High Performance Reconstruction Framework for Straight Ray Tomography:from Micro to Nano Resolution Imaging

We develop a high-performance scheme to reconstruct straight-ray tomographic scans. We preserve the quality of the state-of-the-art schemes typically found in traditional computed tomography but reduce the computational cost substantially. Our approach is based on 1) a rigorous discretization of the forward model using a generalized sampling scheme; 2) a variational formulation of the reconstruction problem; and 3) iterative reconstruction algorithms that use the alternating-direction method of multipliers. To improve the quality of the reconstruction, we take advantage of total-variation regularization and its higher-order variants. In addition, the prior information on the support and the positivity of the refractive index are both considered, which yields significant improvements. The two challenging applications to which we apply the methods of our framework are grating-based \mbox{x-ray} imaging (GI) and single-particle analysis (SPA). In the context of micro-resolution GI, three complementary characteristics are measured: the conventional absorption contrast, the differential phase contrast, and the small-angle scattering contrast. While these three measurements provide powerful insights on biological samples, up to now they were calling for a large-dose deposition which potentially was harming the specimens ({\textit{e.g.}}, in small-rodent scanners). As it turns out, we are able to preserve the image quality of filtered back-projection-type methods despite the fewer acquisition angles and the lower signal-to-noise ratio implied by a reduction in the total dose of {\textit{in-vivo}} grating interferometry. To achieve this, we first apply our reconstruction framework to differential phase-contrast imaging (DPCI). We then add Jacobian-type regularization to simultaneously reconstruct phase and absorption. The experimental results confirm the power of our method. This is a crucial step toward the deployment of DPCI in medicine and biology. Our algorithms have been implemented in the TOMCAT laboratory of the Paul Scherrer Institute. In the context of near-atomic-resolution SPA, we need to cope with hundreds or thousands of noisy projections of macromolecules onto different micrographs. Moreover, each projection has an unknown orientation and is blurred by some space-dependent point-spread function of the microscope. Consequently, the determination of the structure of a macromolecule involves not only a reconstruction task, but also the deconvolution of each projection image. We formulate this problem as a constrained regularized reconstruction. We are able to directly include the contrast transfer function in the system matrix without any extra computational cost. The experimental results suggest that our approach brings a significant improvement in the quality of the reconstruction. Our framework also provides an important step toward the application of SPA for the {\textit{de novo}} generation of macromolecular models. The corresponding algorithms have been implemented in Xmipp

Fast multiscale reconstruction for Cryo-EM

We present a multiscale reconstruction framework for single-particle analysis (SPA). The representation of three-dimensional (3D) objects with scaled basis functions permits the reconstruction of volumes at any desired scale in the real-space. This multiscale approach generates interesting opportunities in SPA for the stabilization of the initial volume problem or the 3D iterative refinement procedure. In particular, we show that reconstructions performed at coarse scale are more robust to angular errors and permit gains in computational speed. A key component of the proposed iterative scheme is its fast implementation. The costly step of reconstruction, which was previously hindering the use of advanced iterative methods in SPA, is formulated as a discrete convolution with a cost that does not depend on the number of projection directions. The inclusion of the contrast transfer function inside the imaging matrix is also done at no extra computational cost. By permitting full 3D regularization, the framework is by itself a robust alternative to direct methods for performing reconstruction in adverse imaging conditions (e.g., heavy noise, large angular misassignments, low number of projections). We present reconstructions obtained at different scales from a dataset of the 2015/2016 EMDataBank Map Challenge. The algorithm has been implemented in the Scipion package

Compressed sensing for STEM tomography

A central challenge in scanning transmission electron microscopy (STEM) is to reduce the electron radiation dosage required for accurate imaging of 3D biological nano-structures. Methods that permit tomographic reconstruction from a reduced number of STEM acquisitions without introducing significant degradation in the final volume are thus of particular importance. In random-beam STEM (RB-STEM), the projection measurements are acquired by randomly scanning a subset of pixels at every tilt view. In this work, we present a tailored RB-STEM acquisition-reconstruction framework that fully exploits the compressed sensing principles. We first demonstrate that RB-STEM acquisition fulfills the "incoherence" condition when the image is expressed in terms of wavelets. We then propose a regularized tomographic reconstruction framework to recover volumes from RB-STEM measurements. We demonstrate through simulations on synthetic and real projection measurements that the proposed framework reconstructs high-quality volumes from strongly downsampled RB-STEM data and outperforms existing techniques at doing so. This application of compressed sensing principles to STEM paves the way for a practical implementation of RB-STEM and opens new perspectives for high-quality reconstructions in STEM tomography. (C) 2017 The Authors. Published by Elsevier B.V

Differential Phase-Contrast X-Ray Computed Tomography: From Model Discretization To Image Reconstruction

Our contribution in this paper is two fold. First, we propose a novel discretization of the forward model for differential phase-contrast imaging that uses B-spline basis functions. The approach yields a fast and accurate algorithm for implementing the forward model, which is based on the first derivative of the Radon transform. Second, as an alternative to the FBP-like approaches that are currently used in practice, we present an iterative reconstruction algorithm that remains more faithful to the data when the number of projections dwindles. Since the reconstruction is an ill-posed problem, we impose a total-variation (TV) regularization constraint. We propose to solve the reconstruction problem using the alternating direction method of multipliers (ADMM). A specificity of our system is the use of a preconditioner that improves the convergence rate of the linear solver in ADMM. Our experiments on test data suggest that our method can achieve the same quality as the standard direct reconstruction, while using only one-third of the projection data. We also find that the approach is much faster than the standard algorithms (ISTA and FISTA) that are typically used for solving linear inverse problems subject to the TV regularization constraint

A Box Spline Calculus for the Discretization of Computed Tomography Reconstruction Problems

B-splines are attractive basis functions for the continuous-domain representation of biomedical images and volumes. In this paper, we prove that the extended family of box splines are closed under the Radon transform and derive explicit formulae for their transforms. Our results are general; they cover all known brands of compactly-supported box splines (tensor-product B-splines, separable or not) in any number of dimensions. The proposed box spline approach extends to non-Cartesian lattices used for discretizing the image space. In particular, we prove that the 2-D Radon transform of an N-direction box spline is generally a (nonuniform) polynomial spline of degree N - 1. The proposed framework allows for a proper discretization of a variety of tomographic reconstruction problems in a box spline basis. It is of relevance for imaging modalities such as X-ray computed tomography and cryo-electron microscopy. We provide experimental results that demonstrate the practical advantages of the box spline formulation for improving the quality and efficiency of tomographic reconstruction algorithms

Optimized Kaiser-Bessel Window Functions for Computed Tomography

Kaiser-Bessel window functions are frequently used to discretize tomographic problems because they have two desirable properties: 1) their short support leads to a low computational cost and 2) their rotational symmetry makes their imaging transform independent of the direction. In this paper, we aim at optimizing the parameters of these basis functions. We present a formalism based on the theory of approximation and point out the importance of the partition-of-unity condition. While we prove that, for compact-support functions, this condition is incompatible with isotropy, we show that minimizing the deviation from the partition of unity condition is highly beneficial. The numerical results confirm that the proposed tuning of the Kaiser-Bessel window functions yields the best performance