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

2. 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

3. A mathematical exercise
Since the canonical form is defined up to
isometry, we can arbitrarily set

4. A mathematical exercise
Element of
a matrix
Element of an
matrix
Conclusion: if points are isometrically embeddable into then
Note: can be defined in different ways!

5. Gram matrices A matrix of inner products of the form
is called a Gram matrix
Properties:
(positive semidefinite)
Jørgen Pedersen Gram
( )

6. Back to our problem… 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
( )
[Schoenberg, 1935]: Points with the metric can be isometrically embedded into a Euclidean space if and only if

7. Keep m largest eignevalues
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)

8. Properties of classic MDS
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)

9. MATLAB® intermezzo
Classic MDS
Canonical forms

10. Classical scaling exampleB 1 1 1 A B 1 1 2 D A C C 1 D A 1 B C D 2

11. Local methods
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

12. Think globally, act locally
Local methods “ ”
Think globally, act locally
David Brower
Local criterion how far apart the embedding takes neighboring points
Global criterion
where

13. Recall stress derivation
Laplacian matrix
Recall stress derivation
in LS-MDS
Matrix formulation
where is an matrix with elements
is called the Laplacian matrix
has zero eigenvalue

14. Introduce a constraint avoiding trivial solution
Local methods
Compute canonical form by solving the optimization problem
Introduce a constraint avoiding trivial solution
Trivial solution ( ): points can collapse to a single point

15. Minimum eigenvalue problems
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

16. Minimum eigenvalue problems
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)

17. 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

18. Laplacian eigenmaps example
Classic MDS
Laplacian eigenmap

19. Inner product on tangent space (metric tensor)
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)

20. 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:

21. 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

22. Laplace-Beltrami operator
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)

23. Pierre Simon de Laplace
Laplace-Beltrami
Pierre Simon de Laplace
( )
Eugenio Beltrami
( )

24. 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:

25. Laplace-Beltrami operator
Continuous vs discrete problem
Continuous:
Laplace-Beltrami operator
Discrete:
Laplacian

26. Chladni’s experimental setup allowing to visualize acoustic waves
To see the sound
Ernst Chladni ['kladnɪ]
( )
Chladni’s experimental setup allowing to visualize acoustic waves
E. Chladni, Entdeckungen über die Theorie des Klanges

27. Chladni plates
Patterns seen by Chladni are solutions to stationary Helmholtz equation
Solutions of this equation are eigenfunction of Laplace-Beltrami operator

28. The first eigenfunctions of the Laplace-Beltrami operator

29. 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

30. Laplace-Beltrami spectrum
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

31. Laplace-Beltrami spectrum
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.

32. Shape similarity using Laplace-Beltrami spectrum
Shape DNA
Shape similarity using Laplace-Beltrami spectrum
Images: Reuter et al.

33. ISOMETRIC SHAPES ARE ISOSPECTRAL ARE ISOSPECTRAL SHAPES ISOMETRIC?
Uniqueness of representation
ISOMETRIC SHAPES ARE ISOSPECTRAL
ARE ISOSPECTRAL SHAPES ISOMETRIC?

34. “ ” 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?

35. 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?

36. Counter-example of isospectral but not isometric shapes
One cannot hear the shape of the drum!
[Gordon et al. 1991]:
Counter-example of isospectral but not isometric shapes

37. 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

38. 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

39. Discrete vs discretized
Continuous surface
Laplace-Beltrami operator
Discretize the surface
Discretize Laplace-Beltrami
operator, preserving some
of the continuous properties
Construct graph Laplacian
Discrete Laplace-Beltrami
operator
Discretized Laplace-Beltrami
operator

40. 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

41. 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