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
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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. We introduce a method to infer a variational approximation to the posterior distribution of solutions in computational imaging inverse problems. Machine learning methods applied to computational imaging have proven very successful, but have so far largely focused on retrieving a single optimal solution for a given task. Such retrieval is arguably an incomplete description of the solution space, as in ill-posed inverse problems there may be many similarly likely reconstructions. We minimise an upper bound on the divergence between our approximate distribution and the true intractable posterior, thereby obtaining a probabilistic description of the solution space in imaging inverse problems with empirical prior. We demonstrate the advantage of our technique in quantitative simulations with the CelebA dataset and common image reconstruction tasks. We then apply our method to two of the currently most challenging problems in experimental optics: imaging through highly scattering media and imaging through multi-modal optical fibres. In both settings we report state of the art reconstructions, while providing new capabilities, such as estimation of error-bars and visualisation of multiple likely reconstructions. 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 KrzakalaEtAl , message passing algorithms can effectively solve the reconstruction problem for spatially coupled measurements with undersampling rates close to the fraction of non-zero 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 @math exceeds the (upper) R 'enyi information dimension of the signal, @math . More precisely, for a sequence of signals of diverging dimension @math whose empirical distribution converges to @math , reconstruction is with high probability successful from @math measurements taken according to a band diagonal matrix. For sparse signals, i.e., sequences of dimension @math and @math non-zero entries, this implies reconstruction from @math measurements. For discrete' signals, i.e., signals whose coordinates take a fixed finite set of values, this implies reconstruction from @math 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 @math . Accurate signal recovery or image reconstruction from indirect and possibly undersampled data is a topic of considerable interest; for example, the literature in the recent field of compressed sensing is already quite immense. This paper applies a smoothing technique and an accelerated first-order algorithm, both from Nesterov [Math. Program. Ser. A, 103 (2005), pp. 127-152], and demonstrates that this approach is ideally suited for solving large-scale compressed sensing reconstruction problems as (1) it is computationally efficient, (2) it is accurate and returns solutions with several correct digits, (3) it is flexible and amenable to many kinds of reconstruction problems, and (4) it is robust in the sense that its excellent performance across a wide range of problems does not depend on the fine tuning of several parameters. Comprehensive numerical experiments on realistic signals exhibiting a large dynamic range show that this algorithm compares favorably with recently proposed state-of-the-art methods. We also apply the algorithm to solve other problems for which there are fewer alternatives, such as total-variation minimization and convex programs seeking to minimize the @math norm of @math under constraints, in which @math is not diagonal. The code is available online as a free package in the MATLAB language. We treat the problem of reconstructing a signal from its nonideal samples where the sampling and reconstruction spaces as well as the class of input signals can be arbitrary subspaces of a Hilbert space. Our formulation is general, and includes as special cases reconstruction from finitely many samples as well as uniform-sampling of continuous-time signals, which are not necessarily bandlimited. To obtain a good approximation of the signal in the reconstruction space from its samples, we suggest two design strategies that attempt to minimize the squared-norm error between the signal and its reconstruction. The approaches we propose differ in their assumptions on the input signal: If the signal is known to lie in an appropriately chosen subspace, then we propose a method that achieves the minimal squared error. On the other hand, when the signal is not restricted, we show that the minimal-norm reconstruction cannot generally be obtained. Instead, we suggest minimizing the worst-case squared error between the reconstructed signal, and the best possible (but usually unattainable) approximation of the signal within the reconstruction space. We demonstrate both theoretically and through simulations that the suggested methods can outperform the consistent reconstruction approach previously proposed for this problem. We present a theory, based on statistical mechanics, to evaluate analytically the performance of uncoded, fully synchronous, randomly spread code-division multiple-access (CDMA) multiuser detectors with additive white Gaussian noise (AWGN) channel, under perfect power control, and in the large-system limit. Application of the replica method, a tool developed in the literature of statistical mechanics, allows us to derive analytical expressions for the bit-error rate, as well as the multiuser efficiency, of the individually optimum (IO) and jointly optimum (JO) multiuser detectors over the whole range of noise levels. The information-theoretic capacity of the randomly spread CDMA channel and the performance of decorrelating and linear minimum mean-square error (MMSE) detectors are also derived in the same replica formulation, thereby demonstrating validity of the statistical-mechanical approach. In dynamic computed tomography (CT) reconstruction, the data acquisition speed limits the spatio-temporal resolution. Recently, compressed sensing theory has been instrumental in improving CT reconstruction from far few-view projections. In this paper, we present an adaptive method to train a tensor-based spatio-temporal dictionary for sparse representation of an image sequence during the reconstruction process. The correlations among atoms and across phases are considered to capture the characteristics of an object. The reconstruction problem is solved by the alternating direction method of multipliers. To recover fine or sharp structures such as