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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 ..."
Abstract

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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.
continuous extension and wavelet approximation
"... an efficient sparse representation of objects of general shape via ..."
Objectoriented image compression via continuous extension and wavelet approximation
"... In objectoriented image analysis, one needs to develop a set of tools to manipulate the detected objects, for example, to compress them, catalog them, analyze their characteristics, and filter their certain spatial frequency features in an individual manner. In this paper, we discuss the continuous ..."
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In objectoriented image analysis, one needs to develop a set of tools to manipulate the detected objects, for example, to compress them, catalog them, analyze their characteristics, and filter their certain spatial frequency features in an individual manner. In this paper, we discuss the continuous extension and wavelet approximation of the detected object on a general domain Ω of R 2. We first extend continuously the image to a square T and such that it vanishes on boundary ∂T. On T \ Ω, the extension has a simple and clear representation which is determined by the equation of the boundary ∂Ω. We expand the extension into wavelet series on R 2. Since the extension tool is polynomials, by the moment theorem, we know that the sequence of wavelet coefficients obtained by us is sparse, so it can compress image efficiently. Moreover, using these wavelet coefficients, we can represent the object compactly for analysis, interpretation, discrimination, and so on. Key words: objectoriented image analysis, continuous extension, wavelet approximation 1