1Regularized partial similarity of shapes NORDIA – CVPR 2008

Not only size matters:Regularized partial similarity of shapes

Alexander Bronstein, Michael Bronstein© 2008 All rights reserved. Web: tosca.cs.technion.ac.il

2Regularized partial similarity of shapes NORDIA – CVPR 2008

I am a centaur.Am I human? Am I equine?Yes, I’m partially human.

Yes, I’m partially equine.

Identity crisis

Partial similarity

3Regularized partial similarity of shapes NORDIA – CVPR 2008

Partial similarity

Shapes are partially similar if they

have

significa

nt

similar

parts

Illustration: Herluf Bidstrup

4Regularized partial similarity of shapes NORDIA – CVPR 2008

Shape

Shape: metric-measure space

Set of points

Metric

Measure

Part: -measurable subset

Represented as indicator function

Part of a shape is a shape

5Regularized partial similarity of shapes NORDIA – CVPR 2008

Significance

= significance of part

Alternatively:

Many ways to define measure

Area of the part

Normalized area

Statistical weighting a là tf-idf

6Regularized partial similarity of shapes NORDIA – CVPR 2008

Metric

Many ways to define metric

Restricted metric: as the bird flies

Intrinsic metric: shortest path length

7Regularized partial similarity of shapes NORDIA – CVPR 2008

Metric

Extrinsic

geometry

Intrinsic

geometry

Invariant to rigid deformation

Invariant to inelastic nonrigid deformation

8Regularized partial similarity of shapes NORDIA – CVPR 2008

Similarity

Extrinsic similarityHow congruent are and

?

Best rigid alignment

Intrinsic similarityHow similar are and ?

Min. distortion embeddingMinimum-distortion

correspondence

9Regularized partial similarity of shapes NORDIA – CVPR 2008

Correspondence and (dis)similarity

Correspondence: binary relation

Belongs to some class of correspondences

Rigid correspondences

Non-rigid correspondences

Given correspondence , quantify mismatch

it creates between part of and part of

Dissimilarity between two parts

10Regularized partial similarity of shapes NORDIA – CVPR 2008

Rigid shape similarity

Closest pointEuclidean isometry

ICP

11Regularized partial similarity of shapes NORDIA – CVPR 2008

Non-rigid shape similarity

GMDS

12Regularized partial similarity of shapes NORDIA – CVPR 2008

Multicriterion optimization

Shapes are partially similar if they have significant

similar parts

Simultaneously minimize dissimilarity and maximize

significance

over all possible pairs of parts

Criteria are competing: find a trade-off

13Regularized partial similarity of shapes NORDIA – CVPR 2008

What is better?

Many small parts… …or one large

part?

Not only size matters!

14Regularized partial similarity of shapes NORDIA – CVPR 2008

Regularization

Shapes are partially similar if they

have

significant, regular, and similar

partsAdd regularization term

is the degree of regularity of part

15Regularized partial similarity of shapes NORDIA – CVPR 2008

Boundary length regularization

Irregular Regular

Part regularity inversely proportional to boundary length

Isoperimetric inequality (planar case): for fixed area

(significance),

the most regular shape is a circle.

16Regularized partial similarity of shapes NORDIA – CVPR 2008

Numerical framework

Minimization over all parts

(binary indicator functions)

is NP-hard!

Relax: replace crisp part

by fuzzy part

Continuous optimization

17Regularized partial similarity of shapes NORDIA – CVPR 2008

Problem “fuzzification”

Substitute binary with a continuous one

Mumford-Shah regularization on curved surfaces

Similarity: no change

Significance: no change

Regularity: fuzzy part has no boundary!

18Regularized partial similarity of shapes NORDIA – CVPR 2008

Minimization algorithm

Freeze parts and solve for

correspondence

Freeze correspondence and solve for

parts

1

2Step 1: weighted ICP (rigid) or weighted GMDS (nonrigid)

Step 2: Newton descent

Discretization & numerical details in the paper

19Regularized partial similarity of shapes NORDIA – CVPR 2008

Rigid partial similarity

SIGNIFICANCE

REG

ULA

RIT

Y

20Regularized partial similarity of shapes NORDIA – CVPR 2008

Without regularization

With regularization

21Regularized partial similarity of shapes NORDIA – CVPR 2008

Conclusions

Framework for regularized partial similarity

Control over part significance and regularity

Optimal geometry-driven weighting scheme for ICP and

GMDS

22Regularized partial similarity of shapes NORDIA – CVPR 2008

Published by Springer

To appear in August 2008

~350 pages

Over 50 illustrations

Color figures

tosca.cs.technion.ac.il

Shameless advertisement

Additional information

23Regularized partial similarity of shapes NORDIA – CVPR 2008

Backup slides

24Regularized partial similarity of shapes NORDIA – CVPR 2008

Pareto optimality

INSIGNIFICANCE

DIS

SIM

ILA

RIT

Y

UTOPIA

Attainable set

Pareto frontier

Solution is Pareto optimal if there exist no

with

and

simultaneously.

25Regularized partial similarity of shapes NORDIA – CVPR 2008

Set-valued dissimilarity

INSIGNIFICANCE

DIS

SIM

ILA

RIT

Y

Pareto frontier = set-valued dissimilarity

Analogy: rate-distortion curve in communication

26Regularized partial similarity of shapes NORDIA – CVPR 2008

Order relations

BLUE < RED

GREEN ? RED

INSIGNIFICANCE

DIS

SIM

ILA

RIT

Y

Is this crocodile greener than ?

Not total ordering of set-valued dissimilarities

Some dissimilarities are not commensurable

27Regularized partial similarity of shapes NORDIA – CVPR 2008

Scalar-valued dissimilarity

INSIGNIFICANCE

DIS

SIM

ILA

RIT

Y

Convert Pareto frontier into

scalarFixed significance:

Fixed dissimilarity:

Area under the curve

Distance from utopia:

28Regularized partial similarity of shapes NORDIA – CVPR 2008

Topology regularization

Euler characteristic as regularity

Gauss-Bonnet identity

No isoperimetric inequality for

general curved surfaces

Same boundary length,

different

number of connected

components

Small boundary length does not

necessarily mean regularity

29Regularized partial similarity of shapes NORDIA – CVPR 2008

Dave Victoria Horse Centaur Seahorse

Data: tosca.cs.technion.ac.il

30Regularized partial similarity of shapes NORDIA – CVPR 2008

Nonrigid partial similarity

Partial similarityFull similarity Scalar partial similarity

Human

Human-like

Horse-like

Hu

man

-like

31Regularized partial similarity of shapes NORDIA – CVPR 2008

Regularized partial similarity

INSIGNIFICANCE

DISSIMILARITY

IRREGULARITY