arXiv research

Survey on broken ray transforms and open problems.

We reduce the broken ray transform on some Riemannian manifolds (with corners) to the geodesic ray transform on another manifold, which is obtained from the original one by reflection. We give examples of this idea and present injectivity results for the broken ray transform using corresponding earlier results for the …

We consider the broken ray transform on Riemann surfaces in the presence of an obstacle, following earlier work of Mukhometov. If the surface has nonpositive curvature and the obstacle is strictly convex, we show that a function is determined by its integrals over broken geodesic rays that reflect on the boundary of th…

Study proves uniqueness for ray transform on surfaces with obstacles.

This PhD thesis studies the broken ray transform, a generalization of the geodesic X-ray transform where geodesics are replaced with broken rays that reflect on a part of the boundary. The fundamental question is whether this transform is injective. We employ four different methods to approach this question, and each o…

New proof of injectivity for broken non-abelian X-ray transform in Minkowski space.

We study the integral transform over a general family of broken rays in $\mathbb{R}^2$. It is natural for broken rays to have conjugate points, for example, when they are reflected from a curved boundary. If there are conjugate points, we show that the singularities cannot be recovered from local data and therefore art…

We reduce boundary determination of an unknown function and its normal derivatives from the (possibly weighted and attenuated) broken ray data to the injectivity of certain geodesic ray transforms on the boundary. For determination of the values of the function itself we obtain the usual geodesic ray transform, but for…

We show injectivity of the X-ray transform and the $d$-plane Radon transform for distributions on the $n$-torus, lowering the regularity assumption in the recent work by Abouelaz and Rouvière. We also show solenoidal injectivity of the X-ray transform on the $n$-torus for tensor fields of any order, allowing the tensor…

We study ray transforms on spherically symmetric manifolds with a piecewise $C^{1,1}$ metric. Assuming the Herglotz condition, the X-ray transform is injective on the space of $L^2$ functions on such manifolds. We also prove injectivity results for broken ray transforms (with and without periodicity) on such manifolds …

We consider Calderon's inverse problem with partial data in dimensions $n \geq 3$. If the inaccessible part of the boundary satisfies a (conformal) flatness condition in one direction, we show that this problem reduces to the invertibility of a broken geodesic ray transform. In Euclidean space, sets satisfying the flat…

We show that a tensor field of any rank integrates to zero over all broken rays if and only if it is a symmetrized covariant derivative of a lower order tensor which satisfies a symmetry condition at the reflecting part of the boundary and vanishes on the rest. This is done in a geometry with non-positive sectional cur…

Study extends geodesic ray transform results to orientable surfaces.

Paper shows invertibility of tensor X-ray transform on certain manifolds.

The article recovers tensor fields from partial data using weighted divergent ray transforms.

Regularity results for geodesic X-ray transform on nonsmooth manifolds

Local X-ray transform works well near boundaries in hyperbolic spaces.

Local invertibility of ray transforms on convex manifolds.

Study light ray transform in pseudo-Euclidean space, derive inversion formula, and prove stability.

Study shows stability in X-ray transform on specific hyperbolic manifolds.

Injective X-ray transform on Heisenberg group for regular functions.

Paper proves injectivity of non-abelian X-ray transform on certain spaces.

Local invertibility of higher order tensor transforms on compact manifolds.

In this paper we consider the local X-ray transform for general flows. We extend the results on the local and global invertibility of the geodesic ray transform proved by Uhlmann and Vasy \cite{UV} to the X-ray transform for a general flow. The key improvement is that our argument for the ellipticity of the conjugated …

We describe the range of the attenuated ray transform of a unitary connection on a simple surface acting on functions and 1-forms. We use this to determine the range of the ray transform acting on symmetric tensor fields.

Given a Lorentzian manifold, the light ray transform of a function is its integrals along null geodesics. This paper is concerned with the injectivity of the light ray transform on functions and tensors, up to the natural gauge for the problem. First, we study the injectivity of the light ray transform of a scalar func…

Injectivity result for light ray transform on Lorentzian manifolds.

Injectivity of geodesic ray transform on specific Finsler manifolds proven.

On simple geodesic disks of constant curvature, we derive new functional relations for the geodesic X-ray transform, involving a certain class of elliptic differential operators whose ellipticity degenerates normally at the boundary. We then use these relations to derive sharp mapping properties for the X-ray transform…

New method eliminates domain size restrictions for X-ray transform inversion.

X-ray transform on H-type groups solved, revealing function injectivity.

Study proper sampling for X-ray transforms on simple surfaces.

Study X-ray transform on manifolds, desingularize, and improve mapping properties.

We consider the mixed ray transform of tensor fields on a three-dimensional compact simple Riemannian manifold with boundary. We prove the injectivity of the transform, up to natural obstructions, and establish stability estimates for the normal operator on generic three dimensional simple manifold in the case of 1+1 a…

Paper solves injectivity of X-ray transform on surfaces.

Paper shows X-ray transform invertible on certain curved spaces.

We study the weighted light ray transform $L$ of integrating functions on a Lorentzian manifold over lightlike geodesics. We analyze $L$ as a Fourier Integral Operator and show that if there are no conjugate points, one can recover the spacelike singularities of a function $f$ from its the weighted light ray transform …

In this paper we study the local magnetic ray transform of symmetric tensor fields up to rank two on a Riemannian manifold of dimension $\geq 3$ with boundary. In particular, we consider the magnetic ray transform of the combinations of tensors of different orders due to the nature of magnetic flows. We show that such …

We show that the existence of a function in $L^{1}$ with constant geodesic X-ray transform imposes geometrical restrictions on the manifold. The boundary of the manifold has to be umbilical and in the case of a strictly convex Euclidean domain, it must be a ball. Functions with constant geodesic X-ray transform always …

The paper studies ray transforms on surfaces with negative curvature, proving injectivity and determining connections and Higgs fields.

Study normal operators of double fibration transforms with conjugate points.

We study the geodesic X-ray transform $X$ on compact Riemannian surfaces with conjugate points. Regardless of the type of the conjugate points, we show that we cannot recover the singularities and therefore, this transform is always unstable (ill-posed). We describe the microlocal kernel of $X$ and relate it to the con…

Guillarmou extends X-ray transform to magnetic and thermostat flows.

New insights into X-ray transform on hyperbolic disk, with functional relations and range characterizations.

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.

Researchers study injectivity of magnetic and thermostatic nonabelian ray transforms on compact surfaces.

We characterize the kernel of the mixed ray transform on simple $2$-dimensional Riemannian manifolds, that is, on simple surfaces for tensors of any order.

We prove a sharp $L^2\to H^{1/2}$ stability estimate for the geodesic X-ray transform of tensor fields of order $0$, $1$ and $2$ on a simple Riemannian manifold with a suitable chosen $H^{1/2}$ norm. We show that such an estimate holds for a family of such $H^{1/2}$ norms, not topologically equivalent, but equivalent o…