 (top right) shows a typical cigar shape of distribution. Chapter 7 116 Table 18, Results for the second half of samples (mesh 1 and breaking at 0.2 and no filtering) compared to the one in Chapter 5 Coefficient of determination r 2 From Chapter 5 Coefficient of determination r 2 MSC-Feature 5 0.44 0.55 Weibull feature with n=3.5 corresponding to feature 5 M S C alg. 0.46 0.56 Summation Alg. 0.48 0.57 A l 0.66 0.71 A2 0.67 0.73 A3 0.67 0.71 A4 0.67 0.72 0 20 40 60 Estimated Sjrjmgth [MPa] 0 20 40 60 Estimated Strength [MPa] 0 20 40 60 Estimated Strength [MPa] 0 20 40 60 Estimated Strength [MPa] Figure 7-15, Results for second half of the samples (compared to results in Chapter 5 (mesh 1 and breaking at "0.2 * radi" and no filtering)) Chapter 7 117 7.6 C o n c l u s i o n Using maximum diameter, erosion, and 15% cut methods does not increase the coefficient of determination. We also tried different threshold levels and different methods for calculating threshold level. Different features were also defined and tried. However, none of these methods result in a better estimation. As can be seen there is a very significant increase in the coefficient of determination in the second half of the samples results compared to using all samples due to the probable sources of errors. Chapter 8 118 8. Conclusions and Future Study 8.1 Conclusions This thesis has demonstrated that a physical model can be used as the basis for calculating board strength and that this approach yields results that are at least equivalent to results obtained from estimation-based methods (such as regression, functional approximation, or neural networks) requiring a training set of representative boards (from one or more species, different harvesting sites, a range of knot configurations etc.). By using a deterministic Non-Destructive Lumber Grading system we reduced the cost of tuning by reducing the calibration factors to only one (or two) clear-wood parameters. The "Maximum Stress Concentration" (MSC) as a result of the presence of knots plays an important role in the strength of a board. It was found that using only X-Ray-derived "longitudinal maximum stress concentration" provides a feature that is well correlated (r = 0.42) with the experimentally-measured board strength, and could be used to provide a measure of the board strength through a simple linear transformation. A Weibull-based feature was also introduced and used to estimate the strength of lumber. It is a better estimator of strength of lumber than the MSC-based feature. The application of the Weibull-based feature by including all points in the entire stress field, improves the estimation of the strength of lumber (r2 = 0.47) compared to the strength estimation with MSC-based features alone. A mixed system using the "Modulus of Elasticity" (MOE) and X-Ray was also used to estimate the strength of a board (r2 = 0.72), using only one combined feature extracted from M O E profiles and the X-Ray image. It has two major benefits compared to the previous work [31],[56]. First by using only one mixed feature (compared to 32 in the neural network based system of [31],[56]) to estimate the strength, it reduced the cost of fine tuning and can make it more practical for use in industry. A knowledge based system can provide approximately the same estimate of board strength as an F E M processor and is capable of doing so in real-time. It is practical without losing accuracy. A system was developed to estimate the location of breakage in tension. From the results one can conclude that X-Ray based features can predict the location of fractures 56% of the