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Paretian similarity for partial comparison of non-rigid objects

Paretian similarity for partial comparison of non-rigid objects (or how to compare a centaur to a horse). Michael M. Bronstein. Department of Computer Science Technion – Israel Institute of Technology. Non-rigid world. 3D OBJECTS (Riemannian 2-manifolds). 2D OBJECTS (silhouettes).

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Paretian similarity for partial comparison of non-rigid objects

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  1. Paretian similarity for partial comparison of non-rigid objects (or how to compare a centaur to a horse) Michael M. Bronstein Department of Computer Science Technion – Israel Institute of Technology

  2. Non-rigid world 3D OBJECTS (Riemannian 2-manifolds) 2D OBJECTS (silhouettes)

  3. Rock, scissors, paper ROCK PAPER SCISSORS

  4. Extrinsic vs intrinsic similarity EXTRINSIC SIMILARITY INTRINSIC SIMILARITY • Are the shapes congruent? • Invariant only to rigid • transformations • Classical solution: ICP • Is the metric structure of the • shape similar? • Invariant to isometric • deformations

  5. Intrinsic similarity • Metric structure described by the geodesic distances • Isometry – deformation that preserves the geodesic distances • Intrinsically similar = isometric • Approximate similarity: and are -isometric if

  6. Gromov-Hausdorff distance where: M. Gromov, 1981, F. Memoli, G. Sapiro, 2005, BBK, PNAS, 2006

  7. Gromov-Hausdorff distance where: M. Gromov, 1981, F. Memoli, G. Sapiro, 2005, BBK, PNAS, 2006

  8. Gromov-Hausdorff distance (cont) • Metric of the space of non-rigid shapes (up to isometry) • If , then and are -isometric • If and are -isometric, then • iff and are isometric • Efficient computation using generalized multidimensional scaling M. Gromov, 1981, F. Memoli, G. Sapiro, 2005, BBK, PNAS, 2006

  9. If it doesn’t fit, you must acquit OJ Simpson measuring the glove that appeared as an evidence in the court

  10. Is a centaur similar to a horse or a man? Example: Jacobs et al.

  11. Partial similarity Horse is similar to centaur Man is similar to centaur Horse is not similar to man • Partial similarity is an intransitive relation • Non-metric (no triangle inequality) • Weaker than full similarity (shapes may be partially but not fully similar)

  12. Recognition by parts • Divide the shapes into meaningful parts and • Compare each part separately using full similarity criterion • Merge the partial similarities, Pentland, et al., Basri et al.

  13. Problems • Problem 1: how to divide the shapes into meaningful parts? • Problem 2: are all parts equally important? • Solution 1: find the most similar pair out of the sets and of all • possible parts of shapes and : • Solution 2: define partiality measuring how significant the • selected parts are w.r.t. entire shapes (larger parts = smaller partiality)

  14. Multicriterion optimization • Minimize the vector objective function over • Competing criteria – impossible to minimize and simultaneously ATTAINABLE CRITERIA DISSIMILARITY PARTIALITY UTOPIA BBBK, IJCV, submitted

  15. Pareto optimality Pareto optimum Minimum of scalar function Pareto optimum: a point at which no criterion can be improved without compromising the other V. Pareto, 1901

  16. Pareto distance • Pareto distance: set of all Pareto optima (Pareto frontier), acting as a • set-valued criterion of partial dissimilarity • Only partial orderrelation exists between set-valued distances: not • always possible to compare • Infinite possibilities to convert Pareto distance into a scalar-valued one • One possibility: select a point on the • Pareto frontier closest to the utopia • point, BBBK, IJCV, submitted

  17. Fuzzy approximation • Optimization over subsets is an NP-hard combinatorial problem • Relaxed problem (fuzzy approximation): optimize over membership • functions Crisp part Fuzzy part BBBK, IJCV, submitted

  18. Example I – mythological creatures Large Gromov-Hausdorff distance Small partial dissimilarity Large Gromov-Hausdorff distance Large partial dissimilarity BBBK, IJCV, submitted

  19. Example I – mythological creatures (cont.) BBBK, IJCV, submitted

  20. Example I – mythological creatures (cont.) Gromov-Hausdorff distance Partial similarity BBBK, IJCV, submitted

  21. Example II – 3D partially missing objects 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Pareto frontiers, representing partial dissimilarities between objects BBBK, SSVM

  22. Example II – 3D partially missing objects Partial dissimilarities between objects BBBK, SSVM

  23. Conclusions • Intrinsic similarity of non-rigid shapes based on the Gromov-Hausdorff • distance • Generic definition of partial similarity and set-valued Pareto distance • Other applications beyond shape recognition (e.g. text sequences)

  24. Grazie

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