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

Does a “natural” correspondence exist?

Correspondence

accurate

‘‘

‘‘ makes sense

‘‘

‘‘ beautiful

‘‘

Geometric Semantic Aesthetic

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.

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

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.

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?

Compute minimal distortion embeddings

Define intrinsic parametrizations

Find rigid motion between parametrizations

Define correspondence between shapes

Canonical forms, bis

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.

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

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

Minimization problem

is ill-posed!

Add a regularization term

Tikhonov

Total variation

Healthy solution to ill-posed problems

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

Physical insight

Cauchy-Green deformation tensor

Square of local change of distance due

elastic deformation.

measures average distance

deformation.

= elastic energy

(a.k.a. Dirichlet energy) of thin rubber

sheet pressed against a mold .

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

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

“Harmonic stress”:

gives the norm of the Cauchy-Green tensor

Our good old L2 stress

gives “as isometric as possible” correspondence.

Generalized stress

Defined up to intrinsic symmetry of and .

Partial correspondence

MATLAB® intermezzo

TIMEReference Transferred texture

Texture transfer

Virtual body painting

Texture substitution

I’m Alice. I’m Bob.I’m Alice’s texture

on Bob’s geometry

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

Calculus of shapes in shape space

Extrapolation Interpolation

Temporal super-resolution

Motion-compensated interpolation

Metamorphing

75% Alice

25% Bob

50% Alice

50% Bob

75% Alice

50% Bob

Face caricaturization

0 1 1.5

EXAGGERATED

EXPRESSION

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

is a valid shape:

Not a manifold

Self-intersecting

Even if shape is valid, it is not

necessarily isometric to .

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

Define deformation field

Tangent to the trajectory

In order for intermediate shapes

to be isometric to ,

must hold for all

Deformation field

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)

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

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

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.

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.

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.

1 Numerical geometry of non-rigid shapes Nonrigid Correspondence & Calculus of Shapes Non-Rigid...

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