1Michael Bronstein Computational metric geometry

Computational metric geometry

Michael Bronstein

Department of Computer ScienceTechnion – Israel Institute of Technology

2Michael Bronstein Computational metric geometry

What is metric geometry?

Metric space

Similarity of metric spaces

Metric representation

?

3Michael Bronstein Computational metric geometry

information retrieval shape analysis

object detection inverse problems medical imaging

Similarity

4Michael Bronstein Computational metric geometry

Non-rigid world from macro to nano

Animals

Organs

Micro-organisms

ProteinsNano-machines

5Michael Bronstein Computational metric geometry

Rock

Paper

Scissors

Rock, paper, scissors

6Michael Bronstein Computational metric geometry

Hands

Rock

Paper

Scissors

Rock, paper, scissors

7Michael Bronstein Analysis of non-rigid shapes

Invariant similarity

Similarity

Transformation

8Michael Bronstein Computational metric geometry

Metric model

Shape

metric space

Similarity

Distance between metric

spaces and .

Invariance

isometry w.r.t.

9Michael Bronstein Computational metric geometry

Isometry

Two metric spaces and are isometric if there exists a

bijective distance preserving map such that

Two metric spaces and are -isometric if there exists a

map which is

distance preserving

surjective

-isometric

‘‘

-similar =‘‘ In which metric?

10Michael Bronstein Computational metric geometry

Examples of metrics

GeodesicEuclidean Diffusion

11Michael Bronstein Computational metric geometry

Rigid similarity

CongruenceIsometry between metric spaces

Min Hausdorff distance over

Euclidean isometries

Unknown correspondence!

12Michael Bronstein Computational metric geometry

Non-rigid similarity

Non-rigid similarityRigid similarity

Part of same metric space Different metric spaces

SOLUTION: Find a representation of and

in a common metric space

13Michael Bronstein Computational metric geometry

Canonical forms

Elad, Kimmel 2003

Non-rigid shape similarity

= Rigid similarity of canonical forms

Compute canonical formsCompare canonical forms as rigid shapes

14Michael Bronstein Computational metric geometry

Multidimensional scaling

2350

7200

31001900

2200

1630

Find a configuration of points in the plane best representing

distances between the cities

SFNY

Rio

TAParis1800

4000 5200

15Michael Bronstein Computational metric geometry

Best possible embedding with minimum distortion

Multidimensional scaling

Non-linear non-convex optimization problem in variables

16Michael Bronstein Computational metric geometry

Interpolate

Multigrid MDS

B et al. 2005

Fine grid

Decimate

Solution

Coarse grid Improved solutionRelax

17Michael Bronstein Computational metric geometry

Multigrid MDS

B et al. 2005, 2006

Complexity (MFLOPs)

Str

es

s

Execution time (sec)

Multigrid MDS

Standard MDS

18Michael Bronstein Computational metric geometry

Examples of canonical forms

19Michael Bronstein Computational metric geometry

Embedding distortion limits discriminative power!

20Michael Bronstein Computational metric geometry

Min distortion

embedding

Min distortion

embedding

Fix some metric space

No fixed (data-independent) embedding space will give

distortion-less canonical forms!

Canonical forms, revisited

Compute canonical forms (defined up to an isometry in )Compute Hausdorff distance between canonical forms

21Michael Bronstein Computational metric geometry

Metric coupling

Disjoint union

Isometric embedding Isometric embedding

?

?

How to choose the metric?

22Michael Bronstein Computational metric geometry

Gromov-Hausdorff distance

Gromov 1981

Find the smallest possible metric

Distance between metric spaces (how isometric two spaces are)

Generalization of the Hausdorff distance

Gromov-Hausdorff distance

23Numerical geometry of non-rigid shapes A journey to non-rigid world

Canonical forms Gromov-Hausdorff

Fixed embedding space Optimal data-dependent

embedding space

Approximate metric

(error dependent on the data)

Metric on equivalence classes of

isometric shapes

-isometric

Consistent to sampling

-isometric

for shapes sampled at radius

24Michael Bronstein Computational metric geometry

Gromov-Hausdorff distance

Gromov 1981

Optimization over all possible correspondences is NP-hard problem!

is a correspondence satisfying

for every there exists s.t.

for every there exists s.t.

Theorem: for compact spaces,

25Michael Bronstein Computational metric geometry

Best possible embedding with minimum distortion

Multidimensional scaling

26Michael Bronstein Computational metric geometry

Generalized multidimensional scaling

Best possible embedding with minimum distortion

B et al. 2006

Geodesic distances have no closed-form expression

No global representation for optimization variables

How to perform optimization on a manifold?

