Related papers: Broken ray transform on a Riemann surface with a c…
The significance of the broken ray transform (BRT) is due to its occurrence in a number of modalities spanning optical, x-ray, and nuclear imaging. When data are indexed by the scatter location, the BRT is both linear and shift invariant.…
We survey recent results on inverse problems for geodesic X-ray transforms and other linear and non-linear geometric inverse problems for Riemannian metrics, connections and Higgs fields defined on manifolds with boundary.
Under a convexity assumption on the boundary we solve a local inverse problem, namely we show that the geodesic X-ray transform can be inverted locally in a stable manner; one even has a reconstruction formula. We also show that under an…
Consider a compact Riemannian manifold in dimension $n$ with strictly convex boundary. We show the local invertibility near a boundary point of the transverse ray transform of $2$ tensors for $n\geq 3$ and the mixed ray transform of $2+2$…
This paper discusses the geometry of a surface endowed with a slope metric. We obtain necessary and sufficient conditions for any surface of revolution to admit a strongly convex slope metric. Such conditions involve certain inequalities…
Let $(M,g)$ be a simple Riemannian manifold with boundary and consider the geodesic ray transform of symmetric 2-tensor fields. Let the integral of $f$ along maximal geodesics vanish on an appropriate open subset of the space of geodesics…
We consider the X-ray transform in a projective space over a finite field. It is well known (after E. Bolker) that this transform is injective. We formulate an analog of I.M. Gelfand's admissibility problem for the Radon transform, which…
Several years ago the authors started looking at some problems of convex geometry from a more general point of view, replacing volume by an arbitrary measure. This approach led to new general properties of the Radon transform on convex…
We describe all pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta. As an application, we solve the Beltrami problem on closed surfaces and prove the nonexistence of…
It has been shown in [Pa1] that on a simple, compact Riemannian 2-manifold the attenuated geodesic ray transform, with attenuation given by a connection and Higgs field, is injective on functions and 1-forms modulo the natural obstruction.…
The present article proposes a partial answer to the explicit inversion of the tensor tomography problem in two dimensions, by proving injectivity over certain kinds of tensors and providing reconstruction formulas for them. These tensors…
We show that the geodesic ray transform is injective on scalar functions on spherically symmetric reversible Finsler manifolds where the Finsler norm satisfies a Herglotz condition. We use angular Fourier series to reduce the injectivity…
We study a particular broken ray transform on the Euclidean unit square and establish injectivity and stability for $C_{0}^{2}$ perturbations of the constant unit weight. Given an open subset $E$ of the boundary, we measure the attenuation…
We study the boundary and lens rigidity problems on domains without assuming the convexity of the boundary. We show that such rigidities hold when the domain is a simply connected compact Riemannian surface without conjugate points. For the…
We give reconstruction formulas inverting the geodesic X-ray transform over functions (call it $I_0$) and solenoidal vector fields on surfaces with negative curvature and strictly convex boundary. These formulas generalize the…
Conformal transformation optics provides a simple scheme for manipulating light rays with inhomogeneous isotropic dielectrics. However, there is usually discontinuity for refractive index profile at branch cuts of different virtual Riemann…
We show that any grafting ray in Teichm\"{u}ller space is (strongly) asymptotic to some Teichm\"{u}ller geodesic ray. As an intermediate step we introduce surfaces that arise as limits of these degenerating Riemann surfaces. Given a…
We show that the attenuated geodesic ray transform on two dimensional simple surfaces is injective. Moreover we give a stability estimate and develop a reconstruction procedure.
We describe all pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta. As an application, we solve the Beltrami problem on closed surfaces, prove the nonexistence of…
We theoretically study negative refraction of inhomogeneous waves at the interface of lossy isotropic media. We obtain explicit (up to the sign) expressions for the parameters of a wave transmitted through the interface between two lossy…