use of laplacian eigenfunctions and eigenvalues for
TRANSCRIPT
Use of Laplacian Eigenfunctions and Eigenvalues forAnalyzing Data on a Domain of Complicated Shape
Naoki Saito1
Department of MathematicsUniversity of California, Davis
May 23, 2007
1 Partially supported by grants from National Science Foundation and Office ofNaval Research. I thank Leo Chalupa (UCD, Neurobiology), Raphy Coifman(Yale), Julie Coombs (UCD, Neurobiology), Dave Donoho (Stanford), JohnHunter (UCD), Peter Jones (Yale), Linh Lieu (UCD), Stan Osher (UCLA), AllenXue (UCD), Katsu Yamatani (Meijo Univ.) for the fruitful [email protected] (UC Davis) Image Analysis via Laplacian Eigenfunctions IPAM RSWS4 1 / 68
Outline
1 Motivations
2 Laplacian Eigenfunctions
3 Integral Operators Commuting with Laplacian
4 Examples1D Example2D Example3D Example
5 Discretization of the Problem
6 ApplicationsImage ApproximationStatistical Image Analysis; Comparison with PCASolving the Heat Equation on a Complicated DomainClustering Mouse Retinal Ganglion Cells
7 Fast Algorithms for Computing Eigenfunctions
8 Conclusions
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Outline
1 Motivations
2 Laplacian Eigenfunctions
3 Integral Operators Commuting with Laplacian
4 Examples1D Example2D Example3D Example
5 Discretization of the Problem
6 ApplicationsImage ApproximationStatistical Image Analysis; Comparison with PCASolving the Heat Equation on a Complicated DomainClustering Mouse Retinal Ganglion Cells
7 Fast Algorithms for Computing Eigenfunctions
8 Conclusions
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Motivations
Consider a bounded domain of general (may be quite complicated)shape Ω ⊂ R
d .
Want to analyze the spatial frequency information inside of the objectdefined in Ω =⇒ need to avoid the Gibbs phenomenon due toΓ = ∂Ω.
Want to represent the object information efficiently for analysis,interpretation, discrimination, etc. =⇒ fast decaying expansioncoefficients relative to a meaningful basis.
Want to extract geometric information about the domain Ω =⇒shape clustering/classification.
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Motivations . . . Data Analysis on a Complicated Domain
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Motivations . . . Clustering Complicated Objects
µ m
µ m
0 50 100 150 200 250 300
0
50
100
150
200
250
300
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Motivations . . . Clustering Complicated Objects . . .
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Outline
1 Motivations
2 Laplacian Eigenfunctions
3 Integral Operators Commuting with Laplacian
4 Examples1D Example2D Example3D Example
5 Discretization of the Problem
6 ApplicationsImage ApproximationStatistical Image Analysis; Comparison with PCASolving the Heat Equation on a Complicated DomainClustering Mouse Retinal Ganglion Cells
7 Fast Algorithms for Computing Eigenfunctions
8 Conclusions
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Eigenfunctions of Laplacian
Our previous attempt was to extend the object to the outsidesmoothly and then bound it nicely with a rectangular box followed bythe ordinary Fourier analysis.
Why not analyze (and synthesize) the object using genuine basisfunctions tailored to the domain?
After all, sines (and cosines) are the eigenfunctions of the Laplacianon the rectangular domain with Dirichlet (and Neumann) boundarycondition.
Spherical harmonics, Bessel functions, and Prolate Spheroidal WaveFunctions, are part of the eigenfunctions of the Laplacian (viaseparation of variables) for the spherical, cylindrical, and spheroidaldomains, respectively.
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Eigenfunctions of Laplacian . . .
Consider an operator L = −∆ in L2(Ω) with appropriate boundarycondition.
Analysis of L is difficult due to unboundedness, etc.
Much better to analyze its inverse, i.e., the Green’s operator becauseit is compact and self-adjoint.
Thus L−1 has discrete spectra (i.e., a countable number ofeigenvalues with finite multiplicity) except 0 spectrum.
