1 numerical geometry of non-rigid shapes spectral methods spectral methods © alexander &...

1Numerical Geometry of Non-Rigid Shapes Spectral methods

Spectral methods

© Alexander & Michael Bronstein, 2006-2009© Michael Bronstein, 2010tosca.cs.technion.ac.il/book

048921 Advanced topics in visionProcessing and Analysis of Geometric Shapes

EE Technion, Spring 2010


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



Since the canonical form is defined up to

isometry, we can arbitrarily set



Conclusion: if points are isometrically embeddable into

then

Element of

a matrix

Element of an

matrix

Note: can be defined in different ways!


Gram matrices

A matrix of inner products of the form

is called a Gram matrix

Jørgen Pedersen Gram(1850-1916)

Properties:

(positive semidefinite)


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


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)


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


MATLAB® intermezzoClassic MDS

Canonical forms


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


Deformation Deformation

+Topology

Topological invariance


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


Local methods

Think globally, act locally

Local criterion how far apart the embedding takes neighboring points

“ ”David Brower

Global criterion

where


Laplacian matrix

where is an matrix with elements

Matrix formulationRecall stress

derivation

in LS-MDS

is called the Laplacian matrix

has zero eigenvalue


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


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


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


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


Laplacian eigenmaps example

Classic MDS Laplacian eigenmap


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)


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:


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


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


Laplace-Beltrami

Pierre Simon de Laplace (1749-1827)

Eugenio Beltrami(1835-1899)


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:


Continuous vs discrete problem

Continuous:

Discrete:

Laplace-Beltrami

operator

Laplacian


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


Chladni plates

Patterns seen by Chladni are solutions to stationary Helmholtz equation

Solutions of this equation are eigenfunction of Laplace-Beltrami operator


The first eigenfunctions of the Laplace-Beltrami operator



Laplace-Beltrami eigenfunctions

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

Laplace-Beltrami operator to isometries


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


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.


Shape DNA

Shape similarity using Laplace-Beltrami spectrum

Images: Reuter et al.


Uniqueness of representation

ISOMETRIC SHAPES ARE ISOSPECTRAL

ARE ISOSPECTRAL SHAPES ISOMETRIC?


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?


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?


One cannot hear the shape of the drum!

[Gordon et al. 1991]:

Counter-example of isospectral but not isometric shapes


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

where


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 eigenfunctions


Discrete vs discretized

Continuous surface


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” FEM


1. Tutte 1963; Zhang 20042. Pinkall 1993; Meyer 2003

Cotangent weight2

sum of areas of triangles sharing vertex

Discrete Laplacian1

(umbrella operator); orvalence of vertex (Tutte)

Discrete Laplace-Beltrami operator


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


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 et al. 2007]: there is no discretization of the Laplace-

Beltrami operator satisfying simultaneously all the desired properties


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

1 numerical geometry of non-rigid shapes spectral methods spectral methods © alexander &...

Documents

canonical form slide

strain slide

positive semidefinite

classical scaling slide

gram matrix of rank

matrix element

matrix note

local properties