1Michael Bronstein Shapes as metric spaces: deformation-invariant similarity

Michael Bronstein

Computational metric geometry:an old new tool in image sciences

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

3Michael Bronstein Shapes as metric spaces: deformation-invariant similarity

retrieval categorization tracking

detection/recognition restoration alignment

Similarity

4Michael Bronstein Shapes as metric spaces: deformation-invariant similarity

Raffaello Santi, School of Athens, Vatican

5Michael Bronstein Shapes as metric spaces: deformation-invariant similarity

Shape similarity and correspondence

Metric space Metric space

Correspondence

Correspondence quality = metric distortion

Similarity

Gromov-Hausdorff distance = Minimum possible correspondence distortion

6Michael Bronstein Shapes as metric spaces: deformation-invariant similarity

Invariance

Rigid Inelastic TopologyScale Elastic

Choice of the metric prescribes the invariance

7Michael Bronstein Shapes as metric spaces: deformation-invariant similarity

Non-rigid shape analysis and synthesis

BBK

Correspondence Morphing

Retrieval

8Michael Bronstein Shapes as metric spaces: deformation-invariant similarity

Self-similarity and symmetry

Permutation

Raviv & BBK 2007

9Michael Bronstein Shapes as metric spaces: deformation-invariant similarity

Metric learning

Data space Embedding space

Min distortion on training set of examples with known

10Michael Bronstein Shapes as metric spaces: deformation-invariant similarity

Video copy detection

Luke vs Vader – Starwars classic

Lightsaber

Star Wars DVD copy Star Wars pirated copy

BBK 2010

11Michael 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