## Data Analysis and Representation on a General Domain using Eigenfunctions of Laplacian (2007)

Citations: | 5 - 0 self |

### BibTeX

@MISC{Saito07dataanalysis,

author = {Naoki Saito},

title = { Data Analysis and Representation on a General Domain using Eigenfunctions of Laplacian},

year = {2007}

}

### OpenURL

### Abstract

We propose a new method to analyze and represent data recorded on a domain of gen-eral shape in R d by computing the eigenfunctions of Laplacian defined over there and expanding the data into these eigenfunctions. Instead of directly solving the eigenvalue problem on such a domain via the Helmholtz equation (which can be quite complicated and costly), we find the integral operator commuting with the Laplacian and diagonal-ize that operator. Although our eigenfunctions satisfy neither the Dirichlet nor the Neu-mann boundary condition, computing our eigenfunctions via the integral operator is simple and has a potential to utilize modern fast algorithms to accelerate the compu-tation. We also show that our method is better suited for small sample data than the Karhunen-Loève Transform/Principal Component Analysis. In fact, our eigenfunctions depend only on the shape of the domain, not the statistics of the data. As a further ap-plication, we demonstrate the use of our Laplacian eigenfunctions for solving the heat equation on a complicated domain.

### Citations

2056 |
Handbook of mathematical functions
- Abramowitz, Stegun
- 1965
(Show Context)
Citation Context ...wn as nonharmonic or almost-periodic cosines). and approximate the Green’s operator for the Dirichlet boundary condition by the gridpoint sampling (i.e., sampling at � x j = j /N �N over the interval =-=[0,1]-=-) and j=0 the trapezoidal rule, then the eigenvectors are the so-called Discrete Sine Transform Type I (DST-I for short). The same discretization scheme for the Neumann boundary condition leads to the... |

1722 | Ten Lectures on Wavelets - Daubechies - 1992 |

734 | Laplacian Eigenmaps for Dimensionality Reduction and
- Belkin
(Show Context)
Citation Context ...with the Laplacian in general may become a useful tool for machine learning problems, in particular, clustering of high dimensional data. Popular procedures for such tasks include Laplacian eigenmaps =-=[3]-=- and diffusion maps [5]. Both start with constructing a graph from available data. Then, the former forms a graph Laplacian or a diffusion kernel while the latter form a normalized diffusion kernel. F... |

685 | Rokhlin: A Fast algorithm for particle simulations
- Greengard, V
- 1987
(Show Context)
Citation Context ...pproach based on the integral operator commuting with the Laplacian is computationally quite stable although it uses dense matrices. Using an approach using the celebrated Fast Multipole Method (FMM) =-=[15]-=- that we shall discuss in the next section and that was implemented and tested in [33], we should be able to speed up the eigenvector computations despite of the denseness of the kernel matrices. As f... |

538 |
Sirovich 'Application of the Karhunen-Loeve Procedure for the Charac:erisation of Hunan Face
- Kirby, L
(Show Context)
Citation Context ...alized by KLT/PCA. The dataset we use for demonstration is the so-called “Rogue’s Gallery” dataset that we obtained through the courtesy of Prof. Larry Sirovich at Mount Sinai School of Medicine. See =-=[18,23]-=- for more about this dataset. Out of 143 face images in the dataset, 72 are used as a training dataset from which we compute the 26sFig. 11. Three samples (or realizations) of the eye data. autocorrel... |

222 |
Introductory Functional Analysis with Applications
- Kreyszig
- 1978
(Show Context)
Citation Context ...we will be more specific about it later). The direct analysis of L is difficult due to its unboundedness that is well known and often covered 4sin any elementary functional analysis course (see e.g., =-=[20]-=-). A much better approach is to analyze its inverse L −1 , which is referred to as the Green’s operator because it is a compact and self-adjoint operator and consequently we can have a good grip of it... |

