Explicit Inversion Formula Research Articles

Uniqueness and stability for a seismic-type generalized Radon transform

In this paper, we consider a generalized seismic Radon transform that maps a given function to its integrals over a certain family of curves in the plane. Such transforms arise in many areas of mathematics, geophysics, and imaging science. This paper contains new explicit inversion formulas of this generalized seismic Radon transform for various families of curves based on their monotonicity. We derive an analogue of the Fourier slice theorem and present a Sobolev space to obtain a stability estimate for this Radon transform. Finally, we consider local uniqueness results for which we prove a sufficiency theorem.

Inversion formulas for space-fractional Bessel heat diffusion through Tikhonov regularization

This article explores the generalized Gauss-Weierstrass transform associated with the space-fractional Bessel diffusion equation. Explicit inversion formulae for this transform are developed using best approximation methods and reproducing kernel theory. To address the inherent ill-posedness of this transform, Tikhonov regularization is implemented. Furthermore, the convergence rate of the regularized solutions is rigorously established.

On the hemispherical transform in the half-space and related Radon transforms

We consider an integral transform which maps functions on the Euclidean half-space to integrals of these functions over hemispheres centered on the boundary hyperplane. The main results include sharp [Formula: see text]-[Formula: see text] estimates for this transform and new explicit inversion formulas under minimal assumptions for functions. The main idea is an intriguing connection between the hemispherical transform, the Radon transform over paraboloids, and the transversal Radon transform over hyperplanes meeting the last coordinate axis.

Weighted Radon transforms of vector fields, with applications to magnetoacoustoelectric tomography

Currently, theory of ray transforms of vector and tensor fields is well developed, but the Radon transforms of such fields have not been fully analyzed. We thus consider linearly weighted and unweighted longitudinal and transversal Radon transforms of vector fields. As usual, we use the standard Helmholtz decomposition of smooth and fast decreasing vector fields over the whole space. We show that such a decomposition produces potential and solenoidal components decreasing at infinity fast enough to guarantee the existence of the unweighted longitudinal and transversal Radon transforms of these components. It is known that reconstruction of an arbitrary vector field from only longitudinal or only transversal transforms is impossible. However, for the cases when both linearly weighted and unweighted transforms of either one of the types are known, we derive explicit inversion formulas for the full reconstruction of the field. Our interest in the inversion of such transforms stems from a certain inverse problem arising in magnetoacoustoelectric tomography (MAET). The connection between the weighted Radon transforms and MAET is exhibited in the paper. Finally, we demonstrate performance and noise sensitivity of the new inversion formulas in numerical simulations.

On the exactness of the universal backprojection formula for the spherical means Radon transform

The spherical means Radon transform is defined by the integral of a function f in over the sphere of radius r centered at a x, normalized by the area of the sphere. The problem of reconstructing f from the data where x belongs to a hypersurface and has important applications in modern imaging modalities, such as photo- and thermo- acoustic tomography. When Γ coincides with the boundary of a bounded (convex) domain , a function supported within Ω can be uniquely recovered from its spherical means known on Γ. We are interested in explicit inversion formulas for such a reconstruction. If , such formulas are only known for the case when Γ is an ellipsoid (or one of its partial cases). This gives rise to a question: can explicit inversion formulas be found for other closed hypersurfaces Γ? In this article we prove, for the so-called ‘universal backprojection inversion formulas’, that their extension to non-ellipsoidal domains Ω is impossible, and therefore ellipsoids constitute the largest class of closed convex hypersurfaces for which such formulas hold.

Strichartz’s Radon transforms for mutually orthogonal affine planes and fractional integrals

In this paper, we study the general orthogonal Radon transform $${R}_{j,k}^p$$ first studied by R.S. Strichartz in [21]. The main conclusions include the sharp existence conditions for $${R}_{j,k}^pf$$ on Lebesgue spaces, the relation formulas connecting our transforms with the fractional integrals and Semyanistyi integrals, through which a number of explicit inversion formulas are obtained when f restricted in the range of j-plane transforms.

