Related papers: An inversion formula for the spherical mean transf…
We study the problem of recovering the initial data of the two dimensional wave equation from values of its solution on the boundary $\partial \Om$ of a smooth convex bounded domain $\Om \subset \R^2$. As a main result we establish…
An explicit series solution is proposed for the inversion of the spherical mean Radon transform. Such an inversion is required in problems of thermo- and photo- acoustic tomography. Closed-form inversion formulae are currently known only…
The paper contains a simple proof of the Finch-Patch-Rakesh inversion formula for the spherical mean Radon transform in odd dimensions. This transform arises in thermoacoustic tomography. Applications are given to the Cauchy problem for the…
The article suggests a new approach what is called a consistency method for the inversion of the spherical Radon transform in 2D with detectors on a line. It is known that there is not an exact inversion formula in 2D. By means of the…
In this paper we refer to the reconstruction formulas given in L.-E. Andersson's On the determination of a function from spherical averages, which are often used in applications such as SAR and SONAR. We demonstrate that the first one of…
In this article we study the problem of recovering the initial data of the two-dimensional wave equation from Neumann measurements on a convex domain with smooth boundary in the plane. We derive an explicit inversion formula of a so-called…
We begin by studying the surface area of an ellipsoid in n-dimensional Euclidean space as the function of the lengths of the semi-axes. We write down an explicit formula as an integral over the unit sphere in n-dimensions and use this…
We obtain new inversion formulas for the Funk type transforms of two kinds associated to spherical sections by hyperplanes passing through a common point $A$ which lies inside the n-dimensional unit sphere or on the sphere itself.…
The spherical mean transform associates to a function $f$ its integral averages over all spheres. We consider the spherical mean transform for functions supported in the unit ball $\mathbb{B}$ in $\mathbb{R}^n$ for odd $n$, with the centers…
We introduce a technique for recovering a sufficiently smooth function from its ray transform over a wide class of curves in a general region of Euclidean space. The method is based on a complexification of the underlying vector fields…
It is known that the Funk transform (the Funk-Radon transform) is invertible in the class of even (symmetric) continuous functions defined on the unit 2-sphere S^2. In this article, for the reconstruction of f from C(S^2) (can be non-even),…
We derive an explicit inversion algorithm for the spherical Radon transform in odd dimensions with partial radial data. We prove that the reconstruction of the unknown function can be reduced to solving ordinary differential equations,…
We study inversion of the spherical Radon transform with centers on a sphere (the data acquisition set). Such inversions are essential in various image reconstruction problems arising in medical, radar and sonar imaging. In the case of…
We introduce fractional integrals on the $n$-dimensional spherical cap, study their boundednes in weighted $L^p$ spaces and obtain explicit inversion formulas. The results are applied to the inversion problem for Riesz potentials on a…
We present an approach to sums of random Hermitian matrices via the theory of spherical functions for the Gelfand pair $(\mathrm{U}(n) \ltimes \mathrm{Herm}(n), \mathrm{U}(n))$. It is inspired by a similar approach of Kieburg and K\"osters…
We study the spherical mean transform on $\rN^n$. The transform is characterized by the Euler-Poisson-Darboux equation. By looking at the spherical harmonic expansions, we obtain a system of 1+1-dimension hyperbolic equations, which provide…
This article provides a novel and simple range description for the spherical mean transform of functions supported in the unit ball of an odd dimensional Euclidean space. The new description comprises a set of symmetry relations between the…
The aim of this article is to introduce a method for recovering functions, defined on the $n - 1$ dimensional unit sphere $\Bbb S^{n - 1}$, using their spherical transform, which integrates functions on $n - 2$ dimensional subspheres, on a…
Let $\mR$ be the restriction of the spherical Radon transform to the set of spheres centered on a hypersurface $\mS$. We study the inversion of $\mR$ by a closed-form formula. We approach the problem by studying an oscillatory integral,…
We investigate discrete spin transformations, a geometric framework to manipulate surface meshes by controlling mean curvature. Applications include surface fairing -- flowing a mesh onto say, a reference sphere -- and mesh extrusion --…