1 numerical geometry of non-rigid shapes nonrigid correspondence & calculus of shapes non-rigid...
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1Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Non-Rigid Correspondenceand Calculus of Shapes
Of bodies changed to various forms, I sing.
Ovid, Metamorphoses
Alexander Bronstein, Michael Bronstein© 2008 All rights reserved. Web: tosca.cs.technion.ac.il
2Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Does a “natural” correspondence exist?
3Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Correspondence
accurate
‘‘
‘‘ makes sense
‘‘
‘‘ beautiful
‘‘
‘‘
Geometric Semantic Aesthetic
4Numerical geometry of non-rigid shapes Nonrigid 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 nonrigid deformations of the same object.
5Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
1D motivation: correspondence between curves
Two curves ,
Arclength parametrization
Unique up to initial point.
Reparametrize and to canonical
parametrization.
Find correspondence between intervals
Correspondence between and
6Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
The curse of higher dimension
We relied on existence of a “canonical” arclength parametrization.
Was possible due to existence of total ordering of points in 1D.
Surfaces (2D objects) do not have a total ordering.
Hence, no analogy of arclength parametrization for surfaces.
We can still find an invariant parametrization.
7Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Invariant parametrization
8Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Invariant parametrization
Ingredients:
Parametrization domain .
Group of deformations .
Shape .
Parametrization procedure, constructing
given the shape .
Desideratum: commutativity of the parametrization procedure with
the deformation:
How to construct such an invariant parametrization procedure?
9Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Compute minimal distortion embeddings
Define intrinsic parametrizations
Find rigid motion between parametrizations
Define correspondence between shapes
Canonical forms, bis
10Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Canonical forms
Embedding into the plane is not distortionless.
Invariance of parametrization holds only approximately
Generally, there exists no rigid motion bringing and
into perfect correspondence.
Relax assumptions on : allow to be any
bijection.
11Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Image processing insight
Given two grayscale images and , find the optical flow
(a.k.a. disparity map, motion field, etc.) minimizing the error
Local image misalignment
Given two shapes parametrized by and
, find minimizing
measures mismatch between
and .
12Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Problem: functional depends on parametrization .
Make it intrinsic replacing with .
has also to be parametrization-independent.
Example: normal misalignment
Problem: not isometry-invariant.
Make an intrinsic quantity, e.g.,
Not limited to geometric quantities.
May include photometric information.
Image processing approach
13Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Minimization problem
is ill-posed!
Add a regularization term
Tikhonov
Total variation
Healthy solution to ill-posed problems
14Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Regularizer has to be parametrization-invariant.
Frobenius norm is replaced by the Hilbert-Schmidt norm
is an intrinsic quantity in parametrization coordinates
is correspondence between shapes.
is the intrinsic gradient on .
is the norm in the tangent space of .
Intrinsic regularization
15Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Physical insight
Cauchy-Green deformation tensor
Square of local change of distance due
to
elastic deformation.
measures average distance
deformation.
= elastic energy
(a.k.a. Dirichlet energy) of thin rubber
sheet pressed against a mold .
16Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Dirichlet energy
We have been looking for a regularizer…
…but found a good measure for shape mismatch!
is an intrinsic quantity.
Minimizing gives a minimum deformation correspondence.
Minimizer is a harmonic map of to .
Some harmony
17Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Define a general energy functional
is intrinsic, hence can be expressed in terms of the metric
Correspondence problem becomes
GMDS with generalized stress.
Minimum distortion correspondence
18Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Minimum distortion correspondence
19Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
“Harmonic stress”:
gives the norm of the Cauchy-Green tensor
Our good old L2 stress
gives “as isometric as possible” correspondence.
Generalized stress
20Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Minimum distortion correspondence
Minimum distortion correspondence
Defined up to intrinsic symmetry of and .
21Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Partial correspondence
22Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Minimum distortion correspondence
MATLAB® intermezzo
23Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
TIMEReference Transferred texture
Texture transfer
24Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Virtual body painting
25Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Texture substitution
I’m Alice. I’m Bob.I’m Alice’s texture
on Bob’s geometry
26Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Given two shapes and , and a
correspondence .
We can define a convex combination of the two shapes
as a new shape, where the extrinsic location of each point is given by
Alternatively
Define deformation field transforming
into and express .
We can create new shapes by adding or subtracting other shapes.
We have a calculus of shapes.
Calculus of shapes
27Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Calculus of shapes in shape space
Extrapolation Interpolation
28Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Temporal super-resolution
TIME
29Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Motion-compensated interpolation
30Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Metamorphing
100%
Alice
100%
Bob
75% Alice
25% Bob
50% Alice
50% Bob
75% Alice
50% Bob
31Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Face caricaturization
0 1 1.5
EXAGGERATED
EXPRESSION
32Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
The quest for trajectory
In our definition of
linear trajectory between
corresponding points was used.
If and are extrinsically
similar, this gives good result.
Generally, there is no guarantee
that
is a valid shape:
Not a manifold
Self-intersecting
Even if shape is valid, it is not
necessarily isometric to .
33Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Given two shapes and , and a correspondence
, we want to find intermediate shapes .
For each point , define a trajectory for
such that
The big question:
How to select trajectories?
No self-intersections of intermediate meshes.
No distortion of intrinsic geometry in intermediate meshes.
The quest for trajectory
34Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Define deformation field
Tangent to the trajectory
In order for intermediate shapes
to be isometric to ,
must hold for all
and .
Deformation field
35Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Killing field: deformation field preserving
the metric.
Satisfies
for all and
May not exist, even if and
are isometric!
Remember: not every nonrigid shape
is continuously bendable…
The Killing field
Wilhelm Karl Joseph Killing(1847-1923)
36Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
As isometric as possible deformation field.
Define inner product between deformation fields of
Induces a norm
Problem: vanishes for being a rigid motion.
Solution: add stiffening term:
Metric for deformation fields
37Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
We have a Riemannian metric on the space of shapes.
Find a minimal geodesic connecting between and .
Boundary conditions .
Minimum deviation from Killing field along the path.
As isometric as possible morph.
As isometric as possible morph
38Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Summary and suggested reading
Invariant surface parametrization
G. Zigelman, R. Kimmel, and N. Kiryati, Texture mapping using surface flatteningvia multi-dimensional scaling, IEEE TVCG 9 (2002), no. 2, 198–207.
An image processing insight to correspondence
B. K. P. Horn and B. G. Schunck, Determining optical flow, Artificial Intelligence17 (1981), no. 1-3, 185–203.
Harmonic embeddings
N. Litke, M. Droske, M. Rumpf, and P. Schroder, An image processing approachto surface matching.
Minimum distortion correspondence
Calculus of shapes
A.M. Bronstein, M.M. Bronstein, R. Kimmel, Calculus of non-rigid surfaces forgeometry and texture manipulation, IEEE TVCG 13 (2007), no. 5, 903–913.
39Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes
Summary and suggested reading
Morphing
M. Alexa, Recent advances in mesh morphing, Computer Graphics Forum 21(2002), no. 2, 173–196.
V. Surazhsky and C. Gotsman, Controllable morphing of compatible planartriangulations, ACM Trans. Graphics 20 (2001), no. 4, 203–231.
M. Kilian, N. J. Mitra, and H. Pottmann, Geometric modeling in shape space,ACM Trans. Graphics 26 (2007), no. 3.