Reconstruction of a polynomial from its Radon projections
Borislav Bojanov, Yuan Xu
本文证明，一个二元n次多项式可由其在单位圆上(2[(n+1)/2]+1)个等距方向中每个方向的[n/2]+1条平行线上的Radon投影唯一确定。
一个二元$n$次多项式被证明可由其在单位圆上$(2[(n+1)/2]+1)$个等距方向中每个方向的$[n/2]+1$条平行线上的Radon投影唯一确定。
A polynomial of degree $n$ in two variables is shown to be uniquely determined by its Radon projections taken over $[n/2]+1$ parallel lines in each of the $(2[(n+1)/2]+1)$ equidistant directions along the unit circle.
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A new approach to the reconstruction of images from Radon projections
提出一种基于正交多项式傅里叶展开的新算法，用于从Radon投影重建图像，并在温和条件下证明其收敛性。
Approximation and Reconstruction from Attenuated Radon Projections
本文揭示了加权函数W_μ下的衰减Radon投影与双变量正交展开的紧密联系，并据此提出了从衰减Radon投影重建二维函数（图像）的算法，同时将类似结果推广到球面、三维球体和圆柱域上的函数重建。
Exact reconstruction formulas for a Radon transform over cones
本文研究了考虑康普顿散射的锥形Radon变换，推导出一种滤波反投影类型的精确重建公式，用于逆变换。
Fast OPED algorithm for reconstruction of images from Radon data
提出并测试了一种基于Radon数据重建的OPED算法的快速实现，该实现利用FFT进行离散正弦变换和插值步骤，在函数适度光滑条件下证明收敛性，且数值测试表明精度损失很小。
Fast algorithms and efficient GPU implementations for the Radon transform and the back-projection operator represented as convolution operators
本文提出利用GPU高效实现Radon变换和反投影算子的快速算法，通过log-polar坐标下的卷积运算，实现比CPU和其它GPU软件快10到1000倍的加速效果。
Local Characterization of Noise in Iterative Reconstruction of the Generalized Radon Transform
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Image reconstruction from radially incomplete spherical Radon data
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Irregular Sampling of the Radon Transform of Bandlimited Functions
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Let $f$ be an unknown function in $\mathbb R^2$, and $f_ε$ be its reconstruction from discrete Radon transform data, where $ε$ is the data sampling rate. We study the resolution of reconstruction when $f$ has a jump discontinuity along a nonsmooth curve $\mathcal S_ε$. The assumptions are that (a) $\mathcal S_ε$ is an $O(ε)$-size perturbation of a smooth curve $\mathcal S$, and (b) $\mathcal S_ε$ is Holder continuous with some exponent $γ\in(0,1]$. We compute the Discrete Transition Behavior (or, DTB) defined as the limit $\text{DTB}(\check x):=\lim_{ε\to0}f_ε(x_0+ε\check x)$, where $x_0$ is g…
Analysis of reconstruction of functions with rough edges from discrete Radon data in $\mathbb R^2$
本文分析了从离散Radon数据重建带粗糙边缘的二维函数的精度，通过摄动建模粗糙边界，推导了基于插值和连续反演公式的重建误差阶为$O(ε^{1/2}\ln(1/ε))$。
Analysis of reconstruction from discrete Radon transform data in $\mathbb R^3$ when the function has jump discontinuities
本文研究了在三维空间中从具有跃变断点的函数的离散Radon变换数据进行重建的问题，通过分析离散重建结果的渐进行为，揭示了在高维空间中如何准确重建具有不连续性函数的原理，并通过数值实验验证了理论的正确性。
This revisit gives a survey on the analytical methods for the inverse exponential Radon transform which has been investigated in the past three decades from both mathematical interests and medical applications such as nuclear medicine emission imaging. The derivation of the classical inversion formula is through the recent argument developed for the inverse attenuated Radon transform. That derivation allows the exponential parameter to be a complex constant, which is useful to other applications such as magnetic resonance imaging and tensor field imaging. The survey also includes the new techn…
Local analysis of iterative reconstruction from discrete generalized Radon transform data in the plane
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Local reconstruction analysis of inverting the Radon transform in the plane from noisy discrete data
本文分析二维含噪离散Radon变换数据经滤波反投影重构后的局部误差，证明其极限为高斯随机场并给出协方差，通过数值模拟验证理论。
Medical Image Reconstruction Using Kernel Based Methods
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Analytic inversion of a conical Radon transform arising in application of Compton cameras on the cylinder
本文研究了在圆柱体上顶点位于圆柱轴上的锥形Radon变换的解析反演方法，并探讨了V线变换在特定条件下的应用。
RAPToR: A Resampling Algorithm for Pseudo-Polar based Tomographic Reconstruction
本文提出了一种稳定的快速重建技术，利用离散伪极坐标（PP）Radon变换对平行束X射线成像进行重建，通过现代采样理论实现快速准确的重采样，并利用几何特性去噪，提升成像质量。
A number of practically important imaging problems involve inverting the generalized Radon transform (GRT) $\mathcal R$ of a function $f$ in $\mathbb R^3$. On the other hand, not much is known about the spatial resolution of the reconstruction from discretized data. In this paper we study how accurately and with what resolution the singularities of $f$ are reconstructed. The GRT integrates over a fairly general family of surfaces $\mathcal S_y$ in $\mathbb R^3$. Here $y$ is the parameter in the data space, which runs over an open set $\mathcal V\subset\mathbb R^3$. Assume that the data $g(y)=(…
