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Matching 2D articulated shapes using Generalized Multidimensional Scaling

Matching 2D articulated shapes using Generalized Multidimensional Scaling. Michael M. Bronstein. Department of Computer Science Technion – Israel Institute of Technology. Co-authors. Alex Bronstein. Ron Kimmel. Main problems. Comparison of articulated shapes.

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Matching 2D articulated shapes using Generalized Multidimensional Scaling

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  1. Matching 2D articulated shapes using Generalized Multidimensional Scaling Michael M. Bronstein Department of Computer Science Technion – Israel Institute of Technology

  2. Co-authors Alex Bronstein Ron Kimmel

  3. Main problems Comparison of articulated shapes Comparison of partially-missing articulated shapes Local differences between shapes Correspondence between articulated shapes

  4. Ideal articulated shape ISOMETRY • Two-dimensional shape • Geodesic distances induced by the boundary • Consists of rigid parts and point joins • Space of ideal articulated shapes • Articulation: an isometric deformation such that

  5. - articulated shape -ISOMETRY • Rigid parts and joints with • Space of -articulated shapes: • Articulation: an -isometry such that

  6. Articulation-invariant distance A distance between articulated shapes should satisfy: • Non-negativity: • Symmetry: • Triangle inequality: • Articulation invariance: for all and all • articulations of • Dissimilarity: if , then there do not exist and • two articulations of such that and • Consistency to sampling: if and are finite -coverings of • and , then • Efficiency: can be efficiently computed

  7. Canonical forms distance (I) Embed and into a common metric space by minimum-distortion embeddings and compare the images (“canonical forms”) A. Elad, R. Kimmel, CVPR 2001 H. Ling, D. Jacobs, CVPR 2005

  8. Canonical forms distance (II) • Approximately articulation invariant • 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

  9. Gromov-Hausdorff distance Allow for an arbitrary embedding space • A metric on the space • Consistent to sampling: if and are -coverings of and , • Computation:untractable M. Gromov, 1981

  10. Computing the Gromov-Hausdorff distance (I) Equivalent definition in terms of metric distortions: Where: Mémoli & Sapiro (2005) • Replace with a simpler expression • Probabilistic bound on the error • Combinatorial problem F. Mémoli, G. Sapiro, Foundations Comp. Math, 2005

  11. Computing the Gromov-Hausdorff distance (II) The Gromov-Hausdorff distance is essentially a problem of finding minimum-distortion maps between and , and can be computed in an MDS-like spirit Given the samplings and , the minimum-distortion embeddings are found by optimizing over the images and B2K, PNAS 2006

  12. Generalized multidimensional scaling (GMDS) G MDS: MDS: • The distances have no analytic expression and must be • approximated numerically • Multiresolution scheme to prevent local convergence • -norm can be used instead of for a more robust computation B2K, PNAS 2006

  13. Example I – comparison of shapes

  14. Example I – comparison of shapes Similarity patterns between different articulated shapes

  15. Adding another axiom… • Partial matching: If is a convex cut of , then Convex cut guarantees Partial matching is non-symmetric: some properties must be sacrificed

  16. Triangle inequality

  17. The danger of partial matching does not necessarily imply that But: if is an -covering of , then Illustration: Herluf Bidstrup

  18. Partial embedding distance (I) Use the distortion as a measure of partial similarity • Non-symmetric • Allows for partial matching • Consistent to sampling • In the discrete setting, posed as a • GMDS problem • Computationally efficient B2K, PNAS 2006

  19. Example – partial matching

  20. Local comparison Use the contribution of a single point to the distortion as a measure of local difference between the shapes, or local distortion B2K, PNAS 2006

  21. Example – local differences

  22. Summary • Isometric model of articulated shape • Axiomatic approach to comparison of shapes • Partial matching and correspondence • GMDS - a generic tool for shape recognition and matching

  23. 3D example B2K, SIAM J. Sci. Comp, to appear

  24. 3D example Canonical forms distance (MDS, 500 points) Gromov-Hausdorff distance (GMDS, 50 points) B2K, SIAM J. Sci. Comp, to appear

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