27Michael Bronstein Computational metric geometry

GMDS: some technical details

B et al. 2005

Use local barycentric coordinates

Interpolate distances from those

pre-computed on the mesh

Perform path unfolding to go across triangles

No global system of coordinates

No closed-form distances

How to perform optimization?

28Michael Bronstein Computational metric geometry

Canonical forms (MDS based on 500 points)

Gromov-Hausdorff distance(GMDS based on 50 points)

BBK, SIAM J. Sci. Comp 2006

29Numerical Geometry of Non-Rigid Shapes Expression-invariant face recognition

Application to face recognition

x

x’

y

y’

Euclidean metric

30Numerical Geometry of Non-Rigid Shapes Expression-invariant face recognition

Application to face recognition

x

x’

y

y’

Geodesic metric

Distance distortion distribution

31Michael Bronstein Computational metric geometry

32Michael Bronstein Computational metric geometry

Eikonal vs heat equation

Kimmel & Sethian 1998Weber, Devir, B2, Kimmel 2008

Viscosity solution: arrival time

(geodesic distance from source)

Boundary conditions: Initial conditions:

Solution : heat

distribution at time t

33Michael Bronstein Computational metric geometry: a new tool in image sciences

Heat equation on manifolds

1D 3D

34Michael Bronstein Computational metric geometry: a new tool in image sciences

1D 3D

Heat kernel

Heat equation on manifolds

35Michael Bronstein Computational metric geometry: a new tool in image sciences

1D 3D

Heat kernel

“Convolution”

Heat equation on manifolds

36Michael Bronstein Computational metric geometry

Diffusion distance

Geodesic = minimum-length path

Diffusion distance = “average” length path (less sensitive to bottlenecks)

“Connectivity rate” from to by paths of length

Small if there are many paths

Large if there are a few paths

Berard, Besson, Gallot, 1994; Coifman et al. PNAS 2005

37Michael Bronstein Computational metric geometry

Invariance: Euclidean metric

Rigid Scale Inelastic Topology

Wang, B, Paragios 2010

38Michael Bronstein Computational metric geometry

Invariance: geodesic metric

Rigid Scale Inelastic Topology

Wang, B, Paragios 2010

39Michael Bronstein Computational metric geometry

Invariance: diffusion metric

Rigid Scale Inelastic Topology

Wang, B, Paragios 2010

40Michael Bronstein Computational metric geometry

41Michael Bronstein Computational metric geometry

shape analysis

object detection inverse problems medical imaging

Similarity

information retrieval

42Michael Bronstein Computational metric geometry

Metric learning

Representation space

“Similar”

“Dissimilar”

Data space

Metric learning: on training set

Sampling of

Generalization

43Michael Bronstein Computational metric geometry

Similarity-sensitive hashing

Hamming spaceData space

Shakhnarovich 2005B2, Kimmel 2010; Strecha, B, Fua 2010

0001

1111

0100

0011

0111

44Michael Bronstein Computational metric geometry

Luke vs Vader – Starwars classic

Lightsaber

Original copy Pirated copy

Video copy detection

45Michael Bronstein Computational metric geometry

C A A A T T G C C

Substitution In/Del

C C A A T T G C CC C A A T T A G C C

B2, Kimmel 2010

Mutation

Substitution In/Del

So, what do you think?

Biological DNA “Video DNA”

46Michael Bronstein Computational metric geometry

So, what do you think?

T

positive

negative

So, what do you think?

So, what do you think?

So, what do you think?

So, what do you think?

Mutation-invariant metric

B2, Kimmel 2010

47Michael Bronstein Computational metric geometry: a new tool in image sciences

Gap

Gap

Gap continuation

Pairwise cost

Dynamic programming sequence alignment with gaps to account

for In/Del mutations (Smith-WATerman algorithm)

Optimal alignment = minimum-cost path

Learned mutation-invariant pairwise matching cost

Video DNA alignment

B2, Kimmel 2010

48Michael Bronstein Computational metric geometry

B2, Kimmel 2010

49Michael Bronstein Computational metric geometry

B2, Kimmel 2010

50Michael Bronstein Computational metric geometry

Object similarity is also a metric space

Summary

Metric space

Gromov-Hausdorffdistance + GMDS

MDS Metric learning

Metric choice=invarianceExamples of similarity(metric sampling)

00011001

1110

11110111

51Michael Bronstein Computational metric geometry

Thank you