L has a complete orthonormal basis of L2(Ω), and this allows us todo eigenfunction expansion in L2(Ω).
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Eigenfunctions of Laplacian . . . Difficulties
The key difficulty is to compute such eigenfunctions; directly solvingthe Helmholtz equation (or eigenvalue problem) on a general domainis tough.
Unfortunately, computing the Green’s function for a general Ωsatisfying the usual boundary condition (i.e., Dirichlet, Neumann) isalso very difficult.
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Outline
1 Motivations
2 Laplacian Eigenfunctions
3 Integral Operators Commuting with Laplacian
4 Examples1D Example2D Example3D Example
5 Discretization of the Problem
6 ApplicationsImage ApproximationStatistical Image Analysis; Comparison with PCASolving the Heat Equation on a Complicated DomainClustering Mouse Retinal Ganglion Cells
7 Fast Algorithms for Computing Eigenfunctions
8 Conclusions
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Integral Operators Commuting with Laplacian
The key idea is to find an integral operator commuting with theLaplacian without imposing the strict boundary condition a priori.
Then, we know that the eigenfunctions of the Laplacian is the sameas those of the integral operator, which is easier to deal with, due tothe following
Theorem (G. Frobenius 1878?; B. Friedman 1956)
Suppose K and L commute and one of them has an eigenvalue with finitemultiplicity. Then, K and L share the same eigenfunction correspondingto that eigenvalue. That is, Lϕ = λϕ and Kϕ = µϕ.
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Integral Operators Commuting with Laplacian . . .
Let’s replace the Green’s function G (x, y) by the fundamentalsolution of the Laplacian:
K (x, y) =
−12 |x − y | if d = 1,
− 12π
log |x − y| if d = 2,|x−y|2−d
(d−2)ωdif d > 2.
The price we pay is to have rather implicit, non-local boundarycondition although we do not have to deal with this condition directly.
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Integral Operators Commuting with Laplacian . . .
Let K be the integral operator with its kernel K (x, y):
Kf (x)∆=
∫
ΩK (x, y)f (y)dy, f ∈ L2(Ω).
Theorem (NS 2005)
The integral operator K commutes with the Laplacian L = −∆ with thefollowing non-local boundary condition:
∫
ΓK (x, y)
∂ϕ
∂νy
(y)ds(y) = −1
2ϕ(x) + pv
∫
Γ
∂K (x, y)
∂νy
ϕ(y)ds(y),
for all x ∈ Γ, where ϕ is an eigenfunction common for both operators.
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Integral Operators Commuting with Laplacian . . .
Corollary (NS 2005)
The integral operator K is compact and self-adjoint on L2(Ω). Thus, thekernel K (x, y) has the following eigenfunction expansion (in the sense ofmean convergence):
K (x, y) ∼∞
∑
j=1
µjϕj(x)ϕj(y),
and ϕjj forms an orthonormal basis of L2(Ω).
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Outline
1 Motivations
2 Laplacian Eigenfunctions
3 Integral Operators Commuting with Laplacian
4 Examples1D Example2D Example3D Example
5 Discretization of the Problem
6 ApplicationsImage ApproximationStatistical Image Analysis; Comparison with PCASolving the Heat Equation on a Complicated DomainClustering Mouse Retinal Ganglion Cells
7 Fast Algorithms for Computing Eigenfunctions
8 Conclusions
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Outline
1 Motivations
2 Laplacian Eigenfunctions
3 Integral Operators Commuting with Laplacian
4 Examples1D Example2D Example3D Example
5 Discretization of the Problem
6 ApplicationsImage ApproximationStatistical Image Analysis; Comparison with PCASolving the Heat Equation on a Complicated DomainClustering Mouse Retinal Ganglion Cells
7 Fast Algorithms for Computing Eigenfunctions
8 Conclusions
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1D Example
Consider the unit interval Ω = (0, 1).
Then, our integral operator K with the kernel K (x , y) = −|x − y |/2gives rise to the following eigenvalue problem:
−ϕ′′ = λϕ, x ∈ (0, 1);
ϕ(0) + ϕ(1) = −ϕ′(0) = ϕ′(1).
