Study extends geodesic ray transform results to orientable surfaces.
problem Characterize and stabilize mixed and transverse ray transforms on surfaces.
method Algebraic arguments applied to various geometries and ray transforms.
result Characterization of kernel and stability for mixed and transverse ray transforms on orientable surfaces.
Local invertibility of ray transforms on convex manifolds.
problem Invertibility of ray transforms on compact Riemannian manifolds with strictly convex boundary.
method Local invertibility results for transverse and mixed ray transforms of 1 and 1+1 tensors.
result Local invertibility of ray transforms near boundary points, leading to global results.
Support theorem for tensor fields on Riemannian manifolds.
problem Determining the support of tensor fields from ray transforms.
method Support theorem for transverse ray transform of rank 2 tensor fields.
result Support of tensor fields vanishes on geodesics where ray transform vanishes.
Local invertibility of higher order tensor transforms on compact manifolds.
problem Invertibility of higher order tensor transforms on compact manifolds.
method Local invertibility of transverse and mixed ray transforms of tensors on compact Riemannian manifolds.
result Local invertibility of transverse and mixed ray transforms of tensors for specific dimensions.
Two new methods for injective ray transforms of tensor fields on surfaces are introduced.
problem Injective ray transforms for tensor fields on surfaces.
method Two approaches: weights varying along geodesic flow and transverse ray transforms.
result Constructive solutions for tensor tomography on simple surfaces.
Study shows boundary measurements can determine transversal singularities in anisotropic geometries.
problem Determining transversal singularities in anisotropic geometries from boundary measurements.
method Geometric condition on transversal manifold and FBI type transform.
result Recovery of transversal singularities in the linearized problem.
We consider the anisotropic Calderon problem of recovering a conductivity matrix or a Riemannian metric from electrical boundary measurements in three and higher dimensions. In the earlier work \cite{DKSaU}, it was shown that a metric in a fixed conformal class is uniquely determined by boundary measurements under two …
In this paper we consider the problem of identifying a connection ∇ on a vector bundle up to gauge equivalence from the Dirichlet-to-Neumann map of the connection Laplacian ∇∗∇ over conformally transversally anisotropic (CTA) manifolds. This was proved in \cite{LCW} for line bundles in the case of t…
Study geodesic X-ray transforms on hyperbolic surfaces, proposing new reconstruction methods.
problem Inverting geodesic X-ray transforms for symmetric tensor fields on asymptotically hyperbolic surfaces.
method Developed a decomposition theorem for m-tensor fields, used Guillemin-Kazhdan operators and 0-calculus, and provided explicit reconstruction methods.
result Explicit reconstruction methods for even tensor fields from their X-ray transform or normal operator.
We study the asymptotic geometry of Teichmueller geodesic rays. We show that when the transverse measures to the vertical foliations of the quadratic differentials determining two different rays are topologically equivalent, but are not absolutely continuous with respect to each other, then the rays diverge in Teichmue…
Survey on broken ray transforms and open problems.
problem Understanding broken ray transforms and their applications.
method Survey of recent developments and open problems.
result Discussion of open problems in broken ray transforms.
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 …
Study characterizes kernel of mixed ray transform on simple surfaces.
problem Characterizing the kernel of mixed ray transform on simple surfaces.
method Characterization of the kernel through analysis of mixed ray transform on simple 2D Riemannian manifolds.
result Characterization of the kernel of the mixed ray transform on simple surfaces.
Paper shows invertibility of tensor X-ray transform on certain manifolds.
problem Invertibility of tensor X-ray transform on asymptotically conic manifolds.
method Used 1-cusp pseudodifferential operator algebra and modified solenoidal gauge condition.
result Invertibility of tensor X-ray transform up to natural obstruction.