179 |
Introduction to Partial Differential Equations, 2nd ed
- Folland
- 1995
(Show Context)
Citation Context ...(y)= f (x) � + Γ ∂K (x+ tνx− y)ds(y) Γ ∂νy ∂K (x+ tνx− y)(f (y)− f (x))ds(y). ∂νy The first term in the righthand side is−f (x) thanks to the following 45 (A.2)sLemma 16 (a variant of Lemma (3.19) in =-=[12]-=-) ⎧ −1 if x ∈ Ω; � ⎪⎨ ∂K (x− y)ds(y)= − Γ ∂νy 1 if x ∈ Γ; 2 ⎪⎩ 0 if x ∉ Ω. As for the second integral in Eq. (A.2), because ψ(y) ∆ = f (y)− f (x) is continuous and ψ(x)=0 for x ∈ Γ, we can use Lemma (... |

156 |
Diffusion maps
- Coifman, Lafon
- 2006
(Show Context)
Citation Context ...eneral may become a useful tool for machine learning problems, in particular, clustering of high dimensional data. Popular procedures for such tasks include Laplacian eigenmaps [3] and diffusion maps =-=[5]-=-. Both start with constructing a graph from available data. Then, the former forms a graph Laplacian or a diffusion kernel while the latter form a normalized diffusion kernel. Finally, both compute th... |

130 | Partial Differential Equations - John - 1981 |

123 |
The fast gauss transform
- Greengard, Strain
- 1991
(Show Context)
Citation Context ...trices. As for computing eigenvectors of diffusion kernels, one may want to use the improve fast Gauss transform [22], which is an improved version of the fast Gauss transform of Greengard and Strain =-=[16]-=-, which in turn is based on FMM again. But if one wants to compute the evolution of the diffusion process by varying the time parameter t, then this method seems less effective since it requires compu... |

100 | Image quality assessment: From error measurement to structural similarity
- Wang, Bovik, et al.
- 2004
(Show Context)
Citation Context ... fidelity of the former is far better than the latter as can be seen in Figures 5(a) and 7(a). It is important to realize that the visual fidelity may not be measured well by the ℓ 2 error; see e.g., =-=[32]-=- for the details about this intricate problem. For a variety of applications, we wish to prove the following conjecture: 19sRelative L 2 error 10 0 10 −5 10 −10 Laplacian Eigenbasis 1D Symmlet 8 2D Sy... |

88 |
Spectral theory and differential operators
- Davies
- 1995
(Show Context)
Citation Context ...multiplicity) except 0 spectrum [8, Chap. 6, 7]. Moreover, thanks to this spectral property, L has a complete orthonormal basis of L 2 (Ω), and this allows us to do eigenfunction expansion in L 2 (Ω) =-=[8,21]-=-. The key difficulty is to compute such eigenfunctions. Directly solving the Helmholtz equation on a general domain, i.e., finding non-trivial solutions of −∆φ = λφ that satisfy Bφ=0 (where B is an op... |

73 | The discrete cosine transform - Strang - 1999 |

60 | Partial differential equations. An introduction - Strauss - 1992 |

50 |
Some comments on fourier analysis, uncertainty and modeling
- Slepian
- 1983
(Show Context)
Citation Context ...function of the first kind of order d/2 and B > 0 is the bandwidth. The eigenfunctions of the integral operator with this kernel are the direct generalization of the prolate spheroidal wave functions =-=[27]-=- 1 ; and 2) the 1 Although such bandlimited kernels do not commute with the usual Laplacian, they may commute with more general elliptic operators. For example, for d = 1, KB (x, y) be7spopular Gaussi... |

48 |
Nonlinear approximation, Acta Numerica
- DeVore
- 1998
(Show Context)
Citation Context ...scillations around the linear lines are of course completely suppressed and the decay rates get faster than those in the linear canonical order. This is one of the reasons why nonlinear approximation =-=[10]-=- provides better approximation than linear ones. Although it is interesting and important to investigate what is a good approximation space for nonlinear approximation using our Laplacian eigenfunctio... |