The kernel K (x, y) is of Toeplitz form =⇒ Eigenvectors must haveeven and odd symmetry (Cantoni-Butler ’76).
In this case, we have the following explicit solution.
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1D Example . . .
λ0 ≈ −5.756915, which is a solution of tanh√−λ02 = 2√
−λ0,
ϕ0(x) = A0 cosh√
−λ0
(
x − 1
2
)
;
λ2m−1 = (2m − 1)2π2, m = 1, 2, . . .,
ϕ2m−1(x) =√
2 cos(2m − 1)πx ;
λ2m, m = 1, 2, . . ., which are solutions of tan√
λ2m
2 = − 2√λ2m
,
ϕ2m(x) = A2m cos√
λ2m
(
x − 1
2
)
,
where Ak , k = 0, 1, . . . are normalization constants.
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First 5 Basis Functions
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.06
−0.04
−0.02
0
0.02
0.04
0.06
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1D Example: Bases on Disconnected Intervals
0 0.5 1 1.5 2 2.5 3−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
x
φ k(x)
(a) Union
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1D Example: Bases on Disconnected Intervals
0 0.5 1 1.5 2 2.5 3−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
x
φ k(x)
(a) Union
0 0.5 1 1.5 2 2.5 3−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
xφ k(x
)
(b) Separated
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Outline
1 Motivations
2 Laplacian Eigenfunctions
3 Integral Operators Commuting with Laplacian
4 Examples1D Example2D Example3D Example
5 Discretization of the Problem
6 ApplicationsImage ApproximationStatistical Image Analysis; Comparison with PCASolving the Heat Equation on a Complicated DomainClustering Mouse Retinal Ganglion Cells
7 Fast Algorithms for Computing Eigenfunctions
8 Conclusions
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2D Example
Consider the unit disk Ω. Then, our integral operator K with thekernel K (x, y) = − 1
2πlog |x − y| gives rise to:
−∆ϕ = λϕ, in Ω;
∂ϕ
∂ν
∣
∣
∣
Γ=
∂ϕ
∂r
∣
∣
∣
Γ= −∂Hϕ
∂θ
∣
∣
∣
Γ,
where H is the Hilbert transform for the circle, i.e.,
Hf (θ)∆=
1
2πpv
∫ π
−π
f (η) cot
(
θ − η
2
)
dη θ ∈ [−π, π].
Let βk,ℓ is the ℓth zero of the Bessel function of order k,Jk(βk,ℓ) = 0. Then,
ϕm,n(r , θ) =
Jm(βm−1,n r)(
cossin
)
(mθ) if m = 1, 2, . . . , n = 1, 2, . . .,
J0(β0,n r) if m = 0, n = 1, 2, . . .,
λm,n =
β2m−1,n, if m = 1, . . . , n = 1, 2, . . .,
β20,n if m = 0, n = 1, 2, . . ..
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First 25 Basis Functions
(a) Our Basis (b) Dirichlet-Laplace
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Outline
1 Motivations
2 Laplacian Eigenfunctions
3 Integral Operators Commuting with Laplacian
4 Examples1D Example2D Example3D Example
5 Discretization of the Problem
6 ApplicationsImage ApproximationStatistical Image Analysis; Comparison with PCASolving the Heat Equation on a Complicated DomainClustering Mouse Retinal Ganglion Cells
7 Fast Algorithms for Computing Eigenfunctions
8 Conclusions
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3D Example
Consider the unit ball Ω in R3. Then, our integral operator K with
the kernel K (x, y) = 14π|x−y| .
Top 9 eigenfunctions cut at the equator viewed from the south:
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Outline
1 Motivations
2 Laplacian Eigenfunctions
3 Integral Operators Commuting with Laplacian
4 Examples1D Example2D Example3D Example
5 Discretization of the Problem
6 ApplicationsImage ApproximationStatistical Image Analysis; Comparison with PCASolving the Heat Equation on a Complicated DomainClustering Mouse Retinal Ganglion Cells
7 Fast Algorithms for Computing Eigenfunctions
8 Conclusions
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Discretization of the Problem
Assume that the whole dataset consists of a collection of datasampled on a regular grid, and that each sampling cell is a box of size∏d
i=1 ∆xi .
