Tutorial 10 Laplace-Beltrami operator Diffusion geometry

236861 Numerical Geometry of Images
Tutorial 10
Laplace-Beltrami operator
Diffusion geometry
Anastasia Dubrovina
c⃝2012
Anastasia Dubrovina CS 236861 - Tutorial 10 - Diffusion geometry

Applications for shape analysis
Shape descriptor, Reuter’06 Point descriptors, Sun’09
Shape matching, Ovsjanikov’10 Shape segmentation, Skraba’10
Anastasia Dubrovina CS 236861 - Tutorial 10 - Diffusion geometry

Heat diffusion in Euclidean domain R m
Heat diffusion on R m is governed by the heat equation
(
∆ + ∂
)
u(x; t) = 0; u(x; 0) = u0(x), u(∂Ω) = ....
∂t
◮ u : Ω ∈ R m × R + → R - heat distribution at x ∈ Ω at time
t > 0.
◮ u0(x) - initial heat distribution.
◮ ∆ - the Laplacian. It is defined as
∆u(x) = ∇ 2 u(x) = div∇u(x) =
m∑ ∂2u .
∂x
i=1
2 i
Anastasia Dubrovina CS 236861 - Tutorial 10 - Diffusion geometry

Heat kernel in Euclidean domain
For an initial distribution u0(x) = δ (x − x0) the solution is called
the heat kernel
(we saw it in HW1).
ht (x − x0) =
1
(4πt) m/2 e−∥x−x0∥ 2 /4t
The solution u(x; t) for a general initial distribution u0(x) is given
by
∫
u(x; t) = ht (y − x) u0(y)dy.
R m
Anastasia Dubrovina CS 236861 - Tutorial 10 - Diffusion geometry

Spectral decomposition
The Laplacian eigenvalue problem (the Helmholtz equation)
∆ϕ = −λϕ.
λ is an eigenvalue of the Laplacian, and ϕ is its corresponding
eigenfunction.
In 1D Euclidean domain R, the eigenfunctions of ∆ are the Fourier
basis functions
ϕk(x) = e ifkx .
Anastasia Dubrovina CS 236861 - Tutorial 10 - Diffusion geometry

Spectral decomposition in 1D: example
In 1D, the Laplacian can be discretized as follows (finite differences
scheme, with periodic boundary condition)
⎛
⎜
L = ⎜
⎝
2 −1 0 0 . . . 0 −1
−1 2 −1 0 . . . 0 0
0 −1 2 −1 . . . 0 0
.
−1 0 0 0 . . . −1 2
Question: what are the eigenvectors of L?
Next: what happens on surfaces?
⎞
⎟
⎠
Anastasia Dubrovina CS 236861 - Tutorial 10 - Diffusion geometry

Heat diffusion
Function defined on a surface X : with each x ∈ X we associate a
function value u(x)
u : X → R.
Heat diffusion on X is governed by the heat equation
(
∆X + ∂
)
u(x; t) = 0; u(x; 0) = u0(x).
∂t
◮ u(x; t) - heat distribution at x ∈ X at time t > 0.
◮ u0(x) - initial heat distribution.
◮ ∆X - the Laplace-Beltrami operator defined on X .
Anastasia Dubrovina CS 236861 - Tutorial 10 - Diffusion geometry

The Laplace-Beltrami operator ∆X
Generalization of the Laplacian for Riemannian manifolds -
surfaces in our case.
∆X f = divX (∇X f ) = 1 (√ ) ij
√ ∂i gg ∂i ,
g
where the gradient and the divergence are calculated on the
manifold X .
Reminder: in local coordinates (v, w) the first fundamental form is
given by
(
⟨Xv , Xv ⟩
G = {gij} =
⟨Xw , Xv ⟩
)
⟨Xv , Xw ⟩
,
⟨Xw , Xw ⟩
and g � det(gij), and g ij = (g −1 )ij.
Anastasia Dubrovina CS 236861 - Tutorial 10 - Diffusion geometry

Derivation of ∆X (Aflalo’12)
For X : R n → R m and some dv, dw ∈ R n , the scalar product
induced by the metric G on the tangent plane of X is
dv T Gdw.
This scalar product implies a new definition of the gradient of
f : X → R induced by G, ∇G f ,
Thus we obtain
f (v + dv) = f (u) + ⟨∇G f , dv⟩G + o(∥dv∥)
= f (u) + ∇G f T Gdv + o(∥dv∥)
= f (v) + ∇f T dv + o(∥dv∥).
∇G f = G −1 ∇f .
Anastasia Dubrovina CS 236861 - Tutorial 10 - Diffusion geometry

