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Lecture 6

© Alexander & Michael Bronstein

tosca.cs.technion.ac.il/book

Numerical geometry of non-rigid shapes

Stanford University, Winter 2009

Assume points with the metric are isometrically embeddable into

Then, there exists a canonical form such that

for all

We can also write

Element of

a matrix

Element of an

matrix

Conclusion: if points are isometrically embeddable into then

Note: can be defined in different ways!

A matrix of inner products of the form

is called a Gram matrix

Properties:

- (positive semidefinite)

Jørgen Pedersen Gram

(1850-1916)

- 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

Isaac Schoenberg

(1903-1990)

[Schoenberg, 1935]: Points with the metric can be isometrically embedded into a Euclidean space if and only if

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)

- Nested dimensions: the first dimensions of an -dimensional

canonical form are equal to an -dimensional canonical form

- The error introduced by taking instead of can be quantified as

- Classic MDS minimizes the strain

- 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)

Make the embedding preserve local properties of the shape

Map neighboring points to neighboring points

If , then is small. We want the corresponding distance in the embedding space to be small

“

”

Think globally, act locally

David Brower

Local criterion how far apart the embedding takes neighboring points

Global criterion

where

Recall stress derivation

in LS-MDS

Matrix formulation

where is an matrix with elements

is called the Laplacian matrix

- has zero eigenvalue

Compute canonical form by solving the optimization problem

Introduce a constraint avoiding trivial solution

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

Lets look at a simplified case: one-dimensional embedding

Express the problem using eigendecomposition

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

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)

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

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)

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:

is the space of square-integrable functions on

We can rewrite

Stokes theorem

The operator is called Laplace-Beltrami operator

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

Laplace-Beltrami operator is a generalization of Laplacian to manifolds

In the Euclidean plane,

In coordinate notation

Intrinsic property of the shape (invariant to isometries)

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:

Ernst Chladni ['kladnɪ]

(1715-1782)

Chladni’s experimental setup allowing to visualize acoustic waves

E. Chladni, Entdeckungen über die Theorie des Klanges

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

An eigenfunction of the Laplace-Beltrami operator computed on

different deformations of the shape, showing the invariance of the

Laplace-Beltrami operator to isometries

Eigendecomposition of Laplace-Beltrami operator of a compact shape gives a discrete set of eigenvalues and eigenfunctions

Since the Laplace-Beltrami operator is symmetric, eigenfunctions

form an orthogonal basis for

The eigenvalues and eigenfunctions are isometry invariant

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

Laplace-Beltrami spectrum

Images: Reuter et al.

”

Can one hear the shape of the drum?

Mark Kac

(1914-1984)

More prosaically: can one reconstruct the shape

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

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

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

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

Continuous surface

Laplace-Beltrami operator

Discretize Laplace-Beltrami

operator, preserving some

of the continuous properties

Discretize the surface

Construct graph Laplacian

Discretized Laplace-Beltrami

operator

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

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

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