Assume that an object of our interest Ω consists of a subset of theseboxes whose centers arexiN
i=1.
Under these assumptions, we can approximate the integral eigenvalueproblem Kϕ = µϕ with a simple quadrature rule with node-weightpairs (xj , wj) as follows.
N∑
j=1
wjK (xi , xj)ϕ(xj) = µϕ(xi ), i = 1, . . . , N , wj =d
∏
i=1
∆xi .
Let Ki ,j∆= wjK (xi , xj), ϕi
∆= ϕ(xi ), and ϕ
∆= (ϕ1, . . . , ϕN)T ∈ R
N .Then, the above equation can be written in a matrix-vector formatas: Kϕ = µϕ, where K = (Kij) ∈ R
N×N . Under our assumptions,the weight wj does not depend on j , which makes K symmetric.
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Outline
1 Motivations
2 Laplacian Eigenfunctions
3 Integral Operators Commuting with Laplacian
4 Examples1D Example2D Example3D Example
5 Discretization of the Problem
6 ApplicationsImage ApproximationStatistical Image Analysis; Comparison with PCASolving the Heat Equation on a Complicated DomainClustering Mouse Retinal Ganglion Cells
7 Fast Algorithms for Computing Eigenfunctions
8 Conclusions
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Outline
1 Motivations
2 Laplacian Eigenfunctions
3 Integral Operators Commuting with Laplacian
4 Examples1D Example2D Example3D Example
5 Discretization of the Problem
6 ApplicationsImage ApproximationStatistical Image Analysis; Comparison with PCASolving the Heat Equation on a Complicated DomainClustering Mouse Retinal Ganglion Cells
7 Fast Algorithms for Computing Eigenfunctions
8 Conclusions
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Image Approximation; Comparison with Wavelets
10 20 30 40 50 60 70 80 90 100
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90
(a) What data?
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Image Approximation; Comparison with Wavelets
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(a) What data?
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(b) χJ · Barbara
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First 25 Basis Functions
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Next 25 Basis Functions
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Reconstruction with Top 100 Coefficients
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(a) Reconstruction
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Reconstruction with Top 100 Coefficients
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(a) Reconstruction
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(b) Error
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Reconstruction with Top 100 2D Wavelets (Symmlet 8)
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(a) Reconstruction
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Reconstruction with Top 100 2D Wavelets (Symmlet 8)
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(a) Reconstruction
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(b) Error
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Reconstruction with Top 100 1D Wavelets (Symmlet 8)
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(a) Reconstruction
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Reconstruction with Top 100 1D Wavelets (Symmlet 8)
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(a) Reconstruction
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(b) Error
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Comparison of Coefficient Decay
0 1000 2000 3000 4000 5000 6000 7000 800010
−8
10−6
10−4
10−2
100
102
104
Laplacian Eigenbasis1D Symmlet 82D Symmlet 82D Symmlet 8 Tensor
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A Real Challenge: Kernel matrix is of 387924 × 387924.
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Conjecture on the Coefficient Decay Rate
Conjecture (NS 2005)
For f ∈ C (Ω) with ∇f ∈ BV (Ω) defined on C 2-domain Ω, the expansioncoefficients 〈f , ϕk〉 w.r.t. the Laplacian eigenbasis decay as O(k−2). Thus,the N-term approximation error measured in the L2-norm should have adecay rate of O(N−1.5).
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Outline
1 Motivations
2 Laplacian Eigenfunctions
3 Integral Operators Commuting with Laplacian
4 Examples1D Example2D Example3D Example
5 Discretization of the Problem
6 ApplicationsImage ApproximationStatistical Image Analysis; Comparison with PCASolving the Heat Equation on a Complicated DomainClustering Mouse Retinal Ganglion Cells
7 Fast Algorithms for Computing Eigenfunctions
8 Conclusions
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Comparison with PCA
Consider a stochastic process living on a domain Ω.