time Chapter 8 119 (for both knots, Chapter 6, Table 11 and Figure 6-11) compared to M O E based features only 34% of the time (for PI, P2 or Pm, Chapter 6, Table 11 and Figure 6-11). The results also show that the presence of knots plays a significant role in the generation of fractures. It also shows that other factors (such as wood local density, humidity, wane, split, resin canal, holes, shakes, check, 3D grain angle,...) besides knots play a role in the fracture mechanism. This study shows that both mechanics and the Weibull based system are very accurate for strength estimation of a board under tension. 8.2 Future Directions One suggestion for increasing the accuracy is to add other components to the strength estimation system. One added component could be the "Slope of Grain" (SOG) information. However, the present measuring system does not produce the required resolution. There is another way to reach the needed resolution. In this way we can photograph the board surface and resulting image can be processed to obtain the slope of grain with the needed resolution. The other added component can be ultrasound information. There is a possibility of correlating the ultrasound with the density and M O E (or other mechanical characteristics). Then we may able to evaluate the sound wood M O E locally. This information can be added to the described systems in this thesis to increase the coefficient of determination. 120 References [I] National Lumber Grades Authority, Canadian Lumber Grading Manual, 1989, Vancouver, B.C., Canada. [2] National Lumber Grades Authority, N L G A Standard Grading Rules for Canadian Lumber, 1991, Vancouver, B.C., Canada. [3] U.S. Department of Agriculture, "Machine Stress Rating: Practical Concerns for Lumber Producers," USDA Forest Service General Technical Report, FPL 7. [4] National Lumber Grades Authority, SPS 2-94: N L G A Special Products Standard for Machine Stress-rated Lumber and Machine Evaluated Lumber, 1994, Burnaby, B.C., Canada. [5] Machine Grading Of Lumber, Practical Concerns for Lumber Prducers, William L. Galligan, Kent A. McDonald, General Technical Report FPL-GTR-7, Madision, WLU.S. Department of Agriculture, Forest Service, Forest Products Laboratory, September 2000, pp 1-39. ( also available at http://www.fpl.fs.fed.us/documnts/fplgtr/fplgtr7.pdf) [6] Bending Stress in the Metriguard Model 7200 HCLT and the CLT, James D. Logan, Metriguard Inc. PO Box 399, Pullman W A 99163.(also available at http://www.metriguard.com/bendstrs.htm) [7] "Computermatic Product Information Sheet,", Plessey Machine Co., Seattle, W A , USA. 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M . , "Economic Feasibility of Strength and Stiffness Prediction of M E L and MSR Lumber," Forest Product Journal, 1997, v. 47 (11/12), pp. 8 5 - 9 1 . [51] Ross, R.J., Brashaw, B.K. , and Pellerin, R.F., "Nondestructive Evaluation of Wood," Forest Products Journal, 1998, v. 48(1), pp.14 - 19. [52] Lam, F., Varoglu, E., "Effect of Length on the Tensile Strength of Lumber," Forest Product Journal, May 1990, v. 40 (5), pp. 37 - 42. [53] Pham, D.T., and Alcock, R.J., "Automated Visual Inspection of Wood Boards: Selection of Features for Defect Classification by a Neural Network", Proceedings of the Institution of Mechanical Engineers' Part E-Journal of Process Mechanical Engineering, 1999,213: (E4) pp. 231 -245. [54] The M A T H WORKS Inc., " M A T L A B Neural Networks Toolbox User Manual," 1999. [55] Bishop,C.M., "Neural Networks for Pattern Recognition", Oxford University Press, November 1995. [56] Saboksayr H.S., Saravi A . A. , Lawrence P. 