1 Regularized partial similarity of shapes NORDIA – CVPR 2008 Not only size matters: Regularized partial similarity of shapes Alexander Bronstein, Michael Bronstein © 2008 All rights reserved. Web: tosca.cs.technion.ac.il

2 Regularized partial similarity of shapes NORDIA – CVPR 2008 I am a centaur.Am I human ?Am I equine ? Yes, I’m partially human. Yes, I’m partially equine. Identity crisis Partial similarity

3 Regularized partial similarity of shapes NORDIA – CVPR 2008 Partial similarity Shapes are partially similar if they have significantsimilar parts Illustration: Herluf Bidstrup

4 Regularized partial similarity of shapes NORDIA – CVPR 2008 Shape Shape: metric-measure space Set of points Metric Measure Part: -measurable subset Represented as indicator function Part of a shape is a shape

5 Regularized partial similarity of shapes NORDIA – CVPR 2008 Significance = significance of part Alternatively: Many ways to define measure Area of the part Normalized area Statistical weighting a là tf-idf

6 Regularized partial similarity of shapes NORDIA – CVPR 2008 Metric Many ways to define metric Restricted metric : as the bird flies Intrinsic metric : shortest path length

7 Regularized partial similarity of shapes NORDIA – CVPR 2008 Metric Extrinsic geometry Intrinsic geometry Invariant to rigid deformation Invariant to inelastic nonrigid deformation

8 Regularized partial similarity of shapes NORDIA – CVPR 2008 Similarity Extrinsic similarity How congruent are and ? Best rigid alignment Intrinsic similarity How similar are and ? Min. distortion embedding Minimum-distortion correspondence

9 Regularized partial similarity of shapes NORDIA – CVPR 2008 Correspondence and ( dis ) similarity Correspondence: binary relation Belongs to some class of correspondences Rigid correspondences Non-rigid correspondences Given correspondence, quantify mismatch it creates between part of and part of Dissimilarity between two parts

10 Regularized partial similarity of shapes NORDIA – CVPR 2008 Rigid shape similarity Closest pointEuclidean isometry ICP

11 Regularized partial similarity of shapes NORDIA – CVPR 2008 Non-rigid shape similarity GMDS

12 Regularized partial similarity of shapes NORDIA – CVPR 2008 Multicriterion optimization Shapes are partially similar if they have significant similar parts Simultaneously minimize dissimilarity and maximize significance over all possible pairs of parts Criteria are competing: find a trade-off

13 Regularized partial similarity of shapes NORDIA – CVPR 2008 What is better? Many small parts… …or one large part? Not only size matters!

14 Regularized partial similarity of shapes NORDIA – CVPR 2008 Regularization Shapes are partially similar if they have significant, regular, and similar parts Add regularization term is the degree of regularity of part

15 Regularized partial similarity of shapes NORDIA – CVPR 2008 Boundary length regularization IrregularRegular Part regularity inversely proportional to boundary length Isoperimetric inequality (planar case): for fixed area (significance), the most regular shape is a circle.

16 Regularized partial similarity of shapes NORDIA – CVPR 2008 Numerical framework Minimization over all parts (binary indicator functions) is NP-hard !  Relax: replace crisp part by fuzzy part Continuous optimization

17 Regularized partial similarity of shapes NORDIA – CVPR 2008 Problem “fuzzification” Substitute binary with a continuous one Mumford-Shah regularization on curved surfaces Similarity: no change Significance: no change Regularity: fuzzy part has no boundary!

18 Regularized partial similarity of shapes NORDIA – CVPR 2008 Minimization algorithm Freeze parts and solve for correspondence Freeze correspondence and solve for parts 12 Step 1: weighted ICP (rigid) or weighted GMDS (nonrigid) Step 2: Newton descent Discretization & numerical details in the paper

19 Regularized partial similarity of shapes NORDIA – CVPR 2008 Rigid partial similarity SIGNIFICANCE REGULARITY

20 Regularized partial similarity of shapes NORDIA – CVPR 2008 Without regularization With regularization

21 Regularized partial similarity of shapes NORDIA – CVPR 2008 Conclusions Framework for regularized partial similarity Control over part significance and regularity Optimal geometry-driven weighting scheme for ICP and GMDS

22 Regularized partial similarity of shapes NORDIA – CVPR 2008 Published by Springer To appear in August 2008 ~350 pages Over 50 illustrations Color figures tosca.cs.technion.ac.il Shameless advertisement Additional information

23 Regularized partial similarity of shapes NORDIA – CVPR 2008 Backup slides

24 Regularized partial similarity of shapes NORDIA – CVPR 2008 Pareto optimality INSIGNIFICANCE DISSIMILARITY UTOPIA Attainable set Pareto frontier Solution is Pareto optimal if there exist no with and simultaneously.

25 Regularized partial similarity of shapes NORDIA – CVPR 2008 Set-valued dissimilarity INSIGNIFICANCE DISSIMILARITY Pareto frontier = set-valued dissimilarity Analogy: rate-distortion curve in communication

26 Regularized partial similarity of shapes NORDIA – CVPR 2008 Order relations BLUE < RED GREEN ? RED INSIGNIFICANCE DISSIMILARITY Is this crocodile greener than ? Not total ordering of set-valued dissimilarities Some dissimilarities are not commensurable

27 Regularized partial similarity of shapes NORDIA – CVPR 2008 Scalar-valued dissimilarity INSIGNIFICANCE DISSIMILARITY Convert Pareto frontier into scalar Fixed significance : Fixed dissimilarity : Area under the curve Distance from utopia:

28 Regularized partial similarity of shapes NORDIA – CVPR 2008 Topology regularization Euler characteristic as regularity Gauss-Bonnet identity No isoperimetric inequality for general curved surfaces Same boundary length, different number of connected components Small boundary length does not necessarily mean regularity

29 Regularized partial similarity of shapes NORDIA – CVPR 2008 DaveVictoriaHorseCentaurSeahorse Data: tosca.cs.technion.ac.il

30 Regularized partial similarity of shapes NORDIA – CVPR 2008 Nonrigid partial similarity Partial similarity Full similarity Scalar partial similarity Human Human- like Horse-like Human- like

31 Regularized partial similarity of shapes NORDIA – CVPR 2008 Regularized partial similarity INSIGNIFICANCE DISSIMILARITY IRREGULARITY