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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
2Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
“Natural” correspondence?
3Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Correspondence
accurate
‘‘
‘‘ makes sense
‘‘
‘‘ beautiful
‘‘
‘‘Geometric Semantic Aesthetic
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.
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 ,
6Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
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
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:
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
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
11Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
INVARIANT SIMILARITY
= INVARIANT REPRESENTATION + RIGID SIMILARITY
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
13Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
14Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
INVARIANT CORRESPONDENCE
= INVARIANT PARAMETRIZATION + RIGID CORRESPONDENCE
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
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
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
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
19Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Physical interpretation
METAL MOULD
RUBBER SURFACE
= ELASTIC ENERGY CONTAINED IN THE RUBBER
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
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 ,
22Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Minimum-distortion correspondence
CONGRUENCES CONFORMAL ISOMETRIES
Dirichlet energy Quadratic stressEuclidean norm
23Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Minimum distortion correspondence
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
25Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Partial correspondence
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)
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
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
29Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
TIMEReference Transferred texture
Texture transfer
30Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Virtual body painting
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
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
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
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
35Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Affine calculus of shapes
?A. Bronstein, M. Bronstein, R. Kimmel, IEEE TVCG, 2006
36Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Temporal super-resolution
TIME
37Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Motion-compensated interpolation
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
39Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Face caricaturization
0 1 1.5
EXAGGERATED
EXPRESSION
40Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
Affine calculus of shapes
41Numerical Geometry of Non-Rigid Shapes Invariant Correspondence & Calculus of Shapes
What happened?
SHAPE SPACE IS NON-EUCLIDEAN!
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
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
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
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
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