Flett potentials associated with differential-difference Laplace operators

A large family of Flett potentials is investigated. Formally, these potentials are negative powers of the operators id + |x|1−1/mΔk, where Δk is the Dunkl Laplace (differential and difference) operator on R. Here, k ≥ 0 and m∈N\{0}. In the (k = 0, m = 1) case, our family of potentials reduces to the classical one studied by Flett [Proc. London Math. Soc. s3-22, 385–451 (1971)]. An explicit inversion formula of the Flett potentials is obtained for functions belonging to C0(R) and weighted Lp spaces, 1 ≤ p < ∞. As a tool, we use a wavelet-like transforms generated by a Poisson type semigroup and signed Borel measures. In this context, a fundamental theorem proving an almost everywhere convergence of a convolution operator for an approximate identity was given. The k = 0 case is already new.

On the Radon transform and the Dirac delta distribution in superspace

In this paper, we obtain a plane wave decomposition for the delta distribution in superspace, provided that the superdimension is not odd and negative. This decomposition allows for explicit inversion formulas for the super Radon transform in these cases. Moreover, we prove a more general Radon inversion formula valid for all possible integer values of the superdimension. The proof of this result comes along with the study of fractional powers of the super Laplacian, their fundamental solutions, and the plane wave decompositions of super Riesz kernels.

Inversion formula and range conditions for a linear system related with the multi‐interval finite Hilbert transform in L2

AbstractGiven n disjoint intervals on together with n functions , , and an matrix , the problem is to find an L2 solution , , to the linear system , where , is a matrix of finite Hilbert transforms with defined on , and is a matrix of the corresponding characteristic functions on . Since we can interpret , as a generalized multi‐interval finite Hilbert transform, we call the formula for the solution as “the inversion formula” and the necessary and sufficient conditions for the existence of a solution as the “range conditions”. In this paper we derive the explicit inversion formula and the range conditions in two specific cases: a) the matrix Θ is symmetric and positive definite, and; b) all the entries of Θ are equal to one. We also prove the uniqueness of solution, that is, that our transform is injective. In the case a), that is, when the matrix Θ is positive definite, the inversion formula is given in terms of the solution of the associated matrix Riemann–Hilbert Problem. In the case b) we reduce the multi interval problem to a problem on n copies of and then express our answers in terms of the Fourier transform. We also discuss other cases of the matrix Θ.

Parabolic-Like Wavelet Transforms and Relevant Reproducing Formulas

We introduce new anisotropic wavelet-type transforms generated by two components: a wavelet measure (or a wavelet function) and a kernel function that naturally generalizes the Gauss and Poisson kernels. The analogues of Calderon’s reproducing formula are established in the framework of the $$L_{p}(\mathbb {R}^{n+1})$$ -theory. These wavelet-type transforms have close connection with a significant generalization of the classical parabolic-Riesz and parabolic-Bessel potentials and can be used to find explicit inversion formulas for the generalized parabolic-type potentials.

Riesz Potentials and Orthogonal Radon Transforms on Affine Grassmannians

Abstract We establish intertwining relations between Riesz potentials associated with fractional powers of minus-Laplacian and orthogonal Radon transforms 𝓡 j,k of the Gonzalez-Strichartz type. The latter take functions on the Grassmannian of j-dimensional affine planes in ℝ n to functions on a similar manifold of k-dimensional planes by integration over the set of all j-planes that meet a given k-plane at a right angle. The main results include sharp existence conditions of 𝓡 j,k f on L p -functions, Fuglede type formulas connecting 𝓡 j,k with Radon-John k-plane transforms and Riesz potentials, and explicit inversion formulas for 𝓡 j,k f under the assumption that f belongs to the range of the j-plane transform. The method extends to another class of Radon transforms defined on affine Grassmannians by inclusion.

An inversion formula for the horizontal conical Radon transform

In this paper, we consider the conical Radon transform on all one-sided circular cones in $$\mathbf{R}^3$$ with horizontal central axis whose vertices are on a vertical line. We derive an explicit inversion formula for such transform. The inversion makes use of the vertical slice transform on a sphere and V-line transform on a plane.

Erdélyi–Kober Fractional Integrals and Radon Transforms for Mutually Orthogonal Affine Planes

Abstract We apply Erdélyi–Kober fractional integrals to the study of Radon type transforms that take functions on the Grassmannian of j-dimensional affine planes in ℝ n to functions on a similar manifold of k-dimensional planes by integration over the set of all j-planes that meet a given k-plane at a right angle. We obtain explicit inversion formulas for these transforms in the class of radial functions under minimal assumptions for all admissible dimensions. The general (not necessarily radial) case, but for j + k = n − 1, n odd, was studied by S. Helgason [8] and F. Gonzalez [4, 5] on smooth compactly supported functions.

