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
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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
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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