Related papers: Spherical Radon transforms with smoothly varying r…
In this paper we deal with the problem of recovering functions from their spherical mean transform $\mathcal{R}$, which integrates functions on circles in the plane, in case where the centers of the circles of integration are located on a…
Given a real valued function on R^n we study the problem of recovering the function from its spherical means over spheres centered on a hyperplane. An old paper of Bukhgeim and Kardakov derived an inversion formula for the odd n case with…
We implement numerically formulas of [Isaev, Novikov, arXiv:2107.07882] for finding a compactly supported function $v$ on $\mathbb{R}^d$, $d\geq 1$, from its Fourier transform $\mathcal{F} [v]$ given within the ball $B_r$. For the…
In image reconstruction there are techniques that use analytical formulae for the Radon transform to recover an image from a continuum of data. In practice, however, one has only discrete data available. Thus one often resorts to sampling…
The work develops further the theory of the following inversion problem, which plays the central role in the rapidly developing area of thermoacoustic tomography and has intimate connections with PDEs and integral geometry: {\it Reconstruct…
We present new necessary and sufficient conditions for a function on $\partial\Omega\times S^1$ to be in the range of the attenuated Radon transform of a sufficiently smooth function support in the convex set…
We present explicit filtration/backprojection-type formulae for the inversion of the spherical (circular) mean transform with the centers lying on the boundary of some polyhedra (or polygons, in 2D). The formulae are derived using the…
In [J. Bures, R. Lavicka, V. Soucek, Elements of quaternionic analysis and Radon transform, Textos de Matematica 42, Departamento de Matematica, Universidade de Coimbra, 2009], the authors describe a link between holomorphic functions…
The light field reconstruction from the focal stack can be mathematically formulated as an ill-posed integral equation inversion problem. Although the previous research about this problem has made progress both in practice and theory, its…
A central objective in inverse problems arising in integral geometry is to understand the kernel characterization, inversion formulas, stability estimates, range characterization, and unique continuation properties of integral transforms.…
The paper studies various properties of the V-line transform (VLT) in the plane and conical Radon transform (CRT) in $\mathbb{R}^n$. VLT maps a function to a family of its integrals along trajectories made of two rays emanating from a…
This article presents extensions of the Cram{\'e}r-Wold theorem to measures that may have infinite mass near the origin. Corresponding results for sequences of measures are presented together with examples showing that the assumptions…
We present spherical analysis of electron backscatter diffraction (EBSD) patterns with two new algorithms: (1) band localisation and band profile analysis using the spherical Radon transform; (2) orientation determination using spherical…
The conical Radon transform is an integral transform that maps a given function $f$ to its integral over a conical surface. In this study, we invesgate the conical Radon transform with a fixed central axis and opening angle, considering the…
We consider the inverse problem for the $2$-dimensional weighted local Radon transform $R_m[f]$, where $f$ is supported in $y\geq x^2$ and $R_m[f](\xi,\eta)=\int f(x, \xi x + \eta) m(\xi, \eta, x)\,\text{d} x$ is defined near…
This article covers polyhomogeneous mapping properties of the Radon transform $R$ of smooth functions on the open unit ball $\Omega\subset\mathbb{R}^n$ and the back-projection operator $R^*$ on $Z=(-1,1)\times S^{n-1}\subset\mathbb{R}\times…
The direct method used for calculating smooth radial basis function (RBF) interpolants in the flat limit becomes numerically unstable. The RBF-QR algorithm bypasses this ill-conditioning using a clever change of basis technique. We extend…
The cone-beam transform consists of integrating a function defined on the three-dimensional space along every ray that starts on a certain scanning set. Based on Grangeat's formula, Louis [2016, Inverse Problems 32 115005] states…
In many applications data are measured or defined on a spherical manifold; spherical harmonic transforms are then required to access the frequency content of the data. We derive algorithms to perform forward and inverse spin spherical…
The well known formula of B. Fuglede expresses the mean value of the Radon k-plane transform on $R^n$ as a Riesz potential. We extend this formula to the space of $n \times m$ real matrices and show that the corresponding matrix k-plane…