Construction of a Large Class of Deterministic Sensing Matrices that Satisfy a Statistical Isometry Property

Compressed Sensing aims to capture attributes of $k$-sparse signals using very few measurements. In the standard Compressed Sensing paradigm, the \m\times \n measurement matrix \A is required to act as a near isometry on the set of all $k$-sparse signals (Restricted Isometry Property or RIP). Although it is known that certain probabilistic processes generate \m \times \n matrices that satisfy RIP with high probability, there is no practical algorithm for verifying whether a given sensing matrix \A has this property, crucial for the feasibility of the standard recovery algorithms. In contrast this paper provides simple criteria that guarantee that a deterministic sensing matrix satisfying these criteria acts as a near isometry on an overwhelming majority of $k$-sparse signals; in particular, most such signals have a unique representation in the measurement domain. Probability still plays a critical role, but it enters the signal model rather than the construction of the sensing matrix. We require the columns of the sensing matrix to form a group under pointwise multiplication. The construction allows recovery methods for which the expected performance is sub-linear in \n, and only quadratic in \m; the focus on expected performance is more typical of mainstream signal processing than the worst-case analysis that prevails in standard Compressed Sensing. Our framework encompasses many families of deterministic sensing matrices, including those formed from discrete chirps, Delsarte-Goethals codes, and extended BCH codes.Comment: 16 Pages, 2 figures, to appear in IEEE Journal of Selected Topics in Signal Processing, the special issue on Compressed Sensin

Empirical recovery performance of fourier-based deterministic compressed sensing

Compressed sensing is a novel technique where one can recover sparse signals from the undersampled measurements. Mathematically, measuring an N-dimensional signal..

Convolutional compressed sensing using deterministic sequences

This is the author's accepted manuscript (with working title "Semi-universal convolutional compressed sensing using (nearly) perfect sequences"). The final published article is available from the link below. Copyright @ 2012 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works.In this paper, a new class of orthogonal circulant matrices built from deterministic sequences is proposed for convolution-based compressed sensing (CS). In contrast to random convolution, the coefficients of the underlying filter are given by the discrete Fourier transform of a deterministic sequence with good autocorrelation. Both uniform recovery and non-uniform recovery of sparse signals are investigated, based on the coherence parameter of the proposed sensing matrices. Many examples of the sequences are investigated, particularly the Frank-Zadoff-Chu (FZC) sequence, the m-sequence and the Golay sequence. A salient feature of the proposed sensing matrices is that they can not only handle sparse signals in the time domain, but also those in the frequency and/or or discrete-cosine transform (DCT) domain

Degraded Visual Environment Tracker

Compressive Sensing (CS) has proven its ability to reduce the number of measurements required to reproduce images with similar quality to those reconstructed by observing the Shannon-Nyquest sampling criteria. By exploiting spatial redundancies, it was shown that CS can be used to denoise and enhance image quality. In this thesis we propose a method that incorporates an efficient use of CS to locate a specific object in zero-visibility environments. This method was developed after multiple implementations of dictionary learning, reconstruction, detection, and tracking algorithms in order to identify the shortcomings of existing techniques and enhance our results. We show that with the use of an over-complete dictionary of the target our technique can perceive the location of the target from hidden information in the scene. This thesis will summarize the previously implemented algorithms, detail the shortcomings evident in their outputs, explain our setups, and present quantified results to support its efficacy in the results section

Global Geometric Conditions on Sensing Matrices for the Success of L1 Minimization Algorithm

Compressed Sensing concerns a new class of linear data acquisition protocols that are more efficient than the classical Shannon sampling theorem when targeting at signals with sparse structures. In this thesis, we study the stability of a Statistical Restricted Isometry Property and show how this property can be further relaxed while maintaining its sufficiency for the Basis Pursuit algorithm to recover sparse signals. We then look at the dictionary extension of Compressed Sensing where signals are sparse under a redundant dictionary and reconstruction is achieved by the $\ell_1$ synthesis method. By establishing a necessary and sufficient condition for the stability of $\ell_1$ synthesis, we are able to predict this algorithm's performances under different dictionaries. Last, we construct a class of deterministic sensing matrix for the Dirac-Fourier joint dictionary