38 |
A class of bases in L 2 for the sparse representation of integral operators
- ALPERT
- 1993
(Show Context)
Citation Context ...k decays logarithmically away from the diagonal. Potentially, we may get a better (i.e., smoother) kernel matrix by rearranging its entries. Therefore, one possibility is to use the “Alpert wavelets” =-=[2]-=- to sparsify this matrix (possibly with some rearrangement), and then use the sparse eigenvalue solver for the resulting matrix. Another more promising possibility is to use the Krylov subspace method... |

30 |
Eigenvalues and eigenvectors of symmetric centrosymmetric matrices
- Cantoni, Butler
(Show Context)
Citation Context ... shows these Laplacian eigenfunctions of the lowest five frequencies. Remark 7 The kernel K (x, y) is of Toeplitz form in this case, and consequently, the eigenvectors must have even and odd symmetry =-=[4]-=-, which is indeed the case. Remark 8 The Laplacian eigenfunctions for the Dirichlet boundary condition on the unit interval satisfy−φ ′′ = λφ, φ(0)=φ(1)=0, and they are sines. The Green’s function in ... |

17 | Principles and Techniques of Applied Mathematics - Friedman - 1956 |

16 |
Integral Equations: a practical treatment, from spectral theory to applications
- Porter, Stirling
(Show Context)
Citation Context ...multiplicity) except 0 spectrum [8, Chap. 6, 7]. Moreover, thanks to this spectral property, L has a complete orthonormal basis of L 2 (Ω), and this allows us to do eigenfunction expansion in L 2 (Ω) =-=[8,21]-=-. The key difficulty is to compute such eigenfunctions. Directly solving the Helmholtz equation on a general domain, i.e., finding non-trivial solutions of −∆φ = λφ that satisfy Bφ=0 (where B is an op... |

16 |
The improved fast Gauss transform with applications to machine learning
- Raykar, Duraiswami
- 1752
(Show Context)
Citation Context ...able to speed up the eigenvector computations despite of the denseness of the kernel matrices. As for computing eigenvectors of diffusion kernels, one may want to use the improve fast Gauss transform =-=[22]-=-, which is an improved version of the fast Gauss transform of Greengard and Strain [16], which in turn is based on FMM again. But if one wants to compute the evolution of the diffusion process by vary... |

10 | Image approximation and modeling via least statistically dependent bases
- Saito
- 2001
(Show Context)
Citation Context ...alized by KLT/PCA. The dataset we use for demonstration is the so-called “Rogue’s Gallery” dataset that we obtained through the courtesy of Prof. Larry Sirovich at Mount Sinai School of Medicine. See =-=[18,23]-=- for more about this dataset. Out of 143 face images in the dataset, 72 are used as a training dataset from which we compute the 26sFig. 11. Three samples (or realizations) of the eye data. autocorrel... |

8 |
Geometric harmonics
- Coifman, Lafon
- 2006
(Show Context)
Citation Context ...e basis {φj }j∈N to expand and represent the data supported on Ω. Remark 4 These eigenfunctions of the Laplacian are closely related to the so-called Geometric Harmonics proposed by Coifman and Lafon =-=[6]-=-. After all, our eigenfunctions are a specific example of the geometric harmonics with a specific kernel (1). Nevertheless, there are some important differences between their objectives and methods wi... |

8 | The polyharmonic local sine transform: a new tool for local image analysis and synthesis without edge effect
- Saito, Remy
(Show Context)
Citation Context ...s should be 20scontrasted with the Fourier sine or complex (i.e., periodic) Fourier bases for data on a rectangular domain, which provide only α=1 in the above approximation statement in general. See =-=[25,34]-=- for the details about the boundary effect of the conventional Fourier/trigonometric bases. This conjecture was derived from our numerous numerical experiments and we shall show some of them here to s... |

