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Numerical geometry of non-rigid shapes: deformation-invariant similarities

Numerical geometry of non-rigid shapes: deformation-invariant similarities. Michael M. Bronstein. Department of Computer Science Technion – Israel Institute of Technology Ph.D under the supervision of Prof. Ron Kimmel. Co-authors. Alex Bronstein. Alfred Bruckstein. Ron Kimmel.

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Numerical geometry of non-rigid shapes: deformation-invariant similarities

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  1. Numerical geometry of non-rigid shapes: deformation-invariant similarities Michael M. Bronstein Department of Computer Science Technion – Israel Institute of Technology Ph.D under the supervision of Prof. Ron Kimmel

  2. Co-authors Alex Bronstein Alfred Bruckstein Ron Kimmel Irad Yavneh BBK = Bronstein, Bronstein, Kimmel BBBK = Bronstein, Bronstein, Bruckstein, Kimmel BBKY = Bronstein, Bronstein, Kimmel, Yavneh

  3. Similarity between non-rigid shapes (deformation-invariant distance) Main problems ANALYSIS SYNTHESIS Non-rigid correspondence and calculus of shapes

  4. Intrinsic vs. extrinsic similarity EXTRINSIC SIMILARITY INTRINSIC SIMILARITY • Similarity = isometry • and are isometric if • and are -isometric if

  5. 3D example: expression-invariant face recognition • Facial expressions are approximate isometries of the facial surface • Identity = intrinsic geometry • Expression = extrinsic geometry BBK, IJCV, 2005

  6. 2D example: articulated shapes • Articulated shape – shape consisting of rigid disjoints parts and non- • rigid joints, • -articulated shape if the joints size is bounded • -articulation: a deformation mapping isometrically rigid parts and • preserving the joints size: H. Ling, D. Jacobs, CVPR 2005 BBBK, AMDO 2006

  7. Articulations versus isometries • -articulation is an -isometry, but not necessarily vice versa -articulated shape -articulation -isometry • A composition of two -articulations is an -articulation • A composition of two -isometries is a -isometry BBBK, IJCV, submitted

  8. Isometry-invariant similarity between non-rigid shapes

  9. Isometry-invariant distance construction Let be a class of non-rigid shapes. A distance measuring dissimilarity of shapes should satisfy: • Non-negativity: • Symmetry: • Triangle inequality: • Similarity: if then and are -isometric • if and are -isometric, then • Consistency to sampling: if is a finite -covering of , then • Efficiency: can be efficiently approximated numerically BBK, PNAS, 2006

  10. Canonical forms distance • Embed and into a common metric space by • minimum-distortion embeddings and . • Compare the images (canonical forms) as rigid objects A. Elad, R. Kimmel, CVPR 2001

  11. Canonical forms distance (cont.) • Satisfies the metric axioms only approximately • Approximately consistent to sampling • Efficient computation using multidimensional scaling (MDS) Given a sampling the minimum-distortion embedding is found by optimizing over the images and not on itself A. Elad, R. Kimmel, CVPR 2001

  12. Multigrid MDS Time (sec) Stress Complexity (MFLOPs) Convergence of our MG MDS algorithm (x10 faster than state-of-the-art) BBKY, NLAA 2006

  13. How to choose the embedding space ? • Schwartz et al. 1989: • Elad & Kimmel 2001: • Elad & Kimmel 2002: • BBK 2005: • Walter & Ritter 2002: Euclidean Spherical Hyperbolic Problem: using non-Euclidean embedding spaces, it is possible to reduce the representation error, but not avoid it completely.

  14. Gromov-Hausdorff distance Allow for arbitrary embedding space where are isometric embeddings. • Satisfies the metric axioms with • Consistent to sampling: if is an -covering of , then • Computation: intractable M. Gromov, 1981

  15. Gromov-Hausdorff distance (cont.) For compact surfaces, there exists an equivalent definition in terms of metric distortions: where:

  16. Gromov-Hausdorff distance (cont.) measures how isometrically can be embedded into

  17. Gromov-Hausdorff distance (cont.) measures how isometrically can be embedded into

  18. Gromov-Hausdorff distance (cont.) measures how far and are from being one the inverse of the other

  19. Computing the Gromov-Hausdorff distance Mémoli & Sapiro (2005) • Drop the terms , • Replace with a simpler expression • Probabilistic bound on the error • Combinatorial problem F. Mémoli, G. Sapiro, Foundations Comp. Math, 2005

