1. Lecture IV – Invariant Correspondence
Numerical geometry
of shapes
non-rigid
Lecture IV – Invariant Correspondence
and Calculus of Shapes
Alex Bronstein 1

2. “Natural” correspondence?

3. ‘ ‘ ‘ Geometric Semantic Aesthetic accurate makes sense beautiful
Correspondence
Geometric
Semantic
Aesthetic
accurate ‘
makes sense ‘
beautiful ‘

4. 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.

5. 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 ,

7. Rigid similarity Class of deformations: congruences
Congruence-invariant (rigid) similarity:
Closest point correspondence between , parametrized by
Its distortion
Minimize distortion over all possible congruences

8. Rigid correspondence Class of deformations: congruences
Congruence-invariant similarity:
Congruence-invariant correspondence:
INVARIANT SIMILARITY  INVARIANT CORRESPONDENCE
RIGID SIMILARITY  RIGID CORRESPONDENCE

9. 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
Representation procedure is -invariant if it translates into
an isometry in , i.e., for every and , there exists
such that

10. INVARIANT SIMILARITY
= INVARIANT REPRESENTATION + RIGID SIMILARITY

11. 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

13. INVARIANT CORRESPONDENCE
= INVARIANT PARAMETRIZATION + RIGID CORRESPONDENCE

14. 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

15. Dirichlet energy Minimize Dirchlet energy functional
Equivalent to solving the Laplace equation
Boundary conditions
Solution (minimizer of Dirichlet energy) is a harmonic function.

16. 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

17. 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.

18. Physical interpretation
RUBBER SURFACE
METAL MOULD
= ELASTIC ENERGY CONTAINED IN THE RUBBER

19. 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

20. Minimum-distortion correspondence
Correspondence procedure is -invariant if distortion is
-invariant, i.e., for every , and ,
Proof:

21. Minimum-distortion correspondence
Euclidean norm
Dirichlet energy
Quadratic stress
CONGRUENCES
CONFORMAL
ISOMETRIES

22. Minimum distortion correspondence

23. Uniqueness MINIMUM-DISTORTION CORRESPONDENCE IS NOT UNIQUE
IS MINIMUM-DISTORTION CORRESPONDENCE UNIQUE?

24. Symmetry Shape is symmetric, if there exists a congruence such that
Yes, I am symmetric.
Am I symmetric?

25. Symmetry
I am symmetric.
What about us?

26. Symmetry Shape is symmetric, if there exists
a non-trivial automorphism
which is metric-preserving, i.e.,
Shape is symmetric, if there
exists a congruence
such that
Symmetry group = self-isometry group

27. Symmetry: extrinsic vs. intrinsic
Extrinsic symmetry
Intrinsic symmetry

28. Symmetry: extrinsic vs. intrinsic
I am extrinsically symmetric.
We are all intrinsically symmetric.
We are extrinsically asymmetric.

29. Uniqueness & symmetry
The converse in not true, i.e. there might exist two distinct
minimum-distortion correspondences such that
for every
Intrinsic symmetries create distinct isometry-invariant minimum-
distortion correspondences, i.e., for every

30. Partial correspondence

31. Texture transfer
TIME
Reference
Transferred texture

32. Virtual body painting

33. Texture substitution I’m Alice. I’m Bob. I’m Alice’s texture
on Bob’s geometry

34. How to add two dogs? + = 1 2 1 2
CALCULUS OF SHAPES

35. Affine calculus in a linear space
Subtraction
creates direction
Addition
creates displacement
Affine combination
spans subspace
Convex combination ( )
spans polytopes

36. Affine calculus of functions
Affine space of functions
Subtraction
Addition
Affine combination
Possible because functions share a common domain

37. Affine calculus of shapes?

38. 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

39. Affine calculus of shapes
CALCULUS OF SHAPES = CALCULUS OF VECTOR FIELDS
Addition:
Subtration:
Combination:

40. Temporal super-resolution (frame rate up-conversion)
TIME
Image processing: motion-compensated interpolation
Geometry processing: deformation-compensated interpolation

41. Metamorphing 100% Alice 75% Alice 25% Bob 50% Alice 50% Bob 75% Alice

42. Face caricaturization
EXAGGERATED
EXPRESSION 1
1.5

43. Calculus of shapes

44. What happened?
SHAPE SPACE IS NON-EUCLIDEAN!

45. 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

46. 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

47. 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

48. Choice of metric Inner product on Induces norm
measures deviation of from Killing field
– defined modulo congruence
Add stiffening term

49. Minimum-distortion trajectory
Geodesic trajectory
Shapes along are as isometric as possible to
Guaranteeing no self-intersections is an open problem

50. 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 50