1 numerical geometry of non-rigid shapes spectral embedding spectral embedding lecture 6 ©...

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1 ical geometry of non-rigid shapes Spectral embedding Spectral embedding Lecture 6 © Alexander & Michael Bronstein tosca.cs.technion.ac.il/book Numerical geometry of non-rigid shapes Stanford University, Winter 2009

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1Numerical geometry of non-rigid shapes Spectral embedding

Spectral embeddingLecture 6

© Alexander & Michael Bronsteintosca.cs.technion.ac.il/book

Numerical geometry of non-rigid shapesStanford University, Winter 2009

2Numerical geometry of non-rigid shapes Spectral embedding

A mathematical exercise

Assume points with the metric are isometrically

embeddable into

Then, there exists a canonical form such that

for all

We can also write

3Numerical geometry of non-rigid shapes Spectral embedding

A mathematical exercise

Since the canonical form is defined up to

isometry, we can arbitrarily set

4Numerical geometry of non-rigid shapes Spectral embedding

A mathematical exercise

Conclusion: if points are isometrically embeddable into

then

Element of

a matrix

Element of an

matrix

Note: can be defined in different ways!

5Numerical geometry of non-rigid shapes Spectral embedding

Gram matrices

A matrix of inner products of the form

is called a Gram matrix

Jørgen Pedersen Gram(1850-1916)

Properties:

(positive semidefinite)

6Numerical geometry of non-rigid shapes Spectral embedding

Back to our problem…

Isaac Schoenberg(1903-1990)

[Schoenberg, 1935]: Points with the metric can

be isometrically embedded into a Euclidean space if and only if

If points with the metric

can be isometrically embedded into , then

can be realized as a Gram matrix of rank ,

which is positive semidefinite

A positive semidefinite matrix of rank

can be written as

giving the canonical form

7Numerical geometry of non-rigid shapes Spectral embedding

Classic MDS

Usually, a shape is not isometrically embeddable into a Eucludean space,

implying that (has negative eignevalues)

We can approximate by a Gram matrix of rank

Keep m largest eignevalues

Canonical form computed as

Method known as classic MDS (or classical scaling)

8Numerical geometry of non-rigid shapes Spectral embedding

Properties of classic MDS

Nested dimensions: the first dimensions of an -

dimensional

canonical form are equal to an -dimensional canonical form

Global optimization problem – no local convergence

Requires computing a few largest eigenvalues of a real symmetric

matrix,

which can be efficiently solved numerically (e.g. Arnoldi and Lanczos)

The error introduced by taking instead of can be quantified as

Classic MDS minimizes the strain

9Numerical geometry of non-rigid shapes Spectral embedding

MATLAB® intermezzoClassic MDS

Canonical forms

10Numerical geometry of non-rigid shapes Spectral embedding

Classical scaling example

1

B

D

A

C1

1

1

1

B

A C2

A

A 1

B C D

B

C

D

2 1

1 1 1

2 1 1

1 1 1

D

1

11Numerical geometry of non-rigid shapes Spectral embedding

Local methods

Make the embedding preserve local properties of the shape

If , then is small. We want the corresponding

distance in the embedding space to be small

Map neighboring points to neighboring points

12Numerical geometry of non-rigid shapes Spectral embedding

Local methods

Think globally, act locally

Local criterion how far apart the embedding takes neighboring points

“ ”David Brower

Global criterion

where

13Numerical geometry of non-rigid shapes Spectral embedding

Laplacian matrix

where is an matrix with elements

Matrix formulationRecall stress

derivation

in LS-MDS

is called the Laplacian matrix

has zero eigenvalue

14Numerical geometry of non-rigid shapes Spectral embedding

Local methods

Compute canonical form by solving the optimization problem

Trivial solution ( ): points can

collapse to a single point

Introduce a constraint

avoiding trivial solution

15Numerical geometry of non-rigid shapes Spectral embedding

Minimum eigenvalue problems

Lets look at a simplified case: one-dimensional embedding

Geometric intuition: find a unit vector shortened the most by the action

of the matrix

Express the problem using eigendecomposition

16Numerical geometry of non-rigid shapes Spectral embedding

Solution of the problem

is given as the smallest non-trivial eigenvectors of

The smallest eigenvalue is zero and the corresponding eigenvector is

constant (collapsing to a point)

Minimum eigenvalue problems

17Numerical geometry of non-rigid shapes Spectral embedding

Laplacian eigenmaps

Compute the canonical form by finding the smallest non-trivial

eigenvectors of

Method called Laplacian eigenmap [Belkin&Niyogi]

is sparse (computational advantage for eigendecomposition)

We need the lower part of the spectrum of

Nested dimensions like in classic MDS

18Numerical geometry of non-rigid shapes Spectral embedding

Laplacian eigenmaps example

Classic MDS Laplacian eigenmap

19Numerical geometry of non-rigid shapes Spectral embedding

Continuous case

Consider a one-dimensional embedding (due to nested dimension property,

each dimension can be considered separately)

We were trying to find a map that maps neighboring

points to neighboring points

In the continuous case, we have a smooth map on surface

Let be a point on and be a point obtained by an infinitesimal

displacement from by a vector in the tangent plane

By Taylor expansion,

Inner product on tangent space (metric tensor)

20Numerical geometry of non-rigid shapes Spectral embedding

Continuous case

By the Cauchy-Schwarz inequality

implying that is small if is small: i.e., points

close to are mapped close to

Continuous local criterion:

Continuous global criterion:

21Numerical geometry of non-rigid shapes Spectral embedding

Continuous analog of Laplacian eigenmaps

Canonical form computed as the minimization problem

where:

