1 bronstein 2 and kimmel extrinsic and intrinsic similarity of nonrigid shapes michael m. bronstein...

1Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes

Michael M. Bronstein

Department of Computer ScienceTechnion – Israel Institute of Technologycs.technion.ac.il/~mbron Technion

1 January 2008

Extrinsic and intrinsic similarityof shapesnonrigid


Collaborators

AlexanderBronstein

Ron Kimmel


Welcome to nonrigid world!


?

SIMILARITYCORRESPONDENCE

Applications


Rock

Paper

Scissors

Rock, scissors, paper


Rock

Paper

Scissors

Hands

Rock, scissors, paper


Extrinsic vs. intrinsic

Are the shapes

congruent?

Invariance to rigid motion

Are the shapes

isometric?

Invariance to inelastic deformations

EXTRINSIC SIMILARITY INTRINSIC SIMILARITY


Metric model

Euclidean metric

Isometry = rigid motion

Geodesic metric

Isometry = inelastic deformation

EXTRINSIC SIMILARITY INTRINSIC SIMILARITY

Similarity = isometry

Shape = metric space


Extrinsic similarity – Iterative closest point (ICP)

Chen & Medioni, 1991; Besl & McKay, PAMI 1992

Find the best rigid alignment of two shapes

Hausdorff distance

In Euclidean space


Extrinsic similarity – limitations

Suitable for nearly rigid shapes Unsuitable for nonrigid shapes

EXTRINSICALLY SIMILAR EXTRINSICALLY DISSIMILAR


Canonical forms

A. Elad, R. Kimmel, CVPR 2001

Multidimensional scaling (MDS)Isometric embedding


Intrinsic similarity – canonical forms

A. Elad, R. Kimmel, CVPR 2001

Compute canonical formsEXTRINSIC SIMILARITY OF CANONICAL FORMS

INTRINSIC SIMILARITY

= INTRINSIC SIMILARITY OF SHAPES


Intrinsic similarity – limitations

Intrinsically similar

Intrinsically dissimilar

Suitable for near-isometric

shape deformations

Unsuitable for deformations

modifying shape topology


Extrinsically dissimilarIntrinsically similar

Extrinsically similarIntrinsically dissimilar

Extrinsically dissimilarIntrinsically dissimilar

A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007

THIS IS THE SAME SHAPE!

Desired result:


Joint extrinsic/intrinsic similarity

DEFORM X TO MATCH Y

EXTRINSICALLY

CONSTRAIN THE DEFORMATION TO BE AS ISOMETRIC AS POSSIBLE




Glove fitting example

Stretching = Intrinsic dissimilarity

Misfit = Extrinsic dissimilarity

17Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapesIf it doesn’t fit, you must acquit!If it doesn’t fit, you must acquit!

Image: Associated Press


Intrinsic dissimilarity

Ext

rinsi

c di

ssim

ilarit

y



Computation of the joint similarity


Optimization variable: the deformed shape vertex coordinates

Assuming has the connectivity of

Split into computation of and

Gradients w.r.t. are required for optimization


Computation of the extrinsic term


Find and fix correspondence between current and

Can be e.g. the closest points

Compute an L2 variant of a one-sided Hausdorff distance

and its gradient

Similar in spirit to ICP


Computation of the intrinsic term


Fix trivial correspondence between and

Compute L2 distortion of geodesic distances

and gradient

is a fixed matrix of all pair-wise geodesic distances on

Can be precomputed using Dijkstra’s algorithm or fast marching


Computation of the intrinsic term


is function of the optimization variables and needs to be

recomputed

First option: modify the Dijkstra’s algorithm or fast marching to compute

the gradient in addition to the distance itself

Second option: compute and fix the path of the geodesic

is a matrix of Euclidean distances between adjacent vertices

is a linear operator integrating the path length along fixed path


Computation of the joint similarity


Alternating minimization algorithm

Compute corresponding points

Compute shortest paths and assemble

Update to sufficiently decrease

If change is small, stop; otherwise, go to Step

1

2

3

4 1


Numerical example – dataset

= topology changeData: tosca.cs.technion.ac.il


Numerical example – intrinsic similarity

no topological changes


Numerical example – intrinsic similarity

= topology change= topology-preserving

Insensitive to strong

deformations

Sensitive to topological

changes


Numerical example – extrinsic similarity


Insensitive to topological

changes

Sensitive to strong

deformations


Numerical example – joint similarity


Insensitive to topologicalchanges...

…and to strong deformations


Numerical example – ROC curves

0.1 1 10 100

0.1

1

10

100

False acceptance rate (FAR), %

Fal

se r

ejec

tion

rate

(F

RR

), %

Intrinsic

Extrinsic

Joint

Intrinsic,no topological

changes

EER=7.7%

EER=10.3%

EER=1.6%

EER=1.1%


Intrinsic dissimilarity

Ext

rinsi

c di

ssim

ilarit

y

Set-valued joint similarity

Dissi

mila

r

Simila

r


Shape morphing

Stronger intrinsic similarity (larger λ)

Stronger extrinsicsimilarity (smaller λ)


Conclusion

Extrinsic similarity is insensitive to topology changes, but sensitive to

nonrigid deformations

Intrinsic similarity is insensitive to nearly-isometric nonrigid

deformations, but sensitive to topology changes

Joint similarity is insensitive to both nonrigid deformations and topology

changes

Can be thought of as nonrigid ICP

Can be used to produce as isometric as possible morphs


Open issues

Efficient minimization (good initialization, multiresolution)

Only topology of one shape can change: topology of Z = topology of X

Mesh validity not enforced: self intersections may occur (may be

important in computer graphics applications)


Published by Springer

To appear in early 2008

~350 pages

Over 50 illustrations

Color figures

tosca.cs.technion.ac.il

Shameless advertisement

Additional information


Workshop on Nonrigid Shape Analysisand Deformable Image Alignment

(NORDIA)

June 2008, Anchorage, Alaska

in conjunction with

CVPR’08

1 bronstein 2 and kimmel extrinsic and intrinsic similarity of nonrigid shapes michael m. bronstein...

Documents

kimmel extrinsic

shapes isometric

shapes congruent

nonrigid world

rigid shapes unsuitable

paper slide

shapes hausdorff distance

metric space slide