1 Michael M. Bronstein Partial similarity of objects 17 December 2006 Partial similarity of objects, or how to compare a centaur to a horse Michael M. Bronstein Department of Computer Science Technion – Israel Institute of Technology

2 Michael M. Bronstein Partial similarity of objects 17 December 2006 Co-authors Ron KimmelAlex Bronstein BBK = Bronstein, Bronstein, Kimmel BBBK = Bronstein, Bronstein, Bruckstein, Kimmel Alfred Bruckstein

3 Michael M. Bronstein Partial similarity of objects 17 December 2006 Intrinsic vs. extrinsic similarity INTRINSIC SIMILARITY EXTRINSIC SIMILARITY

4 Michael M. Bronstein Partial similarity of objects 17 December 2006 Non-rigid objects: basic terms Isometry – deformation that preserves the geodesic distances is -isometrically embeddable into if and are -isometric if, and is -surjective

5 Michael M. Bronstein Partial similarity of objects 17 December 2006 Examples of near-isometric shapes

6 Michael M. Bronstein Partial similarity of objects 17 December 2006 Canonical forms and MDS A. Elad, R. Kimmel, CVPR 2001 Embed and into a common metric space by minimum-distortion embeddings and. Compare the images (canonical forms) as rigid objects Efficient computation using multidimensional scaling (MDS)

7 Michael M. Bronstein Partial similarity of objects 17 December 2006 Generalized MDS Generalized MDS: embed one surface into another Measure of similarity: embedding error Related to the Gromov-Hausdorff distance F. Memoli, G. Sapiro, 2005 BBBK, PNAS, 2006

8 Michael M. Bronstein Partial similarity of objects 17 December 2006 Semantic definition of partial similarity Two objects are partially similar if they have “large” “similar” “parts”. Example: Jacobs et al.

9 Michael M. Bronstein Partial similarity of objects 17 December 2006 More precise definitions Part: subset with restricted metric (technically, the set of all parts of is a -algebra) Dissimilarity: intrinsic distance criterion defined on the set of parts (Gromov-Hausdorff distance) Partiality: size of the object parts cropped off, where is the measure of area on

10 Michael M. Bronstein Partial similarity of objects 17 December 2006 Full versus partial similarity Full similarity Full similarity: and are -isometric Partial similarity: and are -isometric, i.e., have parts which are -isometric, and Partial similarity BBBK, IJCV, submitted

11 Michael M. Bronstein Partial similarity of objects 17 December 2006 Multicriterion optimization BBBK, IJCV, submitted UTOPIA Minimize the vector objective function over Competing criteria – impossible to minimize and simultaneously ATTAINABLE CRITERIA

12 Michael M. Bronstein Partial similarity of objects 17 December 2006 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

13 Michael M. Bronstein Partial similarity of objects 17 December 2006 Fuzzy computation Optimization over subsets turns into an NP-hard combinatorial problem when discretized Fuzzy optimization: optimize over membership functions BBBK, IJCV, submitted Crisp partFuzzy part

14 Michael M. Bronstein Partial similarity of objects 17 December 2006 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

15 Michael M. Bronstein Partial similarity of objects 17 December 2006BBBK, IJCV, submitted Example II – mythological creatures Large Gromov-Hausdorff distance Small Salukwadze distance Large Gromov-Hausdorff distance Large Salukwadze distance

16 Michael M. Bronstein Partial similarity of objects 17 December 2006 Example II – mythological creatures (cont.) BBBK, IJCV, submitted

17 Michael M. Bronstein Partial similarity of objects 17 December 2006BBBK, IJCV, submitted Example II – mythological creatures (cont.) Gromov-Hausdorff distance Salukwadze distance (using L 1 -norm)

18 Michael M. Bronstein Partial similarity of objects 17 December 2006

19 Michael M. Bronstein Partial similarity of objects 17 December 2006 Example II – 3D partially missing objects BBBK, ScaleSpace, submitted Pareto frontiers, representing partial dissimilarities between partially missing objects

20 Michael M. Bronstein Partial similarity of objects 17 December 2006 Example II – 3D partially missing objects Salukwadze distance between partially missing objects (using L 1 -norm) BBBK, ScaleSpace, submitted

21 Michael M. Bronstein Partial similarity of objects 17 December 2006 Partial similarity of strings