Shape Representation and Similarity

1. Stretching Stretching Shape Representation and Similarity Occlusion Articulation Shape similarities should be preserved under occlusions and articulation. Two equal stretching along contours can affect shape similarity differently. Shapes can be represented by contours, or by a set of interior points, or other ways. What is the shape representation that can be best used for shape similarity? Computer Vision

2. Symmetry Axis Representation Shape Shape Axis (SA) SA-Tree Computer Vision

3. Shape Representation via Self-Similarity Based on work of Liu, Kohn and Geiger • A variationalshape representation model based on self-similarity of shapes. • For each shape contour, first compute its shape axis then derive a unique shape-axis-tree (SA-tree) or shape-axis-forest (SA-forest) representation. Shape Contour Shape Axis (SA) SA-Tree Computer Vision

4. counterclockwise clockwise The Insight • Use two different parameterizations to compute the represention of a shape. • Construct a cost functional to measure the goodness of a match between the two parameterizations. • The cost functional is decided by the self-similarity criteria of symmetry. Computer Vision

5. Parameterized Shapes • Two different parameterizations: counterclockwise clockwise • When the curve is closed we have • Matching the curves is described as the match of functions Computer Vision

6. A Global Optimization Approach • We seek “good” matches over the possible correspondences Computer Vision

7. Similarity criteria: Symmetry Mirror Symmetry Co-Circularity Computer Vision

8. A Global Optimization Approach Computer Vision

9. Constraints on the form of F Computer Vision

10. Constraints on the form of F (cont.) Computer Vision

11. Similarity criteria: Symmetry Computer Vision

12. Similarity criteria: Symmetry II (a) (b) (c) (d) Computer Vision

13. Summary: Cost Functional/Energy Density • Structural properties • Symmetric • Parameterization Independent • Geometrical properties • Translation invariant • Rotation invariant • Scale “almost” invariance • Self-Similarity properties Computer Vision

14. A Dynamic Programming Solution Computer Vision

15. bifurcation ! tt ! … ! ! ! ! bifurcation ! 1-tt A Dynamic Programming Solution The recurrence is on the “square boxes”. Each box is either linked to one inside square box or linked to a pair of inside and well aligned square boxes (leading to bifurcations). Assumption, that will not be needed later: The final state is a match (0,0). 1-s ! ! ! ! ! t+1/N ? t s s+1/N 1-t Computer Vision

16. A Dynamic Programming Solution 1-s tt t+1/N t t ! ! … ! ! ! ! ! ? s s 1-tt s+1/N 1-t Computer Vision

17. A Dynamic Programming Solution t A few more steps … Starting states (matching candidates). All corresponding to “self-matches” iteration direction s Computer Vision

18. A General Dynamic Programming Solution t Extend the parameters s and t such that negative values x  s,t=1+x. s x Extended Graph x No need to force a match at (0,0) or any particular node/match. Starting states (matching candidates). All corresponding to “self-matches” Ending states Computer Vision

19. A Dynamic Programming Solution Computer Vision

20. Shape Shape Axis SA-Tree Experimental Results for Closed Shapes Computer Vision

21. Shape Shape Axis (SA) SA-Tree Experimental Results for Open Shapes (the first and last points are assumed to be a match) Computer Vision

22. Experimental Results for Open Shapes Shape Axis (SA) SA-Forest Computer Vision

23. Matching Trees with deletions and merges for articulation and occlusions Computer Vision

24. Experimental Results for Open Shapes Computer Vision

25. Convexity v.s. Symmetry White Convex Region Black Convex Region Computer Vision