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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
2Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
Collaborators
AlexanderBronstein
Ron Kimmel
3Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
Welcome to nonrigid world!
4Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
?
SIMILARITYCORRESPONDENCE
Applications
5Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
Rock
Paper
Scissors
Rock, scissors, paper
6Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
Rock
Paper
Scissors
Hands
Rock, scissors, paper
7Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
Extrinsic vs. intrinsic
Are the shapes
congruent?
Invariance to rigid motion
Are the shapes
isometric?
Invariance to inelastic deformations
EXTRINSIC SIMILARITY INTRINSIC SIMILARITY
8Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
Metric model
Euclidean metric
Isometry = rigid motion
Geodesic metric
Isometry = inelastic deformation
EXTRINSIC SIMILARITY INTRINSIC SIMILARITY
Similarity = isometry
Shape = metric space
9Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
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
10Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
Extrinsic similarity – limitations
Suitable for nearly rigid shapes Unsuitable for nonrigid shapes
EXTRINSICALLY SIMILAR EXTRINSICALLY DISSIMILAR
11Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
Canonical forms
A. Elad, R. Kimmel, CVPR 2001
Multidimensional scaling (MDS)Isometric embedding
12Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
Intrinsic similarity – canonical forms
A. Elad, R. Kimmel, CVPR 2001
Compute canonical formsEXTRINSIC SIMILARITY OF CANONICAL FORMS
INTRINSIC SIMILARITY
= INTRINSIC SIMILARITY OF SHAPES
13Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
Intrinsic similarity – limitations
Intrinsically similar
Intrinsically dissimilar
Suitable for near-isometric
shape deformations
Unsuitable for deformations
modifying shape topology
14Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
Extrinsically dissimilarIntrinsically similar
Extrinsically similarIntrinsically dissimilar
Extrinsically dissimilarIntrinsically dissimilar
A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007
THIS IS THE SAME SHAPE!
Desired result:
15Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
Joint extrinsic/intrinsic similarity
DEFORM X TO MATCH Y
EXTRINSICALLY
CONSTRAIN THE DEFORMATION TO BE AS ISOMETRIC AS POSSIBLE
A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007
16Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007
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
18Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
Intrinsic dissimilarity
Ext
rinsi
c di
ssim
ilarit
y
A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007
19Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
Computation of the joint similarity
A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007
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
20Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
Computation of the extrinsic term
A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007
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
21Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
Computation of the intrinsic term
A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007
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
22Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
Computation of the intrinsic term
A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007
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
23Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
Computation of the joint similarity
A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007
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
24Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
Numerical example – dataset
= topology changeData: tosca.cs.technion.ac.il
25Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
Numerical example – intrinsic similarity
no topological changes
26Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
Numerical example – intrinsic similarity
= topology change= topology-preserving
Insensitive to strong
deformations
Sensitive to topological
changes
27Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
Numerical example – extrinsic similarity
= topology change= topology-preserving
Insensitive to topological
changes
Sensitive to strong
deformations
28Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
Numerical example – joint similarity
= topology change= topology-preserving
Insensitive to topologicalchanges...
…and to strong deformations
29Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
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%
30Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
Intrinsic dissimilarity
Ext
rinsi
c di
ssim
ilarit
y
Set-valued joint similarity
Dissi
mila
r
Simila
r
31Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
Shape morphing
Stronger intrinsic similarity (larger λ)
Stronger extrinsicsimilarity (smaller λ)
32Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
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
33Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
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)
34Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
Published by Springer
To appear in early 2008
~350 pages
Over 50 illustrations
Color figures
tosca.cs.technion.ac.il
Shameless advertisement
Additional information
35Bronstein2 and Kimmel Extrinsic and intrinsic similarity of nonrigid shapes
Workshop on Nonrigid Shape Analysisand Deformable Image Alignment
(NORDIA)
June 2008, Anchorage, Alaska
in conjunction with
CVPR’08
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