Partial similarity of objects, or how to compare a centaur to a horse

1. 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. Co-authors Alex Bronstein Alfred Bruckstein Ron Kimmel BBK = Bronstein, Bronstein, Kimmel BBBK = Bronstein, Bronstein, Bruckstein, Kimmel

3. Intrinsic vs. extrinsic similarity EXTRINSIC SIMILARITY INTRINSIC SIMILARITY

4. 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. Examples of near-isometric shapes

6. Canonical forms and MDS • 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) A. Elad, R. Kimmel, CVPR 2001

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

9. 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. 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

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

12. 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. 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

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

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

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

19. 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

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

21. Partial similarity of strings