1. Invariant correspondence
and calculus of shapes
© Alexander & Michael Bronstein,
tosca.cs.technion.ac.il/book
VIPS Advanced School on
Numerical Geometry of Non-Rigid Shapes
University of Verona, April 2010 1 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. Invariant similarity (reminder)
Ingredients:
Class of shapes
Class of deformations
Distance
Distance is -invariant if for every and every

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

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

10. 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

11. INVARIANT SIMILARITY
= INVARIANT REPRESENTATION + RIGID SIMILARITY

12. 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

14. INVARIANT CORRESPONDENCE
= INVARIANT PARAMETRIZATION + RIGID CORRESPONDENCE

15. 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

16. 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

17. 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

18. 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

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

20. 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

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

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

23. Minimum distortion correspondence

24. 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

25. Partial correspondence

26. Measure coupling Let be probability measures defined on and
(a metric space with measure is called a metric measure or mm space)
A measure on is a coupling of and if
for all measurable sets
The measure can be considered as a fuzzy correspondence
Mémoli, 2007

27. Intrinsic similarity Hausdorff Wasserstein Gromov-Hausdorff
Distance between subsets
of a metric space
Distance between subsets
of a metric measure space
Gromov-Hausdorff
Gromov-Wasserstein
Distance between metric spaces
Distance between metric measure spaces
Mémoli, 2007

28. Minimum-distortion correspondence
Gromov-Hausdorff
Minimum-distortion correspondence between metric spaces
Gromov-Wasserstein
Minimum-distortion fuzzy correspondence between metric measure spaces
Mémoli, 2007

29. Texture transfer
TIME
Reference
Transferred texture

30. Virtual body painting

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

32. 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

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

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

35. ? Affine calculus of shapes
A. Bronstein, M. Bronstein, R. Kimmel, IEEE TVCG, 2006

36. Temporal super-resolution
TIME

37. Motion-compensated interpolation

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

39. Face caricaturization
EXAGGERATED
EXPRESSION 1
1.5

40. Affine calculus of shapes

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

42. 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

43. 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

44. 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

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

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