Topics in Compressed Sensing

Compressed sensing has a wide range of applications that include error correction, imaging, radar and many more. Given a sparse signal in a high dimensional space, one wishes to reconstruct that signal accurately and efficiently from a number of linear measurements much less than its actual dimension. Although in theory it is clear that this is possible, the difficulty lies in the construction of algorithms that perform the recovery efficiently, as well as determining which kind of linear measurements allow for the reconstruction. There have been two distinct major approaches to sparse recovery that each present different benefits and shortcomings. The first, L1-minimization methods such as Basis Pursuit, use a linear optimization problem to recover the signal. This method provides strong guarantees and stability, but relies on Linear Programming, whose methods do not yet have strong polynomially bounded runtimes. The second approach uses greedy methods that compute the support of the signal iteratively. These methods are usually much faster than Basis Pursuit, but until recently had not been able to provide the same guarantees. This gap between the two approaches was bridged when we developed and analyzed the greedy algorithm Regularized Orthogonal Matching Pursuit (ROMP). ROMP provides similar guarantees to Basis Pursuit as well as the speed of a greedy algorithm. Our more recent algorithm Compressive Sampling Matching Pursuit (CoSaMP) improves upon these guarantees, and is optimal in every important aspect

Structured sampling and fast reconstruction of smooth graph signals

This work concerns sampling of smooth signals on arbitrary graphs. We first study a structured sampling strategy for such smooth graph signals that consists of a random selection of few pre-defined groups of nodes. The number of groups to sample to stably embed the set of $k$-bandlimited signals is driven by a quantity called the \emph{group} graph cumulative coherence. For some optimised sampling distributions, we show that sampling $O(k\log(k))$ groups is always sufficient to stably embed the set of $k$-bandlimited signals but that this number can be smaller -- down to $O(\log(k))$ -- depending on the structure of the groups of nodes. Fast methods to approximate these sampling distributions are detailed. Second, we consider $k$-bandlimited signals that are nearly piecewise constant over pre-defined groups of nodes. We show that it is possible to speed up the reconstruction of such signals by reducing drastically the dimension of the vectors to reconstruct. When combined with the proposed structured sampling procedure, we prove that the method provides stable and accurate reconstruction of the original signal. Finally, we present numerical experiments that illustrate our theoretical results and, as an example, show how to combine these methods for interactive object segmentation in an image using superpixels

Radar Signal Recovery using Compressive Sampling Matching Pursuit Algorithm

In this study, we propose compressive sampling matching pursuit (CoSaMP) algorithm for sub-Nyquist based electronic warfare (EW) receiver system. In compressed sensing (CS) theory time-frequency plane localisation and discretisation into a N×N grid in union of subspaces is established. The train of radar signals are sparse in time and frequency can be under sampled with almost no information loss. The CS theory may be applied to EW digital receivers to reduce sampling rate of analog to digital converter; to improve radar parameter resolution and increase input bandwidth. Simulated an efficient approach for radar signal recovery by CoSaMP algorithm by using a set of various sample and different sparsity level with various radar signals. This approach allows a scalable and flexible recovery process. The method has been satisfied with data in a wide frequency range up to 40 GHz. The simulation shows the feasibility of our method

Methods for Quantized Compressed Sensing

In this paper, we compare and catalog the performance of various greedy quantized compressed sensing algorithms that reconstruct sparse signals from quantized compressed measurements. We also introduce two new greedy approaches for reconstruction: Quantized Compressed Sampling Matching Pursuit (QCoSaMP) and Adaptive Outlier Pursuit for Quantized Iterative Hard Thresholding (AOP-QIHT). We compare the performance of greedy quantized compressed sensing algorithms for a given bit-depth, sparsity, and noise level

One-Bit Compressive Sensing with Partial Support Information

This work develops novel algorithms for incorporating prior-support information into the field of One-Bit Compressed Sensing. Traditionally, Compressed Sensing is used for acquiring high-dimensional signals from few linear measurements. In applications, it is often the case that we have some knowledge of the structure of our signal(s) beforehand, and thus we would like to leverage it to attain more accurate and efficient recovery. Additionally, the Compressive Sensing framework maintains relevance even when the available measurements are subject to extreme quantization. Indeed, the field of One-Bit Compressive Sensing aims to recover a signal from measurements reduced to only their sign-bit. This work explores avenues for incorporating partial-support information into existing One-Bit Compressive Sensing algorithms. We provide both a rich background to the field of compressed sensing and in particular the one-bit framework, while also developing and testing new algorithms for this setting. Experimental results demonstrate that newly proposed methods of this work yield improved signal recovery even for varying levels of accuracy in the prior information. This work is thus the first to provide recovery mechanisms that efficiently use prior signal information in the one-bit reconstruction setting

