Video Compressive Sensing for Dynamic MRI

We present a video compressive sensing framework, termed kt-CSLDS, to accelerate the image acquisition process of dynamic magnetic resonance imaging (MRI). We are inspired by a state-of-the-art model for video compressive sensing that utilizes a linear dynamical system (LDS) to model the motion manifold. Given compressive measurements, the state sequence of an LDS can be first estimated using system identification techniques. We then reconstruct the observation matrix using a joint structured sparsity assumption. In particular, we minimize an objective function with a mixture of wavelet sparsity and joint sparsity within the observation matrix. We derive an efficient convex optimization algorithm through alternating direction method of multipliers (ADMM), and provide a theoretical guarantee for global convergence. We demonstrate the performance of our approach for video compressive sensing, in terms of reconstruction accuracy. We also investigate the impact of various sampling strategies. We apply this framework to accelerate the acquisition process of dynamic MRI and show it achieves the best reconstruction accuracy with the least computational time compared with existing algorithms in the literature.Comment: 30 pages, 9 figure

Applications Of Compressive Sensing To Surveillance Problems

In many surveillance scenarios, one concern that arises is how to construct an imager that is capable of capturing the scene with high fidelity. This could be problematic for two reasons: first, the optics and electronics in the camera may have difficulty in dealing with so much information; secondly, bandwidth constraints, may pose difficulty in transmitting information from the imager to the user efficiently for reconstruction or realization. In this thesis, we will discuss a mathematical framework that is capable of skirting the two aforementioned issues. This framework is rooted in a technique commonly referred to as compressive sensing. We will explore two of the seminal works in compressive sensing and will present the key theorems and definitions from these two papers. We will then survey three different surveillance scenarios and their respective compressive sensing solutions. The original contribution of this thesis is the development of a distributed compressive sensing model

Compressed Sensing in Multi-Signal Environments.

Technological advances and the ability to build cheap high performance sensors make it possible to deploy tens or even hundreds of sensors to acquire information about a common phenomenon of interest. The increasing number of sensors allows us to acquire ever more detailed information about the underlying scene that was not possible before. This, however, directly translates to increasing amounts of data that needs to be acquired, transmitted, and processed. The amount of data can be overwhelming, especially in applications that involve high-resolution signals such as images or videos. Compressed sensing (CS) is a novel acquisition and reconstruction scheme that is particularly useful in scenarios when high resolution signals are difficult or expensive to encode. When applying CS in a multi-signal scenario, there are several aspects that need to be considered such as the sensing matrix, the joint signal model, and the reconstruction algorithm. The purpose of this dissertation is to provide a complete treatment of these aspects in various multi-signal environments. Specific applications include video, multi-view imaging, and structural health monitoring systems. For each application, we propose a novel joint signal model that accurately captures the joint signal structure, and we tailor the reconstruction algorithm to each signal model to successfully recover the signals of interest.PHDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/98007/1/jaeypark_1.pd

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

Structured Sub-Nyquist Sampling with Applications in Compressive Toeplitz Covariance Estimation, Super-Resolution and Phase Retrieval

Sub-Nyquist sampling has received a huge amount of interest in the past decade. In classical compressed sensing theory, if the measurement procedure satisfies a particular condition known as Restricted Isometry Property (RIP), we can achieve stable recovery of signals of low-dimensional intrinsic structures with an order-wise optimal sample size. Such low-dimensional structures include sparse and low rank for both vector and matrix cases. The main drawback of conventional compressed sensing theory is that random measurements are required to ensure the RIP property. However, in many applications such as imaging and array signal processing, applying independent random measurements may not be practical as the systems are deterministic. Moreover, random measurements based compressed sensing always exploits convex programs for signal recovery even in the noiseless case, and solving those programs is computationally intensive if the ambient dimension is large, especially in the matrix case. The main contribution of this dissertation is that we propose a deterministic sub-Nyquist sampling framework for compressing the structured signal and come up with computationally efficient algorithms. Besides widely studied sparse and low-rank structures, we particularly focus on the cases that the signals of interest are stationary or the measurements are of Fourier type. The key difference between our work from classical compressed sensing theory is that we explicitly exploit the second-order statistics of the signals, and study the equivalent quadratic measurement model in the correlation domain. The essential observation made in this dissertation is that a difference/sum coarray structure will arise from the quadratic model if the measurements are of Fourier type. With these observations, we are able to achieve a better compression rate for covariance estimation, identify more sources in array signal processing or recover the signals of larger sparsity. In this dissertation, we will first study the problem of Toeplitz covariance estimation. In particular, we will show how to achieve an order-wise optimal compression rate using the idea of sparse arrays in both general and low-rank cases. Then, an analysis framework of super-resolution with positivity constraint is established. We will present fundamental robustness guarantees, efficient algorithms and applications in practices. Next, we will study the problem of phase-retrieval for which we successfully apply the sparse array ideas by fully exploiting the quadratic measurement model. We achieve near-optimal sample complexity for both sparse and general cases with practical Fourier measurements and provide efficient and deterministic recovery algorithms. In the end, we will further elaborate on the essential role of non-negative constraint in underdetermined inverse problems. In particular, we will analyze the nonlinear co-array interpolation problem and develop a universal upper bound of the interpolation error. Bilinear problem with non-negative constraint will be considered next and the exact characterization of the ambiguous solutions will be established for the first time in literature. At last, we will show how to apply the nested array idea to solve real problems such as Kriging. Using spatial correlation information, we are able to have a stable estimate of the field of interest with fewer sensors than classic methodologies. Extensive numerical experiments are implemented to demonstrate our theoretical claims

A Comprehensive Review of Distributed Coding Algorithms for Visual Sensor Network (VSN)

Since the invention of low cost camera, it has been widely incorporated into the sensor node in Wireless Sensor Network (WSN) to form the Visual Sensor Network (VSN). However, the use of camera is bringing with it a set of new challenges, because all the sensor nodes are powered by batteries. Hence, energy consumption is one of the most critical issues that have to be taken into consideration. In addition to this, the use of batteries has also limited the resources (memory, processor) that can be incorporated into the sensor node. The life time of a VSN decreases quickly as the image is transferred to the destination. One of the solutions to the aforementioned problem is to reduce the data to be transferred in the network by using image compression. In this paper, a comprehensive survey and analysis of distributed coding algorithms that can be used to encode images in VSN is provided. This also includes an overview of these algorithms, together with their advantages and deficiencies when implemented in VSN. These algorithms are then compared at the end to determine the algorithm that is more suitable for VSN