PCA/Karhunen-Loeve Transform is often used.
PCA/KLT incorporate geometric information of the measurement (orpixel) location through the data correlation, i.e., implicitly.
Our Laplacian eigenfunctions use explicit geometric informationthrough the harmonic kernel ϕ(x, y).
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Comparison with PCA: Example
“Rogue’s Gallery” dataset from Larry Sirovich
72 training dataset; 71 test dataset
Left & right eye regions
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Comparison with PCA: Basis Vectors
(a) KLB/PCA 1:9
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Comparison with PCA: Basis Vectors
(a) KLB/PCA 1:9 (b) Laplacian Eigenfunctions 1:9
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Comparison with PCA: Basis Vectors . . .
(a) KLB/PCA 10:18 (b) Laplacian Eigenfunctions 10:18
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Comparison with PCA: Kernel Matrix
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(a) Covariance
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120
140
160
180
(b) Harmonic kernel
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Comparison with PCA: Energy Distribution overCoordinates
Magnitude of Top 50 Principal Components (Log scale)
face index
PC
inde
x
20 40 60 80 100 120 140
5
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15
20
25
30
35
40
45
50
(a) KLB/PCA
Magnitude of 50 Lowest Frequency Coefficients (Log scale)
face index
freq
uenc
y in
dex
20 40 60 80 100 120 140
5
10
15
20
25
30
35
40
45
50
(b) Laplacian Eigenfunctions
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Comparison with PCA: Basis Vector #7 . . .
c7:large c7:large
ϕ7
c7:small c7:small
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Comparison with PCA: Basis Vector #13 . . .
c13:large c13:large
ϕ13
c13:small c13:small
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Outline
1 Motivations
2 Laplacian Eigenfunctions
3 Integral Operators Commuting with Laplacian
4 Examples1D Example2D Example3D Example
5 Discretization of the Problem
6 ApplicationsImage ApproximationStatistical Image Analysis; Comparison with PCASolving the Heat Equation on a Complicated DomainClustering Mouse Retinal Ganglion Cells
7 Fast Algorithms for Computing Eigenfunctions
8 Conclusions
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Solving the Heat Equation on a Complicated Domain
It is well known that the semigroup et∆ can be diagonalized using theLaplacian eigenbasis, i.e., for any initial heat distributionu0(x) ∈ L2(Ω), we have the heat distribution at time t formally as
u(x, t) = et∆u0 =∞
∑
j=1
e−tλj 〈u0, ϕj〉ϕj(x),
which is based on the expansion of the Green’s function for the heatequation pt(x, y) via the Laplacian eigenfunctions as follows:
pt(x, y) =∞
∑
j=1
e−λj tϕj(x)ϕj(y) (t, x, y) ∈ (0,∞) × Ω × Ω.
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Solving the Heat Equation on a Complicated Domain
Due to the discretization of the problem, we can write et∆ in thematrix-vector notation as
Φ e−tΛ ΦT = Φ diag(
e−tλ1 , . . . , e−tλN
)
ΦT =N
∑
j=1
e−λj tϕjϕTj ,
where Φ = (ϕ1, . . . , ϕN) is the Laplacian eigenbasis matrix of sizeN × N , and Λ is the diagonal matrix consisting of the eigenvalues ofthe Laplacian, which are the inverse of the eigenvalues of the kernelmatrix, i.e., Λk,k = λk = 1/µk .
Given an initial heat distribution over the domain, u0 ∈ RN , we can
compute the heat distribution at time t as
u(t) = Φ e−tΛ ΦTu0.
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Solving the Heat Equation on a Complicated Domain . . .
t=0 t=1 t=10
t=100 t=250 t=500
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Solving the Heat Equation on a Complicated Domain . . .
It is well known that the eigenvalues of the Laplacian with theDirichlet (or the Neumann/Robin) boundary condition are positive (ornon-negative).
