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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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