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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 ..."
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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.
Tertiary 42B35.
, 188
"... Abstract. Following the previous study on the unit ball of Delanghe et al, halfDirichlet problems for the upperhalf space are presented and solved. The solutions further lead to decompositions of the Poisson kernels, and the fact that the classical Dirichlet problems may be solved merely by using ..."
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Abstract. Following the previous study on the unit ball of Delanghe et al, halfDirichlet problems for the upperhalf space are presented and solved. The solutions further lead to decompositions of the Poisson kernels, and the fact that the classical Dirichlet problems may be solved merely by using Cauchy transformation in the respective two contexts. We show that the only domains for which the halfDirichlet problems are solvable in the same pattern are balls and halfspaces.
Local Structure Analysis by Isotropic Hilbert Transforms
"... Abstract. This work presents the isotropic and orthogonal decomposition of 2D signals into local geometrical and structural components. We will present the solution for 2D image signals in four steps: signal modeling in scale space, signal extension by higher order generalized Hilbert transforms, s ..."
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Abstract. This work presents the isotropic and orthogonal decomposition of 2D signals into local geometrical and structural components. We will present the solution for 2D image signals in four steps: signal modeling in scale space, signal extension by higher order generalized Hilbert transforms, signal representation in classical matrix form, followed by the most important step, in which the matrixvalued signal will be mapped to a so called multivector. We will show that this novel multivectorvalued signal representation is an interesting space for complete geometrical and structural signal analysis. In practical computer vision applications lines, edges, corners, and junctions as well as local texture patterns can be analyzed in one unified algebraic framework. Our novel approach will be applied to parameterfree multilayer decomposition. 1
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"... We consider Hölder continuous circulant (2 × 2) matrix functions G12 defined on the fractal boundary Γ of a Jordan domain Ω in R2n. The main goal is to establish a Hilbert transform for such functions, within the framework of Hermitean Clifford analysis. This is a higher dimensional function theor ..."
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We consider Hölder continuous circulant (2 × 2) matrix functions G12 defined on the fractal boundary Γ of a Jordan domain Ω in R2n. The main goal is to establish a Hilbert transform for such functions, within the framework of Hermitean Clifford analysis. This is a higher dimensional function theory centered around the simultaneous null solutions of two first order vector valued differential operators, called Hermitean Dirac operators. In [10] a Hermitean Cauchy integral was constructed by means of a matrix approach using circulant (2 × 2) matrix functions, from which a Hilbert transform was derived in [8], all of this for the case of domains with smooth boundary. However, crucial parts of the method used are not extendable to the case where the boundary of the considered domain is fractal. At present we propose an alternative approach which will enable us to define a new Hermitean Hilbert transform in that case. As a consequence, we are able to give necessary and sufficient conditions for the Hermitean monogenicity of a circulant matrix function G12 in the interior and exterior of Ω, in terms of its boundary value g12 = G 1 2Γ, extending in this way also results of [4] and [2], where Γ is required to be AhlforsDavid regular.