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D., Lam F., "Identifying Strength of Boards using Mechanical Modeling and a Weibull-Based Feature ", Conference on Control Applications, IEEE-CCA2003, Turkey, pp 54-59, June 23-25, 2003. [78] Saravi A . A. , Lawrence P. D., Lam F., "Implementation of a Mechanics Based System for Estimating the Strength of a Board Using Mixed Signals of M O E and X-Ray Images ", IEEE-PACRIM 2003 conference, Victoria, BC. Canada, pp 413-417, August 28-30, 2003. [79] Saravi A . A. , Lawrence P. D., Lam F., "Identifying the Strength of Boards using Mixed Signals of M O E and X-Ray Image", IEEE-ISPA2003 conference, Rome, Italy, pp 1003-1008, September 18-20, 2003. [80] Saravi A . A. , Lawrence P. D., Lam F., "Real-Time-Intelligent System for Estimating the Strength of Lumber using X-Ray Image", The 4th IASTED International Conference on Visualization, Imaging, and Image ProcessingCVTJP 2004), Marbella, Spain, pp 31-36, September 6-8, 2004. [81] Nikolaidis N . , Pitas I., 2001, "3-D Image Processing Algorithms", John Wiley & Sons, Inc., pp 65-79. [82] M A T L A B help in The Math Works [on line] available at: http://www.mathworks.com/access/helpdesk/help/toolbox/images/morph3.shtml. [83] HyperStat Online Contents [on line] available at: http://davidmlane.com/hyperstat/B8544.html [84] Spiegel, M . R., Stephens, L . J., 1998, "Theory and Problems of Statistics", Schaum's Outline Series, McGraw-Hill, p 337. [85] "Correlation and Regression", Dr Johannes M . Zanker web page [on line] available at: http://www.pc.rhul.ac.uk/zanker/teach_previous/PS101/regress/sld013.htm [86] Armitage, P., Berry, G., Matthews, J.N.S., 2002," Statistical Methods in Medical Research", Forth Edition, Blackwell Scientific Publications. 125 Appendix I: Correlation Confidence Intervals Since the sampling distribution of correlation ( r ) is not normally distributed, correlation ( r ) is converted to Fisher's z' and the confidence interval is computed using Fisher's z'[83]. The values of Fisher's z' in the confidence interval are then converted back to r's. For our work, we want to construct a 99% confidence interval on the correlation between Measured Strength and Estimated Strength (Using different Methods). We obtained data from 1080 boards chosen at random. We found the r value of 0.67, 0.68, 0.81, 0.82, 0.82, 0.82 for different features of MSC-Feature 5, Weibull feature, A l , A2, A3, A4 respectively. Here the calculation for r=0.82 (r2 =0.67) for the A2 method is explained first, then Confidence Interval for the other was computed and put in table form. The first step in computing the confidence interval is to convert 0.82 to a value of z' using the r to z' table [App. III]. The value is: z' = 1.1568. The sampling distribution of z' is known to be approximately normal with a standard error of: 1 V i V - 3 where N=1080 is the number of pairs of scores. Then the value is: 6z'=l/V(1077)=0.03047. From the general formula for a confidence interval, the formula for a confidence interval on z' is: Z'±ZaZ' = Z'±Z . 1 V N - 3 A z table [App. II] can be used to find that for the 99% confidence interval one should use a z of 2.5762. For N = 1080, the standard error of z' is 0.03047. The confidence interval for z' can be computed as: Lower limit = 1.1568 - (2.5762.)( 0.03047) = 1.0783 Upper limit = 1.1568 + (2.5762.)