On inversion and Sobolev estimate results about the weighted Radon transform

In this paper, we derive an inversion of the weighted Radon transform by Fourier transform, Riesz potential, and integral transform. We extend results of Rigaud and Lakhal to the n‐dimensional Euclidean space. Furthermore, we obtain some properties of the weighted Radon transform. Finally, we develop some estimate results of the weighted Radon transform under Sobolev space.     

An inversion formula for the spherical mean transform with data on an ellipsoid in two and three dimensions

In the articles [1] and [2] of D. Finch, M. Haltmeier, S. Patch and D. Rakesh, inversion formulas were found in any dimension greater than or equal to 2 for recovering a smooth function with compact support in the unit ball from spherical means centered on the unit sphere. The aim of this article is to show that the methods used in  and  can be modified in order to get similar inversion formulas from spherical means centered on an ellipsoid in two and three dimensional spaces.     

Remarks on the degenerate Radon transform in

The aim of this note is to prove endpoint boundedness of the generalized Radon transform which was introduced by Phong and Stein. M. Christ's combinatorial method is used to obtain restricted weak type at the endpoints. Also we show that the results of this note are essentially optimal.

     

New range theorems for the dual Radon transform

Three new range theorems are established for the dual Radon transform : on functions that do not decay fast at infinity (and admit an asymptotic expansion), on , and on . Here , and acts on even functions .

     

Bilinear restriction estimates for surfaces with curvatures of different signs

Recently, the sharp -bilinear (adjoint) restriction estimates for the cone and the paraboloid were established by Wolff and Tao, respectively. Their results rely on the fact that for the cone and the paraboloid, the nonzero principal curvatures have the same sign. We generalize those bilinear restriction estimates to surfaces with curvatures of different signs.

     

Fourier transform and related integral transforms in superspace

In this paper extensions of the classical Fourier, fractional Fourier and Radon transforms to superspace are studied. Previously, a Fourier transform in superspace was already studied, but with a different kernel. In this work, the fermionic part of the Fourier kernel has a natural symplectic structure, derived using a Clifford analysis approach. Several basic properties of these three transforms are studied. Using suitable generalizations of the Hermite polynomials to superspace (see [H. De Bie, F. Sommen, Hermite and Gegenbauer polynomials in superspace using Clifford analysis, J. Phys. A 40 (2007) 10441-10456]) an eigenfunction basis for the Fourier transform is constructed.     

A characterization of inverse Radon transform on the Laguerre hypergroup

Let K=[0,∞)×R be the Laguerre hypergroup which is the fundamental manifold of the radial function space for the Heisenberg group. In this note we give another characterization for a subspace of S(K) (Schwartz space) such that the Radon transform Rα on K is a bijection. We show that this characterization is equivalent to that in [M.M. Nessibi, K. Trimèche, Inversion of the Radon transform on the Laguerre hypergroup by using generalized wavelets, J. Math. Anal. Appl. 208 (1997) 337-363]. In addition, we establish an inversion formula of the Radon transform Rα in the weak sense.     

Stability of inverse problems in an infinite slab with partial data

In this paper, we study the stability of two inverse boundary value problems in an infinite slab with partial data. These problems have been studied by Li and Uhlmann for the case of the Schrödinger equation and by Krupchyk, Lassas, and Uhlmann for the case of the magnetic Schrödinger equation. Here, we quantify the method of uniqueness proposed by Li and Uhlmann and prove a log–log stability estimate for the inverse problems associated to the Schrödinger equation. The boundary measurements considered in these problems are modeled by partial knowledge of the Dirichlet-to-Neumann map: in the first inverse problem, the corresponding Dirichlet and Neumann data are known on different boundary hyperplanes of the slab; in the second inverse problem, they are known on the same boundary hyperplane of the slab.     

Invariant differential operators and the range of the matrix Radon transform

In this paper we characterize the range of the matrix Radon transform by invariant differential operators. This generalizes analogous results for the d-plane transform in Rn.     

The generalized totally geodesic Radon transform and its application to texture analysis

The generalized totally geodesic Radon transform associates the mean values over spherical tori to a function f defined on ??³??, where the elements of ??³ are considered as quaternions representing rotations. It is introduced into the analysis of crystallographic preferred orientation and identified with the probability density function corresponding to the angle distribution function W. Eventually, this communication suggests a new approach to recover an approximation of f from data sampling W. At the same time it provides additional clarification of a recently suggested method applying reproducing kernels and radial basis functions by instructive insight into its involved geometry. The focus is on the correspondence of geometrical and group features rather than on the mapping of functions and their spaces. Copyright © 2008 John Wiley & Sons, Ltd.     