  20. Computing the Gromov-Hausdorff distance (cont.) BBK (2006) • Generalized MDS problem • Continuous optimization • Deterministic approximation (exact up to numerical accuracy / local • convergence) BBK, PNAS, 2006

  21. Generalized multidimensional scaling (GMDS) G MDS: MDS: • The distances have no analytic expression and must be • approximated numerically • Points represented in barycentric coordinates • Optimization with a modified line search • Multiresolution scheme to prevent local convergence • -norm can be used instead of BBK, PNAS, 2006

  22. Gromov-Hausdorff distance via GMDS • Sampling: , • Optimization over images and BBK, PNAS, 2006

  23. Gromov-Hausdorff vs. canonical forms GROMOV-HAUSDORFF CANONICAL FORMS • Two stages: embedding and • comparison • Embedding error is a problem • degrading accuracy • Many points (~1000) are • required for accurate comparison • Computational core: MDS • One stage: generalized • embedding • Embedding error is the • measure of similarity • Few points (~100) are required • to compute accurate distortion • Computational core: GMDS

  24. Example I – 3D objects BBK, SIAM J. Sci. Comp, 2006

  25. Example I – 3D objects (cont.) Canonical forms distance (MDS, 500 points) Gromov-Hausdorff distance (GMDS, 50 points) BBK, SIAM J. Sci. Comp, 2006

  26. Example II – Articulated shapes BBBK, IJCV, submitted

  27. Example II – Articulated shapes (cont.) Gromov-Hausdorff distance between articulated shapes BBBK, IJCV, submitted

  28. Face recognition project • Authentication of car driver based on 3D face recognition • Collaboration with General Motors • Current accuracy: <3% error rate Raja Giryes Alon Salzman Daniel Vainsencher Vladimir Zdornov Yaron Honen

  29. Partial similarity between non-rigid shapes, or how to compare a centaur and a horse?

  30. Example from real life Can we compare parts of objects? Conclusion: objects may have similar parts, while being dissimilar. Illustration: Herluf Bidstrup

  31. Semantic definition of partial similarity Two objects are partially similar if they have “large” “similar” “parts”. Example: Jacobs et al.

  32. More precise definitions • Part: subset with restricted metric • (technically, the set of all parts of is a • -algebra) • Dissimilarity: Gromov-Hausdorff distance defined on the set of parts, • Partiality: size of the object parts cropped off, • where is the measure of area on

  33. Full versus partial similarity • Full similarity: and are -isometric • Partial similarity: and are -isometric, i.e., have parts • which are -isometric, and Partial similarity Full similarity BBBK, IJCV, submitted

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

  35. Vector optimality • No total order relation in - impossible to say which point is “better” • Partial order: only when both criteria are better

  36. Scalar versus vector optimality Multicriterion optimization Traditional (scalar) optimization V. Pareto, 1901

  37. Pareto optimum • Pareto optimum: point at which no criterion can be improved without • compromising the other • Pareto frontier: set of all Pareto optima, acting as a set-valued • criterion of partial dissimilarity • Only partial order relation exists between set-valued distances: not • always possible to compare BBBK, IJCV, submitted

  38. Fuzzy computation • Optimization over subsets turns into an NP-hard combinatorial • problem when discretized • Fuzzy optimization: optimize over membership functions Crisp part Fuzzy part BBBK, IJCV, submitted

  39. Salukwadze distance • The set-valued distance can be converted into a scalar valued one by • selecting a single point on the Pareto frontier. • Naïve selection: fixed value of or . • Smart selection: closest to the utopia point (Salukwadze optimum) Salukwadze distance: M. E. Salukwadze, 1979 BBBK, IJCV, submitted

  40. Example II – mythological creatures Large Gromov-Hausdorff distance Small Salukwadze distance Large Gromov-Hausdorff distance Large Salukwadze distance BBBK, IJCV, submitted

  41. Example II – mythological creatures (cont.) BBBK, IJCV, submitted

  42. Example II – mythological creatures (cont.) Gromov-Hausdorff distance Salukwadze distance (using L1-norm) BBBK, IJCV, submitted

  43. 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 partially missing objects BBBK, ScaleSpace, submitted

  44. Example II – 3D partially missing objects Salukwadze distance between partially missing objects (using L1-norm) BBBK, ScaleSpace, submitted

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