On the range of the Attenuated Radon Transform in strictly convex sets.

In the present dissertation, we characterize the range of the attenuated Radon transform of zero, one, and two tensor fields, supported in strictly convex set. The approach is based on a Hilbert transform associated with A-analytic functions of A. Bukhgeim. We first present new necessary and sufficient conditions for a function to be in the range of the attenuated Radon transform of a sufficiently smooth function supported in the convex set. The approach is based on an explicit Hilbert transform associated with traces of the boundary of A-analytic functions in the sense of A. Bukhgeim. We then uses the range characterization of the Radon transform of functions to characterize the range of the attenuated Radon transform of vector fields as they appear in the medical diagnostic techniques of Doppler tomography. As an application we determine necessary and sufficient conditions for the Doppler and X-ray data to be mistaken for each other. We also characterize the range of real symmetric second order tensor field using the range characterization of the Radon transform of zero tensor field

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

In this paper we study the attenuated $X$-ray transform of 2-tensors supported in strictly convex bounded subsets in the Euclidean plane. We characterize its range and reconstruct all possible 2-tensors yielding identical $X$-ray data. The characterization is in terms of a Hilbert-transform associated with $A$-analytic maps in the sense of Bukhgeim.Comment: 1 figure. arXiv admin note: text overlap with arXiv:1411.492

On the XX-ray transform of symmetric higher order tensors

In this article we characterize the range of the attenuated and non-attenuated $X$-ray transform of compactly supported symmetric tensor fields in the Euclidean plane. The characterization is in terms of a Hilbert-transform associated with $A$-analytic maps in the sense of Bukhgeim.Comment: 29 pages. arXiv admin note: text overlap with arXiv:1503.0432

An inverse source problem for linearly anisotropic radiative sources in absorbing and scattering medium

We consider in a two dimensional absorbing and scattering medium, an inverse source problem in the stationary radiative transport, where the source is linearly anisotropic. The medium has an anisotropic scattering property that is neither negligible nor large enough for the diffusion approximation to hold. The attenuating and scattering properties of the medium are assumed known. For scattering kernels of finite Fourier content in the angular variable, we show how to recover the anisotropic radiative sources from boundary measurements. The approach is based on the Cauchy problem for a Beltrami-like equation associated with $A$-analytic maps. As an application, we determine necessary and sufficient conditions for the data coming from two different sources to be mistaken for each other.Comment: 16 pages. arXiv admin note: text overlap with arXiv:2105.0463

On the range of the XX-ray transform of symmetric tensors compactly supported in the plane

We find the necessary and sufficient conditions on the Fourier coefficients of a function $g$ on the torus to be in the range of the $X$-ray transform of a symmetric tensor of compact support in the plane.Comment: 27 pages, 3 figures. arXiv admin note: text overlap with arXiv:2201.1092

Numerical Reconstruction of Radiative Sources from Partial Boundary Measurements

We consider an inverse source problem in the stationary radiative transport through an absorbing and scattering medium in two dimensions. Using the angularly resolved radiation measured on an arc of the boundary, we propose a numerical algorithm to recover the source in the convex hull of this arc. The method involves an unstable step of inverting a bounded operator whose range is not closed. We show that the continuity constant of the discretized inverse grows at most linearly with the discretization step, thus stabilizing the problem. Numerical examples presented show the effectiveness of the proposed method

ON A LOCAL INVERSION OF THE XX-RAY TRANSFORM FROM ONE SIDED DATA (Recent developments on inverse problems for partial differential equations and their applications)

We explain how the theory of A-analytic maps of A. Bukhgeim can apply to a local CT inversion problem, in which the data is restricted to lines leaning on a given arc