Discrete and Continuous-time Soft-Thresholding with Dynamic Inputs

There exist many well-established techniques to recover sparse signals from compressed measurements with known performance guarantees in the static case. However, only a few methods have been proposed to tackle the recovery of time-varying signals, and even fewer benefit from a theoretical analysis. In this paper, we study the capacity of the Iterative Soft-Thresholding Algorithm (ISTA) and its continuous-time analogue the Locally Competitive Algorithm (LCA) to perform this tracking in real time. ISTA is a well-known digital solver for static sparse recovery, whose iteration is a first-order discretization of the LCA differential equation. Our analysis shows that the outputs of both algorithms can track a time-varying signal while compressed measurements are streaming, even when no convergence criterion is imposed at each time step. The L2-distance between the target signal and the outputs of both discrete- and continuous-time solvers is shown to decay to a bound that is essentially optimal. Our analyses is supported by simulations on both synthetic and real data.Comment: 18 pages, 7 figures, journa

Adaptive-Rate Compressive Sensing Using Side Information

We provide two novel adaptive-rate compressive sensing (CS) strategies for sparse, time-varying signals using side information. Our first method utilizes extra cross-validation measurements, and the second one exploits extra low-resolution measurements. Unlike the majority of current CS techniques, we do not assume that we know an upper bound on the number of significant coefficients that comprise the images in the video sequence. Instead, we use the side information to predict the number of significant coefficients in the signal at the next time instant. For each image in the video sequence, our techniques specify a fixed number of spatially-multiplexed CS measurements to acquire, and adjust this quantity from image to image. Our strategies are developed in the specific context of background subtraction for surveillance video, and we experimentally validate the proposed methods on real video sequences

Reconstrução de sinais por Compressive Sensing dinâmico e filtragem de Kalman com estudo de caso em eletrocardiografia

Trabalho de Conclusão de Curso (graduação)—Universidade de Brasília, Faculdade UnB Gama, 2019.A aquisição de sinais digitais com uma quantidade reduzida de medidas é possibilitada por Compressive Sensing (CS). O Filtro de Kalman Adaptativo Baseado em CS é um exemplo de algoritmo que foi elaborado no contexto de streaming. Seu processo de reconstrução considera que os sinais são recebidos de forma contínua e realiza estimativas de suporte para melhorar seus resultados. Entretanto, seu funcionamento foi observado apenas para sinais simulados e esparsos no domínio de Fourier. A aplicação desse algoritmo considerando sinais reais foi investigada no presente trabalho. Para tanto, modificações foram feitas com o objetivo de se obter melhores resultados no cenário específico Para estudo de caso, decidiu-se por adotar sinais de eletrocardiografia. Inicialmente, foram estudadas transformadas esparsificantes para essa nova classe de sinais. Além do domínio de Fourier, foram avaliadas reconstruções utilizando a transformada de Daubechies 4 e uma criada com Análise de Componentes Principais. A observação de resultados parciais permitiram que se propusesse: (i) a atualização iterativa da matriz de covariância do modelo e (ii) modificações na etapa de estimação de suporte. Nas reconstruções, observouse um nível médio de relação sinal ruído de 15, 6 \u1d451\u1d435, porém atingiu-se, nos melhores casos, valores próximos a 40 \u1d451\u1d435.Compressive Sensing (CS) allows a digital signal acquisition with a small amount of measurements. Adaptive Kalman Filter Based on CS is an algorithm created for streaming signals. Its reconstruction approach assumes that the signals are continuously received and support estimations are made to enhance the results. However, its behavior was analyzed only for simulated signals sparse on Fourier domain. The use of this algorithm with real signals was investigated at the present work. Thus, some modifications were made in order to get better results in the new specific scenario. As a case study, electrocardiography signals was chosen. Firstly, sparsifying transforms for the new class of signals were studied. Daubechies 4 transform and one defined by Principal Component Analysis was evaluated, besides the Fourier domain. Partial results enabled us to propose: (i) iterative update of model covariance matrix and (ii) a new method to estimate the support. The reconstructions showed 15, 6 \u1d451\u1d435 as average signal to noise ratio, however the best situations achieved values close to 40 \u1d451\u1d435

Dynamic Compressive Sensing of Time-Varying Signals via Approximate Message Passing

In this work the dynamic compressive sensing (CS) problem of recovering sparse, correlated, time-varying signals from sub-Nyquist, non-adaptive, linear measurements is explored from a Bayesian perspective. While there has been a handful of previously proposed Bayesian dynamic CS algorithms in the literature, the ability to perform inference on high-dimensional problems in a computationally efficient manner remains elusive. In response, we propose a probabilistic dynamic CS signal model that captures both amplitude and support correlation structure, and describe an approximate message passing algorithm that performs soft signal estimation and support detection with a computational complexity that is linear in all problem dimensions. The algorithm, DCS-AMP, can perform either causal filtering or non-causal smoothing, and is capable of learning model parameters adaptively from the data through an expectation-maximization learning procedure. We provide numerical evidence that DCS-AMP performs within 3 dB of oracle bounds on synthetic data under a variety of operating conditions. We further describe the result of applying DCS-AMP to two real dynamic CS datasets, as well as a frequency estimation task, to bolster our claim that DCS-AMP is capable of offering state-of-the-art performance and speed on real-world high-dimensional problems.Comment: 32 pages, 7 figure