In this work we investigate numerically the reconstruction approach proposed in Goncharov, Novikov, 2016, for weighted ray transforms (weighted Radon transforms along oriented straight lines) in 3D. In particular, the approach is based on a geometric reduction of the data modeled by weighted ray transforms to new data modeled by weighted Radon transforms along two-dimensional planes in 3D. Such reduction could be seen as a preprocessing procedure which could be further completed by any preferred reconstruction algorithm. In a series of numerical tests on modelized and real SPECT (single photon…
Analysis of reconstruction from noisy discrete generalized Radon data
该研究分析含噪离散广义Radon数据的重建问题，证明在一定条件下重建误差是零均值高斯随机场并显式计算其协方差，数值实验验证了理论与模拟的高度吻合。
Let $\mathcal R$ denote the generalized Radon transform (GRT), which integrates over a family of $N$-dimensional smooth submanifolds $\mathcal S_{\tilde y}\subset\mathcal U$, $1\le N\le n-1$, where an open set $\mathcal U\subset\mathbb R^n$ is the image domain. The submanifolds are parametrized by points $\tilde y\subset\tilde{\mathcal V}$, where an open set $\tilde{\mathcal V}\subset\mathbb R^n$ is the data domain. The continuous data are $g={\mathcal R} f$, and the reconstruction is $\check f=\mathcal R^*\mathcal B g$. Here $\mathcal R^*$ is a weighted adjoint of $\mathcal R$, and $\mathcal …
A Kernel Method for CT Reconstruction: a Fast Implementation using Circulant Matrices
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Regularized Newton methods for simultaneous Radon inversion and phase retrieval in phase contrast tomography
本文提出正则化牛顿方法解决相位对比断层成像中的非线性病态反演问题，通过分析数学结构，结合近场和远场极限，实现同时进行断层和相位重建，提高稳定性。
Three-dimensional multiscale discrete Radon and John transforms
本文提出三维离散Radon与John变换的多尺度算法，无需插值，仅用整数运算实现线性对数复杂度，并证明其快速精确逆变换，适用于实时三维体数据处理。
X-ray imaging from nonlinear waves: numerical reconstruction of a cubic nonlinearity
针对2+1维非线性波动方程的反边界值问题，提出基于Radon变换的直接数值重建方法，通过谱正则化稳定数值微分，实现从边界测量恢复势函数。
Joint Image Reconstruction and Segmentation Using the Potts Model
本文提出一种新的非光滑非凸Potts问题求解方法，用于逆成像问题，无需先验知识即可实现图像重建与分割，避免几何阶梯效应，并在有限数据下成功恢复Shepp-Logan幻影所有段。
Reconstruction Algorithms for Positron Emission Tomography and Single Photon Emission Computed Tomography and their Numerical Implementation
本文利用非线性可积方程理论中的数学技术，推导了Radon变换和衰减Radon变换的解析逆变换公式，并基于三次样条近似提出了数值实现算法，通过Shepp-Logan体模测试验证了算法的准确性。
Image representation by blob and its application in CT reconstruction from few projections
本文利用局部径向对称函数（blob）建立图像表示模型，并通过最小化总变差或blob系数的1范数，实现少投影数据的CT重建。
The recent application of Fourier Based Iterative Reconstruction Method (FIRM) has made it possible to achieve high-quality 2D images from a fan beam Computed Tomography (CT) scan with a limited number of projections in a fast manner. The proposed methodology in this article is designed to provide 3D Radon space in linogram fashion to facilitate the use of FIRM with cone beam projections (CBP) for the reconstruction of 3D images in a low dose Cone Beam CT (CBCT).
Sampling and Reconstruction in Different Subspaces by Using Oblique Projections
本文提出一种利用斜投影从线性测量中重建函数的新方法，通过不同子空间实现更优的近似常数。
Image reconstruction from limited range projections using orthogonal moments
本文提出一组正交多项式用于图像重建，探讨了投影矩与图像矩的关系，并通过仿真验证了方法的有效性。
Computed tomography is a method for synthesizing volumetric or cross-sectional images of an object from a collection of projections. Popular reconstruction methods for computed tomography are based on idealized models and assumptions that may not be valid in practice. One such assumption is that the exact projection geometry is known. The projection geometry describes the relative location of the radiation source, object, and detector for each projection. However, in practice, the geometric parameters used to describe the position and orientation of the radiation source, object, and detector a…
In this paper we consider 3D axis-symmetric (AS) object reconstruction from single cone-beam x-ray projection data. Traditional x-ray CT fails to capture fleeting state of material due to the long time for data acquisition at all angles. Therefore, AS object is devised to investigate the instant deformation of material under pulse change of environment because single projection data is enough to reconstruct its inner structure. Previous reconstruction methods are layer by layer, and ignore the longitudinal tilt of x-ray paths. We propose a regularization method using adaptive tight frame to re…
In this paper, we address an alternative formulation for the exact inverse formula of the Radon transform on circle arcs arising in a modality of Compton Scattering Tomography in translational geometry proposed by Webber and Miller (Inverse Problems (36)2, 025007, 2020). The original study proposes a first method of reconstruction, using the theory of Volterra integral equations. The numerical realization of such a type of inverse formula may exhibit some difficulties, mainly due to stability issues. Here, we provide a suitable formulation for exact inversion that can be straightforwardly impl…
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