A multiscattering series for impedance tomography in layered media

Borcea, Liliana; Ortiz, Michael

doi:10.1088/0266-5611/15/2/011

Cited by 5 publications

(4 citation statements)

References 29 publications

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“…where all functions U (n) (z, ξ,t) are real. Using decompositions (4), (9), it is easily seen that U (n) (z, ξ,t) are even functions with respect to ξ for even n and odd for odd n.…”

Section: Formulation Of the Problemmentioning

confidence: 99%

“…The theory of one-dimensional inverse problems which goes back to the classical results obtained by Borg, Levinson, Gel'fand-Levitan, Marchenko and Krein of late 40-th -50-th is still an area of active research with new theoretical results and numerical algorithms appearing regularly in mathematical and geophysical literature, see e.g. [9], [12], [16], [23], [25], [28]- [32] to mention just a few.…”

Section: Introductionmentioning

confidence: 99%

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Dynamic inverse problem in a weakly laterally inhomogeneous medium: theory and numerical experiment

Blagovestchenskii

Kurylev²,

Zalipaev³

2006

j inv ill-posed problems

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An inverse problem of wave propagation into a weakly laterally inhomogeneous medium occupying a half-space is considered in the acoustic approximation. The halfspace consists of an upper layer and a semi-infinite bottom separated with an interface. An assumption of a weak lateral inhomogeneity means that the velocity of wave propagation and the shape of the interface depend weakly on the horizontal coordinates, x = (x 1 , x 2 ), in comparison with the strong dependence on the vertical coordinate, z, giving rise to a small parameter ε << 1. Expanding the velocity in power series with respect to ε, we obtain a recurrent system of 1D inverse problems. We provide algorithms to solve these problems for the zero and first-order approximations. In the zero-order approximation, the corresponding 1D inverse problem is reduced to a system of non-linear Volterra-type integral equations. In the first-order approximation, the corresponding 1D inverse problem is reduced to a system of coupled linear Volterra integral equations. These equations are used for the numerical reconstruction of the velocity in both layers and the interface up to O(ε 2 ).

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“…where all functions U (n) (z, ξ,t) are real. Using decompositions (4), (9), it is easily seen that U (n) (z, ξ,t) are even functions with respect to ξ for even n and odd for odd n.…”

Section: Formulation Of the Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dynamic inverse problem in a weakly laterally inhomogeneous medium: theory and numerical experiment

Blagovestchenskii

Kurylev²,

Zalipaev³

2006

j inv ill-posed problems

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show abstract

“…The study of the inverse problem for the wave equation in layered media plays a crucial role in understanding and modelling complex wave propagation phenomena. It enables the estimation of important material parameters in various practical applications such as seismology, geophysics, and medical imaging [4,7,8,10,26,28].…”

Section: Introductionmentioning

confidence: 99%

Wave scattering in 1D: D'Alembert-type representations and a reconstruction method

Kalimeris

Mindrinos

2023

Preprint

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We derive the extension of the classical d'Alembert formula for the wave equation, which provides the analytical solution for the direct scattering problem for a medium with constant refractive index; this is achieved by employing results obtained via the Fokas method. This methodology is further extended to a medium with piecewise constant refractive index, providing the apparatus for the solution of the associated inverse scattering problem. Hence, we provide an exact reconstruction method which is valid for both full and phaseless data.

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“…where the mapping A : x → y is referred to as the forward map (problem). However, in several imaging modalities including optical coherence tomography (OCT) [35,36], ultrasound [4,23], synthetic aperture radar (SAR) imaging [9,34], and electrical impedance tomography (EIT) [2,37], noise can be proportional to the data. In such a case, we have…”

Section: Introductionmentioning

confidence: 99%

An Additive Approximation to Multiplicative Noise

Nicholson¹,

Kaipio²

2018

Preprint

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Multiplicative noise models are often used instead of additive noise models in cases in which the noise variance depends on the state. Furthermore, when Poisson distributions with relatively small counts are approximated with normal distributions, multiplicative noise approximations are straightforward to implement. There are a number of limitations in existing approaches to marginalize over multiplicative errors, such as positivity of the multiplicative noise term. The focus in this paper is in large dimensional (inverse) problems for which sampling type approaches have too high computational complexity. In this paper, we propose an alternative approach to carry out approximative marginalization over the multiplicative error by embedding the statistics in an additive error term. The approach is essentially a Bayesian one in that the statistics of the additive error is induced by the statistics of the other unknowns. As an example, we consider a deconvolution problem on random fields with different statistics of the multiplicative noise. Furthermore, the approach allows for correlated multiplicative noise. We show that the proposed approach provides feasible error estimates in the sense that the posterior models support the actual image.

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A multiscattering series for impedance tomography in layered media

Cited by 5 publications

References 29 publications

Dynamic inverse problem in a weakly laterally inhomogeneous medium: theory and numerical experiment

Dynamic inverse problem in a weakly laterally inhomogeneous medium: theory and numerical experiment

Wave scattering in 1D: D'Alembert-type representations and a reconstruction method

An Additive Approximation to Multiplicative Noise

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