We study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least 5. These metrics come in rays of transversal homothety due to the possible rescaling of the Reeb vector fields. We prove that there exist Reeb vector fields for which…
The article recovers tensor fields from partial data using weighted divergent ray transforms.
problem Recovering tensor fields from partial data.
method Weighted divergent ray transforms, unique continuation property of fractional Laplacian, explicit reconstruction formulas.
result Recovery of symmetric m-tensor fields and unique continuation for vector fields and symmetric 2-tensor fields. Sharp mapping properties and regularization for X-ray transform on disks of constant curvature.
problem Sharp mapping properties and regularization of X-ray transform.
method Derive functional relations and mapping properties using elliptic differential operators.
result Theoretical possibility of regularized inversions for X-ray transform.
Regularity results for geodesic X-ray transform on nonsmooth manifolds
problem Geodesic X-ray transform on nonsmooth simple manifolds
method Symbol smoothing arguments and pseudodifferential operators with low regularity symbols
result Improved injectivity results for Lp functions Study light ray transform on Lorentzian manifolds without conjugate points.
problem Recovering spacelike singularities from weighted light ray transforms.
method Fourier Integral Operator analysis and filtered back-projection.
result Recovery of spacelike singularities from weighted light ray transforms without conjugate points.
Local X-ray transform works well near boundaries in hyperbolic spaces.
problem Injectivity of X-ray transform near boundaries for hyperbolic metrics.
method Local injectivity proof for geodesic X-ray transform on asymptotically hyperbolic manifolds.
result Local injectivity near a boundary point for X-ray transform in dimensions 3 and higher, up to O(ρ5). Artifacts appear in broken ray transform due to conjugate points.
problem Artifacts in broken ray transform due to conjugate points.
method Integral transform over broken rays, analysis of conjugate points.
result Singularities cannot be recovered from local data, leading to artifacts.
Functions with constant geodesic X-ray transform are restricted to manifolds with specific geometrical properties.
problem Existence of functions with constant geodesic X-ray transform on manifolds.
method Analyzing the geometrical properties of manifolds based on the existence of such functions.
result Functions with constant geodesic X-ray transform impose specific geometrical restrictions on the manifold.
The paper explores the injectivity of the light ray transform on Lorentzian manifolds.
problem Injectivity of the light ray transform on functions and tensors.
method Analyzes injectivity conditions for scalar and tensor fields on stationary and static Lorentzian manifolds.
result Injectivity of the light ray transform on functions and tensors is proven under specific conditions.
Study light ray transform in pseudo-Euclidean space, derive inversion formula, and prove stability.
problem Analyzing light ray transform in pseudo-Euclidean space.
method Investigate normal operator, derive inversion formula, analyze as Fourier Integral Operator.
result Derive an inversion formula and prove stability estimates.
Researchers prove injectivity and stability for mixed ray transform on simple manifolds.
problem Injectivity and stability of mixed ray transform for tensor fields.
method Analyzing tensor fields on 3D compact simple Riemannian manifolds with boundary.
result Injectivity and stability estimates for normal operator on generic 3D simple manifolds.
Study shows X-ray transform injectivity for hyperbolic manifolds.
problem Recovering metrics from boundary measurements on hyperbolic manifolds.
method Injectivity of X-ray transform in several cases.
result Injectivity of X-ray transform proven for asymptotically hyperbolic manifolds.
Study proves uniqueness for ray transform on surfaces with obstacles.
problem Uniqueness of functions and 1-forms on surfaces with reflecting obstacles.
method Broken ray transform on twisted geodesics with nonpositive curvature and reflecting boundary.
result Proves uniqueness result for sums of functions and 1-forms.
Study shows stability in X-ray transform on specific hyperbolic manifolds.
problem Stability of X-ray transform on asymptotically hyperbolic manifolds.
method Constructed a parametrix for the normal operator in 0-pseudodifferential calculus.
result Showed a stability estimate for the X-ray transform.
Study X-ray transform on Anosov manifolds with improved stability estimates.
problem Analyzing the geodesic X-ray transform on Anosov manifolds.
method Refined Livsic theorem for Anosov flows, new quantitative finite time Livsic theorem.
result New stability estimates for the X-ray transform.
