arXiv research

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

Injectivity result for light ray transform on Lorentzian manifolds.

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…

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 …

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

Study extends geodesic ray transform results to orientable surfaces.

We study the weighted ray transform of integrating functions on a Lorentzian manifold over lightlike geodesics. We prove support theorems if the manifold and the weight are analytic.

We consider the following problem: given two parallel and identically oriented bundles of light rays in n-dimensional Euclidean space and given a diffeomorphism between the rays of the former bundle and the rays of the latter one, is it possible to realize this diffeomorphism by means of several mirror reflections? We …

Study normal operators of double fibration transforms with conjugate points.

Recovering matrix valued potentials from wave equation data on stationary spacetimes.

The natural topological, differentiable and geometrical structures on the space of light rays of a given spacetime are discussed. The relation between the causality properties of the original spacetime and the natural structures on the space of light rays are stressed. Finally, a symplectic geometrical approach to the …

In this paper we prove several multiplicity results of $t$-periodic light rays in conformally stationary spacetimes using the Fermat metric and the extensions of the classical theorems of Gromoll-Meyer and Bangert-Hingston to Finsler manifolds. Moreover, we exhibit some stationary spacetimes with a finite number of $t$…

Optical interpretation of Euler's angle problem for caustics of light rays.

Study on recovering Lorentzian metrics from scattering data.

Solves open problem on simple surfaces with novel twistor correspondence.

Ray-marching method visualizes 8 Thurston geometries in real-time.

Study scattering rigidity for Hamiltonian systems, proving lens rigidity for non-trapping Finsler manifolds.

A reconstruction theorem in terms of the topology and geometrical structures on the spaces of light rays and skies of a given space-time is discussed. This result can be seen as part of Penrose and Low's programme intending to describe the causal structure of a space-time $M$ in terms of the topological and geometrical…

Ray-Singer torsion measures light degrees of freedom in black hole entropy.

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 …

A closed Riemannian manifold is said to have cross blocking if whenever distinct points p and q are at distance less than the diameter, all light rays from p can be shaded away from q with at most two point shades. Similarly, a closed Riemannian manifold is said to have sphere blocking if for each point p, all the ligh…

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.

New system studies trapped light paths in Euclidean space.

Local invertibility of ray transforms on convex manifolds.

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

Study proves uniqueness for ray transform on surfaces with obstacles.

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.

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…

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.

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…

Injectivity of geodesic ray transform on specific Finsler manifolds proven.

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…

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…

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 …

The tools of presymplectic geometry are used to study light rays trajectories in anisotropic media.

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 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 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…