Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userExact inversion of the conical Radon transform with a fixed opening angle
116
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We study a new class of Radon transforms defined on circular cones called the conical Radon transform. In $mathbb{R}^3$ it maps a function to its surface integrals over circular cones, and in $mathbb{R}^2$ it maps a function to its integrals along two rays with a common vertex. Such transforms appear in various mathematical models arising in medical imaging, nuclear industry and homeland security. This paper contains new results about inversion of conical Radon transform with fixed opening angle and vertical central axis in $mathbb{R}^2$ and $mathbb{R}^3$. New simple explicit inversion formulae are presented in these cases. Numerical simulations were performed to demonstrate the efficiency of the suggested algorithm in 2D.
rate research
Read More
The article presents an efficient image reconstruction algorithm for single scattering optical tomography (SSOT) in circular geometry of data acquisition. This novel medical imaging modality uses photons of light that scatter once in the body to recover its interior features. The mathematical model of SSOT is based on the broken ray (or V-line Radon) transform (BRT), which puts into correspondence to an image function its integrals along V-shaped piecewise linear trajectories. The process of image reconstruction in SSOT requires inversion of that transform. We implement numerical inversion of a broken ray transform in a disc with partial radial data. Our method is based on a relation between the Fourier coefficients of the image function and those of its BRT recently discovered by Ambartsoumian and Moon. The numerical algorithm requires solution of ill-conditioned matrix problems, which is accomplished using a half-rank truncated singular value decomposition method. Several numerical computations validating the inversion formula are presented, which demonstrate the accuracy, speed and robustness of our method in the case of both noise-free and noisy data.
The purpose of this paper is to study the sparse bound of the operator of the form $f mapsto psi(x) int f(gamma_t(x))K(t)dt$, where $gamma_t(x)$ is a $C^infty$ function defined on a neighborhood of the origin in $(x, t) in mathbb R^n times mathbb R^k$, satisfying $gamma_0(x) equiv x$, $psi$ is a $C^infty$ cut-off function supported on a small neighborhood of $0 in mathbb R^n$ and $K$ is a Calderon-Zygmund kernel suppported on a small neighborhood of $0 in mathbb R^k$. Christ, Nagel, Stein and Wainger gave conditions on $gamma$ under which $T: L^p mapsto L^p (1<p<infty)$ is bounded. Under the these same conditions, we prove sparse bounds for $T$, which strengthens their result. As a corollary, we derive weighted norm estimates for such operators.
We show how to calculate the dual horospherical Radon transform of a polynomial in terms of the Harish-Chandra c-function.
In this article, we prove a stability estimate going from the Radon transform of a function with limited angle-distance data to the $L^p$ norm of the function itself, under some conditions on the support of the function. We apply this theorem to obtain stability estimates for an inverse boundary value problem with partial data.
Using the tensor Radon transform and related numerical methods, we study how bulk geometries can be explicitly reconstructed from boundary entanglement entropies in the specific case of $mathrm{AdS}_3/mathrm{CFT}_2$. We find that, given the boundary entanglement entropies of a $2$d CFT, this framework provides a quantitative measure that detects whether the bulk dual is geometric in the perturbative (near AdS) limit. In the case where a well-defined bulk geometry exists, we explicitly reconstruct the unique bulk metric tensor once a gauge choice is made. We then examine the emergent bulk geometries for static and dynamical scenarios in holography and in many-body systems. Apart from the physics results, our work demonstrates that numerical methods are feasible and effective in the study of bulk reconstruction in AdS/CFT.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