( 0.03047) = 1.2353 126 Using the r to z' table [App. Ill] to convert the values of 1.0783 and 1.2353 back to Pearson r's, it turns out that the confidence interval for the population value of Pearson's correlation (r) is: 0.79 <r< 0.84 Therefore, the correlation between Measured Strength and Estimated Strength (Using A2 Method) is highly likely to be somewhere between 0.79 and 0.84. Converting to r 2 the results will be: 0.62 < r 2 =0.67 < 0.71. Results for other features are: N=1080 SD=0.03047 Z=2.5762 r 2 r Z'-SD*z SD*z= 0.0785 Z'+ SD*z Confidence Interval for P=99% Z' min r max min r2 max MSC-Feature 5 0.44 0.67 0.7322 0.8107 0.8892 0.62 0.67 0.71 0.38 0.44 0.50 Weibull feature 0.46 0.68 0.7506 0.8291 0.9076 0.64 0.68 0.72 0.41 0.46 0.52 A1 0.66 0.81 1.0485 1.1270 1.2055 0.78 0.81 0.84 0.61 0.66 0.71 A2 0.67 0.82 1.0783 1.1568 1.2353 0.79 0.82 0.84 0.62 0.67 0.71 A3 0.67 0.82 1.0783 1.1568 1.2353 0.79 0.82 0.84 0.62 0.67 0.71 A4 0.67 0.82 1.0783 1.1568 1.2353 0.79 0.82 0.84 0.62 0.67 0.71 There is another way to solve the problem, using the direct equation for the z' table [84]: Z ' = 1 . 1 5 1 3 I o _ 1 0 ^ 1- r We recalculated the same example. The first step in computing the confidence interval is to convert 0.82 to a value of z' using the r to z' equation. The value is: z' = 1.1568. The rest the same except at the end, to convert the values of 1.0783 and 1.2353 back to Pearson r's, using the same equation in z' to r form: 1 0 # - i 5 i 3 _ i r = y 10M.1513 + 1 By using the z' to r equation we reach the value 0.7926, 0.8441 which is almost the same as using the table (0.79 and 0.84). Appendix II: Z Table Z can be found from normal-curve area. C o n f i d e n c e % Z 1 i n f i n i t e 99 . 9 9 9 9 9 9 9 9 9 6 . 8 0 6 1 9 9 . 9 9 9 9 9 9 9 9 6 . 4 6 6 5 9 9 . 9 9 9 9 9 9 9 6 . 1 0 9 9 9 . 9 9 9 9 9 9 5 . 7 3 0 4 9 9 . 9 9 9 9 9 5 . 3 2 6 4 9 9 . 9 9 9 9 4 . 8 9 1 4 9 9 . 9 9 9 4 . 4 1 7 1 9 9 . 9 9 3 . 8 9 0 6 9 9 . 9 3 . 2 9 0 8 9 9 . 7 3 3 . 00 9 9 . 5 2 . 8 0 7 4 99 2 . 5 7 6 2 98 2 . 3 2 6 8 97 2 . 1 7 0 5 96 2 . 0 5 4 2 9 5 . 4 5 2 . 00 95 1 . 9 6 0 4 90 1 . 6 4 5 2 80 1 . 2 8 1 7 70 1 . 0 3 6 4 68 . 2 7 1 . 0 0 60 0 . 8 4 1 5 50 0 . 6 7 4 5 40 0 . 5 2 4 30 0 . 3 8 4 9 20 0 . 2 5 2 9 10 0 . 1 2 5 4 0 0 128 Appendix III: Correlation r to Fisher's z' Table r z ' r z ' 0.0000 0 . 0000 0.5000 0.5493 0.0100 0.0100 0.5100 0.5627 0.0200 0 . 0200 0.5200 0.5763 0.0300 0 . 0300 0.5300 0.5901 0.0400 0 . 0400 0.5400 0.6042 0.0500 0 . 0500 0.5500 0.6184 0.0600 0 . 0601 0.5600 0.6328 0 . 0700 0 . 0701 0.5700 0.6475 0.0800 0.0802 0.5800 0.6625 0 . 0900 0.0902 0.5900 0.6777 0.1000 0.1003 0 . 6000 0 . 6931 0.1100 0.1104 0 . 6100 0 .7089 0.1200 0.1206 0 . 6200 0 .7250 0.1300 0.1307 0.6300 0.7414 0.1400 0.1409 0.6400 0.7582 0.1500 0.1511 0.6500 0.7753 0.1600 0.1614 0 . 6600 0 .7928 0.1700 0.1717 0.6700 0.8107 0.1800 0.1820 0.6800 0.8291 0.1900 0.1923 0.6900 0.8480 0.2000 0.2027 0.7000 0.8673 0.2100 0.2132 0 .7100 0.8872 0.2200 0.2237 0 .7200 0.9076 0 .2300 0.2342 0 .7300 0.9287 0 .2400 0.2448 0 .7400 0.9505 0 .2500 0.2554 0.7500 0.9730 0 .2600 0.2661 0 .7600 0.9962 0 .2700 0.2769 0.7700 1.0203 0 .2800 0.2877 0.7800 1.0454 0 .2900 0.2986 0.7900 1.0714 0 .3000 0.3095 0.8000 1.0986 0 .3100 0.3205 0.8100 1.1270 0 .3200 0.3316 0 . 8200 1.1568 0 .3300 0.3428 0.8300 1.1881 0 .3400 0.3541 0.8400 1.2212 0.3500 0.3654 0.8500 1.2562 0.3600 0.3769 0.8600 1.2933 0.3700 0.3884 0.8700 1.3331 0.3800 0.4001 0.8800 1.3758 0.3900 0.4118 0.8900 1.4219 0.4000 0.4236 0.9000 1.4722 0.4100 0.4356 0.9100 1.5275 0 .4200 0.4477 0 . 9200 1.5890 0 .4300 0.4599 0.9300 1.6584 0.4400 0.4722 0.9400 1.7380 0 .4500 0.4847 0 . 9500 1.8318 0.4600 0.4973 0.9600 1.9459 0.4700 0 . 5101 0.9700 2.0923 0.4800 0.5230 0.9800 2.2976 0.4900 0.5361 0.9900 2.6467 Using the following equation the r to z' table was constructed. 1 + r Z' = 1.1513Log 10 1-r 

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