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

Deep learning methods for solving linear inverse problems: Research directions and paradigms

The linear inverse problem is fundamental to the development of various scientific areas. Innumerable attempts have been carried out to solve different variants of the linear inverse problem in different applications. Nowadays, the rapid development of deep learning provides a fresh perspective for solving the linear inverse problem, which has various well-designed network architectures results in state-of-the-art performance in many applications. In this paper, we present a comprehensive survey of the recent progress in the development of deep learning for solving various linear inverse problems. We review how deep learning methods are used in solving different linear inverse problems, and explore the structured neural network architectures that incorporate knowledge used in traditional methods. Furthermore, we identify open challenges and potential future directions along this research line

Compressive Sensing with Side Information: Analysis, Measurements Design and Applications

Compressive sensing is a breakthrough technology in view of the fact that it enables the acquisition and reconstruction of certain signals with a number of measurements much lower than those dictated by the Shannon-Nyquist paradigm. It has also been recognised in the last few years that it is possible to improve compressive sensing systems by leveraging additional knowledge – so-called side information – that may be available about the signal of interest. The goal of this thesis is to investigate how to improve the acquisition and reconstruction process in compressive sensing systems in the presence of side information. In particular, by assuming that both the signal of interest and the side information obey a joint Gaussian mixture model (GMM), the thesis focuses on the analysis and the design of linear measurements for two different scenarios: i) the scenario where one wishes to design a linear projection matrix to capture the signal of interest; and ii) the scenario where one wishes to design a linear projection matrix to capture the side information. In both cases, we derive sufficient and (occasionally) necessary conditions on the number of measurements needed for the reliable reconstruction in the low-noise regime and we also derive linear measurement designs that are close to optimal. Numerical results are presented with synthetic data from both Gaussian and GMM distributions and with real world imaging data that confirm that analysis is well aligned with practice. We also showcase our measurement design scheme can lead to significant improvement on the application example associated with the reconstruction of high-resolution RGB images from gray scale images using low-resolution, compressive, hyperspectral measurements as side information

Unsupervised training of deep learning based image denoisers from undersampled measurements

Department of Electrical EngineeringCompressive sensing is a method to recover the original image from undersampled measurements. In order to overcome the ill-posedness of this inverse problem, image priors are used such as sparsity, minimal total-variation, or self-similarity of images. Recently, deep learning based compressive image recovery methods have been proposed and have yielded state-of-the-art performances. They used data-driven approaches instead of hand-crafted image priors to regularize ill-posed inverse problems with undersampled data. Ironically, training deep neural networks (DNNs) for them requires ???clean??? ground truth images, but obtaining the best quality images from undersampled data requires well-trained DNNs. To resolve this dilemma, we propose novel methods based on two well-grounded theories: denoiser-approximate message passing (D-AMP) and Stein???s unbiased risk estimator (SURE). Our proposed methods, LDAMP SURE and LDAMP SURE-T, were able to train deep learning based image denoisers from undersampled measurements without ground truth images and without additional image priors and to recover images with state-of-the-art qualities from undersampled data. We evaluated our methods for various compressive sensing recovery problems with Gaussian random, coded diffraction pattern, and compressive sensing MRI (CS-MRI) measurement matrices. Our proposed methods yielded state-of-the-art performances for all cases without ground truth images. Our methods also yielded comparable performances to the approaches with ground truth data. Moreover, we have extended our methods to deal with a Gaussian noise in a measurement domain and further enhance reconstruction quality by developing an image refining method called LDAMP SURE-FT.clos

Sparse Signal Recovery Based on Compressive Sensing and Exploration Using Multiple Mobile Sensors

