Low Complexity Regularization of Linear Inverse Problems

Inverse problems and regularization theory is a central theme in contemporary signal processing, where the goal is to reconstruct an unknown signal from partial indirect, and possibly noisy, measurements of it. A now standard method for recovering the unknown signal is to solve a convex optimization problem that enforces some prior knowledge about its structure. This has proved efficient in many problems routinely encountered in imaging sciences, statistics and machine learning. This chapter delivers a review of recent advances in the field where the regularization prior promotes solutions conforming to some notion of simplicity/low-complexity. These priors encompass as popular examples sparsity and group sparsity (to capture the compressibility of natural signals and images), total variation and analysis sparsity (to promote piecewise regularity), and low-rank (as natural extension of sparsity to matrix-valued data). Our aim is to provide a unified treatment of all these regularizations under a single umbrella, namely the theory of partial smoothness. This framework is very general and accommodates all low-complexity regularizers just mentioned, as well as many others. Partial smoothness turns out to be the canonical way to encode low-dimensional models that can be linear spaces or more general smooth manifolds. This review is intended to serve as a one stop shop toward the understanding of the theoretical properties of the so-regularized solutions. It covers a large spectrum including: (i) recovery guarantees and stability to noise, both in terms of $\ell^2$-stability and model (manifold) identification; (ii) sensitivity analysis to perturbations of the parameters involved (in particular the observations), with applications to unbiased risk estimation ; (iii) convergence properties of the forward-backward proximal splitting scheme, that is particularly well suited to solve the corresponding large-scale regularized optimization problem

A Compact Formulation for the ℓ2,1\ell_{2,1} Mixed-Norm Minimization Problem

Parameter estimation from multiple measurement vectors (MMVs) is a fundamental problem in many signal processing applications, e.g., spectral analysis and direction-of- arrival estimation. Recently, this problem has been address using prior information in form of a jointly sparse signal structure. A prominent approach for exploiting joint sparsity considers mixed-norm minimization in which, however, the problem size grows with the number of measurements and the desired resolution, respectively. In this work we derive an equivalent, compact reformulation of the $\ell_{2,1}$ mixed-norm minimization problem which provides new insights on the relation between different existing approaches for jointly sparse signal reconstruction. The reformulation builds upon a compact parameterization, which models the row-norms of the sparse signal representation as parameters of interest, resulting in a significant reduction of the MMV problem size. Given the sparse vector of row-norms, the jointly sparse signal can be computed from the MMVs in closed form. For the special case of uniform linear sampling, we present an extension of the compact formulation for gridless parameter estimation by means of semidefinite programming. Furthermore, we derive in this case from our compact problem formulation the exact equivalence between the $\ell_{2,1}$ mixed-norm minimization and the atomic-norm minimization. Additionally, for the case of irregular sampling or a large number of samples, we present a low complexity, grid-based implementation based on the coordinate descent method

Multi-contrast reconstruction with Bayesian compressed sensing

Clinical imaging with structural MRI routinely relies on multiple acquisitions of the same region of interest under several different contrast preparations. This work presents a reconstruction algorithm based on Bayesian compressed sensing to jointly reconstruct a set of images from undersampled k-space data with higher fidelity than when the images are reconstructed either individually or jointly by a previously proposed algorithm, M-FOCUSS. The joint inference problem is formulated in a hierarchical Bayesian setting, wherein solving each of the inverse problems corresponds to finding the parameters (here, image gradient coefficients) associated with each of the images. The variance of image gradients across contrasts for a single volumetric spatial position is a single hyperparameter. All of the images from the same anatomical region, but with different contrast properties, contribute to the estimation of the hyperparameters, and once they are found, the k-space data belonging to each image are used independently to infer the image gradients. Thus, commonality of image spatial structure across contrasts is exploited without the problematic assumption of correlation across contrasts. Examples demonstrate improved reconstruction quality (up to a factor of 4 in root-mean-square error) compared with previous compressed sensing algorithms and show the benefit of joint inversion under a hierarchical Bayesian model

Proceedings of the second "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'14)

The implicit objective of the biennial "international - Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST) is to foster collaboration between international scientific teams by disseminating ideas through both specific oral/poster presentations and free discussions. For its second edition, the iTWIST workshop took place in the medieval and picturesque town of Namur in Belgium, from Wednesday August 27th till Friday August 29th, 2014. The workshop was conveniently located in "The Arsenal" building within walking distance of both hotels and town center. iTWIST'14 has gathered about 70 international participants and has featured 9 invited talks, 10 oral presentations, and 14 posters on the following themes, all related to the theory, application and generalization of the "sparsity paradigm": Sparsity-driven data sensing and processing; Union of low dimensional subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph sensing/processing; Blind inverse problems and dictionary learning; Sparsity and computational neuroscience; Information theory, geometry and randomness; Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?; Sparse machine learning and inference.Comment: 69 pages, 24 extended abstracts, iTWIST'14 website: http://sites.google.com/site/itwist1

