Accelerated Proximal Algorithm for Finding the Dantzig Selector and Source Separation Using Dictionary Learning

In most of the applications, signals acquired from different sensors are composite and are corrupted by some noise. In the presence of noise, separation of composite signals into its components without losing information is quite challenging. Separation of signals becomes more difficult when only a few samples of the noisy undersampled composite signals are given. In this paper, we aim to find Dantzig selector with overcomplete dictionaries using Accelerated Proximal Gradient Algorithm (APGA) for recovery and separation of undersampled composite signals. We have successfully diagnosed leukemia disease using our model and compared it with Alternating Direction Method of Multipliers (ADMM). As a test case, we have also recovered Electrocardiogram (ECG) signal with great accuracy from its noisy version using this model along with Proximity Operator based Algorithm (POA) for comparison. With less computational complexity compared with ADMM and POA, APGA has a good clustering capability depicted from the leukemia diagnosis

Compressed Sensing For Functional Magnetic Resonance Imaging Data

This thesis addresses the possibility of applying the compressed sensing (CS) framework to Functional Magnetic Resonance Imaging (fMRI) acquisition. The fMRI is one of the non-invasive neuroimaging technique that allows the brain activity to be captured and analysed in a living body. One disadvantage of fMRI is the trade-off between the spatial and temporal resolution of the data. To keep the experiments within a reasonable length of time, the current acquisition technique sacrifices the spatial resolution in favour of the temporal resolution. It is possible to improve this trade-off using compressed sensing. The main contribution of this thesis is to propose a novel reconstruction method, named Referenced Compressed Sensing, which exploits the redundancy between a signal and a correlated reference by using their distance as an objective function. The compressed video sequences reconstructed using Referenced CS have at least 50% higher in terms of Peak Signal-to-Noise Ratio (PSNR) compared to state-of-the-art conventional reconstruction methods. This thesis also addresses two issues related to Referenced CS. Firstly, the relationship between the reference and the reconstruction performance is studied. To maintain the high-quality references, the Running Gaussian Average (RGA) reference estimator is proposed. The reconstructed results have at least 3dB better PSNR performance with the use of RGA references. Secondly, the Referenced CS with Least Squares is proposed. This study shows that by incorporating the correlated reference, it is possible to perform a linear reconstruction as opposed to the iterative reconstruction commonly used in CS. This approach gives at least 19% improvement in PSNR compared to the state of the art, while reduces the computation time by at most 1200 times. The proposed method is applied to the fMRI data. This study shows that, using the same amount of samples, the data reconstructed using Referenced CS has higher resolution than the conventional acquisition technique and has on average 50% higher PSNR than state-of-the-art reconstructions. Lastly, to enhance the feature of interest in the fMRI data, the baseline independent (BI) analysis is proposed. Using the BI analysis shows up to 25% improvement in the accuracy of the Referenced CS feature

Composite Minimization: Proximity Algorithms and Their Applications

ABSTRACT Image and signal processing problems of practical importance, such as incomplete data recovery and compressed sensing, are often modeled as nonsmooth optimization problems whose objective functions are the sum of two terms, each of which is the composition of a prox-friendly function with a matrix. Therefore, there is a practical need to solve such optimization problems. Besides the nondifferentiability of the objective functions of the associated optimization problems and the larger dimension of the underlying images and signals, the sum of the objective functions is not, in general, prox-friendly, which makes solving the problems challenging. Many algorithms have been proposed in literature to attack these problems by making use of the prox-friendly functions in the problems. However, the efficiency of these algorithms relies heavily on the underlying structures of the matrices, particularly for large scale optimization problems. In this dissertation, we propose a novel algorithmic framework that exploits the availability of the prox-friendly functions, without requiring any structural information of the matrices. This makes our algorithms suitable for large scale optimization problems of interest. We also prove the convergence of the developed algorithms. This dissertation has three main parts. In part 1, we consider the minimization of functions that are the sum of the compositions of prox-friendly functions with matrices. We characterize the solutions to the associated optimization problems as the solutions of fixed point equations that are formulated in terms of the proximity operators of the dual of the prox-friendly functions. By making use of the flexibility provided by this characterization, we develop a block Gauss-Seidel iterative scheme for finding a solution to the optimization problem and prove its convergence. We discuss the connection of our developed algorithms with some existing ones and point out the advantages of our proposed scheme. In part 2, we give a comprehensive study on the computation of the proximity operator of the ℓp-norm with 0 ≤ p \u3c 1. Nonconvexity and non-smoothness have been recognized as important features of many optimization problems in image and signal processing. The nonconvex, nonsmooth ℓp-regularization has been recognized as an efficient tool to identify the sparsity of wavelet coefficients of an image or signal under investigation. To solve an ℓp-regularized optimization problem, the proximity operator of the ℓp-norm needs to be computed in an accurate and computationally efficient way. We first study the general properties of the proximity operator of the ℓp-norm. Then, we derive the explicit form of the proximity operators of the ℓp-norm for p ∈ {0, 1/2, 2/3, 1}. Using these explicit forms and the properties of the proximity operator of the ℓp-norm, we develop an efficient algorithm to compute the proximity operator of the ℓp-norm for any p between 0 and 1. In part 3, the usefulness of the research results developed in the previous two parts is demonstrated in two types of applications, namely, image restoration and compressed sensing. A comparison with the results from some existing algorithms is also presented. For image restoration, the results developed in part 1 are applied to solve the ℓ2-TV and ℓ1-TV models. The resulting restored images have higher peak signal-to-noise ratios and the developed algorithms require less CPU time than state-of-the-art algorithms. In addition, for compressed sensing applications, our algorithm has smaller ℓ2- and ℓ∞-errors and shorter computation times than state-ofthe- art algorithms. For compressed sensing with the ℓp-regularization, our numerical simulations show smaller ℓ2- and ℓ∞-errors than that from the ℓ0-regularization and ℓ1-regularization. In summary, our numerical simulations indicate that not only can our developed algorithms be applied to a wide variety of important optimization problems, but also they are more accurate and computationally efficient than stateof- the-art algorithms