The work in this dissertation is focused on two areas within the general discipline of statistical signal processing. First, several new algorithms are developed and exhaustively tested for solving the inverse problem of compressive sensing (CS). CS is a recently developed sub-sampling technique for signal acquisition and reconstruction which is more efficient than the traditional Nyquist sampling method. It provides the possibility of compressed data acquisition approaches to directly acquire just the important information of the signal of interest. Many natural signals are sparse or compressible in some domain such as pixel domain of images, time, frequency and so forth. The notion of compressibility or sparsity here means that many coefficients of the signal of interest are either zero or of low amplitude, in some domain, whereas some are dominating coefficients. Therefore, we may not need to take many direct or indirect samples from the signal or phenomenon to be able to capture the important information of the signal. As a simple example, one can think of a system of linear equations with N unknowns. Traditional methods suggest solving N linearly independent equations to solve for the unknowns. However, if many of the variables are known to be zero or of low amplitude, then intuitively speaking, there will be no need to have N equations. Unfortunately, in many real-world problems, the number of non-zero (effective) variables are unknown. In these cases, CS is capable of solving for the unknowns in an efficient way. In other words, it enables us to collect the important information of the sparse signal with low number of measurements. Then, considering the fact that the signal is sparse, extracting the important information of the signal is the challenge that needs to be addressed. Since most of the existing recovery algorithms in this area need some prior knowledge or parameter tuning, their application to real-world problems to achieve a good performance is difficult. In this dissertation, several new CS algorithms are proposed for the recovery of sparse signals. The proposed algorithms mostly do not require any prior knowledge on the signal or its structure. In fact, these algorithms can learn the underlying structure of the signal based on the collected measurements and successfully reconstruct the signal, with high probability. The other merit of the proposed algorithms is that they are generally flexible in incorporating any prior knowledge on the noise, sparisty level, and so on. The second part of this study is devoted to deployment of mobile sensors in circumstances that the number of sensors to sample the entire region is inadequate. Therefore, where to deploy the sensors, to both explore new regions while refining knowledge in aleady visited areas is of high importance. Here, a new framework is proposed to decide on the trajectories of sensors as they collect the measurements. The proposed framework has two main stages. The first stage performs interpolation/extrapolation to estimate the phenomenon of interest at unseen loactions, and the second stage decides on the informative trajectory based on the collected and estimated data. This framework can be applied to various problems such as tuning the constellation of sensor-bearing satellites, robotics, or any type of adaptive sensor placement/configuration problem. Depending on the problem, some modifications on the constraints in the framework may be needed. As an application side of this work, the proposed framework is applied to a surrogate problem related to the constellation adjustment of sensor-bearing satellites

Compressive sensing for signal ensembles

Compressive sensing (CS) is a new approach to simultaneous sensing and compression that enables a potentially large reduction in the sampling and computation costs for acquisition of signals having a sparse or compressible representation in some basis. The CS literature has focused almost exclusively on problems involving single signals in one or two dimensions. However, many important applications involve distributed networks or arrays of sensors. In other applications, the signal is inherently multidimensional and sensed progressively along a subset of its dimensions; examples include hyperspectral imaging and video acquisition. Initial work proposed joint sparsity models for signal ensembles that exploit both intra- and inter-signal correlation structures. Joint sparsity models enable a reduction in the total number of compressive measurements required by CS through the use of specially tailored recovery algorithms. This thesis reviews several different models for sparsity and compressibility of signal ensembles and multidimensional signals and proposes practical CS measurement schemes for these settings. For joint sparsity models, we evaluate the minimum number of measurements required under a recovery algorithm with combinatorial complexity. We also propose a framework for CS that uses a union-of-subspaces signal model. This framework leverages the structure present in certain sparse signals and can exploit both intra- and inter-signal correlations in signal ensembles. We formulate signal recovery algorithms that employ these new models to enable a reduction in the number of measurements required. Additionally, we propose the use of Kronecker product matrices as sparsity or compressibility bases for signal ensembles and multidimensional signals to jointly model all types of correlation present in the signal when each type of correlation can be expressed using sparsity. We compare the performance of standard global measurement ensembles, which act on all of the signal samples; partitioned measurements, which act on a partition of the signal with a given measurement depending only on a piece of the signal; and Kronecker product measurements, which can be implemented in distributed measurement settings. The Kronecker product formulation in the sparsity and measurement settings enables the derivation of analytical bounds for transform coding compression of signal ensembles and multidimensional signals. We also provide new theoretical results for performance of CS recovery when Kronecker product matrices are used, which in turn motivates new design criteria for distributed CS measurement schemes

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