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.
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 Metric transforms make Euclidean half lines hyperbolic, preserving geodesic properties.
problem Characterizing metric transforms that make Euclidean half lines hyperbolic.
method Characterization of metric transforms φ such that ([0,∞),∣⋅∣φ) is Gromov hyperbolic. result Metric transform rigidity for roughly geodesic Gromov hyperbolic spaces.
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.
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.
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. 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.
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.
Injectivity theorems for geodesic ray transform on curved 2D manifolds.
problem Understanding geodesic ray transform on curved 2D manifolds.
method Injectivity theorems for geodesic ray transform on Cartan-Hadamard manifolds with non-positive curvature.
result Proved two injectivity theorems for geodesic ray transform on 2D Cartan-Hadamard manifolds.
Geodesic ray transform links solenoidal injectivity to invariant distributions.
problem Injectivity of geodesic ray transform on symmetric tensors.
method Equivalence principle between solenoidal injectivity and invariant distributions.
result Existence of invariant distributions for geodesic ray transform.
Study characterizes geodesic ray transform on surfaces, isolating and separating sub-ranges.
problem Characterizing the range of the attenuated geodesic ray transform on surfaces.
method Isolating and separating sub-ranges of sums of functions and one-forms, deriving new inversion formulas.
result Range characterizations and new inversion formulas for geodesic ray transform.
Study geodesic X-ray transforms on curved manifolds using Carleman estimates.
problem Invertibility of geodesic X-ray transforms on curved manifolds.
method Using Carleman estimates to show invertibility of geodesic vector field.
result Geodesic X-ray transform is invertible on negatively curved simple manifolds.
Reconstructs geodesic ray transforms on simple Riemannian surfaces with connections.
problem Reconstructing geodesic ray transforms on Riemannian surfaces with connections.
method Derives reconstruction formulas and injectivity conditions for geodesic ray transforms with connections.
result Injectivity of geodesic ray transforms in a neighborhood of constant curvature metrics and non-unitary connections.
Study inverse problems for twisted geodesic flows on manifolds.
problem Understanding inverse problems for twisted geodesic flows.
method Generalized ray transforms and tensor tomography.
result New insights into rigidity problems for twisted geodesic flows.
Injectivity and support theorem for X-ray transform on specific Lie groups.
problem Injectivity and support theorem for X-ray transform on 2-step nilpotent Lie groups.
method General reduction principle for manifolds with uniformly escaping geodesics.
result Injectivity and support theorem for X-ray transform on 2-step nilpotent Lie groups.
The paper studies geodesic circles and concircular transformations in Finsler geometry.
problem Characterizing and understanding concircular transformations in Finsler geometry.
method Applying Lie derivatives and the Cartan Y-connection to study geodesic circles and concircular transformations. result Characterization of concircular vector fields and conditions for conformal and concircular transformations.
The paper proves a transformation theorem under a monotone property of almost Euclidean factors of geodesic balls.
problem The non-increasing property of numbers of almost Euclidean factors of geodesic balls.
method Proves a transformation theorem under a non-decreasing property compared to the non-increasing property.
result Shows that for a manifold with nonnegative Ricci curvature, if its universal cover is polar at infinity and the number of almost Euclidean factors is monotone, then its fundamental group is finitely generated and virtually abelian.
Article examines stability of geodesic X-ray transforms and artifacts.
problem Stability of geodesic X-ray transforms and artifacts.
method Analyzes the impact of weight and geometry on stability, and examines artifacts in unstable cases.
result Landweber algorithm cannot provide accurate reconstruction in unstable cases.
Reconstruction formulas for geodesic X-ray transforms on negatively curved surfaces.
problem Inverting geodesic X-ray transforms on surfaces with negative curvature.
method Generalizes Pestov-Uhlmann formulas to surfaces with trapping, using Fredholm equations and new estimates for the normal operator.
result Reconstruction formulas for geodesic X-ray transforms on surfaces with negative curvature.
For a compact Riemannian surface with boundary we study attenuated geodesic transform of functions and differential forms. We generalize several known results on uniqueness and stability of this transform dropping condition of absence of conjugate points.
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.