Sharp stability estimate for geodesic ray transform on simple manifolds.
problem Stability estimate for geodesic ray transform of tensor fields.
method Microlocal analysis and structured range of geodesic ray transform.
result Sharp L2oH1/2 stability estimate for tensor fields of order 0, 1, and 2. Study X-ray transform on conic spaces, proving injectivity under certain conditions.
problem Injectivity of geodesic X-ray transform on conic metrics.
method Injectivity under non-trapping and no conjugate point assumptions.
result Injectivity of geodesic X-ray transform for asymptotically conic metrics.
Injectivity of geodesic ray transform for piecewise constants on compact manifolds.
problem Injectivity of geodesic ray transform for piecewise constant functions.
method Injectivity of geodesic ray transform on piecewise constant functions weighted by a continuous matrix weight.
result Injectivity of the geodesic X-ray transform on piecewise constant functions.
Injectivity of X-ray transform on solenoidal tensors proven for certain manifolds.
problem Injectivity of X-ray transform on solenoidal tensors on manifolds with hyperbolic trapped set.
method Proof of equivalence principle concerning injectivity and surjectivity of operators on symmetric solenoidal tensors.
result Injectivity of X-ray transform on solenoidal tensors of any order for surfaces with hyperbolic trapped set.
Injective X-ray transform on Heisenberg group for regular functions.
problem Injectivity of X-ray transform on sub-Riemannian manifolds.
method Group Fourier Transform and analysis of taming metrics.
result Sufficiently regular functions on Heisenberg group are determined by their line integrals.
Paper proves injectivity of non-abelian X-ray transform on certain spaces.
problem Injectivity of non-abelian X-ray transform on asymptotically hyperbolic spaces.
method Gauge equivalence for unitary connections and skew-Hermitian Higgs fields.
result Injectivity result for non-abelian X-ray transform over skew-Hermitian Higgs fields.
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.
New proof of injectivity for broken non-abelian X-ray transform in Minkowski space.
problem Injectivity of broken non-abelian X-ray transform in Minkowski space.
method Stability estimate considering gauge, light-sink connections, Bayesian inversion.
result Consistent recovery of light-sink connections from noisy data.
Injectivity result for light ray transform on Lorentzian manifolds.
problem Injectivity of light ray transform on Lorentzian manifolds.
method Explicit relationship between geodesic and magnetic vector fields.
result Injectivity up to natural obstruction under certain conditions.
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…
Let X be a compact connected strongly pseudoconvex CR manifold of dimension 2n+1,n≥1 with a transversal CR S1-action on X. We introduce the Fourier components of the Ray-Singer analytic torsion on X with respect to the S1-action. We establish an asymptotic formula for the Fourier components of the an…
Injectivity of geodesic ray transform on specific Finsler manifolds proven.
problem Injectivity of geodesic ray transform on spherically symmetric reversible Finsler manifolds.
method Reduction to invertibility of generalized Abel transforms using angular Fourier series and Taylor expansions of geodesics.
result Injectivity of geodesic ray transform proven on specified Finsler manifolds.
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…
We study ray transforms on spherically symmetric manifolds with a piecewise C1,1 metric. Assuming the Herglotz condition, the X-ray transform is injective on the space of L2 functions on such manifolds. We also prove injectivity results for broken ray transforms (with and without periodicity) on such manifolds …
X-ray transform on H-type groups solved, revealing function injectivity.
problem Injectivity in sub-Riemannian geometry.
method Fourier Slice Theorem adapted to H-type groups.
result Integrable functions on H-type groups are uniquely determined by their integrals over geodesics.
New method eliminates domain size restrictions for X-ray transform inversion.
problem Injectivity and stability of X-ray transform in convex domains.
method Semiclassical analysis to invert X-ray transform without small domain assumptions.
result Elimination of domain size restrictions for injectivity and stability.
Study proper sampling for X-ray transforms on simple surfaces.
problem Proper discretizing and sampling issues related to geodesic X-ray transforms on simple surfaces.
method Provide minimal sampling rates for faithful reconstruction, quantify sampling quality, and predict artifacts.
result Minimal sampling rates and artifact prediction for geodesic X-ray transforms on simple surfaces.