Compressed sensing is designed to measure sparse signals directly in a compressed form. However, most signals of interest are only “approximately sparse”, i.e. even though the signal contains only a small fraction of relevant (large) components the other components are not strictly equal to zero, but are only close to zero. In this paper we model the approximately sparse signal with a Gaussian distribution of small components, and we study its compressed sensing with dense random matrices. We use replica calculations to determine the mean-squared error of the Bayes-optimal reconstruction for such signals, as a function of the variance of the small components, the density of large components and the measurement rate. We then use the G-AMP algorithm and we quantify the region of parameters for which this algorithm achieves optimality (for large systems). Finally, we show that in the region where the G-AMP algorithm for the homogeneous measurement matrices is not optimal, a special “seeding” design of a spatially-coupled measurement matrix allows to restore optimality.
The optimal reconstruction was studied extensively in @cite_11 . The replica method was used to analyse the optimal reconstruction in compressed sensing in e.g. @cite_2 @cite_4 . In the statistical physics point of view the replica method is closely related to the state evolution @cite_20 .
@cite_20 @cite_2 @cite_4 @cite_11
2090842051 2571527823 2024251722 2174133987
The replica method is a nonrigorous but well-known technique from statistical physics used in the asymptotic analysis of large, random, nonlinear problems. This paper applies the replica method, under the assumption of replica symmetry, to study estimators that are maximum a posteriori (MAP) under a postulated prior distribution. It is shown that with random linear measurements and Gaussian noise, the replica-symmetric prediction of the asymptotic behavior of the postulated MAP estimate of an -dimensional vector “decouples” as scalar postulated MAP estimators. The result is based on applying a hardening argument to the replica analysis of postulated posterior mean estimators of Tanaka and of Guo and Verdu. The replica-symmetric postulated MAP analysis can be readily applied to many estimators used in compressed sensing, including basis pursuit, least absolute shrinkage and selection operator (LASSO), linear estimation with thresholding, and zero norm-regularized estimation. In the case of LASSO estimation, the scalar estimator reduces to a soft-thresholding operator, and for zero norm-regularized estimation, it reduces to a hard threshold. Among other benefits, the replica method provides a computationally tractable method for precisely predicting various performance metrics including mean-squared error and sparsity pattern recovery probability. We study the compressed sensing reconstruction problem for a broad class of random, band-diagonal sensing matrices. This construction is inspired by the idea of spatial coupling in coding theory. As demonstrated heuristically and numerically by Krzakala [30], message passing algorithms can effectively solve the reconstruction problem for spatially coupled measurements with undersampling rates close to the fraction of nonzero coordinates. We use an approximate message passing (AMP) algorithm and analyze it through the state evolution method. We give a rigorous proof that this approach is successful as soon as the undersampling rate δ exceeds the (upper) Renyi information dimension of the signal, d(pX). More precisely, for a sequence of signals of diverging dimension n whose empirical distribution converges to pX, reconstruction is with high probability successful from d(pX) n+o(n) measurements taken according to a band diagonal matrix. For sparse signals, i.e., sequences of dimension n and k(n) nonzero entries, this implies reconstruction from k(n)+o(n) measurements. For “discrete” signals, i.e., signals whose coordinates take a fixed finite set of values, this implies reconstruction from o(n) measurements. The result is robust with respect to noise, does not apply uniquely to random signals, but requires the knowledge of the empirical distribution of the signal pX. We propose a method for the reconstruction of volumetric fet al MRI from 2D slices, comprising super-resolution reconstruction of the volume interleaved with slice-to-volume registration to correct for the motion. The method incorporates novel intensity matching of acquired 2D slices and robust statistics which completely excludes identified misregistered or corrupted voxels and slices. The reconstruction method is applied to motion-corrupted data simulated from MRI of a preterm neonate, as well as 10 clinically acquired thick-slice fet al MRI scans and three scan-sequence optimized thin-slice fet al datasets. The proposed method produced high quality reconstruction results from all the datasets to which it was applied. Quantitative analysis performed on simulated and clinical data shows that both intensity matching and robust statistics result in statistically significant improvement of super-resolution reconstruction. The proposed novel EM-based robust statistics also improves the reconstruction when compared to previously proposed Huber robust statistics. The best results are obtained when thin-slice data and the correct approximation of the point spread function is used. This paper addresses the need for a comprehensive reconstruction algorithm of 3D fet al MRI, so far lacking in the scientific literature. This paper presents a novel probabilistic foundation for volumetric 3D reconstruction. We formulate the problem as inference in a Markov random field, which accurately captures the dependencies between the occupancy and appearance of each voxel, given all input images. Our main contribution is an approximate highly parallelized discrete-continuous inference algorithm to compute the marginal distributions of each voxel's occupancy and appearance. In contrast to the MAP solution, marginals encode the underlying uncertainty and ambiguity in the reconstruction. Moreover, the proposed algorithm allows for a Bayes optimal prediction with respect to a natural reconstruction loss. We compare our method to two state-of-the-art volumetric reconstruction algorithms on three challenging aerial datasets with LIDAR ground truth. Our experiments demonstrate that the proposed algorithm compares favorably in terms of reconstruction accuracy and the ability to expose reconstruction uncertainty.