arXiv research

Summary of tensor tomography proofs on manifolds with boundaries.

Sketch Tomography improves quantum state estimation accuracy.

Extends magnetic flow theory results to higher dimensions.

Study inverse problems for twisted geodesic flows on manifolds.

We survey recent progress in the problem of recovering a tensor field from its integrals along geodesics. We also propose several open problems.

We prove a sharp stability estimate for the problem of reconstructing a symmetric 2-tensor from its integrals along all maximal geodesics on a simple manifold.

We show that on simple surfaces the geodesic ray transform acting on solenoidal symmetric tensor fields of arbitrary order is injective. This solves a long standing inverse problem in the two-dimensional case.

In the recent articles \cite{PSU1,PSU3}, a number of tensor tomography results were proved on two-dimensional manifolds. The purpose of this paper is to extend some of these methods to manifolds of any dimension. A central concept is the surjectivity of the adjoint of the geodesic ray transform, or equivalently the exi…

We survey some results on travel time tomography. The question is whether we can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as geometry problems, the boundary rigidity problem and the lens rigidity problem. The boundary r…

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.

Study proves solenoidal injectivity for tensor fields on curved manifolds with low regularity.

Method learns topological states from randomized measurements.

Estimates joint probability distribution from 1-way marginals using low-rank tensors and random projections.

We first give a constructive answer to the attenuated tensor tomography problem on simple surfaces. We then use this result to propose two approaches to produce vector-valued integral transforms which are fully injective over tensor fields. The first approach is by construction of appropriate weights which vary along t…

We study the geodesic X-ray transform on Cartan-Hadamard manifolds, and prove solenoidal injectivity of this transform acting on functions and tensor fields of any order. The functions are assumed to be exponentially decaying if the sectional curvature is bounded, and polynomially decaying if the sectional curvature de…

In this paper we consider the Gaussian thermostat ray transform on both closed Riemannian surfaces and compact Riemannian surfaces with boundary. We establish certain results on the injectivity of the thermostat ray transform and the surjectivity of its adjoint.

The X-ray transform on the periodic slab $[0,1]\times\mathbb T^n$, $n\geq0$, has a non-trivial kernel due to the symmetry of the manifold and presence of trapped geodesics. For tensor fields gauge freedom increases the kernel further, and the X-ray transform is not solenoidally injective unless $n=0$. We characterize t…

Consider a Riemannian manifold in dimension $n\geq 3$ with strictly convex boundary. We prove the local invertibility, up to potential fields, of the geodesic ray transform on tensor fields of rank four near a boundary point. This problem is closely related with elastic \textit{qP}-wave tomography. Under the condition …

We show that a tensor field of any rank integrates to zero over all broken rays if and only if it is a symmetrized covariant derivative of a lower order tensor which satisfies a symmetry condition at the reflecting part of the boundary and vanishes on the rest. This is done in a geometry with non-positive sectional cur…

Study geodesic X-ray transforms on hyperbolic surfaces, proposing new reconstruction methods.

We consider the geodesic X-ray transform acting on solenoidal tensor fields on a compact simply connected manifold with strictly convex boundary and non-positive curvature. We establish a stability estimate of the form $L^2\mapsto H^{1/2}_{T}$, where the $H^{1/2}_{T}$-space is defined using the natural parametrization …

The present article proposes a partial answer to the explicit inversion of the tensor tomography problem in two dimensions, by proving injectivity over certain kinds of tensors and providing reconstruction formulas for them. These tensors are symmetric differentials of any order as well as other types obtained after ta…

We prove uniqueness results for a Calderon type inverse problem for the Hodge Laplacian acting on graded forms on certain manifolds in three dimensions. In particular, we show that partial measurements of the relative-to-absolute or absolute-to-relative boundary value maps uniquely determine a zeroth order potential. T…

Study travel time tomography for transversely isotropic media using modified pseudodifferential calculus.

It has been shown in [Pa1] that on a simple, compact Riemannian 2-manifold the attenuated geodesic ray transform, with attenuation given by a connection and Higgs field, is injective on functions and 1-forms modulo the natural obstruction. Furthermore, the scattering relation determines the connection and Higgs field m…

Paper addresses travel time tomography stability and statistical inversion.

DeepTensor uses deep networks to efficiently decompose tensors with improved performance and robustness.

Study uses machine learning to solve photoacoustic tomography's inverse problem.

These are lecture notes for the course "Analysis and X-ray tomography". The course is a broad overview of various tools in analysis that can be used to study X-ray tomography. The focus is on tools and ideas, not so much on technical details and minimal assumptions. Only very basic functional analysis is assumed as bac…

Deep network improves electrical tomography across multiple frequencies.

A new method speeds up quantum state estimation.

Improved computed tomography reconstruction with deep learning and deep image prior.

We develop a 2D travel time tomography method which regularizes the inversion by modeling groups of slowness pixels from discrete slowness maps, called patches, as sparse linear combinations of atoms from a dictionary. We propose to use dictionary learning during the inversion to adapt dictionaries to specific slowness…

New method uses MRI data to improve PET tomography uncertainty quantification.

We propose a globally convergent alternating minimization (AM) algorithm for image reconstruction in transmission tomography, which extends automatic relevance determination (ARD) to Poisson noise models with Beer's law. The algorithm promotes solutions that are sparse in the pixel/voxel-differences domain by introduci…

TomOpt optimizes muon detector designs using differentiable programming.

Dissertation tackles geodesic ray transform on Riemannian manifolds.

The distribution of absorbed dose in radionuclide therapy with Lu$^{177}$ can be approximated by convolving an image of the time-integrated activity distribution with a dose voxel kernel representing different tissue types. This fast but inaccurate approximation is unsuitable for personalised dosimetry because it negle…

Study online learning of quantum processes, showing feasibility for certain types.

CRC method provides tighter uncertainty intervals for CT images.

Paper develops a new method for solving IBVPs on star-shaped domains.

This work tackles uncertainty quantification in tomography reconstruction.

Study reveals how travel times on cylindrical boundaries can identify spacetime structure.

Framework for confidence estimation in deep CT reconstructions.

Score-based models improve diffuse optical tomography accuracy.

Deep learning is having a profound impact in many fields, especially those that involve some form of image processing. Deep neural networks excel in turning an input image into a set of high-level features. On the other hand, tomography deals with the inverse problem of recreating an image from a number of projections.…

Research on unique continuation principles in medical and seismic imaging.

Unique continuation for X-ray transforms of one-forms with partial data.