Dynamics and correlations in sparse signal acquisition

One of the most important parts of engineered and biological systems is the ability to acquire and interpret information from the surrounding world accurately and in time-scales relevant to the tasks critical to system performance. This classical concept of efficient signal acquisition has been a cornerstone of signal processing research, spawning traditional sampling theorems (e.g. Shannon-Nyquist sampling), efficient filter designs (e.g. the Parks-McClellan algorithm), novel VLSI chipsets for embedded systems, and optimal tracking algorithms (e.g. Kalman filtering). Traditional techniques have made minimal assumptions on the actual signals that were being measured and interpreted, essentially only assuming a limited bandwidth. While these assumptions have provided the foundational works in signal processing, recently the ability to collect and analyze large datasets have allowed researchers to see that many important signal classes have much more regularity than having finite bandwidth. One of the major advances of modern signal processing is to greatly improve on classical signal processing results by leveraging more specific signal statistics. By assuming even very broad classes of signals, signal acquisition and recovery can be greatly improved in regimes where classical techniques are extremely pessimistic. One of the most successful signal assumptions that has gained popularity in recet hears is notion of sparsity. Under the sparsity assumption, the signal is assumed to be composed of a small number of atomic signals from a potentially large dictionary. This limit in the underlying degrees of freedom (the number of atoms used) as opposed to the ambient dimension of the signal has allowed for improved signal acquisition, in particular when the number of measurements is severely limited. While techniques for leveraging sparsity have been explored extensively in many contexts, typically works in this regime concentrate on exploring static measurement systems which result in static measurements of static signals. Many systems, however, have non-trivial dynamic components, either in the measurement system's operation or in the nature of the signal being observed. Due to the promising prior work leveraging sparsity for signal acquisition and the large number of dynamical systems and signals in many important applications, it is critical to understand whether sparsity assumptions are compatible with dynamical systems. Therefore, this work seeks to understand how dynamics and sparsity can be used jointly in various aspects of signal measurement and inference. Specifically, this work looks at three different ways that dynamical systems and sparsity assumptions can interact. In terms of measurement systems, we analyze a dynamical neural network that accumulates signal information over time. We prove a series of bounds on the length of the input signal that drives the network that can be recovered from the values at the network nodes~[1--9]. We also analyze sparse signals that are generated via a dynamical system (i.e. a series of correlated, temporally ordered, sparse signals). For this class of signals, we present a series of inference algorithms that leverage both dynamics and sparsity information, improving the potential for signal recovery in a host of applications~[10--19]. As an extension of dynamical filtering, we show how these dynamic filtering ideas can be expanded to the broader class of spatially correlated signals. Specifically, explore how sparsity and spatial correlations can improve inference of material distributions and spectral super-resolution in hyperspectral imagery~[20--25]. Finally, we analyze dynamical systems that perform optimization routines for sparsity-based inference. We analyze a networked system driven by a continuous-time differential equation and show that such a system is capable of recovering a large variety of different sparse signal classes~[26--30].Ph.D