An Introduction To Compressive Sampling [A sensing/sampling paradigm that goes against the common knowledge in data acquisition]

This article surveys the theory of compressive sampling, also known as compressed sensing or CS, a novel sensing/sampling paradigm that goes against the common wisdom in data acquisition. CS theory asserts that one can recover certain signals and images from far fewer samples or measurements than traditional methods use. To make this possible, CS relies on two principles: sparsity, which pertains to the signals of interest, and incoherence, which pertains to the sensing modality. Our intent in this article is to overview the basic CS theory that emerged in the works [1]–[3], present the key mathematical ideas underlying this theory, and survey a couple of important results in the field. Our goal is to explain CS as plainly as possible, and so our article is mainly of a tutorial nature. One of the charms of this theory is that it draws from various subdisciplines within the applied mathematical sciences, most notably probability theory. In this review, we have decided to highlight this aspect and especially the fact that randomness can — perhaps surprisingly — lead to very effective sensing mechanisms. We will also discuss significant implications, explain why CS is a concrete protocol for sensing and compressing data simultaneously (thus the name), and conclude our tour by reviewing important applications

Structure-Constrained Basis Pursuit for Compressively Sensing Speech

Compressed Sensing (CS) exploits the sparsity of many signals to enable sampling below the Nyquist rate. If the original signal is sufficiently sparse, the Basis Pursuit (BP) algorithm will perfectly reconstruct the original signal. Unfortunately many signals that intuitively appear sparse do not meet the threshold for sufficient sparsity . These signals require so many CS samples for accurate reconstruction that the advantages of CS disappear. This is because Basis Pursuit/Basis Pursuit Denoising only models sparsity. We developed a Structure-Constrained Basis Pursuit that models the structure of somewhat sparse signals as upper and lower bound constraints on the Basis Pursuit Denoising solution. We applied it to speech, which seems sparse but does not compress well with CS, and gained improved quality over Basis Pursuit Denoising. When a single parameter (i.e. the phone) is encoded, Normalized Mean Squared Error (NMSE) decreases by between 16.2% and 1.00% when sampling with CS between 1/10 and 1/2 the Nyquist rate, respectively. When bounds are coded as a sum of Gaussians, NMSE decreases between 28.5% and 21.6% in the same range. SCBP can be applied to any somewhat sparse signal with a predictable structure to enable improved reconstruction quality with the same number of samples

Provably secure and efficient audio compression based on compressive sensing

The advancement of systems with the capacity to compress audio signals and simultaneously secure is a highly attractive research subject. This is because of the need to enhance storage usage and speed up the transmission of data, as well as securing the transmission of sensitive signals over limited and insecure communication channels. Thus, many researchers have studied and produced different systems, either to compress or encrypt audio data using different algorithms and methods, all of which suffer from certain issues including high time consumption or complex calculations. This paper proposes a compressing sensing-based system that compresses audio signals and simultaneously provides an encryption system. The audio signal is segmented into small matrices of samples and then multiplied by a non-square sensing matrix generated by a Gaussian random generator. The reconstruction process is carried out by solving a linear system using the pseudoinverse of Moore-Penrose. The statistical analysis results obtaining from implementing different types and sizes of audio signals prove that the proposed system succeeds in compressing the audio signals with a ratio reaching 28% of real size and reconstructing the signal with a correlation metric between 0.98 and 0.99. It also scores very good results in the normalized mean square error (MSE), peak signal-to-noise ratio metrics (PSNR), and the structural similarity index (SSIM), as well as giving the signal a high level of security

ИНТЕЛЛЕКТУАЛЬНЫЙ числовым программным ДЛЯ MIMD-компьютер

For most scientific and engineering problems simulated on computers the solving of problems of the computational mathematics with approximately given initial data constitutes an intermediate or a final stage. Basic problems of the computational mathematics include the investigating and solving of linear algebraic systems, evaluating of eigenvalues and eigenvectors of matrices, the solving of systems of non-linear equations, numerical integration of initial- value problems for systems of ordinary differential equations.Для більшості наукових та інженерних задач моделювання на ЕОМ рішення задач обчислювальної математики з наближено заданими вихідними даними складає проміжний або остаточний етап. Основні проблеми обчислювальної математики відносяться дослідження і рішення лінійних алгебраїчних систем оцінки власних значень і власних векторів матриць, рішення систем нелінійних рівнянь, чисельного інтегрування початково задач для систем звичайних диференціальних рівнянь.Для большинства научных и инженерных задач моделирования на ЭВМ решение задач вычислительной математики с приближенно заданным исходным данным составляет промежуточный или окончательный этап. Основные проблемы вычислительной математики относятся исследования и решения линейных алгебраических систем оценки собственных значений и собственных векторов матриц, решение систем нелинейных уравнений, численного интегрирования начально задач для систем обыкновенных дифференциальных уравнений