S p e - PowerPoint Presentation

Slide1

S

pectral methods

© Alexander & Michael Bronstein, 2006-2009

© Michael Bronstein, 2010tosca.cs.technion.ac.il/book

048921 Advanced topics in vision

Processing and Analysis of Geometric Shapes

EE

Technion

, Spring 2010Slide2

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 writeSlide3

A mathematical exercise

Since the canonical form is defined up to

isometry, we can arbitrarily setSlide4

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!Slide5

Gram matrices

A matrix of inner products of the formis called a Gram matrix

Jørgen Pedersen Gram(1850-1916)

Properties:

(

positive semidefinite

)

Slide6

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 formSlide7

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

)Slide8

Properties of classic MDS

Nested dimensions: the first dimensions of an -dimensionalcanonical form are equal to an -dimensional canonical form

Global optimization problem

– no local convergenceRequires 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

strainSlide9

MATLAB® intermezzo

Classic MDSCanonical formsSlide10

Classical scaling example

1

B

D

A

C

1

1

1

1

B

A

C

2

A

A

1

B

C

D

B

C

D

2

1

1

1

1

2

1

1

1

1

1

D

1Slide11

Deformation

Deformation

+Topology

Topological invarianceSlide12

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 pointsSlide13

Local methods

Think globally, act locally

Local criterion how far apart the embedding takes neighboring points

“

”

David Brower

Global criterion

whereSlide14

Laplacian matrix

where is an matrix with elementsMatrix formulation

Recall stress derivation

in LS-MDS

is called the

Laplacian matrix

has zero eigenvalueSlide15

Local methods

Compute canonical form by solving the optimization problem

Trivial solution ( ): points can collapse to a single point

Introduce a constraint avoiding trivial solutionSlide16

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 eigendecompositionSlide17

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 problemsSlide18

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 MDSSlide19

Laplacian eigenmaps example

Classic MDS

Laplacian eigenmapSlide20

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)Slide21

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:Slide22

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 onSlide23

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 Slide24

Laplace-Beltrami

Pierre Simon de Laplace (1749-1827)

Eugenio Beltrami

(1835-1899)Slide25

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 pointsEuclidean case: if is Euclidean plane and

then

Positive semidefinite:Slide26

Continuous vs discrete problem

Continuous:

Discrete:

Laplace-Beltrami operator

LaplacianSlide27

To see the sound

Chladni’s experimental setup allowing to visualize acoustic wavesErnst Chladni ['kladnɪ]

(1715-1782)

E. Chladni, Entdeckungen über die Theorie des KlangesSlide28

Chladni plates

Patterns seen by Chladni are solutions to stationary Helmholtz equation

Solutions of this equation are

eigenfunction of Laplace-Beltrami operatorSlide29

The first eigenfunctions of the Laplace-Beltrami operator

Laplace-Beltrami operatorSlide30

Laplace-Beltrami eigenfunctions

An eigenfunction of the Laplace-Beltrami operator computed on different deformations of the shape, showing the invariance of theLaplace-Beltrami operator to isometriesSlide31

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 Slide32

Shape DNA

[Reuter et al. 2006]: use the Laplace-Beltrami spectrum as an isometry-invariant shape descriptor (“shape DNA”)

Laplace-Beltrami spectrum

Images: Reuter

et al.Slide33

Shape DNA

Shape similarity using Laplace-Beltrami spectrum

Images: Reuter

et al.Slide34

Uniqueness of representation

ISOMETRIC SHAPES ARE ISOSPECTRAL

ARE ISOSPECTRAL SHAPES ISOMETRIC?Slide35

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?Slide36

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 curvatureEuler characteristicArea

Can we “hear” the metric?Slide37

One cannot hear the shape of the drum!

[Gordon et al. 1991]:Counter-example of isospectral but not isometric shapesSlide38

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

In matrix notation

whereSlide39

Find the discrete eigenfunctions of the Laplace-Beltrami operator by solving the

generalized eigenvalue problem

Levy 2006Reuter, Biasotti, Giorgi, Patane & Spagnuolo 2009

where is an matrix whose columns are the eigenfunctions

is a diagonal matrix of corresponding eigenvalues

Discrete Laplace-Beltrami eigenfunctionsSlide40

Discrete vs discretized

Continuous surface

Laplace-Beltrami operator

Discretize the surfaceas a graph

Discretize Laplace-Beltrami

operator, preserving some

of the continuous properties

Graph Laplacian

Eigendecomposition

Continuous eigenfunctions and eigenvalues

Eigendecomposition

Discretize

eigenfunctions

and eigenvalues

“Discrete Laplacian”

“Discretized Laplacian”

FEMSlide41

1. Tutte 1963; Zhang 2004

2. Pinkall 1993; Meyer 2003

Cotangent weight

2

sum of areas of triangles

sharing vertex

Discrete Laplacian

1

(umbrella operator); or

valence of vertex (Tutte)

Discrete Laplace-Beltrami operatorSlide42

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 Slide43

No free lunch

Laplacian matrix we used in Laplacian eigenmaps does not converge to the continuous Laplace-Beltrami operatorThere exist many other approximations of the Laplace-Beltrami operator, satisfying different properties

[Wardetzky

et al. 2007]: there is no discretization of the Laplace-Beltrami operator satisfying simultaneously all the desired propertiesSlide44

Finite elements method

Reuter, Biasotti, Giorgi, Patane & Spagnuolo 2009

Eigendecomposition problem in the weak form

for any smooth

Given a finite basis spanning a subspace of

can be expanded as

Write a system of equation

posed as a generalized eigenvalue problem