Radon transforms over lower-dimensional horospheres in real hyperbolic space

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 obtained for these transforms acting on smooth functions and functions belonging to $L^p$. The case $d=n-1$ agrees with the well-known Gelfand-Graev transform.

The V-line transform with some generalizations and cone differentiation

The paper studies various properties of the V-line transform (VLT) in the plane and the conical Radon transform (CRT) in . The VLT maps a function to a family of its integrals along trajectories made of two rays emanating from a common point. The CRT considered in this paper maps a function to a set of its integrals over surfaces of polyhedral cones. These types of operators appear in mathematical models of single scattering optical tomography, Compton camera imaging and other applications. We derive new explicit inversion formulae for the VLT and the CRT, as well as proving some previously known results using more intuitive geometric ideas. Using our inversion formula for the VLT, we describe the range of that transformation when applied to a fairly broad class of functions and prove some support theorems. The efficiency of our method is demonstrated on several numerical examples. As an auxiliary result that plays a big role in this article, we derive a generalization of the fundamental theorem of calculus, which we call the cone differentiation theorem.

Inverse elastic surface scattering with far-field data

A rigorous mathematical model and an efficient computational method are proposed to solving the inverse elastic surface scattering problem which arises from the near-field imaging of periodic structures. We demonstrate how an enhanced resolution can be achieved by using more easily measurable far-field data. The surface is assumed to be a small and smooth perturbation of an elastically rigid plane. By placing a rectangular slab of a homogeneous and isotropic elastic medium with larger mass density above the surface, more propagating wave modes can be utilized from the far-field data which contributes to the reconstruction resolution. Requiring only a single illumination, the method begins with the far-to-near (FtN) field data conversion and utilizes the transformed field expansion to derive an analytic solution for the direct problem, which leads to an explicit inversion formula for the inverse problem. Moreover, a nonlinear correction scheme is developed to improve the accuracy of the reconstruction. Results show that the proposed method is capable of stably reconstructing surfaces with resolution controlled by the slab's density.

Explicit inversion formulas for properly Hessenberg and Toeplitz–Hessenberg matrices

ABSTRACT The nice inversion properties of Toeplitz–Hessenberg matrices from a Hessenbergian representation of Catalan's numbers encourage us to provide explicit inversion formulas (in terms of basic arithmetical operations involving entries from the original matrix) for the non-singular Toeplitz–Hessenberg matrices. Our approach is based on an elementary matrix inflation method, joint with the nested sums formulas. These explicit inversion formulas are then extended to apply to the entries of the inverse of every non-singular properly Hessenberg matrix.

A note on Bernstein-Vandermonde matrices

ABSTRACTThe inverse matrix of a Vandermonde matrix with arbitrary field entries of dimension with is well-studied in the literature. Especially, an explicit inversion formula does exist. However in theory as e.g. for determining (analytically) the eigenvalues of (arbitrary) matrix polynomials, Bernstein-Vandermonde matrices must be inverted. This note derives an explicit low-cost inversion formula for this matrix class with entries over a field whose characteristic equals zero. Its derivation requires at most additions and multiplications.

Single scattering tomography with curved detectors

Classical computed tomography assumes that x-rays propagate without scattering. The data is modeled as the two-dimensional Radon transform of the attenuation coefficient of the scanned object. To reconstruct the image, the Radon transform has to be inverted. Recently, several new imaging methods that use singly scattered rays have been suggested. The data in these cases is modeled by integrals of a function along V-lines (two rays joined at a common vertex). In this paper we study the V-line transform arising in Single Scattering Optical Tomography in the case of a single concave detector. In this setup the direction of one of the rays constituting a V-line depends on the common vertex. We derive an explicit inversion formula which allows the recovery of the absorption coefficient. Our results are illustrated with numerical simulations.

Some generalizations of Bessel and Flett potentials associated to the Laplace–Bessel differential operator

ABSTRACTWe introduce the notion of bi-parametric potential-type operators, which generalize the Bessel and Flett potentials associated to singular Laplace–Bessel differential operator. Explicit inversion formulas for them are obtained by using a special wavelet-like transform, generated by the so-called ‘modified beta-semigroup’.