相关论文: Singularities of the Radon transform

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…

We provide conditions for exact reconstruction of a bandlimited function from irregular polar samples of its Radon transform. First, we prove that the Radon transform is a continuous L2-operator for certain classes of bandlimited signals.…

Let $f$ be an unknown function in $\mathbb R^2$, and $f_\epsilon$ be its reconstruction from discrete Radon transform data, where $\epsilon$ is the data sampling rate. We study the resolution of reconstruction when $f$ has a jump…

A nonlinear scattering transform is studied for the two-dimensional Schrodinger equation at zero energy with a radial potential. First explicit examples are presented, both theoretically and computationally, of potentials with nontrivial…

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…

We revisit the standard representation of the (inverse) Radon transform which is well-known in the mathematical literature. We extend this representation to the case involving the parton distributions. We have found the new additional…

We show that under a certain non-cancellation condition the attenuated Radon transform uniquely determines piecewise constant attenuation $a$ and piecewise $C^2$ source density $f$ with jumps over real analytic boundaries possibly having…

In this paper we consider the so-called crystallographic Radon transform (or crystallographic $X$-ray transform) and totally geodesic Radon transform on the group of rotations SO(3). As we show both of these transforms naturally appear in…

Locally analytically, any isolated double point occurs as a double covering of a smooth surface. It can be desingularized via the canonical resolution, as it is well-known. In this paper we explicitly compute the fundamental cycle of both…

Let $\sigma$ be arc-length measure on $S^1\subset \mathbb R^2$ and $\Theta$ denote rotation by an angle $\theta \in (0, \pi]$. Define a model bilinear generalized Radon transform, $$B_{\theta}(f,g)(x)=\int_{S^1} f(x-y)g(x-\Theta y)\,…

The sonar transform in geometric tomography maps functions on the Euclidean half-space to integrals of those functions over hemispheres centered on the boundary hyperplane. We obtain sharp $L^p$-$L^q$ estimates for this transform and new…

We investigate geometric properties of surfaces given by certain formulae. In particular, we calculate the singular curvature and the limiting normal curvature of such surfaces along the set of singular points consisting of singular points…

In this paper we consider three types of discrete operators stemming from singular Radon transforms. We first extend an $\ell^p$ result for translation invariant discrete singular Radon transforms to a class of twisted operators including…

Recovering a function from its spherical Radon transform with centers of spheres of integration restricted to a hypersurface is at the heart of several modern imaging technologies, including SAR, ultrasound imaging, and photo- and…

We study reconstruction of an unknown function from its $d$-plane Radon transform on the flat $n$-torus when $1 \leq d \leq n-1$. We prove new reconstruction formulas and stability results with respect to weighted Bessel potential norms. We…

An alternative method to invert the Radon transforms without the use of Courant-Hilbert's identities has been proposed and developed independently from the space dimension. For the universal representation of inverse Radon transform, we…

We obtain sharp norm estimates for fractional integrals generated by Radon transforms of three types in the n-dimensional real Euclidean space. The method relies on recent interpolation results for analytic families of operators.

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,…

Properties of general Legendrian cycles $T$ acting in ${\mathbb R}^d\times S^{d-1}$ are studied. In particular, we give short proofs for certain uniqueness theorems with respect to the projections on the first and second component of such…

This work characterizes the range of the single-quadrant approximate discrete Radon transform (ADRT) of square images. The characterization follows from a set of linear constraints on the codomain. We show that for data satisfying these…