1 numerical geometry of non-rigid shapes spectral embedding spectral embedding lecture 6 ©...
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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
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A mathematical exercise
Since the canonical form is defined up to
isometry, we can arbitrarily set
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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!
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Gram matrices
A matrix of inner products of the form
is called a Gram matrix
Jørgen Pedersen Gram(1850-1916)
Properties:
(positive semidefinite)
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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
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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)
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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
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MATLAB® intermezzoClassic MDS
Canonical forms
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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
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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
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Local methods
Think globally, act locally
Local criterion how far apart the embedding takes neighboring points
“ ”David Brower
Global criterion
where
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Laplacian matrix
where is an matrix with elements
Matrix formulationRecall stress
derivation
in LS-MDS
is called the Laplacian matrix
has zero eigenvalue
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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
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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
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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
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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
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Laplacian eigenmaps example
Classic MDS Laplacian eigenmap
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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)
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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:
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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
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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
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Laplace-Beltrami
Pierre Simon de Laplace (1749-1827)
Eugenio Beltrami(1835-1899)
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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:
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Continuous vs discrete problem
Continuous:
Discrete:
Laplace-Beltrami
operator
Laplacian
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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
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Chladni plates
Patterns seen by Chladni are solutions to stationary Helmholtz equation
Solutions of this equation are eigenfunction of Laplace-Beltrami operator
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Laplace-Beltrami operator
The first eigenfunctions of the Laplace-Beltrami operator
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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
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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
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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.
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Shape DNA
Shape similarity using Laplace-Beltrami spectrum
Images: Reuter et al.
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Uniqueness of representation
ISOMETRIC SHAPES ARE ISOSPECTRAL
ARE ISOSPECTRAL SHAPES ISOMETRIC?
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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?
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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?
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One cannot hear the shape of the drum!
[Gordon et al. 1991]:
Counter-example of isospectral but not isometric shapes
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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
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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
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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
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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
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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