At this point, precisely specifying the boundary values is difficultbecause our formulation satisfies neither the Dirichlet nor theNeumann nor the Robin conditions.
Our empirical observation so far has led to the following conjecture:
Conjecture (NS 2007)
The eigenvalues of the Laplacian satisfying our boundary condition anddefined over a bounded domain Ω ∈ R
d are all positive possibly with afinite number of negative ones.
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Outline
1 Motivations
2 Laplacian Eigenfunctions
3 Integral Operators Commuting with Laplacian
4 Examples1D Example2D Example3D Example
5 Discretization of the Problem
6 ApplicationsImage ApproximationStatistical Image Analysis; Comparison with PCASolving the Heat Equation on a Complicated DomainClustering Mouse Retinal Ganglion Cells
7 Fast Algorithms for Computing Eigenfunctions
8 Conclusions
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Clustering Mouse Retinal Ganglion Cells
Objective: To understand how the structural/geometric properties ofmouse retinal ganglion cells (RGCs) relate to the cell types and theirfunctionality
Why mouse? =⇒ great possibilities for genetic manipulation
Data: 3D images of dendrites/axons of RGCs
State of the Art: Process each image via specialized software toextract geometric/morphological parameters (totally 14 suchparameters) followed by a conventional clustering algorithm
These parameters include: somal size; dendric field size; total dendritelength; branch order; mean internal branch length; branch angle;mean terminal branch length, etc. =⇒ takes half a day per cell with alot of human interactions!
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Clustering Mouse Retinal Ganglion Cells . . . 3D Data
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Very Preliminary Study on Mouse Retinal Ganglion Cells
Use 2D plane projection data instead of full 3D
Compute the smallest k Laplacian eigenvalues using our method (i.e.,the largest k eigenvalues of K) for each image
Construct a feature vector per image
Possible feature vectors reflecting geometric information:F1 = (λ1, . . . , λk)T ; F2 = (µ1, . . . , µk)
T ; F3 = (λ1/λ2, . . . , λ1/λk)T ;F4 = (µ1/µ2, . . . , µ1/µk)
T .
Do visualization and clustering
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Very Preliminary Study on Mouse RGCs . . .
−3 −2.5 −2 −1.5 −1 −0.5
x 10−4
1
2
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5
x 10−31
1.5
2
2.5
3
3.5
4
4.5
5
5.5
x 10−3
λ2
λ1
The first three Laplacian eigenvalues
λ 3
(a) F1
0.060.07
0.080.09
0.10.11
0.12 0.04
0.05
0.06
0.07
0.08
0.04
0.045
0.05
0.055
0.06
|λ1|/λ
3
The ratios of the Laplacian eigenvalues
|λ1|/λ
2
|λ1|/λ
4
(b) F3
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5−4
−2
0
2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
Diffusion map of manual features
φ1
φ2
φ 3
(c) Manual
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Laplacian Eigenfunctions on a Mouse RGC
−5000
500−5000
5000.01
0.015
0.02
φ1
−5000
500−5000
500−0.05
0
0.05
φ2
−5000
500−5000
500−0.05
0
0.05
φ3
−5000
500−5000
500−0.05
0
0.05
φ4
−5000
500−5000
500−0.05
0
0.05
φ5
−5000
500−5000
500−0.05
0
0.05
φ6
−5000
500−5000
500−0.05
0
0.05
φ7
−5000
500−5000
500−0.05
0
0.05
φ8
−500
0
500
−5000500
−0.10
0.1
φ200
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Challenges of Mouse Retinal Ganglion Cells
Their shapes are very complicated.Interpretation of our eigenvalues are not yet fully understoodcompared to the usual Dirichlet-Laplacian case that have been wellstudied: the Payne-Polya-Weinberger inequalities; the Faber-Krahninequalities; the Ashbaugh-Benguria results, etc. For Ω ∈ R
d ,
λ(D)1 (Ω) ≥
( |Bd1 |
|Ω|
)2
λ(D)1 (Bd
1 ),λ
(D)k+1(Ω)
λ(D)k (Ω)
≤ λ(D)2 (Bd
1 )
λ(D)1 (Bd
1 ), k = 1, 2, 3.
