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We present a method to reconstruct the dielectric susceptibility (scattering potential) of an inhomogeneous scattering medium, based on the solution to the inverse scattering problem with internal sources. We employ the theory of reproducing kernel Hilbert spaces, together with regularization to recover the susceptibility of two- and three-dimensional scattering media. Numerical examples illustrate the effectiveness of the proposed reconstruction method., Comment: 11 Pages, 6 figures, 1 table

We present a reconstruction method that stably recovers the real valued, symmetric tensors compactly supported in the Euclidean plane, from knowledge of their attenuated momenta ray transform. The problem is recast as an inverse boundary value problem for a system of transport equations, which we solve by an extension of Bukhgeim's $A$-analytic theory. The method of proof is constructive. To illustrate the reconstruction method, we present results obtained in the numerical implementation for the non-attenuated case of 1-tensors. This new version now includes the results of the preprint arXiv: 2307.10758., Comment: 31 pages, 19 figure. This new version now includes the results of the preprint arXiv: 2307.10758. arXiv admin note: text overlap with arXiv:2307.10758

We introduce an analytic method which stably reconstructs both components of a (sufficiently) smooth, real valued, vector field compactly supported in the plane from knowledge of its Doppler transform and its first moment Doppler transform. The method of proof is constructive. Numerical inversion results indicate robustness of the method., Comment: This manuscript has been superseded by the arXiv manuscript arXiv:2309.00499

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:2105.04634

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:1503.04322

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:2201.10926

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.

We find 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 functions with compact support in the plane, and establish the connection between the range characterization based on the Bukhgeim-Hilbert transform and the classical Gelfand-Graev, Helgason, and Ludwig characterization., Comment: 21 pages, 3 figures

We consider an inverse source problem in the stationary radiating transport through a two dimensional absorbing and scattering medium. Of specific interest, the exiting radiation is measured on an arc. The attenuation and scattering properties of the medium are assumed known. For scattering kernels of finite Fourier content in the angular variable, we show how to quantitatively recover the part of the isotropic sources restricted to the convex hull of the measurement arc. The approach is based on the Cauchy problem with partial data for a Beltrami-like equation associated with $A$-analytic maps in the sense of Bukhgeim, and extends authors' previous work to this specific partial data case. The robustness of the method is demonstrated by the results of several numerical experiments., Comment: 15 pages, 4 figures

Single molecule localization microscopy is a recently developed superresolution imaging technique to visualize structural properties of single cells. The basic principle consists in chemically attaching fluorescent dyes to the molecules, which after excitation with a strong laser may emit light. To achieve superresolution, signals of individual fluorophores are separated in time. In this paper we follow the physical and chemical literature and derive mathematical models describing the propagation of light emitted from dyes in single molecule localization microscopy experiments via Maxwell's equations. This forms the basis of formulating inverse problems related to single molecule localization microscopy. We also show that the current status of reconstruction methods is a simplification of more general inverse problems for Maxwell's equations as discussed here., Comment: 38 pages, 10 figures

We consider the two dimensional quantitative imaging problem of recovering a radiative source inside an absorbing and scattering medium from knowledge of the outgoing radiation measured at the boundary. The medium has an anisotropic scattering property that is neither negligible nor large enough for the diffusion approximation to hold. We present the numerical realization of the authors' recently proposed reconstruction method. For scattering kernels of finite Fourier content in the angular variable, the solution is exact. The feasibility of the proposed algorithms is demonstrated in several numerical experiments, including simulated scenarios for parameters meaningful in optical molecular imaging.

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.

In two dimensions, we consider the problem of inversion of the attenuated $X$-ray transform of a compactly supported function from data restricted to lines leaning on a given arc. We provide a method to reconstruct the function on the convex hull of this arc. The attenuation is assumed known. The method of proof uses the Hilbert transform associated with $A$-analytic functions in the sense of Bukhgeim., Comment: 13 pages, 7 figures

While in standard photoacoustic imaging the propagation of sound waves is modeled by the standard wave equation, our approach is based on a generalized wave equation with variable sound speed and material density, respectively. In this paper we present an approach for photoacoustic imaging, which in addition to recovering of the absorption density parameter, the imaging parameter of standard photoacoustics, also allows to reconstruct the spatially varying sound speed and density, respectively, of the medium. We provide analytical reconstruction formulas for all three parameters based in a linearized model based on single plane illumination microscopy (SPIM) techniques.

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:1411.4923

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 functions \`{a} 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.

We present new necessary and sufficient conditions for a function on $\partial\Omega\times S^1$ to be in the range of the attenuated Radon transform of a sufficiently smooth function support in the convex set $\bar\Omega\subset\mathbb{R}^2$. The approach is based on an explicit Hilbert transform associated with traces of the boundary of A-analytic functions in the sense of Bukhgeim., Comment: 2 figures