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In this article, we analyze the microlocal properties of the linearized forward scattering operator F and the normal operator F * F (where F * is the L 2 adjoint of F ) which arises in Synthetic Aperture Radar imaging for the common midpoint acquisition geometry. When F * is applied to the scattered data, artifacts appear. We show that F * F can be decomposed as a sum of four operators, each belonging to a class of distributions associated to two cleanly intersecting Lagrangians, I p,l (Λ 0 , Λ 1 ), thereby explaining the latter artifacts.

Abstract. Let (M, g) be a simple Riemannian manifold. Under the assumption that the metric g is real-analytic, it is shown that if the geodesic ray transform of a function f ∈ L 2 (M ) vanishes on an appropriate open set of geodesics, then f = 0 on the set of points lying on these geodesics. The approach is based on analytic microlocal analysis.

This paper considers the non-linear inverse problem of reconstructing an electric conductivity distribution from the interior power density in a bounded domain. Applications include the novel tomographic method known as acousto-electric tomography, in which the measurement setup in Electrical Impedance Tomography is modulated by ultrasonic waves thus giving rise to a method potentially having both high contrast and high resolution. We formulate the inverse problem as a regularized non-linear optimization problem, show the existence of a minimizer, and derive optimality conditions. We propose a non-linear conjugate gradient scheme for finding a minimizer based on the optimality conditions. All our numerical experiments are done in two-dimensions. The experiments reveal new insight into the non-linear effects in the reconstruction. One of the interesting features we observe is that, depending on the choice of regularization, there is a trade-off between high resolution and high contrast in the reconstructed images. Our proposed non-linear optimization framework can be generalized to other hybrid imaging modalities.

We generalize the inversion formulas obtained by Pestov–Uhlmann for the geodesic ray transform of functions and vector fields on 2-dimensional manifolds with boundary of constant curvature. Our formulas hold for simple 2-dimensional manifolds whose curvatures are close to a constant.