1 numerical geometry of non-rigid shapes invariant correspondence & calculus of shapes invariant...
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1Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Invariant correspondence and calculus of shapes
© Alexander & Michael Bronstein, 2006-2010tosca.cs.technion.ac.il/book
VIPS Advanced School onNumerical Geometry of Non-Rigid Shapes
University of Verona, April 2010
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2Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
“Natural” correspondence?
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3Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Correspondence
accurate
‘‘
‘‘ makes sense
‘‘
‘‘ beautiful
‘‘
‘‘Geometric Semantic Aesthetic
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4Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Correspondence
Correspondence is not a well-defined problem!
Chances to solve it with geometric tools are slim.
If objects are sufficiently similar, we have better chances.
Correspondence between deformations of the same object.
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5Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Invariant correspondence
Ingredients:
Class of shapes
Class of deformations
Correspondence procedure which given two shapes
returns a map
Correspondence procedure is -invariant if it commutes with
i.e., for every and every ,
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6Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
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7Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Invariant similarity (reminder)
Ingredients:
Class of shapes
Class of deformations
Distance
Distance is -invariant if for every and every
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8Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Closest point correspondence between , parametrized by
Its distortion
Minimize distortion over all possible congruences
Rigid similarity
Class of deformations: congruences
Congruence-invariant (rigid) similarity:
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9Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Rigid correspondence
Class of deformations: congruences
Congruence-invariant similarity:
Congruence-invariant correspondence:
RIGID SIMILARITY RIGID CORRESPONDENCEINVARIANT SIMILARITY INVARIANT CORRESPONDENCE
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10Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Representation procedure is -invariant if it translates into
an isometry in , i.e., for every and , there exists
such that
Invariant representation (canonical forms)
Ingredients:
Class of shapes
Class of deformations
Embedding space and its isometry group
Representation procedure which given a shape
returns an embedding
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11Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
INVARIANT SIMILARITY
= INVARIANT REPRESENTATION + RIGID SIMILARITY
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12Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Invariant parametrization
Ingredients:
Class of shapes
Class of deformations
Parametrization space and its isometry group
Parametrization procedure which given a shape
returns a chart
Parametrization procedure is -invariant if it commutes with
up to an isometry in , i.e., for every and ,
there exists such that
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13Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
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14Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
INVARIANT CORRESPONDENCE
= INVARIANT PARAMETRIZATION + RIGID CORRESPONDENCE
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15Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Representation errors
Invariant similarity / correspondence is reduced to finding isometry
in embedding / parametrization space.
Such isometry does not exist and invariance holds approximately
Given parametrization domains and , instead of isometry
find a least distorting mapping .
Correspondence is
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16Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Dirichlet energy
Minimize Dirchlet energy functional
Equivalent to solving the Laplace equation
Boundary conditions
Solution (minimizer of Dirichlet energy) is a harmonic function.
N. Litke, M. Droske, M. Rumpf, P. Schroeder, SGP, 2005
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17Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Dirichlet energy
Caveat: Dirichlet functional is not invariant
Not parametrization-independent
Solution: use intrinsic quantities
Frobenius norm becomes
Hilbert-Schmidt norm
Intrinsic area element
Intrinsic Dirichlet energy functional
N. Litke, M. Droske, M. Rumpf, P. Schroeder, SGP, 2005
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18Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
The harmony of harmonic maps
Intrinsic Dirichlet energy functional
is the Cauchy-Green deformation tensor
Describes square of local change in distances
Minimizer is a harmonic map.
N. Litke, M. Droske, M. Rumpf, P. Schroeder, SGP, 2005
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19Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Physical interpretation
METAL MOULD
RUBBER SURFACE
= ELASTIC ENERGY CONTAINED IN THE RUBBER
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20Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Minimum-distortion correspondence
Ingredients:
Class of shapes
Class of deformations
Distortion function which given a correspondence
between two shapes assigns to it
a non-negative number
Minimum-distortion correspondence procedure
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21Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Minimum-distortion correspondence
Correspondence procedure is -invariant if distortion is
-invariant, i.e., for every , and ,
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22Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Minimum-distortion correspondence
CONGRUENCES CONFORMAL ISOMETRIES
Dirichlet energy Quadratic stressEuclidean norm
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23Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Minimum distortion correspondence
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24Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Intrinsic symmetries create distinct isometry-invariant minimum-
distortion correspondences, i.e., for every
Uniqueness & symmetry
The converse in not true, i.e. there might exist two distinct
minimum-distortion correspondences such that
for every
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25Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Partial correspondence
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26Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Measure coupling
Let be probability measures defined on and
The measure can be considered as a fuzzy correspondence
A measure on is a coupling of and if
for all measurable sets
Mémoli, 2007
(a metric space with measure is called a metric measure or mm space)
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27Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Intrinsic similarity
Hausdorff
Mémoli, 2007
Distance between subsets
of a metric space .
Gromov-Hausdorff
Distance between metric spaces
Wasserstein
Distance between subsets
of a metric measure
space .
Gromov-Wasserstein
Distance between metric
measure spaces
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28Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Minimum-distortion correspondence
Mémoli, 2007
Gromov-Hausdorff
Minimum-distortion correspondence
between metric spaces
Gromov-Wasserstein
Minimum-distortion fuzzy correspondence
between metric measure spaces
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29Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
TIMEReference Transferred texture
Texture transfer
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30Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Virtual body painting
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31Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Texture substitution
I’m Alice. I’m Bob.I’m Alice’s texture
on Bob’s geometry
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32Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
=
How to add two dogs?
+1
2
1
2
C A L C U L U S O F S H A P E S
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33Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Addition
creates displacement
Affine calculus in a linear space
Subtraction
creates direction
Affine combination
spans subspace
Convex combination (
)
spans polytopes
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34Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Affine calculus of functions
Affine space of functions
Subtraction
Addition
Affine combination
Possible because functions share a common domain
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35Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Affine calculus of shapes
?A. Bronstein, M. Bronstein, R. Kimmel, IEEE TVCG, 2006
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36Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Temporal super-resolution
TIME
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37Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Motion-compensated interpolation
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38Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Metamorphing
100%
Alice
100%
Bob
75% Alice
25% Bob
50% Alice
50% Bob
75% Alice
50% Bob
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39Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Face caricaturization
0 1 1.5
EXAGGERATED
EXPRESSION
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40Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Affine calculus of shapes
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41Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
What happened?
SHAPE SPACE IS NON-EUCLIDEAN!
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42Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Shape space
Shape space is an abstract manifold
Deformation fields of a shape are vectors in tangent space
Our affine calculus is valid only locally
Global affine calculus can be constructed by defining
trajectories
confined to the manifold
Addition
Combination
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43Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Choice of trajectory
Equip tangent space with an inner product
Riemannian metric on
Select to be a minimal geodesic
Addition: initial value problem
Combination: boundary value problem
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44Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Choice of metric
Deformation field of is called
Killing field if for every
Infinitesimal displacement by
Killing field is metric preserving
and are isometric
Congruence is always a Killing field
Non-trivial Killing field may not exist
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45Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Choice of metric
Inner product on
Induces norm
measures deviation of from Killing field
– defined modulo congruence
Add stiffening term
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46Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Minimum-distortion trajectory
Geodesic trajectory
Shapes along are as isometric as possible to
Guaranteeing no self-intersections is an open problem