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Diffusion Wavelets
, 2004
"... We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their ..."
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Cited by 72 (12 self)
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We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their fast application. Classes of operators satisfying these conditions include diffusionlike operators, in any dimension, on manifolds, graphs, and in nonhomogeneous media. In this case our construction can be viewed as a farreaching generalization of Fast Multipole Methods, achieved through a different point of view, and of the nonstandard wavelet representation of CalderónZygmund and pseudodifferential operators, achieved through a different multiresolution analysis adapted to the operator. We show how the dyadic powers of an operator can be used to induce a multiresolution analysis, as in classical LittlewoodPaley and wavelet theory, and we show how to construct, with fast and stable algorithms, scaling function and wavelet bases associated to this multiresolution analysis, and the corresponding downsampling operators, and use them to compress the corresponding powers of the operator. This allows to extend multiscale signal processing to general spaces (such as manifolds and graphs) in a very natural way, with corresponding fast algorithms.
Biorthogonal diffusion wavelets for multiscale representations on manifolds and graphs
 59141M. SPIE, 2005. URL HTTP://LINK.AIP.ORG/LINK/?PSI/5914/59141M/1
, 2005
"... Recent work by some of the authors presented a novel construction of a multiresolution analysis on manifolds and graphs, acted upon by a given symmetric Markov semigroup {T t}t≥0, for which T t has low rank for large t. 1 This includes important classes of diffusionlike operators, in any dimension, ..."
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Cited by 8 (6 self)
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Recent work by some of the authors presented a novel construction of a multiresolution analysis on manifolds and graphs, acted upon by a given symmetric Markov semigroup {T t}t≥0, for which T t has low rank for large t. 1 This includes important classes of diffusionlike operators, in any dimension, on manifolds, graphs, and in nonhomogeneous media. The dyadic powers of an operator are used to induce a multiresolution analysis, analogous to classical LittlewoodPaley 14 and wavelet theory, while associated wavelet packets can also be constructed. 2 This extends multiscale function and operator analysis and signal processing to a large class of spaces, such as manifolds and graphs, with efficient algorithms. Powers and functions of T (notably its Green’s function) are efficiently computed, represented and compressed. This construction is related and generalizes certain Fast Multipole Methods, 3 the wavelet representation of CalderónZygmund and pseudodifferential operators, 4 and also relates to algebraic multigrid techniques. 5 The original diffusion wavelet construction yields orthonormal bases for multiresolution spaces {Vj}. The orthogonality requirement has some advantages from the numerical perspective, but several drawbacks in terms of the space and frequency localization of the basis functions. Here we show how to relax this requirement in order to construct biorthogonal bases of diffusion scaling functions and wavelets. This yields more compact representations of the powers of the operator, better localized basis functions. This new construction also applies to non selfadjoint semigroups, arising in many applications.
Diffusiondriven multiscale analysis on manifolds and graphs: topdown and bottomup constructions. volume 5914, page 59141D. SPIE
, 2005
"... Classically, analysis on manifolds and graphs has been based on the study of the eigenfunctions of the Laplacian and its generalizations. These objects from differential geometry and analysis on manifolds have proven useful in applications to partial differential equations, and their discrete counte ..."
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Cited by 6 (4 self)
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Classically, analysis on manifolds and graphs has been based on the study of the eigenfunctions of the Laplacian and its generalizations. These objects from differential geometry and analysis on manifolds have proven useful in applications to partial differential equations, and their discrete counterparts have been applied to optimization problems, learning, clustering, routing and many other algorithms. 1–7 The eigenfunctions of the Laplacian are in general global: their support often coincides with the whole manifold, and they are affected by global properties of the manifold (for example certain global topological invariants). Recently a framework for building natural multiresolution structures on manifolds and graphs was introduced, that greatly generalizes, among other things, the construction of wavelets and wavelet packets in Euclidean spaces. 8,9 This allows the study of the manifold and of functions on it at different scales, which are naturally induced by the geometry of the manifold. This construction proceeds bottomup, from the finest scale to the coarsest scale, using powers of a diffusion operator as dilations and a numerical rank constraint to critically sample the multiresolution subspaces. In this paper we introduce a novel multiscale construction, based on a topdown recursive partitioning induced by the eigenfunctions of the Laplacian. This yields associated local cosine packets on manifolds, generalizing local cosines in Euclidean spaces. 10 We discuss some of the connections with the construction of diffusion wavelets. These constructions have direct applications to the approximation, denoising, compression and learning of functions on a manifold and are promising in view of applications to problems in manifold approximation, learning, dimensionality reduction.
Data Analysis and Representation on a General Domain using Eigenfunctions of Laplacian
, 2007
"... We propose a new method to analyze and represent data recorded on a domain of general 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 ..."
