Stabilized Numerical Analytic Prolongation with Poles

Miller, Keith

doi:10.1137/0118029

Cited by 41 publications

(39 citation statements)

References 11 publications

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“…The order n term in the representation formula (1.4) is physically that of a voltage potential corresponding to m appropriately polarized dipoles located at the points z j , 1 ≤ j ≤ m. Similarly the order term in (1.10) is an "average" of voltage potentials corresponding to appropriately polarized dipoles distributed along the curve σ 0 . Results that address the theoretical feasibility of recovering the location of dipoles from boundary measurements are found various places in the literature; the reader may for example consult [14] and [12]. The special formulae found in [12] Section 3 for the recovery of a single inhomogeneity are very related to our brief discussion in Section 3 of the present paper.…”

Section: Introductionmentioning

confidence: 82%

Boundary integral formulae for the reconstruction of electric and electromagnetic inhomogeneities of small volume

2003

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Abstract. In this paper we discuss the approximate reconstruction of inhomogeneities of small volume. The data used for the reconstruction consist of boundary integrals of the (observed) electromagnetic fields. The numerical algorithms discussed are based on highly accurate asymptotic formulae for the electromagnetic fields in the presence of small volume inhomogeneities.Mathematics Subject Classification. 35J25, 35R30, 65R99.

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Section: Introductionmentioning

confidence: 82%

Boundary integral formulae for the reconstruction of electric and electromagnetic inhomogeneities of small volume

2003

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“…3 Most localization techniques depend on the forward model; i.e., computing the boundary data for a given source configuration and iteratively fitting its parameters by least-squares optimization. 4 However, the non-linear relationship between the sources' positions and the boundary data imply the existence of local minima and make the solution very sensitive to the initial guess. Hence, recovery of the parametric sources is often limited to single-dipole models.…”

Section: Introductionmentioning

confidence: 99%

Analytic sensing: direct recovery of point sources from planar Cauchy boundary measurements

2007

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Inverse problems play an important role in engineering. A problem that often occurs in electromagnetics (e.g. EEG) is the estimation of the locations and strengths of point sources from boundary data.We propose a new technique, for which we coin the term "analytic sensing". First, generalized measures are obtained by applying Green's theorem to selected functions that are analytic in a given domain and at the same time localized to "sense" the sources. Second, we use the finite-rate-of-innovation framework to determine the locations of the sources. Hence, we construct a polynomial whose roots are the sources' locations. Finally, the strengths of the sources are found by solving a linear system of equations. Preliminary results, using synthetic data, demonstrate the feasibility of the proposed method.

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“…The corresponding cost function has many local minima and makes the solution very sensitive to the initial guess. Therefore, successful recovery of the parametric sources is often limited to single-dipole models.Despite the practical difficulties in identifying parametric source models, the mathematical uniqueness of the solution has been proven [7], and stability results are available for the case of dipolar and point sources in 2D [5,16] and 3D [20]. Instead…”

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confidence: 99%

“…Despite the practical difficulties in identifying parametric source models, the mathematical uniqueness of the solution has been proven [7], and stability results are available for the case of dipolar and point sources in 2D [5,16] and 3D [20]. Instead…”

mentioning

confidence: 99%

“…This situation is particularly attractive in the electromagnetic setting for applications such as electroencephalography (EEG); e.g., the localization of some type of epileptic foci can be reasonably modeled by pointwise (dipolar) sources [15]. Almost all known techniques rely on the forward model; i.e., computing the boundary data for a given source configuration and iteratively fitting its parameters by least-squares optimization [16]. The corresponding cost function has many local minima and makes the solution very sensitive to the initial guess.…”

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confidence: 99%

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Analytic Sensing: Noniterative Retrieval of Point Sources from Boundary Measurements

Kandaswamy¹,

Blu²,

Ville³

2009

SIAM J. Sci. Comput.

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Abstract. We consider the problem of locating point sources in the planar domain from overdetermined boundary measurements of solutions of Poisson's equation. In this paper, we propose a novel technique, termed "analytic sensing," which combines the application of Green's theorem to functions with vanishing Laplacian-known as the "reciprocity gap" principle-with the careful selection of analytic functions that "sense" the manifestation of the sources in order to determine their positions and intensities. Using this formalism we express the problem at hand as a generalized sampling problem, where the signal to be reconstructed is the source distribution. To determine the positions of the sources, which is a nonlinear problem, we extend the annihilating-filter method, which reduces the problem to solving a linear system of equations for a polynomial whose roots are the positions of the point sources. Once these positions are found, resolving the according intensities boils down to solving a linear system of equations. We demonstrate the performance of our technique in the presence of noise by comparing the achieved accuracy with the theoretical lower bound provided by Cramér-Rao theory. 1. Introduction. Source imaging from boundary Cauchy data satisfying the Laplace equation is a classical inverse problem that is of high interest to many fields in engineering. Unfortunately, the problem is ill-posed and additional assumptions about the source configuration are needed to make the solution unique. Typically, one can restrict the class of source distributions by imposing smoothness properties (e.g., Tikhonov regularization [19]) or by imposing a parametric source model.It is when the source configurations are expected to be "sparse" that parametric solutions may prove very useful. This situation is particularly attractive in the electromagnetic setting for applications such as electroencephalography (EEG); e.g., the localization of some type of epileptic foci can be reasonably modeled by pointwise (dipolar) sources [15]. Almost all known techniques rely on the forward model; i.e., computing the boundary data for a given source configuration and iteratively fitting its parameters by least-squares optimization [16]. The corresponding cost function has many local minima and makes the solution very sensitive to the initial guess. Therefore, successful recovery of the parametric sources is often limited to single-dipole models.Despite the practical difficulties in identifying parametric source models, the mathematical uniqueness of the solution has been proven [7], and stability results are available for the case of dipolar and point sources in 2D [5,16] and 3D [20]. Instead

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Stabilized Numerical Analytic Prolongation with Poles

Cited by 41 publications

References 11 publications

Boundary integral formulae for the reconstruction of electric and electromagnetic inhomogeneities of small volume

Boundary integral formulae for the reconstruction of electric and electromagnetic inhomogeneities of small volume

Analytic sensing: direct recovery of point sources from planar Cauchy boundary measurements

Analytic Sensing: Noniterative Retrieval of Point Sources from Boundary Measurements

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