Geodesics in positive Lagrangian spaces map to special Lagrangian cylinders.
problem Understanding geodesics in spaces of positive Lagrangian submanifolds.
method Cylindrical transform of geodesics, solving elliptic PDEs.
result Geodesics in positive Lagrangian spaces correspond to one-parameter families of special Lagrangian cylinders.
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.
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.
The geodesic X-ray transform on disks of constant curvature is characterized and decomposed.
problem Characterizing and decomposing the geodesic X-ray transform on disks of constant curvature.
method Explicit construction of a basis for the range and co-kernel, derivation of Singular Value Decomposition.
result Explicit Singular Value Decomposition for the geodesic X-ray transform.
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). 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 in M. Under the assumption that the metric g is real-analytic, it is shown th…
Geodesic X-ray transform proves injective for smooth one-forms on gas giant manifolds.
problem Injectivity of geodesic X-ray transform for one-forms on specific manifolds.
method Pestov identity and asymptotic analysis of short geodesics.
result Geodesic X-ray transform is solenoidally injective for smooth one-forms on gas giant manifolds.
Study characterizes X-ray transform kernel for periodic slabs and related manifolds.
problem Characterizing the kernel of X-ray transform for tensor fields on periodic slabs.
method Characterization of the kernel for L2-regular m-tensors on [0,1]imesTn. result Kernel characterization extends to more general manifolds, including the Möbius strip.
Study geodesic ray transform on 2D manifolds with conjugate points.
problem Understanding geodesic ray transform on manifolds with conjugate points.
method Decomposition into pseudodifferential operator and Fourier integral operators using method of stationary phase.
result Explicit computation of principal symbol and cancellation of singularities.
Projective geodesic extensions for nonholonomic systems are derived under conformal transformations.
problem Deriving conditions for projective geodesic extensions in nonholonomic mechanics.
method Analyzing necessary and sufficient conditions for existence under conformal modifications.
result Conditions for existence of projective geodesic extensions in nonholonomic systems under conformal transformations.
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 assumption on the existence of a global foliation by strictly convex hypersurfaces…
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 geodesic X-ray transform on curved spaces, proving injectivity for decaying functions.
problem Injectivity of geodesic X-ray transform on curved spaces.
method Proving injectivity for decaying functions and tensor fields of any order.
result Injectivity of the geodesic X-ray transform for functions and tensor fields of any order on Cartan-Hadamard manifolds.
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∈L2(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 a micr…
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…
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.
Geodesic algorithms extended to arbitrary ellipsoids.
problem Computing geodesics on ellipsoids of varying eccentricity.
method Implementation of geodesic algorithms using elliptic integrals and discrete sine transform.
result Achieved high accuracy (close to machine precision) for geodesic computations.
Geodesic 3-webs on surfaces are characterized by integrable flow equations.
problem Characterizing surfaces with hexagonal geodesic 3-webs.
method Integrable flow equations and generalized hodograph transform method.
result Geodesic flow on surfaces admits cubic first integrals for hexagonal 3-webs.
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 …
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.
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 geodesic X-ray transform and streaking artifacts on simple surfaces or spaces of constant curvature.
problem Streaking artifacts in CT images due to metal regions.
method Geodesic X-ray transform on nontrapping compact Riemannian manifolds with strictly convex boundaries.
result Streaking artifacts result from conormal singularities along common tangent geodesics.
We show that the Radon transform related to closed geodesics is injective on a Lie group if and only if the connected components are not homeomorphic to S1 nor to S3. This is true for both smooth functions and distributions. The key ingredients of the proof are finding totally geodesic tori and realizing the Rado…
The study provides a criterion for fractional-linear integrals of geodesics on surfaces.
problem Existence and classification of fractional-linear integrals for geodesic flows on Riemannian surfaces.
method Criterion and analysis of moduli space of local integrals.
result The moduli space of such local integrals is either the 2D projective plane or finite points.
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 …
Method reconstructs potential from partial boundary measurements on admissible manifolds.
problem Reconstructing potential from partial data on admissible manifolds.
method Develops a method using the geodesic ray transform and complex geometrical optics solutions.
result Reconstructs potential locally and globally under certain conditions.