Related papers: The Funk-Radon transform for hyperplane sections t…
In this paper, we consider the conical Radon transform on all cones with horizontal central axis whose vertices are on a straight line. We derive an explicit inversion formula for such transform. The inversion makes use of the vertical…
We consider an inverse problem arising in thermo-/photo- acoustic tomography that amounts to reconstructing a function $f$ from its circular or spherical means with the centers lying on a given measurement surface. (Equivalently, these…
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…
Recovering a function from its integrals over circular cones recently gained significance because of its relevance to novel medical imaging technologies such emission tomography using Compton cameras. In this paper we investigate the case…
The object of this study is an integral operator $\mathcal{S}$ which averages functions in the Euclidean upper half-space $\mathbb{R}_{+}^{n}$ over the half-spheres centered on the topological boundary $\partial \mathbb{R}_{+}^{n}$. By…
Many known Radon-type transforms of symmetric (radial or zonal) functions are represented by one-dimensional Riemann-Liouville fractional integrals or their modifications. The present article contains new examples of such transforms in the…
The Funk--Minkowski transform ${\mathcal F}$ associates a function $f$ on the sphere ${\mathbb S}^2$ with its mean values (integrals) along all great circles of the sphere. Thepresented analytical inversion formula reconstruct the unknown…
We study non-geodesic Funk-type transforms associated with cross-sections of the n-sphere by k-dimensional planes passing through an arbitrary fixed point inside the sphere. The main results include injectivity conditions for these…
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…
The conical Radon transform, which assigns to a given function $f$ on $\mathbb R^3$ its integrals over conical surfaces, arises in several imaging techniques, e.g. in astronomy and homeland security, especially when the so-called Compton…
We study integral transforms mapping a function on the Euclidean space to the family of its integration on some hypersurfaces, that is, a function of hypersurfaces. The hypersurfaces are given by the graphs of functions with fixed axes of…
We study horospherical Radon transforms that integrate functions on the $n$-dimensional real hyperbolic space over horospheres of arbitrary fixed dimension $1\le d\le n-1$. Exact existence conditions and new explicit inversion formulas are…
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 Hua-Radon and polarized Hua-Radon transform are two orthogonal projections defined on holomorphic functions in the Lie sphere. Both transformations can be written as integral transforms with respect to a suitable reproducing kernel.…
The star transform is a generalized Radon transform mapping a function of two variables to its integrals along "star-shaped" trajectories, which consist of a finite number of rays emanating from a common vertex. Such operators appear in…
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…
In this paper we consider the generalized Radon transform $\mathcal R$ in the plane. Let $f$ be a piecewise smooth function, which has a jump across a smooth curve $\mathcal S$. We obtain a formula, which accurately describes view aliasing…
The monogenic Hua-Radon transform is defined as an orthogonal projection on holomorphic functions in the Lie sphere. Extending the work of Sabadini and Sommen, J. Geom. Anal. 29 (2019), 2709-2737, we determine its reproducing kernel.…
In recent years, many types of elliptical Radon transforms that integrate functions over various sets of ellipses/ellipsoids have been considered, relating to studies in bistatic synthetic aperture radar, ultrasound reflection tomography,…
The spherical Radon-Dunkl transform $R_\kappa$, associated to weight functions invariant under a finite reflection group, is introduced, and some elementary properties are obtained in terms of $h$-harmonics. Several inversion formulas of…