Derivation of ∆X (Aflalo’12) II
In Euclidean domain Ω ⊂ R m , the Laplacian has the following
property
∫
∫
∆fgda = − ⟨∇f , ∇g⟩da, ∀g|g |∂Ω = 0,
x∈Ω
x∈Ω
where da is the infinitesimal area element.
Hence, a natural extension of the Laplacian for a given scalar
product and a given definition of the gradient would consist of
finding ∆G f such that
∫
v∈Ω
∆G fhda = −
∫
v∈Ω
√
⟨∇G f , ∇G h⟩G gdv1 . . . dvn, ∀h|h |∂Ω = 0
where g = det(G) and da = √ gdv1 . . . dvn is the local infinitesimal
area element according to the metric G.
Anastasia Dubrovina CS 236861 - Tutorial 10 - Diffusion geometry

Derivation of ∆X (Aflalo’12) III
Since ⟨∇G f , ∇G h⟩G = ∇f T G −1 ∇h, we have
∫
∆G fhda =
∫
∇f T G −1 ∇h √ gdu1 . . . dun
v∈Ω
=
∫
u∈Ω
u∈Ω
= −
=
∫
u∈Ω
∫
u∈Ω
(√ gG −1 ∇f ) T ∇hdu1 . . . dun
( n∑
i=1
∂i
∆G fh √ gdu1 . . . dun.
(√ ) −1
gG ∇f i h
)
du1 . . . dun
Using Einstein summation convention ∆G f = 1 (√ ij
√ ∂i gg ∂jf
g ) .
Anastasia Dubrovina CS 236861 - Tutorial 10 - Diffusion geometry

Spectral decomposition of ∆X
The Laplace-Beltrami operator ∆X has a discrete set of
eigenvectors and eigenvalues
where
∆X ϕ = λϕ, ϕ : X → R + ,
0 = λ0 ≤ λ1 ≤ λ2 ≤ ...
There exist λ0 = 0 when X has a boundary, with ϕ0 = const.
The corresponding set of eigenvalues forms an orthogonal basis for
functions defined on X
{ϕi} i≥1 .
Anastasia Dubrovina CS 236861 - Tutorial 10 - Diffusion geometry

Spectral decomposition of ∆X : example
As the Laplace-Beltrami operator ∆X depends only on gij, it is
invariant to isometric transformations of X , and so are its
eigenvalues and eigenfunctions (up to a sign).
X
Y = iso(X)
ϕ1 ϕ2 ϕ3 ϕ4
Anastasia Dubrovina CS 236861 - Tutorial 10 - Diffusion geometry

Applications
Shape DNA, Reuter’06: use {λi} i≥1 as an isometry invariant
descriptor of X .
Reuter’07
Global Point Signature (GPS), Rustamov’07:
Canonical representation, defined up to isometry.
[
1
√λ1 ϕ1, 1
]
√ ϕ2, ...
λ2
Anastasia Dubrovina CS 236861 - Tutorial 10 - Diffusion geometry

Diffusion geometry on surfaces: heat kernel
The heat kernel is given by
ht(x, x ′ ) = ∑
i>0
e −λi t ϕi(x)ϕi(x ′ )
It corresponds to the initial heat distribution u0(x) = δ (x − x0).
The value ht(x, x ′ ) can be interpreted as the probability density of
a random walk of length t from the point x to the point x ′ .
Anastasia Dubrovina CS 236861 - Tutorial 10 - Diffusion geometry

Diffusion geometry on surfaces: diffusion distance
The diffusion distance is defined as
d 2 t (x, x ′ ∫
(
) = ht(x, z) − ht(x ′ , y) ) 2
da
= ∑
i>0
X
e −2λi t ( ϕi(x) − ϕi(x ′ ) ) 2 .
In practice, we sum over N smallest eigenvalues and their
corresponding eigenvectors.
Euclidean distance Geodesic distance Diffusion distance
Anastasia Dubrovina CS 236861 - Tutorial 10 - Diffusion geometry

Comparison between geodesic and diffusion distances
Anastasia Dubrovina CS 236861 - Tutorial 10 - Diffusion geometry

Heat kernel signature and heat kernel map
Heat kernel signature (HKS) and heat kernel map (HKM) are
intrinsic point signatures introduced in Sun’09 and Ovsjanikov’10.
HKS(x) = (ht1 (x, x), ht2 (x, x), ..., htK (x, x))
HKM(x) = (ht1 (x0, x), ht2 (x0, x), ..., htK (x0, x))
To defined latter, we should choose a point x0 ∈ X .
Both can be used for
◮ Shape matching.
◮ Symmetry detection.
◮ Interest point detection.
Anastasia Dubrovina CS 236861 - Tutorial 10 - Diffusion geometry