Stokes theorem

We can rewrite

is the space of square-integrable functions on

22Numerical geometry of non-rigid shapes Spectral embedding

Laplace-Beltrami operator

The operator is called Laplace-Beltrami

operator

Laplace-Beltrami operator is a generalization of Laplacian to manifolds

In the Euclidean plane,

Intrinsic property of the shape (invariant to isometries)

Note: we define Laplace-Beltrami operator with minus, unlike many books

In coordinate notation

23Numerical geometry of non-rigid shapes Spectral embedding

Laplace-Beltrami

Pierre Simon de Laplace (1749-1827)

Eugenio Beltrami(1835-1899)

24Numerical geometry of non-rigid shapes Spectral embedding

Properties of Laplace-Beltrami operator

Let be smooth functions on the surface . Then the

Laplace-Beltrami operator has the following properties

Constant eigenfunction: for any

Symmetry:

Locality: is independent of for any points

Euclidean case: if is Euclidean plane and

then

Positive semidefinite:

25Numerical geometry of non-rigid shapes Spectral embedding

Continuous vs discrete problem

Continuous:

Discrete:

Laplace-Beltrami

operator

Laplacian

26Numerical geometry of non-rigid shapes Spectral embedding

To see the sound

Chladni’s experimental setup allowing to visualize acoustic waves

Ernst Chladni ['kladnɪ] (1715-1782)

E. Chladni, Entdeckungen über die Theorie des Klanges

27Numerical geometry of non-rigid shapes Spectral embedding

Chladni plates

Patterns seen by Chladni are solutions to stationary Helmholtz equation

Solutions of this equation are eigenfunction of Laplace-Beltrami operator

28Numerical geometry of non-rigid shapes Spectral embedding

Laplace-Beltrami operator

The first eigenfunctions of the Laplace-Beltrami operator

29Numerical geometry of non-rigid shapes Spectral embedding

Laplace-Beltrami operator

An eigenfunction of the Laplace-Beltrami operator computed on different deformations of the shape, showing the invariance of the

Laplace-Beltrami operator to isometries

30Numerical geometry of non-rigid shapes Spectral embedding

Laplace-Beltrami spectrum

Eigendecomposition of Laplace-Beltrami operator of a compact shape

gives a discrete set of eigenvalues and eigenfunctions

The eigenvalues and eigenfunctions are isometry invariant

Since the Laplace-Beltrami operator is symmetric, eigenfunctions

form an orthogonal basis for

31Numerical geometry of non-rigid shapes Spectral embedding

Shape DNA

[Reuter et al. 2006]: use the Laplace-Beltrami spectrum as an

isometry-invariant shape descriptor (“shape DNA”)

Laplace-Beltrami spectrumImages: Reuter et al.

32Numerical geometry of non-rigid shapes Spectral embedding

Shape DNA

Shape similarity using Laplace-Beltrami spectrum

Images: Reuter et al.

33Numerical geometry of non-rigid shapes Spectral embedding

Uniqueness of representation

ISOMETRIC SHAPES ARE ISOSPECTRAL

ARE ISOSPECTRAL SHAPES ISOMETRIC?

34Numerical geometry of non-rigid shapes Spectral embedding

Mark Kac(1914-1984)

Can one hear the shape of the drum?“ ”

More prosaically: can one reconstruct the shape

(up to an isometry) from its Laplace-Beltrami spectrum?

35Numerical geometry of non-rigid shapes Spectral embedding

To hear the shape

In Chladni’s experiments, the spectrum describes acoustic characteristics

of the plates (“modes” of vibrations)

What can be “heard” from the spectrum:

Total Gaussian curvature

Euler characteristic

Area

Can we “hear” the metric?

36Numerical geometry of non-rigid shapes Spectral embedding

One cannot hear the shape of the drum!

[Gordon et al. 1991]:

Counter-example of isospectral but not isometric shapes

37Numerical geometry of non-rigid shapes Spectral embedding

GPS embedding

The eigenvalues and the eigenfunctions of the Laplace-Beltrami operator

uniquely determine the metric tensor of the shape

I.e., one can recover the shape up to an isometry from

[Rustamov, 2007]: Global Point Signature (GPS) embedding

An infinite-dimensional canonical form

Unique (unlike MDS-based canonical form, defined up to isometry)

Must be truncated for practical computation

38Numerical geometry of non-rigid shapes Spectral embedding

Discrete Laplace-Beltrami operator

Let the surface be sampled at points and represented as a

triangular mesh , and let

Discrete version of the Laplace-Beltrami operator

Can be expressed as a matrix

Discrete analog of constant eigenfunction property is satisfied by definition

39Numerical geometry of non-rigid shapes Spectral embedding

Discrete vs discretized

Continuous surface

Laplace-Beltrami operator

Discretize the surface

Discrete Laplace-Beltrami operator

Discretize Laplace-Beltrami operator, preserving some

of the continuous properties

Construct graph Laplacian

Discretized Laplace-Beltrami operator

40Numerical geometry of non-rigid shapes Spectral embedding

Properties of discrete Laplace-Beltrami operator

The discrete analog of the properties of the continuous Laplace-Betrami

operator is

Symmetry:

Locality: if are not directly connected

Euclidean case: if is Euclidean plane,

Positive semidefinite:

In order for the discretization to be consistent,

Convergence: solution of discrete PDE with converges to the

solution

of continuous PDE with for

41Numerical geometry of non-rigid shapes Spectral embedding

No free lunch

Laplacian matrix we used in Laplacian eigenmaps does not converge to the

continuous Laplace-Beltrami operator

There exist many other approximations of the Laplace-Beltrami operator,

satisfying different properties

[Wardetzky, 2007]: there is no discretization of the Laplace-Beltrami

operator satisfying simultaneously all the desired properties