1. 1 Michael Bronstein Shapes as metric spaces: deformation-invariant similarity Michael Bronstein Computational metric geometry: an old new tool in image sciences

2. 2 Michael Bronstein Shapes as metric spaces: deformation-invariant similarity

3. 3 retrievalcategorization tracking detection/recognition restorationalignment Similarity

4. 4 Michael Bronstein Shapes as metric spaces: deformation-invariant similarity Raffaello Santi, School of Athens, Vatican

5. 5 Michael Bronstein Shapes as metric spaces: deformation-invariant similarity Shape similarity and correspondence Metric space Correspondence Correspondence quality = metric distortion Similarity Gromov-Hausdorff distance = Minimum possible correspondence distortion

6. 6 Michael Bronstein Shapes as metric spaces: deformation-invariant similarity Invariance RigidInelasticTopologyScaleElastic Choice of the metric prescribes the invariance

7. 7 Michael Bronstein Shapes as metric spaces: deformation-invariant similarity Non-rigid shape analysis and synthesis BBK CorrespondenceMorphing Retrieval

8. 8 Michael Bronstein Shapes as metric spaces: deformation-invariant similarity Self-similarity and symmetry Permutation Raviv & BBK 2007

9. 9 Michael Bronstein Shapes as metric spaces: deformation-invariant similarity Metric learning Data spaceEmbedding space Min distortion on training set of examples with known

10. 10 Michael Bronstein Shapes as metric spaces: deformation-invariant similarity Video copy detection Luke vs Vader – Starwars classic Lightsaber Star Wars DVD copyStar Wars pirated copy BBK 2010

11. 11 Michael Bronstein Shapes as metric spaces: deformation-invariant similarity Challenges Theoretical Approximate symmetry notion: group-like structure Comparing data from different spaces Computational Efficient solution of minimum distortion correspondence problems (Gromov-Hausdorff distance) Efficient algorithms for embedding into interesting metric spaces Applications Problems that can be formulated in terms of metric geometry