1 Bronstein 2 & Kimmel Matching 2D articulated shapes using GMDS AMDO, Puerto de Andratx, 2006 Matching 2D articulated shapes using Generalized Multidimensional Scaling Michael M. Bronstein Department of Computer Science Technion – Israel Institute of Technology

2 Bronstein 2 & Kimmel Matching 2D articulated shapes using GMDS AMDO, Puerto de Andratx, 2006 Co-authors Ron KimmelAlex Bronstein

3 Bronstein 2 & Kimmel Matching 2D articulated shapes using GMDS AMDO, Puerto de Andratx, 2006 Main problems Comparison of articulated shapesComparison of partially-missing articulated shapes Local differences between shapesCorrespondence between articulated shapes

4 Bronstein 2 & Kimmel Matching 2D articulated shapes using GMDS AMDO, Puerto de Andratx, 2006 Ideal articulated shape 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 ISOMETRY

5 Bronstein 2 & Kimmel Matching 2D articulated shapes using GMDS AMDO, Puerto de Andratx, articulated shape Rigid parts and joints with Space of -articulated shapes: Articulation: an -isometry such that -ISOMETRY

6 Bronstein 2 & Kimmel Matching 2D articulated shapes using GMDS AMDO, Puerto de Andratx, 2006 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 Bronstein 2 & Kimmel Matching 2D articulated shapes using GMDS AMDO, Puerto de Andratx, 2006 Canonical forms distance (I) A. Elad, R. Kimmel, CVPR 2001 H. Ling, D. Jacobs, CVPR 2005 Embed and into a common metric space by minimum- distortion embeddings and compare the images (“canonical forms”)

8 Bronstein 2 & Kimmel Matching 2D articulated shapes using GMDS AMDO, Puerto de Andratx, 2006 Approximately articulation invariant Approximately consistent to sampling Efficient computation using multidimensional scaling (MDS) A. Elad, R. Kimmel, CVPR 2001 Given a sampling the minimum-distortion embedding is found by optimizing over the images and not on itself Canonical forms distance (II)

9 Bronstein 2 & Kimmel Matching 2D articulated shapes using GMDS AMDO, Puerto de Andratx, 2006 Gromov-Hausdorff distance M. Gromov, 1981 Allow for an arbitrary embedding space A metric on the space Consistent to sampling: if and are -coverings of and, Computation: untractable

10 Bronstein 2 & Kimmel Matching 2D articulated shapes using GMDS AMDO, Puerto de Andratx, 2006 Computing the Gromov-Hausdorff distance (I) F. Mémoli, G. Sapiro, Foundations Comp. Math, 2005 Equivalent definition in terms of metric distortions: Replace with a simpler expression Probabilistic bound on the error Combinatorial problem Mémoli & Sapiro (2005) Where:

11 Bronstein 2 & Kimmel Matching 2D articulated shapes using GMDS AMDO, Puerto de Andratx, 2006B 2 K, PNAS 2006 Given the samplings and, the minimum-distortion embeddings are found by optimizing over the images and The Gromov-Hausdorff distance is essentially a problem of finding minimum-distortion maps between and, and can be computed in an MDS-like spirit Computing the Gromov-Hausdorff distance (II)

12 Bronstein 2 & Kimmel Matching 2D articulated shapes using GMDS AMDO, Puerto de Andratx, 2006 Generalized multidimensional scaling (GMDS) 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 B 2 K, PNAS 2006 G MDS:

13 Bronstein 2 & Kimmel Matching 2D articulated shapes using GMDS AMDO, Puerto de Andratx, 2006 Example I – comparison of shapes

14 Bronstein 2 & Kimmel Matching 2D articulated shapes using GMDS AMDO, Puerto de Andratx, 2006 Similarity patterns between different articulated shapes Example I – comparison of shapes

15 Bronstein 2 & Kimmel Matching 2D articulated shapes using GMDS AMDO, Puerto de Andratx, 2006 Adding another axiom… Partial matching: If is a convex cut of, then Partial matching is non-symmetric: some properties must be sacrificed Convex cut guarantees

16 Bronstein 2 & Kimmel Matching 2D articulated shapes using GMDS AMDO, Puerto de Andratx, 2006 Triangle inequality

17 Bronstein 2 & Kimmel Matching 2D articulated shapes using GMDS AMDO, Puerto de Andratx, 2006Illustration: Herluf Bidstrup The danger of partial matching does not necessarily imply that But: if is an -covering of, then

18 Bronstein 2 & Kimmel Matching 2D articulated shapes using GMDS AMDO, Puerto de Andratx, 2006 Partial embedding distance (I) Use the distortion as a measure of partial similarity B 2 K, PNAS 2006 Non-symmetric Allows for partial matching Consistent to sampling In the discrete setting, posed as a GMDS problem Computationally efficient

19 Bronstein 2 & Kimmel Matching 2D articulated shapes using GMDS AMDO, Puerto de Andratx, 2006 Example – partial matching

20 Bronstein 2 & Kimmel Matching 2D articulated shapes using GMDS AMDO, Puerto de Andratx, 2006 Local comparison B 2 K, PNAS 2006 Use the contribution of a single point to the distortion as a measure of local difference between the shapes, or local distortion

21 Bronstein 2 & Kimmel Matching 2D articulated shapes using GMDS AMDO, Puerto de Andratx, 2006 Example – local differences

22 Bronstein 2 & Kimmel Matching 2D articulated shapes using GMDS AMDO, Puerto de Andratx, 2006 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 Bronstein 2 & Kimmel Matching 2D articulated shapes using GMDS AMDO, Puerto de Andratx, D example B 2 K, SIAM J. Sci. Comp, to appear

24 Bronstein 2 & Kimmel Matching 2D articulated shapes using GMDS AMDO, Puerto de Andratx, 2006 Canonical forms distance (MDS, 500 points) Gromov-Hausdorff distance (GMDS, 50 points) 3D example B 2 K, SIAM J. Sci. Comp, to appear