Graph-based techniques for compression and reconstruction of sparse sources

The main goal of this thesis is to develop lossless compression schemes for analog and binary sources. All the considered compression schemes have as common feature that the encoder can be represented by a graph, so they can be studied employing tools from modern coding theory. In particular, this thesis is focused on two compression problems: the group testing and the noiseless compressed sensing problems. Although both problems may seem unrelated, in the thesis they are shown to be very close. Furthermore, group testing has the same mathematical formulation as non-linear binary source compression schemes that use the OR operator. In this thesis, the similarities between these problems are exploited. The group testing problem is aimed at identifying the defective subjects of a population with as few tests as possible. Group testing schemes can be divided into two groups: adaptive and non-adaptive group testing schemes. The former schemes generate tests sequentially and exploit the partial decoding results to attempt to reduce the overall number of tests required to label all members of the population, whereas non-adaptive schemes perform all the test in parallel and attempt to label as many subjects as possible. Our contributions to the group testing problem are both theoretical and practical. We propose a novel adaptive scheme aimed to efficiently perform the testing process. Furthermore, we develop tools to predict the performance of both adaptive and non-adaptive schemes when the number of subjects to be tested is large. These tools allow to characterize the performance of adaptive and non-adaptive group testing schemes without simulating them. The goal of the noiseless compressed sensing problem is to retrieve a signal from its lineal projection version in a lower-dimensional space. This can be done only whenever the amount of null components of the original signal is large enough. Compressed sensing deals with the design of sampling schemes and reconstruction algorithms that manage to reconstruct the original signal vector with as few samples as possible. In this thesis we pose the compressed sensing problem within a probabilistic framework, as opposed to the classical compression sensing formulation. Recent results in the state of the art show that this approach is more efficient than the classical one. Our contributions to noiseless compressed sensing are both theoretical and practical. We deduce a necessary and sufficient matrix design condition to guarantee that the reconstruction is lossless. Regarding the design of practical schemes, we propose two novel reconstruction algorithms based on message passing over the sparse representation of the matrix, one of them with very low computational complexity.El objetivo principal de la tesis es el desarrollo de esquemas de compresión sin pérdidas para fuentes analógicas y binarias. Los esquemas analizados tienen en común la representación del compresor mediante un grafo; esto ha permitido emplear en su estudio las herramientas de codificación modernas. Más concretamente la tesis estudia dos problemas de compresión en particular: el diseño de experimentos de testeo comprimido de poblaciones (de sangre, de presencia de elementos contaminantes, secuenciado de ADN, etcétera) y el muestreo comprimido de señales reales en ausencia de ruido. A pesar de que a primera vista parezcan problemas totalmente diferentes, en la tesis mostramos que están muy relacionados. Adicionalmente, el problema de testeo comprimido de poblaciones tiene una formulación matemática idéntica a los códigos de compresión binarios no lineales basados en puertas OR. En la tesis se explotan las similitudes entre todos estos problemas. Existen dos aproximaciones al testeo de poblaciones: el testeo adaptativo y el no adaptativo. El primero realiza los test de forma secuencial y explota los resultados parciales de estos para intentar reducir el número total de test necesarios, mientras que el segundo hace todos los test en bloque e intenta extraer el máximo de datos posibles de los test. Nuestras contribuciones al problema de testeo comprimido han sido tanto teóricas como prácticas. Hemos propuesto un nuevo esquema adaptativo para realizar eficientemente el proceso de testeo. Además hemos desarrollado herramientas que permiten predecir el comportamiento tanto de los esquemas adaptativos como de los esquemas no adaptativos cuando el número de sujetos a testear es elevado. Estas herramientas permiten anticipar las prestaciones de los esquemas de testeo sin necesidad de simularlos. El objetivo del muestreo comprimido es recuperar una señal a partir de su proyección lineal en un espacio de menor dimensión. Esto sólo es posible si se asume que la señal original tiene muchas componentes que son cero. El problema versa sobre el diseño de matrices y algoritmos de reconstrucción que permitan implementar esquemas de muestreo y reconstrucción con un número mínimo de muestras. A diferencia de la formulación clásica de muestreo comprimido, en esta tesis se ha empleado un modelado probabilístico de la señal. Referencias recientes en la literatura demuestran que este enfoque permite conseguir esquemas de compresión y descompresión más eficientes. Nuestras contribuciones en el campo de muestreo comprimido de fuentes analógicas dispersas han sido también teóricas y prácticas. Por un lado, la deducción de la condición necesaria y suficiente que debe garantizar la matriz de muestreo para garantizar que se puede reconstruir unívocamente la secuencia de fuente. Por otro lado, hemos propuesto dos algoritmos, uno de ellos de baja complejidad computacional, que permiten reconstruir la señal original basados en paso de mensajes entre los nodos de la representación gráfica de la matriz de proyección.Postprint (published version