edges, the nonlocal total variation is incorporated into the algorithmic framework. Preclinical examples including a sheep lung perfusion study and a dynamic mouse cardiac imaging demonstrate that the proposed approach outperforms the vectorized dictionary-based CT reconstruction in the case of few-view reconstruction. We consider the problem of next-best view selection for volumetric reconstruction of an object by a mobile robot equipped with a camera. Based on a probabilistic volumetric map that is built in real time, the robot can quantify the expected information gain from a set of discrete candidate views. We propose and evaluate several formulations to quantify this information gain for the volumetric reconstruction task, including visibility likelihood and the likelihood of seeing new parts of the object. These metrics are combined with the cost of robot movement in utility functions. The next best view is selected by optimizing these functions, aiming to maximize the likelihood of discovering new parts of the object. We evaluate the functions with simulated and real world experiments within a modular software system that is adaptable to other robotic platforms and reconstruction problems. We release our implementation open source. In this paper we generalize to the case of diluted spin models and random combinatorial optimization problems a technique recently introduced by Guerra (cond-mat 0205123) to prove that the replica method generates variational bounds for disordered systems. We analyze a family of models that includes the Viana–Bray model, the diluted p-spin model or random XOR-SAT problem, and the random K-SAT problem, showing that the replica cavity method, at the various levels of approximation, provides systematic schemes to obtain lower bounds of the free-energy at all temperatures and of the ground state energy. In the case of K-SAT and XOR-SAT it thus gives upper bounds of the satisfiability threshold. Our analysis underlines deep connections with the cavity method which are not evident in the long range case. We propose and study a new technique for efficiently acquiring and reconstructing signals based on convolution with a fixed FIR filter having random taps. The method is designed for sparse and compressible signals, i.e., ones that are well approximated by a short linear combination of vectors from an orthonormal basis. Signal reconstruction involves a non-linear Orthogonal Matching Pursuit algorithm that we implement efficiently by exploiting the nonadaptive, time-invariant structure of the measurement process. While simpler and more efficient than other random acquisition techniques like Compressed Sensing, random filtering is sufficiently generic to summarize many types of compressible signals and generalizes to streaming and continuous-time signals. Extensive numerical experiments demonstrate its efficacy for acquiring and reconstructing signals sparse in the time, frequency, and wavelet domains, as well as piecewise smooth signals and Poisson processes. We propose a probabilistic formulation of joint silhouette extraction and 3D reconstruction given a series of calibrated 2D images. Instead of segmenting each image separately in order to construct a 3D surface consistent with the estimated silhouettes, we compute the most probable 3D shape that gives rise to the observed color information. The probabilistic framework, based on Bayesian inference, enables robust 3D reconstruction by optimally taking into account the contribution of all views. We solve the arising maximum a posteriori shape inference in a globally optimal manner by convex relaxation techniques in a spatially continuous representation. For an interactively provided user input in the form of scribbles specifying foreground and background regions, we build corresponding color distributions as multivariate Gaussians and find a volume occupancy that best fits to this data in a variational sense. Compared to classical methods for silhouette-based multiview reconstruction, the proposed approach does not depend on initialization and enjoys significant resilience to violations of the model assumptions due to background clutter, specular reflections, and camera sensor perturbations. In experiments on several real-world data sets, we show that exploiting a silhouette coherency criterion in a multiview setting allows for dramatic improvements of silhouette quality over independent 2D segmentations without any significant increase of computational efforts. This results in more accurate visual hull estimation, needed by a multitude of image-based modeling approaches. We made use of recent advances in parallel computing with a GPU implementation of the proposed method generating reconstructions on volume grids of more than 20 million voxels in up to 4.41 seconds. This paper introduces a framework for the reconstruction of magnetic resonance images in the presence of rigid motion. The rationale behind our proposal is to make use of the partial @math -space information provided by multiple receiver coils in order to estimate the position of the imaged object throughout the shots that contribute to the image. The estimated motion is incorporated into the reconstruction model in an iterative manner to obtain a motion-free image. The method is parameter-free, does not assume any prior model for the image to be reconstructed, avoids blurred images due to resampling, does not make use of external sensors, and does not require modifications in the acquisition sequence. Validation is performed using synthetically corrupted data to study the limits for full motion-recovered reconstruction in terms of the amount of motion, encoding trajectories, number of shots and availability of prior information, and to compare with the state of the art. Quantitative and visual results of its application to a highly challenging volumetric brain imaging cohort of @math neonates are also presented, showing the ability of the proposed reconstruction to generally improve the quality of reconstructed images, as evaluated by both sparsity and gradient entropy based metrics. In sparse approximation theory, the fundamental problem is to reconstruct a signal A∈ℝn from linear measurements 〈Aψi〉 with respect to a dictionary of ψi's. Recently, there is focus on the novel direction of Compressed Sensing [9] where the reconstruction can be done with very few—O(k logn)—linear measurements over a modified dictionary if the signal is compressible, that is, its information is concentrated in k coefficients