Formal loops III: Additive functions and the Radon transform

To any algebraic variety X and closed 2-form ω on X, we associate the “symplectic action functional” T(ω) which is a function on the formal loop space LX introduced by the authors earlier. The correspondence ω→T(ω) can be seen as a version of the Radon transform. We give a characterization of the functions of the form T(ω) in terms of factorizability (infinitesimal analog of additivity in holomorphic pairs of pants) as well as in terms of vertex operator algebras.These results will be used in the subsequent paper which will relate the gerbe of chiral differential operators on X (whose lien is the sheaf of closed 2-forms) and the determinantal gerbe of the tangent bundle of LX (whose lien is the sheaf of invertible functions on LX). On the level of liens this relation associates to a closed 2-form ω the invertible function expT(ω).     

An inversion formula for the dual horocyclic Radon transform on the hyperbolic plane

Consider the Poincare unit disk model for the hyperbolic plane H ². Let Ξ be the set of all horocycles in H ² parametrized by (θ, p), where e^iθ is the point where a horocycle ξ is tangent to the boundary |z| = 1, and p is the hyperbolic distance from ξ to the origin. In this paper we invert the dual Radon transform R* : μ(θ, p) → (z) under the assumption of exponential decay of μ and some of its derivatives. The additional assumption is that P_m(d/dp)(μ_m(p)e^p) be even for all m ∈ ?. Here P_m(d/dp) is a family of differential operators introduced by Helgason, and μ_m(p) are the coefficients of the Fourier series expansion of μ(θ, p). (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Inverse problems with partial data for a Dirac system: A Carleman estimate approach

We prove that the material parameters in a Dirac system with magnetic and electric potentials are uniquely determined by measurements made on a possibly small subset of the boundary. The proof is based on a combination of Carleman estimates for first and second order systems, and involves a reduction of the boundary measurements to the second order case. For this reduction a certain amount of decoupling is required. To effectively make use of the decoupling, the Carleman estimates are established for coefficients which may become singular in the asymptotic limit.     

Wavelet transform and radon transform on the quaternion Heisenberg group

Let Q be the quaternion Heisenberg group,and let P be the affine automorphism group of Q.We develop the theory of continuous wavelet transform on the quaternion Heisenberg group via the unitary representations of P on L2(Q).A class of radial wavelets is constructed.The inverse wavelet transform is simplified by using radial wavelets.Then we investigate the Radon transform on Q.A Semyanistyi–Lizorkin space is introduced,on which the Radon transform is a bijection.We deal with the Radon transform on Q both by the Euclidean Fourier transform and the group Fourier transform.These two treatments are essentially equivalent.We also give an inversion formula by using wavelets,which does not require the smoothness of functions if the wavelet is smooth.In addition,we obtain an inversion formula of the Radon transform associated with the sub-Laplacian on Q.     

Partial data inverse problems for Maxwell equations via Carleman estimates

In this article we consider an inverse boundary value problem for the time-harmonic Maxwell equations. We show that the electromagnetic material parameters are determined by boundary measurements where part of the boundary data is measured on a possibly very small set. This is an extension of earlier scalar results of Bukhgeim–Uhlmann and Kenig–Sjöstrand–Uhlmann to the Maxwell system. The main contribution is to show that the Carleman estimate approach to scalar partial data inverse problems introduced in those works can be carried over to the Maxwell system.     

Symmetric subvarieties in compactifications and the Radon transform on Riemannian symmetric spaces of noncompact type

We investigate the totally geodesic Radon transform which assigns a function to its integration over totally geodesic symmetric submanifolds in Riemannian symmetric spaces of noncompact type. Our main concern is focused on the injectivity and support theorem. Our approach is based on the projection slice theorem relating the totally geodesic Radon transform and the Fourier transforms on symmetric spaces. Our approach also uses the study of geometry concerned with the totally geodesic symmetric subvarieties in Riemannian symmetric spaces in terms of the cell structure of the Satake compactifications.     

A new complex approach towards approximate inverse and generalized Radon transform

In this paper, we first consider the framework of Sobolev spaces and derive analytically a reconstruction algorithm by means of the residue theorem of complex analysis, the approximate inverse, Gaussian mollifier and integral equations. And we successfully extend Natterer’s results to the generalized Radon transform with non-uniform attenuation. Finally, we investigate the smoothing properties of the generalized Radon transform.