Bayesian Compressive Sensing

Abstract

The data of interest are assumed to be represented as -dimensional real vectors, and these vectors are compressible in some linear basis B, implying that the signal can be reconstructed accurately using only a small number of basis-function coefficients associated with B. Compressive sensing is a framework whereby one does not measure one of the aforementioned -dimensional signals directly, but rather a set of related measurements, with the new measurements a linear combination of the original underlying -dimensional signal. The number of required compressive-sensing measurements is typically much smaller than , offering the potential to simplify the sensing system. Let denote the unknown underlying -dimensional signal, and a vector of compressive-sensing measurements, then one may approximate accurately by utilizing knowledge of the (under-determined) linear relationship between and , in addition to knowledge of the fact that is compressible in B. In this paper we employ a Bayesian formalism for estimating the underlying signal based on compressive-sensing measurements . The proposed framework has the following properties: i) in addition to estimating the underlying signal , "error bars" are also estimated, these giving a measure of confidence in the inverted signal; ii) using knowledge of the error bars, a principled means is provided for determining when a sufficient number of compressive-sensing measurements have been performed; iii) this setting lends itself naturally to a framework whereby the compressive sensing measurements are optimized adaptively and hence not determined randomly; and iv) the framework accounts for additive noise in the compressive-sensing measurements and provides an estimate of the noise variance. In this paper we present the underlying theory, an associated algorithm, example results, and provide comparisons to other compressive-sensing inversion algorithms in the literature.Index Terms-Adaptive compressive sensing, Bayesian model selection, compressive sensing (CS), experimental design, relevance vector machine (RVM), sparse Bayesian learning.

Bayesian compressive sensing for cluster structured sparse signals

“…Then the following experiments are using the same settings, where the model hyperparameters for the priors in CluSS are set as Then several experiments considered widely in CS literatures are implemented via CluSS, and comparisons are made to the state-of-the-art CS algorithms, respectively, Basis Pursuit (BP) [31], CoSaMP [32], Block-CoSaMP [10], (K, S)-sparse recovery algorithm via Dynamic Programming (Block-DP) [12] and Bayesian Compressive Sensing (BCS) [21]. Without special explanation, the sensing matrix Φ is constructed randomly as in the seminal work [2], i.e., entries are drawn independently from Gaussian distribution…”

“…Unlike the model expressed in [21], the cluster prior is considered in the proposed model via the pattern selection procedure. Meanwhile, it is different from the Ising model expressed in [11], where Markov Random Field is considered and there is no explicit overall prior on sparse coefficients.…”

“…It is usually called Latent Variable Analysis (LVA) [17], and possibly, results in some non-parametric approaches to Bayesian estimators. Exploiting sparsity probabilistic model [18], many algorithms based on the LVA are proposed to solve the sparse decomposition problems [19,11,20,21,22,14]. Moreover, the structures of the sparse coefficients can be conveniently introduced into the LVA framework using Graphical Models [18,11,14].…”

“…LaMP which exploits the MP procedure, the proposed algorithm is based on the Bayesian CS framework, where "deleting" process is also carried out during basis selecting iterations (MP does not have) and thus will not suffer worse case when selecting a wrong basis [21].…”

Adaptive Ratio for the Compressed Sensing of Multi-view Videos

“…For example, the Bayesian CS is proposed on the optimal measurement matrix [4]. Although it is possible to achieve the optimal adaptive measurement, but it requires complicated iterative process to calculate the optimal measurement matrix and is too time-consuming to be used in real-time application.…”