with the original dictionary. In particular, these results [9, 4, 23] prove that there exists a single O(k logn) ×n measurement matrix such that any such signal can be reconstructed from these measurements, with error at most O(1) times the worst case error for the class of such signals. Compressed sensing has generated tremendous excitement both because of the sophisticated underlying Mathematics and because of its potential applications In this paper, we address outstanding open problems in Compressed Sensing. Our main result is an explicit construction of a non-adaptive measurement matrix and the corresponding reconstruction algorithm so that with a number of measurements polynomial in k, logn, 1 e, we can reconstruct compressible signals. This is the first known polynomial time explicit construction of any such measurement matrix. In addition, our result improves the error guarantee from O(1) to 1 + e and improves the reconstruction time from poly(n) to poly(k logn) Our second result is a randomized construction of O(kpolylog (n)) measurements that work for each signal with high probability and gives per-instance approximation guarantees rather than over the class of all signals. Previous work on Compressed Sensing does not provide such per-instance approximation guarantees; our result improves the best known number of measurements known from prior work in other areas including Learning Theory [20, 21], Streaming algorithms [11, 12, 6] and Complexity Theory [1] for this case Our approach is combinatorial. In particular, we use two parallel sets of group tests, one to filter and the other to certify and estimate; the resulting algorithms are quite simple to implement In this paper, we present a new method to determine the “next best view” (NBV) solution for accurate 3D reconstruction of an object with minimum prior information about the object's geometry. The proposed method determines the best visible surface of unknown objects using an adaptive mean shift algorithm which avoids the inaccessible position. The proposed method automatically generates the 3D model of objects in real time with a minimum number of best visible surface patches while the objects are moving on a turntable. By generating a set of potential next views, the proposed method ensures proper avoidance of unreachable positions. The number of views required to reconstruct a 3D model of objects depends upon their complexity. The proposed method is applicable to all kinds of range sensors and experimental results validate the proposed method for 3D modeling of real objects and prove its robustness. We address the problem of reconstructing a multiband signal from its sub-Nyquist pointwise samples, when the band locations are unknown. Our approach assumes an existing multi-coset sampling. To date, recovery methods for this sampling strategy ensure perfect reconstruction either when the band locations are known, or under strict restrictions on the possible spectral supports. In this paper, only the number of bands and their widths are assumed without any other limitations on the support. We describe how to choose the parameters of the multi-coset sampling so that a unique multiband signal matches the given samples. To recover the signal, the continuous reconstruction is replaced by a single finite-dimensional problem without the need for discretization. The resulting problem is studied within the framework of compressed sensing, and thus can be solved efficiently using known tractable algorithms from this emerging area. We also develop a theoretical lower bound on the average sampling rate required for blind signal reconstruction, which is twice the minimal rate of known-spectrum recovery. Our method ensures perfect reconstruction for a wide class of signals sampled at the minimal rate, and provides a first systematic study of compressed sensing in a truly analog setting. Numerical experiments are presented demonstrating blind sampling and reconstruction with minimal sampling rate. In this article, we consider the problem of animation reconstruction, that is, the reconstruction of shape and motion of a deformable object from dynamic 3D scanner data, without using user-provided template models. Unlike previous work that addressed this problem, we do not rely on locally convergent optimization but present a system that can handle fast motion, temporally disrupted input, and can correctly match objects that disappear for extended time periods in acquisition holes due to occlusion. Our approach is motivated by cartography: We first estimate a few landmark correspondences, which are extended to a dense matching and then used to reconstruct geometry and motion. We propose a number of algorithmic building blocks: a scheme for tracking landmarks in temporally coherent and incoherent data, an algorithm for robust estimation of dense correspondences under topological noise, and the integration of local matching techniques to refine the result. We describe and evaluate the individual components and propose a complete animation reconstruction pipeline based on these ideas. We evaluate our method on a number of standard benchmark datasets and show that we can obtain correct reconstructions in situations where other techniques fail completely or require additional user guidance such as a template model. Abstract We present a generic unsupervised method to increase the discriminative power of image vectors obtained from a broad family of deep neural networks for object retrieval. This goal is accomplished by simultaneously selecting and weighting informative deep convolutional features using the replicator equation, commonly used to capture the essence of selection in evolutionary game theory. The proposed method includes three major steps: First, efficiently detecting features within Regions of Interest (ROIs) using a simple algorithm, as well as trivially collecting a subset of background features. Second, assigning unassigned features by optimizing a standard quadratic problem using the replicator equation. Finally, using the replicator equation again in order to partially address the issue of feature burstiness . We provide theoretical time complexity analysis to show that our method is efficient. Experimental results on several common object retrieval benchmarks using both pre-trained and fine-tuned deep networks show that our method compares favorably to the state-of-the-art. We also publish an easy-to-use Matlab implementation of the proposed method for reproducing our results.