Belief Propagation Reconstruction for Discrete Tomography

International audienceWe consider the reconstruction of a two-dimensional discrete image from a set of tomographic measurements corresponding to the Radon projection. Assuming that the image has a structure where neighbouring pixels have a larger probability to take the same value, we follow a Bayesian approach and introduce a fast message-passing reconstruction algorithm based on belief propagation. For numerical results, we specialize to the case of binary tomography. We test the algorithm on binary synthetic images with different length scales and compare our results against a more usual convex optimization approach. We investigate the reconstruction error as a function of the number of tomographic measurements, corresponding to the number of projection angles. The belief propagation algorithm turns out to be more efficient than the convex-optimization algorithm, both in terms of recovery bounds for noise-free projections, and in terms of reconstruction quality when moderate Gaussian noise is added to the projections

A Novel Convex Relaxation for Non-Binary Discrete Tomography

We present a novel convex relaxation and a corresponding inference algorithm for the non-binary discrete tomography problem, that is, reconstructing discrete-valued images from few linear measurements. In contrast to state of the art approaches that split the problem into a continuous reconstruction problem for the linear measurement constraints and a discrete labeling problem to enforce discrete-valued reconstructions, we propose a joint formulation that addresses both problems simultaneously, resulting in a tighter convex relaxation. For this purpose a constrained graphical model is set up and evaluated using a novel relaxation optimized by dual decomposition. We evaluate our approach experimentally and show superior solutions both mathematically (tighter relaxation) and experimentally in comparison to previously proposed relaxations

Non-convex image reconstruction via Expectation Propagation

Tomographic image reconstruction can be mapped to a problem of finding solutions to a large system of linear equations which maximize a function that includes \textit{a priori} knowledge regarding features of typical images such as smoothness or sharpness. This maximization can be performed with standard local optimization tools when the function is concave, but it is generally intractable for realistic priors, which are non-concave. We introduce a new method to reconstruct images obtained from Radon projections by using Expectation Propagation, which allows us to reframe the problem from an Bayesian inference perspective. We show, by means of extensive simulations, that, compared to state-of-the-art algorithms for this task, Expectation Propagation paired with very simple but non log-concave priors, is often able to reconstruct images up to a smaller error while using a lower amount of information per pixel. We provide estimates for the critical rate of information per pixel above which recovery is error-free by means of simulations on ensembles of phantom and real images.Comment: 12 pages, 6 figure

Regularized Newton Methods for X-ray Phase Contrast and General Imaging Problems

Like many other advanced imaging methods, x-ray phase contrast imaging and tomography require mathematical inversion of the observed data to obtain real-space information. While an accurate forward model describing the generally nonlinear image formation from a given object to the observations is often available, explicit inversion formulas are typically not known. Moreover, the measured data might be insufficient for stable image reconstruction, in which case it has to be complemented by suitable a priori information. In this work, regularized Newton methods are presented as a general framework for the solution of such ill-posed nonlinear imaging problems. For a proof of principle, the approach is applied to x-ray phase contrast imaging in the near-field propagation regime. Simultaneous recovery of the phase- and amplitude from a single near-field diffraction pattern without homogeneity constraints is demonstrated for the first time. The presented methods further permit all-at-once phase contrast tomography, i.e. simultaneous phase retrieval and tomographic inversion. We demonstrate the potential of this approach by three-dimensional imaging of a colloidal crystal at 95 nm isotropic resolution.Comment: (C)2016 Optical Society of America. One print or electronic copy may be made for personal use only. Systematic reproduction and distribution, duplication of any material in this paper for a fee or for commercial purposes, or modifications of the content of this paper are prohibite

Continuous-variable optical quantum state tomography

This review covers latest developments in continuous-variable quantum-state tomography of optical fields and photons, placing a special accent on its practical aspects and applications in quantum information technology. Optical homodyne tomography is reviewed as a method of reconstructing the state of light in a given optical mode. A range of relevant practical topics are discussed, such as state-reconstruction algorithms (with emphasis on the maximum-likelihood technique), the technology of time-domain homodyne detection, mode matching issues, and engineering of complex quantum states of light. The paper also surveys quantum-state tomography for the transverse spatial state (spatial mode) of the field in the special case of fields containing precisely one photon.Comment: Finally, a revision! Comments to lvov(at)ucalgary.ca and raymer(at)uoregon.edu are welcom