8 | Improvement of DCT-based compression algorithms using poisson’s equation
- Yamatani, Saito
- 2006
(Show Context)
Citation Context ...s should be 20scontrasted with the Fourier sine or complex (i.e., periodic) Fourier bases for data on a rectangular domain, which provide only α=1 in the above approximation statement in general. See =-=[25,34]-=- for the details about the boundary effect of the conventional Fourier/trigonometric bases. This conjecture was derived from our numerous numerical experiments and we shall show some of them here to s... |

7 | Linear integral equations, 2nd ed - Kress - 1999 |

6 |
Geometric harmonics as a statistical image processing tool for images defined on irregularly-shaped domains
- Saito
- 2005
(Show Context)
Citation Context ...ed the author improve this article. A preliminary version of a part of the material in this paper was presented at the 13th IEEE Workshop on Statistical Signal Processing, July 2005, Bordeaux, France =-=[24]-=-. 44sA Proof of Theorem 2 PROOF. Let L = −∆ and K be defined as (2). Then, for f ∈ C 2 (Ω)∪ C 1 (Ω), we have LKf (x)=−∆xKf (x)= f (x) x ∈ Ω, which is referred to as “Poisson’s formula” [17, p.99]. On ... |

4 |
Generalized polyharmonic trigonometric transform: A tool for object-oriented image analysis and synthesis
- Saito, Yamatani, et al.
(Show Context)
Citation Context ... idea from potential theory and elliptic partial differential equations, we developed the so-called generalized polyharmonic local trigonometric transform to do this extension and subsequent analysis =-=[26]-=-, [35, Chap. 4]. Although this approach can analyze the spatial frequency contents of the object without being bothered by the Gibbs phenomenon, the resulting analysis (e.g., the Fourier cosine coeffi... |

3 |
Eigenvalues of the Laplacian and the heat equation
- Dodziuk
- 1981
(Show Context)
Citation Context ...lly as u(x, t)=e t∆ u0(x)= ∞� e −tλ j 〈u0,φj〉φj (x), j=1 which is based on the expansion of the Green’s function for the heat equation pt (x, y) via the Laplacian eigenfunctions as follows (see e.g., =-=[11]-=-). pt (x, y)= ∞� e −λ j t φj (x)φj (y) (t, x, y)∈(0,∞)×Ω×Ω. j=1 In practice, the domain Ω must be discretized by a finite number (i.e., N ∈ N) of sample points (or pixels) as discussed in Section 2.2.... |

3 | Efficient Approximations: Overcoming Boundary Effects - Zhao - 2006 |

2 |
Engineering Geomorphologic Classification Map
- Wakamatsu, Kubo, et al.
- 2005
(Show Context)
Citation Context ...side. Remark 13 The example of the Japanese Islands discussed in this section was of small size. We recently obtained a digital map called “Japan Engineering Geomorphologic Classification Map” (JEGM) =-=[31]-=-. The number of sampling points in this map is 387,924 over the Japanese Islands. Each point is associated with a vector of length 11 representing a type of geological layer, an elevation, a slope, et... |

1 |
On some properties of the Hilbert transform in Euclidean space
- Delanghe
- 2004
(Show Context)
Citation Context ...oundary condition (like Eq. 10 of the 2D unit disk) for the 3D unit ball. To do so, we may need to employ a higher dimensional analog of the Hilbert transform involving “Clifford Analysis” (see e.g., =-=[9]-=- and the references therein). Second, understanding the physical meaning of our unconventional non-local boundary condition (3). This is important for physical applications such as the heat equation t... |

1 |
On a fast algorithm for computing the Laplacian eigenpairs via commuting integral operators
- Xue
- 2007
(Show Context)
Citation Context ... The three straight lines plotted with the ‘dashdot’ pattern are for the reference: they indicate decay rates of k −1 , k −1.5 , k −2 , respectively. implemented and tested in the Ph.D. thesis of Xue =-=[33]-=- is indispensable to compute the Laplacian eigenfunctions for such a large scale problem. 25 10 4 10 4s5 Comparison with KLT/PCA In this section, we shall discuss the use of the Laplacian eigenfunctio... |