1 Regularized partial similarity of shapes NORDIA – CVPR 2008 Not only size matters: Regularized...
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![Page 1: 1 Regularized partial similarity of shapes NORDIA – CVPR 2008 Not only size matters: Regularized partial similarity of shapes Alexander Bronstein, Michael.](https://reader031.fdocuments.us/reader031/viewer/2022032704/56649d595503460f94a39d15/html5/thumbnails/1.jpg)
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
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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
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3Regularized partial similarity of shapes NORDIA – CVPR 2008
Partial similarity
Shapes are partially similar if they
have
significa
nt
similar
parts
Illustration: Herluf Bidstrup
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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
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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
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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
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7Regularized partial similarity of shapes NORDIA – CVPR 2008
Metric
Extrinsic
geometry
Intrinsic
geometry
Invariant to rigid deformation
Invariant to inelastic nonrigid deformation
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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
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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
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10Regularized partial similarity of shapes NORDIA – CVPR 2008
Rigid shape similarity
Closest pointEuclidean isometry
ICP
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11Regularized partial similarity of shapes NORDIA – CVPR 2008
Non-rigid shape similarity
GMDS
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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
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13Regularized partial similarity of shapes NORDIA – CVPR 2008
What is better?
Many small parts… …or one large
part?
Not only size matters!
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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
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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.
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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
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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!
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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
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19Regularized partial similarity of shapes NORDIA – CVPR 2008
Rigid partial similarity
SIGNIFICANCE
REG
ULA
RIT
Y
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20Regularized partial similarity of shapes NORDIA – CVPR 2008
Without regularization
With regularization
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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
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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
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23Regularized partial similarity of shapes NORDIA – CVPR 2008
Backup slides
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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.
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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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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
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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:
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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
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29Regularized partial similarity of shapes NORDIA – CVPR 2008
Dave Victoria Horse Centaur Seahorse
Data: tosca.cs.technion.ac.il
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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
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31Regularized partial similarity of shapes NORDIA – CVPR 2008
Regularized partial similarity
INSIGNIFICANCE
DISSIMILARITY
IRREGULARITY