1 / 24

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.

adara
Télécharger la présentation

Matching 2D articulated shapes using Generalized Multidimensional Scaling

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related