Compressed sensing using sparse binary measurements: a rateless coding perspective

Compressed Sensing (CS) methods using sparse binary measurement matrices and iterative message-passing re- covery procedures have been recently investigated due to their low computational complexity and excellent performance. Drawing much of inspiration from sparse-graph codes such as Low-Density Parity-Check (LDPC) codes, these studies use analytical tools from modern coding theory to analyze CS solutions. In this paper, we consider and systematically analyze the CS setup inspired by a class of efficient, popular and flexible sparse-graph codes called rateless codes. The proposed rateless CS setup is asymptotically analyzed using tools such as Density Evolution and EXIT charts and fine-tuned using degree distribution optimization techniques

Low-complexity Multiclass Encryption by Compressed Sensing

The idea that compressed sensing may be used to encrypt information from unauthorised receivers has already been envisioned, but never explored in depth since its security may seem compromised by the linearity of its encoding process. In this paper we apply this simple encoding to define a general private-key encryption scheme in which a transmitter distributes the same encoded measurements to receivers of different classes, which are provided partially corrupted encoding matrices and are thus allowed to decode the acquired signal at provably different levels of recovery quality. The security properties of this scheme are thoroughly analysed: firstly, the properties of our multiclass encryption are theoretically investigated by deriving performance bounds on the recovery quality attained by lower-class receivers with respect to high-class ones. Then we perform a statistical analysis of the measurements to show that, although not perfectly secure, compressed sensing grants some level of security that comes at almost-zero cost and thus may benefit resource-limited applications. In addition to this we report some exemplary applications of multiclass encryption by compressed sensing of speech signals, electrocardiographic tracks and images, in which quality degradation is quantified as the impossibility of some feature extraction algorithms to obtain sensitive information from suitably degraded signal recoveries.Comment: IEEE Transactions on Signal Processing, accepted for publication. Article in pres

On Known-Plaintext Attacks to a Compressed Sensing-based Encryption: A Quantitative Analysis

Despite the linearity of its encoding, compressed sensing may be used to provide a limited form of data protection when random encoding matrices are used to produce sets of low-dimensional measurements (ciphertexts). In this paper we quantify by theoretical means the resistance of the least complex form of this kind of encoding against known-plaintext attacks. For both standard compressed sensing with antipodal random matrices and recent multiclass encryption schemes based on it, we show how the number of candidate encoding matrices that match a typical plaintext-ciphertext pair is so large that the search for the true encoding matrix inconclusive. Such results on the practical ineffectiveness of known-plaintext attacks underlie the fact that even closely-related signal recovery under encoding matrix uncertainty is doomed to fail. Practical attacks are then exemplified by applying compressed sensing with antipodal random matrices as a multiclass encryption scheme to signals such as images and electrocardiographic tracks, showing that the extracted information on the true encoding matrix from a plaintext-ciphertext pair leads to no significant signal recovery quality increase. This theoretical and empirical evidence clarifies that, although not perfectly secure, both standard compressed sensing and multiclass encryption schemes feature a noteworthy level of security against known-plaintext attacks, therefore increasing its appeal as a negligible-cost encryption method for resource-limited sensing applications.Comment: IEEE Transactions on Information Forensics and Security, accepted for publication. Article in pres

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