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.
Study of rays and co-rays in Wasserstein space, introducing Busemann functions.
problem Understanding rays and co-rays in Wasserstein space.
method Representation of rays as probability measures, existence of co-rays, introduction of Busemann functions.
result Existence and properties of co-rays and Busemann functions.
Ray-Singer torsion is a mathematical concept with applications in physics.
problem No specific problem stated in the abstract.
method No specific method stated in the abstract.
result No specific key result stated in the abstract.
We show that any grafting ray in Teichmüller space is (strongly) asymptotic to some Teichmüller geodesic ray. As an intermediate step we introduce surfaces that arise as limits of these degenerating Riemann surfaces. Given a grafting ray, the proof involves a Teichmüller ray with a conformally equivalent limit, and bui…
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 …
Three disjoint rays in euclidean 3-space form Borromean rays provided their union is knotted, but the union of any two components is unknotted. We construct infinitely many Borromean rays, uncountably many of which are pairwise inequivalent. We obtain uncountably many Borromean hyperplanes.
Infinite clique of rays in plane minus Cantor set.
problem Understanding the mapping class group of plane minus Cantor set.
method Using a graph of loops and cliques of high-filling rays.
result Construction of an infinite clique of high-filling rays.
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.
Proves symplectomorphism of noncompact manifolds with embedded rays.
problem Symplectic manifolds with embedded rays.
method Constructs explicit symplectomorphisms using excision trick.
result Symplectomorphic to complement of the ray in standard Euclidean space.
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.
Unique geodesics selected by energy minimization in Teichmüller space.
problem Finding a unique geodesic between points in Teichmüller space.
method Energy minimization of harmonic map rays, extending Thurston boundary.
result Selection of a unique Thurston geodesic through points in Teichmüller space.
Lecture notes on analysis tools for X-ray tomography.
problem Understanding X-ray tomography using mathematical analysis.
method Overview of analysis tools and ideas, minimal assumptions.
result Broad overview of analysis tools for X-ray tomography.
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.
The ray graph is a Gromov hyperbolic graph on which the mapping class group of the plane minus a Cantor set acts by isometries. We give a description of the Gromov boundary of the ray graph in terms of cliques of long rays on the plane minus a Cantor set. As a consequence, we prove that the Gromov boundary of the ray g…
CNNs trained on one hospital's x-rays perform poorly on x-rays from other hospitals.
problem Generalization of radiological deep learning models across different hospitals.
method Cross-sectional design using x-rays from three hospitals (NIH, Mount Sinai, Indiana).
result CNNs trained on one hospital's x-rays perform significantly worse on x-rays from other hospitals.
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.
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.
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). New proofs of Donaldson-Uhlenbeck-Yau theorem using geodesic rays.
problem Donaldson-Uhlenbeck-Yau theorem implications
method Geodesic rays of Hermitian metrics
result New proofs of the theorem
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.
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.
The study examines the asymptotic behavior of extremal length in Teichmüller space.
problem Understanding the asymptotic behavior of extremal length along Teichmüller rays.
method Analyzing the limit of extremal length and deriving formulas for limiting Teichmüller distance and detour metric.
result An explicit formula for the limiting Teichmüller distance and a necessary and sufficient condition for Teichmüller rays to be asymptotic.
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.
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.
Bound skinning map diameter on hyperbolic 3-manifold rays.
problem Bounding skinning map diameter on hyperbolic 3-manifold rays.
method Using acylindrical hyperbolic 3-manifolds and thick Teichmüller geodesic rays.
result Diameter of skinning map bounded by a constant.
This article contains a detailed study, in the toric case, of the test configuration geodesic rays defined by Phong-Sturm. We show that the `Bergman approximations' of Phong-Sturm converge in C^1 to the geodesic ray and that the geodesic ray itself is C^{1,1} and no better. The \kahler metrics associated to the geodesi…
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.
Study on Teichmüller rays' asymptotic behavior and distances.
problem Understanding the asymptotic behavior of Teichmüller rays.
method Explicit formula derivation for limiting Teichmüller distance under specific conditions.
result Two Teichmüller rays are asymptotic if their vertical measured foliations are modularly equivalent and their limit surfaces coincide.
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.
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 consider the asymptotic behavior of two Teichmüller geodesic rays determined by Jenkins-Strebel differentials, and we obtain a generalization of a theorem in \cite{Amano14}. We also consider the infimum of the asymptotic distance in shifting base points of the rays along the geodesics. We show that th…
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.
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 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.
New methods model gamma-ray data to better understand Galactic emissions.
problem Uncertain diffuse Galactic gamma-ray emissions bias data interpretation.
method Gaussian processes and variational inference for flexible modeling.
result More robust interpretation of gamma-ray sky, especially dark matter signals.
Deep learning tackles X-ray noise without clean data.
problem Lack of clean X-ray images for deep learning denoising.
method Uses Stein's Unbiased Risk Estimator (SURE) to train a deep neural network.
result SURE-based approach effectively denoises X-ray images.
New Teichmüller geodesic rays found with unique foliations.
problem Capturing generic directions in Teichmüller space.
method Using Chaika-Masur-Wolf and Durham-Zalloum work.
result First sublinearly-Morse geodesic rays with minimal non-uniquely ergodic foliations.
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.
CheXNet detects pneumonia from chest X-rays better than radiologists.
problem Detecting pneumonia from chest X-rays with high accuracy.
method 121-layer convolutional neural network trained on ChestX-ray14 dataset.
result CheXNet outperforms radiologists on F1 metric.
We construct symplectic and Kähler ray reduced spaces and discuss their relation with the Marsden-Weinstein (point) reduction. This Kähler reduction is well defined even when the momentum value is not totally isotropic. The compatibility of the ray reduction with the cone construction and the Boothby-Wang fibration is …
Ray tracing sampler improves neural network sampling efficiency and resilience.
problem Sampling neural network posterior distributions efficiently and robustly.
method Markov Chain Monte Carlo using ray tracing through likelihood space.
result Significantly higher resilience to gradient heating compared to HMC.
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.
Optical interpretation of Euler's angle problem for caustics of light rays.
problem Euler's angle problem for caustics of light rays.
method Optical interpretation of intrinsic equations and study of caustics.
result Characterization of curves under specific conditions and novel pantograph equations.