Gregory Eskin
We consider the inverse problem of the recovery of the gauge field in R^2 modulo gauge transformations from the non-abelian Radon transform.A global uniqueness theorem is proven for the case when the gauge field has a compact support.
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We study the Radon transform in the plane in parallel geometry possibly undersampled in the angular variables. We study resolution, aliasing artifacts, and edge recovery.
This paper is devoted to a Radon-type transform arising in a version of Photoacoustic Tomography that uses integrating circular detectors. We show that the transform can be decomposed into the spherical Radon transform and the Funk-Minkowski transform. An inversion formula and a range description are obtained by combining existing inversion for- mulas and range descriptions for the spherical Radon transform and the Funk-Minkowski transform. Numerical simulations are performed to demonstrate our proposed algorithm.
This note establishes sharp $L^p-L^r$ estimates for $X$-ray transforms and Radon transforms in finite fields.
Inversion of generalized Radon transform over symmetric $m$-tensor fields in $\mathbb{R}^n$
本文将向量场加权 Radon 变换恢复方法推广到对称 $m$-张量场，引入纵向/横向 Radon 变换并证明其组合可唯一反演张量场。
Radon Transform over Tensor Fields: Injectivity, Range, and Unique Continuation Principle
本文研究了作用于对称m-张量场上的Radon变换的核特征、反演公式、稳定性估计、范围特征和唯一连续性性质，揭示了其一致的解析结构，扩展了经典Radon变换和张量射线变换的主要特性。
We present new necessary and sufficient conditions for a function on $\partialΩ\times S^1$ to be in the range of the attenuated Radon transform of a sufficiently smooth function support in the convex set $\barΩ\subset\mathbb{R}^2$. The approach is based on an explicit Hilbert transform associated with traces of the boundary of A-analytic functions in the sense of Bukhgeim.
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 $R_m[f](ξ,η)=\int f(x, ξx + η) m(ξ, η, x)\,\text{d} x$ is defined near $(ξ,η)=(0,0)$. For weight functions satisfying a certain differential equation we give weak estimates of $f$ in terms of $R_m[f]$ for functions $f$ that satisfies an a priori bound.
This paper is devoted to a Radon-type transform arising in Photoacoustic Tomography that uses integrating line detectors. We consider two situations: when the line detectors are tangent to the boundary of a cylindrical domain and when the line detectors are located on a plane. We present the analogue of the Fourier slice theorems for each case of the Radon-type transforms. Also, we provide several new inversion formulas, a support theorem, and stability estimate and necessary range condition results.
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 means can be expressed through the solution $u(t,x)$ of the wave equation with the initial pressure equal to $f$.) An explicit solution of this inverse problem is obtained in 3D for the surface that is the boundary of an open octet, and in 2D for the case when the centers of integration circles lie on two rays starting at the origin and intersecting at the angle…
We derive explicit formulas for the reconstruction of a function from its integrals over a family of spheres, or for the inversion of the spherical mean Radon transform. Such formulas are important for problems of thermo- and photo- acoustic tomography. A closed-form inversion formula of a filtration-backprojection type is found for the case when the centers of the integration spheres lie on a sphere in Rn surrounding the support of the unknown function. An explicit series solution is presented for the case when the centers of the integration spheres lie on a general closed surface.
In the paper we prove the existence results for initial-value boundary value problems for compressible isothermal Navier-Stokes equations. We restrict ourselves to 2D case of a problem with no-slip condition for nonstationary motion of viscous compressible isothermal fluid. However, the technique of modeling and analysis presented here is general and can be used for 3D problems.
In this article, we prove a stability estimate going from the Radon transform of a function with limited angle-distance data to the $L^p$ norm of the function itself, under some conditions on the support of the function. We apply this theorem to obtain stability estimates for an inverse boundary value problem with partial data.
We show injectivity of the X-ray transform and the $d$-plane Radon transform for distributions on the $n$-torus, lowering the regularity assumption in the recent work by Abouelaz and Rouvière. We also show solenoidal injectivity of the X-ray transform on the $n$-torus for tensor fields of any order, allowing the tensors to have distribution valued coefficients. These imply new injectivity results for the periodic broken ray transform on cubes of any dimension.
Weighted Radon transforms of vector fields, with applications to magnetoacoustoelectric tomography
本文研究了向量场的线性加权与未加权纵向及横向 Radon 变换，推导了显式反演公式以完全重建向量场，并将其应用于磁声电断层成像（MAET）的逆问题求解与数值验证。
Let $\mR$ be the restriction of the spherical Radon transform to the set of spheres centered on a hypersurface $\mS$. We study the inversion of $\mR$ by a closed-form formula. We approach the problem by studying an oscillatory integral, which depends on the observation surface $\mS$ as a parameter. We then derive various microlocal analytic properties of the associated closed-form inversion formula.
Remarks on the general Funk-Radon transform and thermoacoustic tomography
研究光滑流形中一族超曲面定义的一般积分变换的性质，包括Sobolev范数估计、值域条件和积分变换核的逼近定理，并讨论在热/光/光声断层成像中出现的球面平均变换的应用。
The circular Radon transform integrates a function over the set of all spheres with a given set of centers. The problem of injectivity of this transform (as well as inversion formulas, range descriptions, etc.) arises in many fields from approximation theory to integral geometry, to inverse problems for PDEs, and recently to newly developing types of tomography. The article discusses known and provides new results that one can obtain by methods that essentially involve only the finite speed of propagation and domain dependence for the wave equation.
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