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.
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 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). 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.
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.
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 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.
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 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.
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.
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.
Study X-ray transform on manifolds, desingularize, and improve mapping properties.
problem Characterize mapping properties of X-ray transform and its adjoint on manifolds with strictly convex boundary.
method Use b-fibrations, desingularize, and apply Melrose's Pushforward Theorem to analyze polyhomogeneous functions.
result Improved mapping properties of X-ray transform and its adjoint, recovering sharp results.
Paper solves injectivity of X-ray transform on surfaces.
problem Injectivity of non-Abelian X-ray transform on surfaces.
method Factorization theorem for Loop Groups, energy methods, scalar holomorphic integrating factors.
result Extends results to arbitrary Lie groups.
Paper shows X-ray transform invertible on certain curved spaces.
problem Invertibility of X-ray transform on curved spaces.
method Introduced new pseudodifferential operator algebra for analysis.
result Stable invertibility near infinity on asymptotically conic manifolds.
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.
Support theorem proved for X-ray transform on certain non-compact manifolds.
problem Support theorem for X-ray transform on non-compact manifolds with conjugate points.
method Use of plane covers and support theorem for simple manifolds by Krishnan.
result Support theorem proved for simply connected 2-step nilpotent Lie groups and some non-homogeneous 3D manifolds.
Guillarmou extends X-ray transform to magnetic and thermostat flows.
problem Stability of magnetic X-ray transforms.
method Generalizes normal operator to thermostat and magnetic flows, proving ellipticity.
result Elliptic pseudodifferential operators of order -1 for generalized normal operators.
New insights into X-ray transform on hyperbolic disk, with functional relations and range characterizations.
problem Understanding the X-ray transform on hyperbolic geometry.
method Derived new singular value decompositions, range characterizations, and intertwining relations with wedge-type differential operators.
result Sharp understanding of boundary behavior and invertibility settings for the X-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.
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.
Study shows certain manifolds uniquely determined by X-ray data.
problem Determining manifolds from X-ray data.
method Injectivity of X-ray transform over symmetric solenoidal 2-tensors.
result Smooth compact manifolds with strictly convex boundary, no conjugate points, and hyperbolic trapped set are locally marked boundary rigid.
Injectivity of X-ray transform proven for non-smooth metrics.
problem Injectivity of X-ray transform on non-smooth metrics.
method Microlocal analysis of the normal operator, establishing ellipticity and smoothing properties.
result Injectivity of X-ray transform on L2 for metrics with finitely differentiable tensor. Injectivity of geodesic X-ray transform on low-regularity manifolds.
problem Injectivity of geodesic X-ray transform on manifolds with low regularity.
method Calculus of differential and curvature operators on non-smooth structures.
result Injectivity of geodesic X-ray transform on simple Riemannian manifolds with C1,1-regularity. 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.
In this paper we consider the geodesic X-ray transform with attenuation coefficient as a combination of smooth complex function and 1-form. We show that attenuated X-ray transform applied to the pair of tensors is injective modulo the natural obstruction.
Unique continuation for X-ray transforms of one-forms with partial data.
problem Proving unique continuation for X-ray transforms of one-forms with limited data.
method Proved unique continuation for the normal operator of X-ray transforms of one-forms, leading to partial data results.
result Unique continuation for X-ray transforms of one-forms with partial data.
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…
We prove injectivity and a support theorem for the X-ray transform on 2-step nilpotent Lie groups with many totally geodesic 2-dimensional flats. The result follows from a general reduction principle for manifolds with uniformly escaping geodesics.
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 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…
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 …
Surveying inverse problems on manifolds with boundaries.
problem Inverse problems for geometric structures on manifolds with boundaries.
method Analyzes linear and non-linear geometric inverse problems.
result Recent results on geodesic X-ray transforms and Riemannian metrics.
Proves X-ray transform injectivity on specific manifolds.
problem Injectivity of X-ray transform on closed Anosov manifolds and spherical boundary.
method Perturbative argument of the 0-eigenvalue of elliptic operators via microlocal analysis.
result Generically injective X-ray transform on specified manifolds.
The X-ray transform on a compact symmetric space M is here inverted by means of an explicit inversion formula. The proof uses the conjugacy of the minimal closed geodesics in M and of the maximally curved totally geodesic spheres in M, proved in Math. Ann. 165 (1966), 309--317.
For Anosov flows preserving a smooth measure on a closed manifold M, we define a natural self-adjoint operator Π which maps into the space of invariant distributions in ∩u<0Hu(M) and whose kernel is made of coboundaries in ∪s>0Hs(M). We describe relations to Liv…
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 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 …
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…
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 benchmarks uncertainty quantification in chest X-ray classification.
problem Reliable uncertainty quantification for medical AI models.
method Evaluation of 13 uncertainty quantification methods on MIMIC-CXR-JPG dataset.
result Insights into effectiveness and disentanglement of epistemic and aleatoric uncertainties.
New method simplifies tomographic reconstruction using RKHS.
problem Tomographic reconstruction challenges.
method RKHS framework for X-ray transform.
result Sharp stability results without Fourier transform.
The paper explains how microlocal analysis solves geometric inverse problems.
problem Recovering geometric information from boundary measurements.
method Microlocal analysis applied to three inverse problems.
result Microlocal techniques solve specific inverse problems in Riemannian geometry.