Promoted by the advent of coherent synchrotron light sources, phase contrast tomography allows to resolve three-dimensional variations of an unknown sample's complex refractive index from scattering intensities recorded at different incident angles of an X-ray beam. By diffractive free-space propagation of the transmitted wave field, this method is sensitive not only to absorption but also to refractive phase shifts induced by the specimen, permitting three-dimensional nanoscale imaging of quasi-transparent samples such as biological cells. However, the reconstruction of the specimen structure from the observed data constitutes an algorithmically challenging nonlinear ill-posed inverse problem, mainly due to the characteristic loss of phase information in the detection of the wave field.
In this work, regularized Newton methods are developed for the solution of this tomographic phase retrieval problem, based on a detailed analysis of its mathematical structure. We consider both the near-field- or Fresnel regime characterized by a moderate propagation length between sample and detector and the far-field limit of large detector distances, where propagation is governed by the Fourier transform. In the former setting, excellent numerical reconstructions are obtained via the chosen Newton-type approach, supplemented by novel theoretical results stating that measurements from a single detector distance are sufficient to uniquely recover both refraction and absorption of a sample. The proposed algorithm simultaneously performs tomographic- and phase reconstruction, which is found to stabilize the latter by exploiting correlations between the diffraction patterns recorded under different incident angles.
In this work we introduce a new Radon transform which arises from a new modality of Compton Scattering Tomography (CST). This new system is made of a single detector rotating around a fixed source. Unlike some previous CST, no collimator is used at the detector. Such a system allows us to collect scattered photons coming from two opposite sides of the source-detector segment, hence the manifold of the associated Radon transform is a family of double circular arcs. As first main theoretical result, an analytic inversion formula is established for this new Radon transform. This is achieved throu…
We investigate the problem of optimal transport in the so-called Kantorovich form, i.e. given two Radon measures on two compact sets, we seek an optimal transport plan which is another Radon measure on the product of the sets that has these two measures as marginals and minimizes a certain cost function. We consider quadratic regularization of the problem, which forces the optimal transport plan to be a square integrable function rather than a Radon measure. We derive the dual problem and show strong duality and existence of primal and dual solutions to the regularized problem. Then we derive …
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…
We study the inversion of the conical Radon which integrates a function in three-dimensional space from integrals over circular cones. The conical Radon recently got significant attention due to its relevance in various imaging applications such as Compton camera imaging and single scattering optical tomography. The unrestricted conical Radon transform is over-determined because the manifold of all cones depends on six variables: the center position, the axis orientation and the opening angle of the cone. In this work, we consider a particular restricted transform using triple line sensors whe…
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…
Objective: To present the first 3D CGO-based absolute EIT reconstructions from experimental tank data. Approach: CGO-based methods for absolute EIT imaging are compared to traditional TV regularized non-linear least squares reconstruction methods. Additional robustness testing is performed by considering incorrect model\textbf{}ing of domain shape. Main Results: The CGO-based methods are fast, and show strong robustness to incorrect domain modeling comparable to classic difference EIT imaging and fewer boundary artefacts than the TV regularized non-linear least squares reference reconstruction…
Institute of Biomathematics and Biometry, Helmholtz Zentrum München, German Research Center for Environmental Health（生物数学与生物计量研究所，赫尔霍兹慕尼黑中心，德国环境健康研究中心）；Technische Universität München（慕尼黑工业大学）
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…
In this paper, we formulate the reconstruction problem in diffuse optical tomography (DOT) in a statistical setting for determining the optical parameters, scattering and absorption, from boundary photon density measurements. A special kind of adaptive Metropolis algorithm for the reconstruction procedure using sparsity and total variation prior is presented. Finally, a simulation study of this technique with different regularization functions and its comparison to the deterministic Iteratively Regularized Gauss Newton method shows the effectiveness and stability of the method.
We present a new computed tomography (CT) method for inverting the Radon transform in 2D. The idea relies on the geometry of the flat torus, hence we call the new method Torus CT. We prove new inversion formulas for integrable functions, solve a minimization problem associated to Tikhonov regularization in Sobolev spaces and prove that the solution operator provides an admissible regularization strategy with a quantitative stability estimate. This regularization is a simple post-processing low-pass filter for the Fourier series of a phantom. We also study the adjoint and the normal operator of…
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）