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Cited by 4 (0 self)
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We propose a new method to analyze and represent data recorded on a domain of general 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 diagonalize that operator. Although our eigenfunctions satisfy neither the Dirichlet nor the Neumann boundary condition, computing our eigenfunctions via the integral operator is simple and has a potential to utilize modern fast algorithms to accelerate the computation. We also show that our method is better suited for small sample data than the KarhunenLoève 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 application, we demonstrate the use of our Laplacian eigenfunctions for solving the heat equation on a complicated domain.
SVM Optimization for Hyperspectral Colon Tissue Cell Classification
 in Medical Image Computing and Computer Assisted Intervention (MICCAI’04), StMalo
, 2004
"... The classification of normal and malginant colon tissue cells is crucial to the diagnosis of colon cancer in humans. Given the right set of feature vectors, Support Vector Machines (SVMs) have been shown to perform reasonably well for the classification [4, 13]. In this paper, we address the fol ..."
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Cited by 4 (1 self)
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The classification of normal and malginant colon tissue cells is crucial to the diagnosis of colon cancer in humans. Given the right set of feature vectors, Support Vector Machines (SVMs) have been shown to perform reasonably well for the classification [4, 13]. In this paper, we address the following question: how does the choice of a kernel function and its parameters a#ect the SVM classification performance in such a system? We show that the Gaussian kernel function combined with an optimal choice of parameters can produce high classification accuracy.
Multiresolution analysis associated to diffusion semigroups: construction and fast algorithms
, 2004
"... Abstract. We introduce a novel multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the inverse, in compressed form, and their f ..."
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Cited by 1 (1 self)
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Abstract. We introduce a novel multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the inverse, in compressed form, and their fast application. Classes of operators satisfying these conditions include important differential operators, in any dimension, on manifolds, and in nonhomogeneous media. In this case our construction can be viewed as a farreaching generalization of Fast Multipole Methods, achieved through a different point of view, and of the nonstandard wavelet representation of CalderónZygmund and pseudodifferential operators, achieved through a different multiresolution analysis adapted to the operator. We show how the dyadic powers of an operator can be used to induce a multiresolution analysis, as in classical LittlewoodPaley and wavelet theory, and we show how to construct, with fast and stable algorithms, scaling function and wavelet bases associated to this multiresolution analysis, and the corresponding downsampling operators, and use them to compress the corresponding powers of the operator. This allows to extend multiscale signal processing to general spaces (such as manifolds and graphs) in a very natural way, with corresponding fast algorithms. 1.
1 Algorithms from Signal and Data Processing Applied to Hyperspectral Analysis: Discriminating Normal and Malignant Microarray Colon Tissue Sections Using a Novel Digital Mirror Device System
"... Abstract — Hyperspectral imaging is an important tool in various fields, notably geosensing and astronomy, and with the development of new devices, it is now also being applied also in medicine. Concepts and tools from signal processing and data analysis need to be employed to analyze these large an ..."
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Abstract — Hyperspectral imaging is an important tool in various fields, notably geosensing and astronomy, and with the development of new devices, it is now also being applied also in medicine. Concepts and tools from signal processing and data analysis need to be employed to analyze these large and complex data sets. In this paper, we present several techniques which generally apply to hyperspectral data, and we use them to analyze a particular data set. With light sources of increasingly broader ranges, spectral analysis of tissue sections has evolved from 2 wavelength image subtraction techniques to Raman near infrared microspectroscopic analysis permitting discrimination of cell types and tissue patterns. We have developed and used a unique tuned light source based on microoptoelectromechanical systems (MOEMS) and applied algorithms for spectral microscopic analysis of normal and malignant colon tissue. We compare the results to our previous studies which used a tunable liquid filter light source. I.
HighDimensional Pattern Recognition using LowDimensional Embedding and Earth Mover’s Distance
"... We propose an algorithm that combines existing techniques in a novel way to do classification of datasets consisting of highdimensional data (e.g., sets of signals or images). Furthermore, our algorithm sets up a framework for application of the Earth Mover’s Distance (EMD) [1, 2] as a discriminant ..."
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We propose an algorithm that combines existing techniques in a novel way to do classification of datasets consisting of highdimensional data (e.g., sets of signals or images). Furthermore, our algorithm sets up a framework for application of the Earth Mover’s Distance (EMD) [1, 2] as a discriminant measure between datasets. We show how to prepare a compact representation – a signature – for each dataset so that computation of EMD between datasets can be done efficiently. This signatureconstruction step requires the tasks of dimension reduction, automatic determination of the data’s intrinsic dimensionality, outofsample extension, and point clustering. We will show how to apply some existing methods (which include Laplacian eigenmaps [3, 4, 5], diffusion maps framework [6, 7, 8], and elongated Kmeans [9]) to perform these tasks successfully. We will also provide two examples of applications of our proposed algorithm. Key words: diffusion maps, Laplacian eigenmaps, principal component analysis, Earth Mover’s Distance, Hausdorff distance