Invariant correspondence - PowerPoint Presentation

Slide1

Invariant correspondence

and calculus of shapes

© Alexander & Michael Bronstein, 2006-2010tosca.cs.technion.ac.il/book

VIPS Advanced School onNumerical Geometry of Non-Rigid Shapes University of Verona, April 2010Slide2

“Natural” correspondence?

?Slide3

Correspondence

accurate

‘

‘

‘

‘

makes sense

‘

‘

‘

‘

beautiful

‘

‘

‘

‘

Geometric

Semantic

AestheticSlide4

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.Slide5

Invariant correspondence

Ingredients:

Class of

shapesClass 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 ,Slide6
Slide7

Invariant similarity (reminder)

Ingredients:

Class of shapes

Class of deformationsDistance

Distance is -

invariant

if for every and everySlide8

Closest point correspondence

between , parametrized by

Its distortion

Minimize distortion over all possible congruencesRigid similarity

Class of deformations:

congruences

Congruence-invariant (rigid)

similarity

:Slide9

Rigid correspondence

Class of deformations: congruences

Congruence-invariant similarity:

Congruence-invariant correspondence:

RIGID SIMILARITY

 RIGID CORRESPONDENCE

INVARIANT SIMILARITY

 INVARIANT CORRESPONDENCESlide10

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 embeddingSlide11

INVARIANT SIMILARITY

= INVARIANT REPRESENTATION + RIGID SIMILARITYSlide12

Invariant parametrization

Ingredients:

Class of shapes

Class of deformationsParametrization 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 thatSlide13
Slide14

INVARIANT CORRESPONDENCE

= INVARIANT PARAMETRIZATION + RIGID CORRESPONDENCESlide15

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 isSlide16

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

, 2005Slide17

Dirichlet energy

Caveat: Dirichlet functional is not invariant

Not parametrization-independent

Solution: use intrinsic quantitiesFrobenius norm becomes

Hilbert-Schmidt norm

Intrinsic area element

Intrinsic Dirichlet energy functional

N. Litke, M. Droske, M. Rumpf, P. Schroeder,

SGP

, 2005Slide18

The harmony of harmonic maps

Intrinsic Dirichlet energy functional

is the

Cauchy-Green deformation tensorDescribes square of local change in distances

Minimizer is a harmonic map.

N. Litke, M. Droske, M. Rumpf, P. Schroeder,

SGP

, 2005Slide19

Physical interpretation

METAL MOULD

RUBBER SURFACE

= ELASTIC ENERGY CONTAINED IN THE RUBBERSlide20

Minimum-distortion correspondence

Ingredients:

Class of shapesClass of deformations

Distortion function which given a correspondence between two shapes assigns to it

a non-negative number Minimum-distortion correspondence

procedureSlide21

Minimum-distortion correspondence

Correspondence procedure is -

invariant

if distortion is

-invariant, i.e., for every , and ,Slide22

Minimum-distortion correspondence

CONGRUENCES

CONFORMAL

ISOMETRIES

Dirichlet energy

Quadratic stress

Euclidean normSlide23

Minimum distortion correspondenceSlide24

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 Slide25

Partial correspondenceSlide26

Measure coupling

Let be probability measures defined on and

The measure can be considered as a fuzzy correspondence

A measure on is a

coupling

of and if

for all measurable sets

Mémoli, 2007

(a metric space with measure is called a

metric measure

or

mm space

)Slide27

Intrinsic similarity

Hausdorff

Mémoli, 2007

Distance between subsets

of a metric space .

Gromov-Hausdorff

Distance between

metric spaces

Wasserstein

Distance between subsets

of a metric measure space .

Gromov-Wasserstein

Distance between

metric measure spacesSlide28

Minimum-distortion correspondence

Mémoli, 2007

Gromov-Hausdorff

Minimum-distortion correspondence between metric spaces

Gromov-Wasserstein

Minimum-distortion fuzzy correspondence between

metric measure spacesSlide29

TIME

Reference

Transferred texture

Texture transferSlide30

Virtual body paintingSlide31

Texture substitution

I’m

Alice

.I’m Bob.

I’m

Alice

’s texture

on

Bob

’s geometrySlide32

=

How to add two dogs?

+

1

2

1

2

C A L C U L U S O F S H A P E SSlide33

Addition

creates displacement

Affine calculus in a linear space

Subtraction

creates direction

Affine combination

spans subspace

Convex combination

( )

spans polytopesSlide34

Affine calculus of functions

Affine space of functions

Subtraction

Addition

Affine combination

Possible because functions share a

common domainSlide35

Affine calculus of shapes

?

A. Bronstein, M. Bronstein, R. Kimmel,

IEEE TVCG

, 2006Slide36

Temporal super-resolution

TIMESlide37

Motion-compensated interpolationSlide38

Metamorphing

100%

Alice

100%Bob

75% Alice

25%

Bob

50%

Alice

50%

Bob

75%

Alice

50%

BobSlide39

Face caricaturization

0

1

1.5

EXAGGERATED

EXPRESSIONSlide40

Affine calculus of shapesSlide41

What happened?

SHAPE SPACE IS NON-EUCLIDEAN!Slide42

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

AdditionCombinationSlide43

Choice of trajectory

Equip tangent space with an inner product

Riemannian metric on Select to be a minimal geodesic

Addition: initial value

problemCombination: boundary value

problemSlide44

Choice of metric

Deformation field of is called

Killing field if for every

Infinitesimal displacement byKilling field is metric preserving and are

isometricCongruence is always a Killing field

Non-trivial

Killing field

may not exist

Slide45

Choice of metric

Inner product on

Induces

norm

measures

deviation

of from Killing field

– defined modulo

congruence

Add

stiffening termSlide46

Minimum-distortion trajectory

Geodesic

trajectoryShapes along are

as isometric as possible to Guaranteeing no self-intersections is an open problem