Topics in Matrix Sampling Algorithms

We study three fundamental problems of Linear Algebra, lying in the heart of various Machine Learning applications, namely: 1)"Low-rank Column-based Matrix Approximation". We are given a matrix A and a target rank k. The goal is to select a subset of columns of A and, by using only these columns, compute a rank k approximation to A that is as good as the rank k approximation that would have been obtained by using all the columns; 2) "Coreset Construction in Least-Squares Regression". We are given a matrix A and a vector b. Consider the (over-constrained) least-squares problem of minimizing ||Ax-b||, over all vectors x in D. The domain D represents the constraints on the solution and can be arbitrary. The goal is to select a subset of the rows of A and b and, by using only these rows, find a solution vector that is as good as the solution vector that would have been obtained by using all the rows; 3) "Feature Selection in K-means Clustering". We are given a set of points described with respect to a large number of features. The goal is to select a subset of the features and, by using only this subset, obtain a k-partition of the points that is as good as the partition that would have been obtained by using all the features. We present novel algorithms for all three problems mentioned above. Our results can be viewed as follow-up research to a line of work known as "Matrix Sampling Algorithms". [Frieze, Kanna, Vempala, 1998] presented the first such algorithm for the Low-rank Matrix Approximation problem. Since then, such algorithms have been developed for several other problems, e.g. Graph Sparsification and Linear Equation Solving. Our contributions to this line of research are: (i) improved algorithms for Low-rank Matrix Approximation and Regression (ii) algorithms for a new problem domain (K-means Clustering).Comment: PhD Thesis, 150 page

Improved compressed sensing algorithm for sparse-view CT

In computed tomography (CT) there are many situations where reconstruction may need to be performed with sparse-view data. In sparse-view CT imaging, strong streak artifacts may appear in conventionally reconstructed images due to the limited sampling rate, compromising image quality. Compressed sensing (CS) algorithm has shown potential to accurately recover images from highly undersampled data. In the past few years, total variation (TV)-base compressed sensing algorithms have been proposed to suppress the streak artifact in CT image reconstruction. In this paper, we formulate the problem of CT imaging under transform sparsity and sparse-view constraints, and propose a novel compressed sensing-based algorithm for CT image reconstruction from few-view data, in which we simultaneously minimize the ℓ1 norm, total variation and a least square measure. The main feature of our algorithm is the use of two sparsity transforms: discrete wavelet transform and discrete gradient transform, both of which are proven to be powerful sparsity transforms. Experiments with simulated and real projections were performed to evaluate and validate the proposed algorithm. The reconstructions using the proposed approach have less streak artifacts and reconstruction errors than other conventional methods

Recognition of the Numbers in the Polish Language, Journal of Telecommunications and Information Technology, 2013, nr 4

Automatic Speech Recognition is one of the hottest research and application problems in today’s ICT technologies. Huge progress in the development of the intelligent mobile systems needs an implementation of the new services, where users can communicate with devices by sending audio commands. Those systems must be additionally integrated with the highly distributed infrastructures such as computational and mobile clouds, Wireless Sensor Networks (WSNs), and many others. This paper presents the recent research results for the recognition of the separate words and words in short contexts (limited to the numbers) articulated in the Polish language. Compressed Sensing Theory (CST) is applied for the first time as a methodology of speech recognition. The effectiveness of the proposed methodology is justified in numerical tests for both separate words and short sentences

Detección y análisis armónico en señales eléctricas usando sensado comprimido para evaluación de la calidad de energía

The measurement and analysis of harmonics are key parts of the evaluation of the quality of the energy; however, measuring the harmonics under the large-scale distributed measurement architecture faces the problem of obtaining measurements outside the sequence that brings latency into communication and data fusion. For this reason, this article proposes as a solution to the problems of measurement and improvement of the resolution of harmonic signals, the compressed sensing (SC) as a technique for the recovery and estimation of signals, a technique that reduces the length of harmonic sampling and complexity of the data procedure compared to other theories. For this, we use restoration algorithms such as least squares (LQ), Basic Pursuit (BP), Orthogonal Matching Pursuit (OMP), performing experiments on three signals with different degree of total harmonic distortion (THD), obtaining as results the error of reconstruction in each algorithm to find the minimum percentage of compression in the recovery of harmonic signals. Finally, the results of the experiment show the accuracy of the detection and the system response can be improved without the need to increase the sampling points, showing the variation of the error as a function of the percentage of compression.La medición y el análisis de armónicos son partes clave de la evaluación de la calidad de la energía; sin embargo, el medir armónicos bajo la arquitectura de medición distribuida a gran escala se enfrenta al problema de obtener mediciones fuera de secuencia que trae latencia en la comunicación y la fusión de datos. Por esta razón, en este artículo se propone como solución a los problemas de medición y mejora de la resolución de las señales armónicas el sensado comprimido (SC) como técnica para la recuperación y estimación de señales, técnica que permite disminuir la longitud de muestreo armónico y complejidad del procedimiento de datos en comparación con otras teorías. Para esto, se utilizan algoritmos de restauración como el least squares (LQ), Basic Pursuit (BP), Ortogonal Matching Pursuit (OMP), realizándose experimentos en tres señales con diferente grado de distorsión armónica total (THD), obteniendo como resultados el error de reconstrucción en cada algoritmo para encontrar el porcentaje mínimo de compresión en la recuperación de señales armónicas. Finalmente, los resultados del experimento muestran la precisión de detección y que la respuesta del sistema se puede mejorar sin la necesidad de aumentar los puntos de muestreo, mostrando la variación del error en función del porcentaje de compresión