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Biological Shape

Matching Shapes Serge Belongie * , Jitendra Malik and Jan Puzicha U.C. Berkeley * Present address: U.C. San Diego. Biological Shape. D’Arcy Thompson: On Growth and Form , 1917 studied transformations between shapes of organisms. Deformable Templates: Related Work.

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Biological Shape

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  1. Matching ShapesSerge Belongie*, Jitendra Malikand Jan Puzicha U.C. Berkeley*Present address: U.C. San Diego

  2. Biological Shape • D’Arcy Thompson: On Growth and Form, 1917 • studied transformations between shapes of organisms

  3. Deformable Templates: Related Work • Fischler & Elschlager (1973) • Grenander et al. (1991) • Yuille (1991) • von der Malsburg (1993)

  4. Matching Framework ... model target • Find correspondences between points on shape • Estimate transformation • Measure similarity

  5. Comparing Pointsets

  6. Shape Context Count the number of points inside each bin, e.g.: Count = 4 ... Count = 10 • Compact representation of distribution of points relative to each point

  7. Shape Context

  8. Comparing Shape Contexts Compute matching costs using Chi Squared distance: Recover correspondences by solving linear assignment problem with costs Cij [Jonker & Volgenant 1987]

  9. Matching Framework ... model target • Find correspondences between points on shape • Estimate transformation • Measure similarity

  10. Thin Plate Spline Model • 2D counterpart to cubic spline: • Minimizes bending energy: • Solve by inverting linear system • Can be regularized when data is inexact Duchon (1977), Meinguet (1979), Wahba (1991)

  11. MatchingExample model target

  12. Outlier Test Example

  13. Synthetic Test Results Fish - deformation + noise Fish - deformation + outliers ICP Shape Context Chui & Rangarajan

  14. Matching Framework ... model target • Find correspondences between points on shape • Estimate transformation • Measure similarity

  15. Terms in Similarity Score • Shape Context difference • Local Image appearance difference • orientation • gray-level correlation in Gaussian window • … (many more possible) • Bending energy We used the weighted sumShape context + 1.6*image appearance + 0.3*bending

  16. Object Recognition Experiments • COIL 3D Objects • Handwritten Digits • Trademarks • Exactly the same algorithm used on all experiments

  17. COIL Object Database

  18. Error vs. Number of Views

  19. Prototypes Selected for 2 Categories Details in Belongie, Malik & Puzicha (NIPS2000)

  20. Error vs. Number of Views

  21. Handwritten Digit Recognition • MNIST 600 000 (distortions): • LeNet 5: 0.8% • SVM: 0.8% • Boosted LeNet 4: 0.7% • MNIST 60 000: • linear: 12.0% • 40 PCA+ quad: 3.3% • 1000 RBF +linear: 3.6% • K-NN: 5% • K-NN (deskewed): 2.4% • K-NN (tangent dist.): 1.1% • SVM: 1.1% • LeNet 5: 0.95% • MNIST 20 000: • K-NN, Shape Context matching: 0.63%

  22. Trademark Similarity

  23. Conclusion • Introduced new matching algorithm matching based on shape contexts and TPS • Robust to outliers & noise • Forms basis of object recognition technique that performs well in a variety of domains using exactly the same algorithm

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