Varying metric g
Reminder:
∆X f = −divX (∇X f ) = 1 (√ ) ij
√ ∂i gg ∂i
g
By changing the metric g we obtain scale-invariant (Aflalo’11) or
affine-invariant (Raviv’11) Laplace-Beltrami operator.
Scale invariant HKS signature Affine invariant matching
Anastasia Dubrovina CS 236861 - Tutorial 10 - Diffusion geometry

Discretization of ∆X
Discrete version of the Laplace-Beltrami operator
(Lu)i = ∑
wij (ui − uj) , ui = u(xi).
j∈Ni
Some properties of smooth Laplacians:
◮ ∆u = 0 for u = const.
◮ Local support: for any x ̸= y ∈ X , ∆(x) is independent of
u(y).
∫
◮ Positive semi-definiteness: u∆uda ≥ 0.
Wardetzky’07: ”discrete Laplacians cannot satisfy all natural
properties”.
X
Anastasia Dubrovina CS 236861 - Tutorial 10 - Diffusion geometry

Discretization of ∆X : existing schemes
There exist various discretization schemes for the Laplace-Beltrami
operator
◮ Graph Laplacian (polygonal meshes, point clouds).
◮ Cotangent weights, mean-value coordinates (triangulated
meshes).
◮ Finite elements (polygonal meshes).
◮ Belkin and Nyogi (triangulated meshes, point clouds).
◮ Many more.
Anastasia Dubrovina CS 236861 - Tutorial 10 - Diffusion geometry

Graph Laplacian
Interpret the surface as a graph (X , E), where X are the graph
vertices, and E are the edges.
(Lu)i = ∑
wij (ui − uj) , Ni = {j : (i, j) ∈ E}
j∈Ni
where wij = 1, or wij ∼ ∥xi − xj∥ −1 , etc.
The matrix L is therefore
⎧
⎪⎨ ∑
−wij, i ̸= j, (i, j) ∈ E,
(L)ij =
⎪⎩
j ′ wij ′, i = j,
∈Ni
0, otherwise.
Anastasia Dubrovina CS 236861 - Tutorial 10 - Diffusion geometry

Cotangent weight scheme (Meyer’02)
A different geometric discretization was suggested by Meyer et al.,
for surfaces given by triangulated meshes
wij = 1
(cot αij + cot βij) , i ̸= j, j ∈ Ni,
2Ai
where Ni is the 1-ring neighborhood of xi, and Ai is the Voronoi
area of the vertex xi
Ai = 1 ∑
(cot αij + cot βij) ∥xi − xj∥
8
j∈Ni
2 .
(Note that for obtuse triangles adjacent to the vertex i we will use
a mixed area defined in the next slide instead).
Anastasia Dubrovina CS 236861 - Tutorial 10 - Diffusion geometry

Cotangent weight scheme (Meyer’02)
Anastasia Dubrovina CS 236861 - Tutorial 10 - Diffusion geometry

Bibliography
• Y. Aflalo and R. Kimmel and M. Zibulevsky. Conformal mapping with as uniform as possible conformal factor.
accepted to SIAM Journal of Imaging Science, 2012.
• M. Reuter, F.-E. Wolter and N. Peinecke. Laplace-Beltrami spectra as ”Shape-DNA” of surfaces and solids.
Computer-Aided Design 38(4):342-366, 2006.
• R. M. Rustamov. Laplace-Beltrami eigenfunctions for deformation invariant shape representation. Proc. of SGP,
225-233, 2007.
• J. Sun, M. Ovsjanikov and L. Guibas. A Concise and Provably Informative Multi-Scale Signature Based on Heat
Diffusion. Computer Graphics Forum 28(5):1383-1392, 2009.
• M. Ovsjanikov, Q. Mérigot, F. Mémoli and L. Guibas. One point isometric matching with the heat kernel.
Computer Graphics Forum 29(5):1555-1564, 2010.
• M. Wardetzky, S. Mathur, F. Kalberer and F. Grinspun. Discrete Laplace operators: no free lunch. ACM
International Conference Proceeding Series 257:33-37, 2007.
• M. Meyer, M. Desbrun, P. Schröder and A.H. Barr. Discrete differential-geometry operators for triangulated
2-manifolds. Visualization and mathematics 3(7):34-57, 2002.
• U. Pinkall, K. Polthier. Computing Discrete Minimal Surfaces and Their Conjugates, Experiment. Math. 2(1):
15-36 (1993).
Anastasia Dubrovina CS 236861 - Tutorial 10 - Diffusion geometry