Recovery of pointwise sources or small inclusions in 2D domains and rational approximation

Baratchart, Laurent; Abda, Amel Ben; Hassen, M. F. Ben; Leblond, Juliette

doi:10.1088/0266-5611/21/1/005

Cited by 50 publications

(59 citation statements)

References 39 publications

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“…Furthermore, if we choose ψ(z) = 1 and use Theorem 1, then it is clear that the following should hold: This parametrization is typical for a "finite rate of innovation" setting [21]. We will see that this setting ensures that P is well-posed-or at least, that if it has a solution, this solution is unique [2]. Thus, the purpose of solving the inverse problem at hand shifts towards the retrieval of the positions and the intensities of the pointwise sources.…”

Section: Homogeneous Mediummentioning

confidence: 99%

“…These methods are planar by their nature, but they can be extended to 3D; e.g., for the axisymmetric setting or by combining the results of multiple cross-sections. In particular, Baratchart et al [2,3] have chosen to approximate the known boundary data for monopolar and dipolar sources by a meromorphic function which, under the source model hypothesis, has poles that can be related to the sources' positions.…”

mentioning

confidence: 99%

“…This problem is not well-posed as there are many possible source fields that solve P with the same measurements [2]. This can be seen by an example when Ω is a disk of radius 1: consider a function V which satisfies 1 V (x) = 0 and x T ∇V (x) = 0 on the boundary x = 1.…”

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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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Section: Homogeneous Mediummentioning

confidence: 99%

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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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“…The statistical approach described in [11] is therefore limited by the number of time samples used to estimate higher order statistics, and its performance depends on the statistical properties of source signals. In [12,13] an analytical approach is presented for the localization of monopole and dipole sources within a disk from given boundary data.…”

Section: Eeg Applicationmentioning

confidence: 99%

Multi-way space–time–wave-vector analysis for EEG source separation

et al. 2012

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Deterministic approaches for source localization and extraction are desirable for short or nonstationary data, as opposed to techniques based on second or higher order statistics. Techniques based on tensor decompositions are recognized to be efficient in this framework, provided some diversity is available, in addition to time and space. With this goal, some authors have proposed to decompose a SpaceTime-Frequency data tensor. In this paper, we propose a new multiway approach based on Space-Time-Wave-Vector (STWV) data which is obtained by a 3D local Fourier transform over space accomplished on the measured data. This method does not only permit to accurately localize sources even in a noisy environment, but simultaneously extracts the temporal behaviour associated with each source. The performance of this STWV analysis is investigated by means of computer simulations in the context of ElectroEncephaloGraphic (EEG) data analysis. 1

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“…Consequently, we apply to these 2D problems a technique inspired by that described in [Baratchart et al, 2005] that relies on approximating f on the boundary circle by a rational function with poles in the disk.…”

Section: Rr N°7704mentioning

confidence: 99%

Source localization using rational approximation on plane sections

et al. 2012

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In functional neuroimaging, a crucial problem is to localize active sources within the brain non-invasively, from knowledge of electromagnetic measurements outside the head. Identification of point sources from boundary measurements is an ill-posed inverse problem. In the case of electroencephalography (EEG), measurements are only available at electrode positions, the number of sources is not known in advance and the medium within the head is inhomogeneous. This paper presents a new method for EEG source localization, based on rational approximation techniques in the complex plane. The method is used in the context of a nested sphere head model, in combination with a cortical mapping procedure. Results on simulated data prove the applicability of the method in the context of realistic measurement configurations.In memory of Line Garnero, and of her communicative dedication to unveiling the mysteries of the brain.

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Recovery of pointwise sources or small inclusions in 2D domains and rational approximation

Cited by 50 publications

References 39 publications

Analytic Sensing: Noniterative Retrieval of Point Sources from Boundary Measurements

Analytic Sensing: Noniterative Retrieval of Point Sources from Boundary Measurements

Multi-way space–time–wave-vector analysis for EEG source separation

Source localization using rational approximation on plane sections

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