On the range of the attenuated Radon transform in strictly convex sets

We revisit the inverse source problem in a two dimensional absorbing and scattering medium and present a non-iterative reconstruction method using measurements of the radiating flux at the boundary. The attenuation and scattering coefficients are known and the unknown source is isotropic. The approach is based on the Cauchy problem for a Beltrami-like equation for the sequence valued maps, and extends the original ideas of A. Bukhgeim from the non-scattering to scattering media. We demonstrate the feasibility of the method in a numerical experiment in which the scattering is modeled by the two dimensional Henyey-Greenstein kernel with parameters meaningful in Optical Tomography.

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“…By applying 4B to (38), the mode u pM q M`1 is then the solution to the Dirichlet problem for the Poisson equation…”

Section: Source Reconstruction For Scattering Of Polynomial Typementioning

confidence: 99%

A Fourier approach to the inverse source problem in an absorbing and anisotropic scattering medium

2019

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“…Moreover, g is the trace on Γ ×S 1 of a solution u ∈ C 1,α (Ω× S 1 ) of the transport equation (20). By [30,Proposition 4…”

Section: Proof (I) Necessitymentioning

confidence: 99%

“…Step 3: The construction of modes u −1 and u 1 . Let ψ ∈ Ψ g as in (30). We define u −1 := ψ, and u 1 := ψ.…”

Section: Proof (I) Necessitymentioning

confidence: 99%

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On the X-ray transform of planar symmetric 2-tensors

Sadiq

Scherzer²,

Tamasan

2016

Journal of Mathematical Analysis and Applications

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“…A separate method to invert the two dimensional attenuated X-ray transform based on the theory of A-analytic functions was originally developed in [3]. In [29] the authors introduce a Hilbert transform corresponding to A-analytic maps, and use it to characterize the range of the attenuated Radon transform of compactly supported functions.…”

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On the Range Characterization of the Two-Dimensional Attenuated Doppler Transform

Sadiq

Tamasan

2015

SIAM J. Math. Anal.

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Abstract. We characterize the range of the attenuated and non-attenuated X-ray transform of compactly supported vector fields in the plane. The characterization is in terms of a Hilbert transform associated with the A-analytic functionsà la Bukhgeim. As an application we determine necessary and sufficient conditions for the attenuated Doppler and X-ray data to be mistaken for each other. The problem of inversion of the X-ray transform of higher order tensor fields have been formulated in [30] in the geometric setting of Riemannian manifolds with boundary; see [33] for the Euclidean setting. Coming from the practical procedure of acquiring data, the X-ray transform of vector fields is also known as the Doppler transform. Various partial results (e.g., [31], [32]) culminated with the inversion formulas for recovering the solenoidal part of one-tensors on simple Riemannian surfaces with boundary [27]; see also [28]. Injectivity in the attenuated case for both 0-and 1-tensors is much more recent [34]; see also [12] for a more general weighted transform. Inversion formulas to tensors of higher orders have been found in [15] for the Euclidean case, and [25] for the Riemannian case. However, these works do not address range characterization.The first range characterizations of the (non-attenuated) X-ray transform is given in [27] for both 0-and 1-tensors supported on simple Riemannian surfaces with boundary. This characterization is given in terms of the scattering relation.We consider here the problem of the range characterization of both attenuated and non-attenuated Doppler transform in a strictly convex bounded domain in the Euclidian plane. Our approach uses the Hilbert transform for A-analytic maps in [29] and relies on new identities enjoyed by such maps, see Lemma 2.6. The characterization concerns the Fourier modes (in the angular variables) of the data and can be

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On the range of the attenuated Radon transform in strictly convex sets

Cited by 25 publications

References 24 publications

A Fourier approach to the inverse source problem in an absorbing and anisotropic scattering medium

A Fourier approach to the inverse source problem in an absorbing and anisotropic scattering medium

On the X-ray transform of planar symmetric 2-tensors

On the Range Characterization of the Two-Dimensional Attenuated Doppler Transform

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