Structured Compressed Sensing Using Deterministic Sequences

The problem of estimating sparse signals based on incomplete set of noiseless or noisy measurements has been investigated for a long time from different perspec- tives. In this dissertation, after the review of the theory of compressed sensing (CS) and existing structured sensing matrices, a new class of convolutional sensing matri- ces based on deterministic sequences are developed in the first part. The proposed matrices can achieve a near optimal bound with O(K log(N)) measurements for non-uniform recovery. Not only are they able to approximate compressible signals in the time domain, but they can also recover sparse signals in the frequency and discrete cosine transform domain. The candidates of the deterministic sequences include maximum length sequence (or called m-sequence), Golay's complementary sequence and Legendre sequence etc., which will be investigated respectively. In the second part, Golay-paired Hadamard matrices are introduced as structured sensing matrices, which are constructed from the Hadamard matrix, followed by diagonal Golay sequences. The properties and performances are analyzed in the following. Their strong structures ensure special isometry properties, and make them be easier applicable to hardware potentially. Finally, we exploit novel CS principles successfully in a few real applications, including radar imaging and dis- tributed source coding. The performance and the effectiveness of each scenario are verified in both theory and simulations

MIMO Radar Waveform Design and Sparse Reconstruction for Extended Target Detection in Clutter

This dissertation explores the detection and false alarm rate performance of a novel transmit-waveform and receiver filter design algorithm as part of a larger Compressed Sensing (CS) based Multiple Input Multiple Output (MIMO) bistatic radar system amidst clutter. Transmit-waveforms and receiver filters were jointly designed using an algorithm that minimizes the mutual coherence of the combined transmit-waveform, target frequency response, and receiver filter matrix product as a design criterion. This work considered the Probability of Detection (P D) and Probability of False Alarm (P FA) curves relative to a detection threshold, τ th, Receiver Operating Characteristic (ROC), reconstruction error and mutual coherence measures for performance characterization of the design algorithm to detect both known and fluctuating targets and amidst realistic clutter and noise. Furthermore, this work paired the joint waveform-receiver filter design algorithm with multiple sparse reconstruction algorithms, including: Regularized Orthogonal Matching Pursuit (ROMP), Compressive Sampling Matching Pursuit (CoSaMP) and Complex Approximate Message Passing (CAMP) algorithms. It was found that the transmit-waveform and receiver filter design algorithm significantly outperforms statically designed, benchmark waveforms for the detection of both known and fluctuating extended targets across all tested sparse reconstruction algorithms. In particular, CoSaMP was specified to minimize the maximum allowable P FA of the CS radar system as compared to the baseline ROMP sparse reconstruction algorithm of previous work. However, while the designed waveforms do provide performance gains and CoSaMP affords a reduced peak false alarm rate as compared to the previous work, fluctuating target impulse responses and clutter severely hampered CS radar performance when either of these sparse reconstruction techniques were implemented. To improve detection rate and, by extension, ROC performance of the CS radar system under non-ideal conditions, this work implemented the CAMP sparse reconstruction algorithm in the CS radar system. It was found that detection rates vastly improve with the implementation of CAMP, especially in the case of fluctuating target impulse responses amidst clutter or at low receive signal to noise ratios (β n). Furthermore, where previous work considered a τ th=0, the implementation of a variable τ th in this work offered novel trade off between P D and P FA in radar design to the CS radar system. In the simulated radar scene it was found that τ th could be moderately increased retaining the same or similar P D while drastically improving P FA. This suggests that the selection and specification of the sparse reconstruction algorithm and corresponding τ th for this radar system is not trivial. Rather, a tradeoff was noted between P D and P FA based on the choice and parameters of the sparse reconstruction technique and detection threshold, highlighting an engineering trade-space in CS radar system design. Thus, in CS radar system design, the radar designer must carefully choose and specify the sparse reconstruction technique and appropriate detection threshold in addition to transmit-waveforms, receiver filters and building the dictionary of target impulse responses for detection in the radar scene