Note the related work on “Shape DNA” by Reuter et al. (2005), andclassification of tree leaves by Khabou et al. (2007).Perhaps original 3D data should be used instead of projected 2D data.Reduce computational burden =⇒ need to develop fast algorithms.Heat propagation on the dendrites may give us interesting and usefulinformation; after all the dendrites are network to disseminateinformation via chemical reaction-diffusion mechanism.Construct actual graphs based on the connectivity and analyze themdirectly via spectral graph theory and diffusion maps.
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Outline
1 Motivations
2 Laplacian Eigenfunctions
3 Integral Operators Commuting with Laplacian
4 Examples1D Example2D Example3D Example
5 Discretization of the Problem
6 ApplicationsImage ApproximationStatistical Image Analysis; Comparison with PCASolving the Heat Equation on a Complicated DomainClustering Mouse Retinal Ganglion Cells
7 Fast Algorithms for Computing Eigenfunctions
8 Conclusions
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A Possible Fast Algorithm for Computing ϕj ’s
Observation: our kernel function K (x, y) is of special form, i.e., thefundamental solution of Laplacian used in potential theory.
Idea: Accelerate the matrix-vector product Kϕ using the FastMultipole Method (FMM).
Convert the kernel matrix to the tree-structured matrix via the FMMwhose submatrices are nicely organized in terms of their ranks.(Computational cost: our current implementation costs O(N2), butcan achieve O(N log N) via the randomized SVD algorithm ofMartinsson-Rokhlin-Tygert.)
Construct O(N) matrix-vector product module fully utilizing rankinformation (See also the work of Bremer (2007) and the “HSS”algorithm of Chandrasekaran et al. (2006)).
Embed that matrix-vector product module in the Krylov subspacemethod, e.g., Lanczos iteration.(Computational cost: O(N) for each eigenvalue/eigenvector).
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Splitting into Subproblems for Faster Computation
200 400 600 800 1000 1200 1400 1600
200
400
600
800
1000
1200
1400
1600
(a) Whole islands
200 400 600 800 1000 1200 1400 1600
200
400
600
800
1000
1200
1400
1600
(b) Separated islands
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Eigenfunctions for Separated Islands
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Outline
1 Motivations
2 Laplacian Eigenfunctions
3 Integral Operators Commuting with Laplacian
4 Examples1D Example2D Example3D Example
5 Discretization of the Problem
6 ApplicationsImage ApproximationStatistical Image Analysis; Comparison with PCASolving the Heat Equation on a Complicated DomainClustering Mouse Retinal Ganglion Cells
7 Fast Algorithms for Computing Eigenfunctions
8 Conclusions
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Conclusions
Allow object-oriented image analysis & synthesis
Can get fast-decaying expansion coefficients
Can decouple geometry/domain information and statistics of data
Can extract geometric information of a domain through theeigenvalues
∃ A variety of applications: interpolation, extrapolation, local featurecomputation, . . .
Fast algorithms are the key for higher dimensions/large domains
Connection to lots of interesting mathematics: spectral geometry,spectral graph theory, isoperimetric inequalities, Toeplitz operators,PDEs, potential theory, almost-periodic functions, . . .Many things to be done:
Synthesize the Dirichlet-Laplacian eigenvalues/eigenfunctions from oureigenvalues/eigenfunctionsHow about higher order, i.e., polyharmonic ?Features derived from heat kernels ?Improve our fast algorithm
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References
The following articles are available athttp://www.math.ucdavis.edu/˜saito/publications/:
N. Saito: “Geometric harmonics as a statistical image processing toolfor images defined on irregularly-shaped domains,” in Proc. IEEEWorkshop on Statistical Signal Processing, Bordeaux, France,Jul. 2005.
N. Saito: “Data analysis and representation using eigenfunctions ofLaplacian on a general domain,” Submitted to Applied &Computational Harmonic Analysis, Mar. 2007.
Thank you very much for your attention!
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