1 Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes...
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![Page 1: 1 Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes Numerical geometry of shapes Lecture IV – Invariant.](https://reader036.fdocuments.us/reader036/viewer/2022062500/56649d485503460f94a23309/html5/thumbnails/1.jpg)
1Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Numerical geometryof shapes
Lecture IV – Invariant Correspondenceand Calculus of Shapes
non-rigid
Alex Bronstein
![Page 2: 1 Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes Numerical geometry of shapes Lecture IV – Invariant.](https://reader036.fdocuments.us/reader036/viewer/2022062500/56649d485503460f94a23309/html5/thumbnails/2.jpg)
2Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
“Natural” correspondence?
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3Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Correspondence
accurate
‘‘
‘‘ makes sense
‘‘
‘‘ beautiful
‘‘
‘‘Geometric Semantic Aesthetic
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4Numerical geometry of non-rigid shapes Lecture IV - 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 Lecture IV - 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 Lecture IV - Invariant Correspondence & Calculus of Shapes
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7Numerical geometry of non-rigid shapes Lecture IV - 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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8Numerical geometry of non-rigid shapes Lecture IV - 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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9Numerical geometry of non-rigid shapes Lecture IV - 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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10Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
INVARIANT SIMILARITY
= INVARIANT REPRESENTATION + RIGID SIMILARITY
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11Numerical geometry of non-rigid shapes Lecture IV - 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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12Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
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13Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
INVARIANT CORRESPONDENCE
= INVARIANT PARAMETRIZATION + RIGID CORRESPONDENCE
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14Numerical geometry of non-rigid shapes Lecture IV - 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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15Numerical geometry of non-rigid shapes Lecture IV - 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.
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16Numerical geometry of non-rigid shapes Lecture IV - 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
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17Numerical geometry of non-rigid shapes Lecture IV - 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.
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18Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Physical interpretation
METAL MOULD
RUBBER SURFACE
= ELASTIC ENERGY CONTAINED IN THE RUBBER
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19Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Minimum-distortion correspondence
Ingredients:
Class of shapes
Class (groupoid) 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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20Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Minimum-distortion correspondence
Correspondence procedure is -invariant if distortion is
-invariant, i.e., for every , and ,
Proof:
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21Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Minimum-distortion correspondence
CONGRUENCES CONFORMAL ISOMETRIES
Dirichlet energy Quadratic stressEuclidean norm
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22Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Minimum distortion correspondence
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23Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Uniqueness
IS MINIMUM-DISTORTION CORRESPONDENCE UNIQUE?
MINIMUM-DISTORTION CORRESPONDENCE IS NOT UNIQUE
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24Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Shape is symmetric, if there
exists a congruence
such that
Am I symmetric?Yes, I am symmetric.
Symmetry
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25Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
What about us?
Symmetry
I am symmetric.
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26Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Symmetry
Shape is symmetric, if there
exists a congruence
such that
Symmetry group = self-isometry group
Shape is symmetric, if there exists
a non-trivial automorphism
which is metric-preserving, i.e.,
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27Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Symmetry: extrinsic vs. intrinsic
Extrinsic symmetry Intrinsic symmetry
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28Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Symmetry: extrinsic vs. intrinsic
I am extrinsically symmetric. We are extrinsically asymmetric.We are all intrinsically symmetric.
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29Numerical geometry of non-rigid shapes Lecture IV - 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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30Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Partial correspondence
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31Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
TIMEReference Transferred texture
Texture transfer
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32Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Virtual body painting
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33Numerical geometry of non-rigid shapes Lecture IV - 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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34Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
=
How to add two dogs?
+1
2
1
2
CALCULUS OF SHAPES
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35Numerical geometry of non-rigid shapes Lecture IV - 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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36Numerical geometry of non-rigid shapes Lecture IV - 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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37Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Affine calculus of shapes
?
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38Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Affine calculus of shapes
Ingredients:
Space of shapes embedded in
Class of correspondences
Space of deformation fields in
Since all shapes are corresponding, they can be jointly parametrized
in some by
Shape = vector field
Correspondences = joint parametrizations
Deformation field = vector field
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39Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Addition:
Subtration:
Combination:
Affine calculus of shapes
CALCULUS OF SHAPES = CALCULUS OF VECTOR FIELDS
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40Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Temporal super-resolution (frame rate up-conversion)
TIME
Image processing: motion-compensated interpolation
Geometry processing: deformation-compensated interpolation
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41Numerical geometry of non-rigid shapes Lecture IV - 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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42Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Face caricaturization
0 1 1.5
EXAGGERATED
EXPRESSION
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43Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Calculus of shapes
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44Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
What happened?
SHAPE SPACE IS NON-EUCLIDEAN!
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45Numerical geometry of non-rigid shapes Lecture IV - 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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46Numerical geometry of non-rigid shapes Lecture IV - 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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47Numerical geometry of non-rigid shapes Lecture IV - 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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48Numerical geometry of non-rigid shapes Lecture IV - 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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49Numerical geometry of non-rigid shapes Lecture IV - 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
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50Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Summary
Invariant correspondence = invariant similarity
Invariant parametrization
Minimum-distortion correspondence
Symmetry – self similarity
Extrinsic – self-congruence
Intrinsic – self-isometry
Affine calculus of shapes
Naïve linear model
Manifold of shapes
As isometric as possible