We present a new approach to solve inverse source problems for the three-dimensional time-harmonic Maxwell's equations using boundary measurements of the radiated fields. The modelling is based on the formulation as a system of integro-differential equations for the electric field. We introduce a method to recast the intertwined vector equations of Maxwell into decoupled scalar problems. The method of the approximate inverse is used both for regularization and the development of fast algorithms. We make the analysis of the method when data are collected on a spherical setting around the object. Based on the singular value decomposition, we study the smoothing properties for the underlying operator and derive an error estimate for the regularized solution in a Sobolev-space framework. Numerical simulations illustrate the efficiency and practical usefulness of the developed method.

The ill-posedness of the attenuated Radon transform is a challenging issue in practice due to the Poisson noise and the high level of attenuation. The investigation of the smoothing properties of the underlying operator is essential for developing a stable inversion. In this paper, we consider the framework of Sobolev spaces and derive analytically a reconstruction algorithm based on the method of the approximate inverse. The derived method inherits the efficiency and stability of the approximate inverse and supplies a method of extraction of contours. These algorithms appear to be efficient for an attenuation of human body type. However, for higher attenuations the ill-posedness increases exponentially what deteriorates accordingly the quality of reconstructions. Nevertheless, a high attenuation map affects less the contour extraction of a high contrast function and so can be neglected. This leads to simplifying the proposed method and circumvents in this case the artifacts due to the attenuation as attested by simulation results.

We present an iterative reconstruction method for an inverse medium scattering problem (IMP) for the three-dimensional time-harmonic Maxwell equations. The goal here is to determine the electromagnetic properties of a nonmagnetic unknown inhomogeneous object. The data are near-field measurements of scattered electric fields for multiple illuminations at a fixed frequency. We use the concept of the generalized induced source (GIS) to recast the intertwined vector equations of Maxwell into decoupled scalar scattering problems. To treat the nonlinearity of the IMP, we apply the localized nonlinear approximation due to Habashy and co-workers. In this framework, we derive a fast reconstruction method based on the Kaczmarz algorithm. Besides, for the underlying approximation we present a uniqueness result for determining the contrast function. Numerical experiments in 3D with synthetic and real data show the scope and limitations of the method.

This paper is concerned with the Radon transform over a family of Cormack-type curves and provides an exact inversion formula. The studied family of curves, called C1, appeared in previous works as a suitable manifold for modeling imaging concepts in conventional and Compton scattering tomography (CST). More specifically, the straight line, integral support of the classical Radon transform used in computed tomography (CT) belongs to C1. In conventional tomography, many reconstruction techniques compute the derivative of the data with the aim of reducing the order of singularity of the reconstruction kernel associated here to the Radon transform in two dimensions. However, differentiating data requires a regularization step (for instance, convolution with a smooth function) which reduces the resolution of reconstructed images. Here, the proposed analytical inversion formula recovers the circular harmonic components of the sought object without differentiation of the data, which leads to an improvement of the final resolution. Furthermore, we deal with the singularity issue of the reconstruction kernel by applying a range property of the Radon transform. Since theoretical results are developed in a quite general context of inverse problems for Radon transforms over C1, the potential applications of our algorithm appear to be numerous in the field of CT and CST. Numerical results in the framework of CT and of one modality on CST reveal the strength of this algorithm in terms of accuracy and stability in comparison with the well-known filtered back-projection.

This paper is concerned with a method of image reconstruction and feature extraction for the attenuated Radon transform in two dimensions based on the decomposition in circular harmonics of the integral kernel in Novikovʼs inversion formula for an arbitrary known attenuation. This analytical decomposition of the reconstruction kernel provides an alternative reconstruction algorithm. Besides, we propose to use our formula to directly extract features of the object with no need for process imaging techniques. Numerical results attest to the strengths and limitations of our reconstruction method in terms of accuracy and robustness for image and feature reconstruction.