1. Joshua I. Cohen Brown University September 2001 – May 2002 “Computational Procedures For Extracting Landmarks In Order To Represent The Geometry Of A Sherd”

2. Introduction  Project  Process - Feature extraction - Reconstruction of the original 3D object using the extracted features  Motivation

3. Data Collection  Sherd - ShapeGrabber - Polyworks -.mat files

4. Data Collection Cont’d  Breakcurve  Breakcurve Algorithm (Xavior Orriols) 1. Subdivides sherd into smaller planes recursively starting from centroid 2. Singular value decomposition  3 orthogonal vectors 3. Project points into 2D plane 4. Find edge points.  Breakcurve Algorithm Pitfall

5. Data Collection Cont’d .iv files (Dongjin Han)

6. Polynomial Approximation of Curves Containing High Curvature Points  Design Decision  f2D Software  Two Cases To Consider 1. Polynomial Approximation of a High Curvature Segment 2. Polynomial Approximation of a Breakcurve

7. Polynomial Approximation of a High Curvature Segment Gradient-1Gradient-1 R.R. Degree 4 Degree 11

8. Polynomial Approximation of a Breakcurve Gradient-1Gradient-1 R.R. Degree 4 Degree 11

9. Advantages of Corners  Good Landmarks  Segmentation of Breakcurve  Better Representation Locally  Lower Degree Polynomial Fit (3 or 4) - Computation Time - Stability

10. Landmark Extraction Algorithm  Pre-Processing 1. Find Normals on Breakcurve - Patch - Eigenvector Associated With Minimum Eigenvalue - Check Direction S = M 1 + M 2 + M 3 + … + M k 2. Order Breakcurve Points

11. Landmark Extraction Algorithm  Corner Detection 1. Concatenate Breakcurve 2. Polynomial Degree One Fitting To Approximate Tangent Vectors - t R and t L - 2 Planes: ax + by + cz + d = 0  x +  y +  z +  = 0 - Eigenvectors associated with 2 smallest eigenvalue - Normalize 3. Ensure Tangent Vectors have the Right Direction Line of Intersection of 2 Planes = [a,b,c] x [ , ,  ] Distance Positive  Distance Negative

12. Landmark Extraction Algorithm  Corner Detection Cont’d 4. Compute Angle - goodness of fit

13. Landmark Extraction Algorithm  Corner Detection Cont’d 5. Find Local Minimum Angles (Corners) - smaller than angle of neighbors - smaller than angle threshold Angle Threshold: 145 degreesAngle Threshold: 135 degrees

14. Landmark Extraction Algorithm  Computing Curvature Extrema 1. Segment Breakcurve at Corners 2. Project Breakcurve into 2D using Local Projection - Global vs Local - B mid, N mid - Rotated perpendicular to [1,0,0], x components are 0

15. Landmark Extraction Algorithm  Computing Curvature Extrema Cont’d 3. Gradient-1 2D Curve Fitting of Projected Breakcurve Segments - ipfit_5.3.0 - gradient-1 and gradient-1 ridge regression w/specified degree - ipfit_5.3.0 vs f2D Degree = 3

16. Landmark Extraction Algorithm  Computing Curvature Extrema Cont’d 4. Compute Curvature of Projected Breakcurve Segments - Obtain points on g(x,y) in [B xmin, B xmax, B ymin, B ymax ] - Order according to contour - Compute Curvature

17. Landmark Extraction Algorithm  Computing Curvature Extrema Cont’d 5. Find Curvature Extrema of Projected Breakcurve Segments - Minima: K < 0, K < Neighbors, K < Threshold - Maxima: K > Neighbors, K > Threshold Threshold = 0.012

18. Landmark Extraction Algorithm  Computing Curvature Extrema Cont’d 6. Obtain Landmarks by Combining Curvature Extrema of Projected Breakcurve Segments with the Corners

19. Analysis of Results  Curvature Extrema Not Always Accurate  Problems 1. The Polynomial Fit is Not Always Very Good 2. Points on g(x,y) are Approximate Without Any K Threshold

20. Conclusion  Correct Curvature Problems - f2D software, g(x,y) = 0 and dotprod(  T,K) = 0  Corners match for p6ed and p10ed  Groundwork of Landmark Detector Established

21. References 1. Linear Algebra and Its Applications, Gilbert Strang, International Thomson Publishing, 3 rd edition, 1988. 2. Numerically Invariant Signature Curves, Mireille Boutin 3. Numerical Recipes http://www.nr.com/http://www.nr.com/ 4. Numerical Recipes in C: The Art of Science, William H. Press, Cambridge University Press, 2nd edition, 1993 5. Scientific Computing An Introduction With Parallel Computing, Gene Golub, Academic Press, 1993 6. Wolfram Research http://mathworld.wolfram.com/http://mathworld.wolfram.com/ Thanks to the following people for all their help: Professor David Cooper Dr. Mireille Boutin Andrew Willis