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
2Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
“Natural” correspondence?
3Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Correspondence
accurate
‘‘
‘‘ makes sense
‘‘
‘‘ beautiful
‘‘
‘‘Geometric Semantic Aesthetic
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.
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 ,
6Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
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:
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
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
10Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
INVARIANT SIMILARITY
= INVARIANT REPRESENTATION + RIGID SIMILARITY
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
12Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
13Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
INVARIANT CORRESPONDENCE
= INVARIANT PARAMETRIZATION + RIGID CORRESPONDENCE
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
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.
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
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.
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
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
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:
21Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Minimum-distortion correspondence
CONGRUENCES CONFORMAL ISOMETRIES
Dirichlet energy Quadratic stressEuclidean norm
22Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Minimum distortion correspondence
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
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
25Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
What about us?
Symmetry
I am symmetric.
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.,
27Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Symmetry: extrinsic vs. intrinsic
Extrinsic symmetry Intrinsic symmetry
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.
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
30Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Partial correspondence
31Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
TIMEReference Transferred texture
Texture transfer
32Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Virtual body painting
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
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
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
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
37Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Affine calculus of shapes
?
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
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
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
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
42Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Face caricaturization
0 1 1.5
EXAGGERATED
EXPRESSION
43Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
Calculus of shapes
44Numerical geometry of non-rigid shapes Lecture IV - Invariant Correspondence & Calculus of Shapes
What happened?
SHAPE SPACE IS NON-EUCLIDEAN!
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
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
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
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
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
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