A fast implementation of the OPED algorithm, a reconstruction algorithm for Radon data introduced recently, is proposed and tested. The new implementation uses FFT for discrete sine transform and an interpolation step. The convergence of the fast implementation is proved under the condition that the function is mildly smooth. The numerical test shows that the accuracy of the OPED algorithm changes little when the fast implementation is used.
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…
Image reconstruction from radially incomplete spherical Radon data
University of Texas at Arlington（德克萨斯大学阿灵顿分校）；American University of Sharjah（沙迦美国大学）；Tata Institute of Fundamental Research - Centre for Applicable Mathematics（塔塔基础研究所应用数学中心）；University of Würzburg（维尔茨堡大学）
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…
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 …
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…
Let $f(x)$, $x\in\mathbb R^2$, be a piecewise smooth function with a jump discontinuity across a smooth surface $\mathcal S$. Let $f_{Λε}$ denote the Lambda tomography (LT) reconstruction of $f$ from its discrete Radon data $\hat f(α_k,p_j)$. The sampling rate along each variable is $\simε$. First, we compute the limit $f_0(\check x)=\lim_{ε\to0}εf_{Λε}(x_0+ε\check x)$ for a generic $x_0\in\mathcal S$. Once the limiting function $f_0(\check x)$ is known (which we call the discrete transition behavior, or DTB for short), the resolution of reconstruction can be easily found. Next, we show that s…
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).
Joint Image Reconstruction and Segmentation Using the Potts Model
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Technische Universität München（慕尼黑工业大学）；Helmholtz Zentrum München（赫尔姆霍兹慕尼黑中心）；Tufts University（塔夫茨大学）
Compton scatter tomography is an emerging technique with attractive applications in several fields in imaging such as non-destructive testing and medical scanning. In this paper, we introduce a novel modality in three dimensions with a fixed source and a single detector that moves on a spherical surface. We also study the Radon transform modeling the data that consists in integrals on toric surfaces. Using spherical harmonics we arrive to a generalized Abel s type equation connecting the coefficients of the expansion of the data with those of the function. We show the uniqueness of its solutio…
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…
LMU Munich（慕尼黑大学）；University of Tromsø（特罗姆瑟大学）；KTH - Royal Institute of Technology（瑞典皇家理工学院）；Uppsala University（乌普萨拉大学）；University of Vienna（维也纳大学）
We present a deep learning-based algorithm to jointly solve a reconstruction problem and a wavefront set extraction problem in tomographic imaging. The algorithm is based on a recently developed digital wavefront set extractor as well as the well-known microlocal canonical relation for the Radon transform. We use the wavefront set information about x-ray data to improve the reconstruction by requiring that the underlying neural networks simultaneously extract the correct ground truth wavefront set and ground truth image. As a necessary theoretical step, we identify the digital microlocal canon…
A Variational Reconstruction Method for Undersampled Dynamic X-ray Tomography based on Physical Motion Models
Department of Computer Science, University College London（伦敦大学学院计算机科学系）；Department of Mathematics and Statistics, University of Helsinki（赫尔辛基大学数学与统计学系）
Belief Propagation Reconstruction for Discrete Tomography
Surface du Verre et Interfaces, UMR 125 CNRS/Saint-Gobain（玻璃表面与界面，CNRS/圣戈班联合研究单位125）；CNRS and ESPCI ParisTech（法国国家科学研究中心与巴黎高等物理化工学院）；Ecole normale supérieure（巴黎高等师范学院）；LPTMS, Univ. Paris-Sud/CNRS（巴黎南大学/法国国家科学研究中心理论与统计物理实验室）；Institut de Physique Théorique, IPhT, CEA Saclay, and URA 2306, CNRS（CEA萨克雷研究所理论物理研究所，CNRS联合研究单位2306）
Convergent Data-driven Regularizations for CT Reconstruction
Helmholtz Imaging（亥姆霍兹成像中心）；Deutsches Elektronen-Synchrotron DESY（德国电子同步加速器研究所（DESY））；University of Siegen（锡根大学）；Queen Mary University of London（伦敦玛丽女王大学）；The Alan Turing Institute（艾伦·图灵研究所）；Universität Hamburg（汉堡大学）