Sparse and Redundant Representations for Inverse Problems and Recognition

Sparse and redundant representation of data enables the description of signals as linear combinations of a few atoms from a dictionary. In this dissertation, we study applications of sparse and redundant representations in inverse problems and object recognition. Furthermore, we propose two novel imaging modalities based on the recently introduced theory of Compressed Sensing (CS). This dissertation consists of four major parts. In the first part of the dissertation, we study a new type of deconvolution algorithm that is based on estimating the image from a shearlet decomposition. Shearlets provide a multi-directional and multi-scale decomposition that has been mathematically shown to represent distributed discontinuities such as edges better than traditional wavelets. We develop a deconvolution algorithm that allows for the approximation inversion operator to be controlled on a multi-scale and multi-directional basis. Furthermore, we develop a method for the automatic determination of the threshold values for the noise shrinkage for each scale and direction without explicit knowledge of the noise variance using a generalized cross validation method. In the second part of the dissertation, we study a reconstruction method that recovers highly undersampled images assumed to have a sparse representation in a gradient domain by using partial measurement samples that are collected in the Fourier domain. Our method makes use of a robust generalized Poisson solver that greatly aids in achieving a significantly improved performance over similar proposed methods. We will demonstrate by experiments that this new technique is more flexible to work with either random or restricted sampling scenarios better than its competitors. In the third part of the dissertation, we introduce a novel Synthetic Aperture Radar (SAR) imaging modality which can provide a high resolution map of the spatial distribution of targets and terrain using a significantly reduced number of needed transmitted and/or received electromagnetic waveforms. We demonstrate that this new imaging scheme, requires no new hardware components and allows the aperture to be compressed. Also, it presents many new applications and advantages which include strong resistance to countermesasures and interception, imaging much wider swaths and reduced on-board storage requirements. The last part of the dissertation deals with object recognition based on learning dictionaries for simultaneous sparse signal approximations and feature extraction. A dictionary is learned for each object class based on given training examples which minimize the representation error with a sparseness constraint. A novel test image is then projected onto the span of the atoms in each learned dictionary. The residual vectors along with the coefficients are then used for recognition. Applications to illumination robust face recognition and automatic target recognition are presented

MR Shuffling: Accelerated Single-Scan Multi-Contrast Magnetic Resonance Imaging

Magnetic resonance imaging (MRI) is an attractive medical imaging modality as it is non-invasive and does not involve ionizing radiation. Routine clinical MRI exams obtain MR images corresponding to different soft tissue contrast by performing multiple scans. When two-dimensional (2D) imaging is used, these scans are often repeated in other scanning planes. As a result, the number of scans comprising an MRI exam leads to prohibitively long exam times as compared to other medical imaging modalities such as computed tomography. Many approaches have been designed to accelerate the MRI acquisition while maintaining diagnostic quality.One approach is to collect multiple measurements while the MRI signal is evolving due to relaxation. This enables a reduction in scan time, as fewer acquisition windows are needed to collect the same number of measurements. However, when the temporal aspect of the acquisition is left unmodeled, artifacts are likely to appear in the reconstruction. Most often, these artifacts manifest as image blurring. The effect depends on the acquisition parameters as well as the tissue relaxation itself, resulting in spatially varying blurring. The severity of the artifacts is directly related to the level of acceleration, and thus presents a tradeoff with scan time. The effect is amplified when imaging in three dimensions, severely limiting scan efficiency. Volumetric variants would be used if not for the blurring, as they are able to reconstruct images at isotropic resolution and support mutli-planar reformatting.Another established acceleration technique, called parallel imaging, takes advantage of spatially sensitive receive coil arrays to collect multiple MRI measurements in parallel. Thus, the acquisition is shortened, and the reconstruction uses the spatial sensitivity information to recover the image. More recently, methods have been developed that leverage image structure such as sparsity and low rank to reduce the required number of samples for a well-posed reconstruction. Compressed sensing and its low rank extensions use these concepts to acquire incoherent measurements below the Nyquist rate. These techniques are especially suited to MRI, as incoherent measurements can be easily achieved through pseudo-random under-sampling. As the mechanisms behind parallel imaging and compressed sensing are fundamentally different, they can be combined to achieve even higher acceleration.This dissertation proposes accelerated MRI acquisition and reconstruction techniques that account for the temporal dynamics of the MR signal. The methods build off of parallel imaging and compressed sensing to reduce scan time and flexibly model the temporal relaxation behavior. By randomly shuffling the sampling in the acquisition stage and imposing low rank constraints in the reconstruction stage, intrinsic physical parameters are modeled and their dynamics are recovered as multiple images of varying tissue contrast. Additionally, blurring artifacts are significantly reduced, as the temporal dynamics are accounted for in the reconstruction.This dissertation first introduces T2 Shuffling, a volumetric technique that reduces blurring and reconstructs multiple T2-weighted image contrasts from a single acquisition. The method is integrated into a clinical hospital environment and evaluated on patients. Next, this dissertation develops a fast and distributed reconstruction for T2 Shuffling that achieves clinically relevant processing time latency. Clinical validation results are shown comparing T2 Shuffling as a single-sequence alternative to conventional pediatric knee MRI. Based off the compelling results, a fast targeted knee MRI using T2 Shuffling is implemented, enabling same-day access to MRI at one-third the cost compared to the conventional exam. To date, over 2,400 T2 Shuffling patient scans have been performed.Continuing the theme of accelerated multi-contrast imaging, this dissertation extends the temporal signal model with T1-T2 Shuffling. Building off of T2 Shuffling, the new method additionally samples multiple points along the saturation recovery curve by varying the repetition time durations during the scan. Since the signal dynamics are governed by both T1 recovery and T2 relaxation, the reconstruction captures information about both intrinsic tissue parameters. As a result, multiple target synthetic contrast images are reconstructed, all from a single scan. Approaches for selecting the sequence parameters are provided, and the method is evaluated on in vivo brain imaging of a volunteer.Altogether, these methods comprise the theme of MR Shuffling, and may open new pathways toward fast clinical MRI