A probabilistic compressive sensing framework with applications to ultrasound signal processing.

Highlights • Paper provides a framework for probabilistic compressive sensing. • The paper aims to address two short-comings of the traditional formulations: the ad-hoc manner for tuning sparsity levels in signal reconstructions, and the lack of uncertainty quantification. • The key idea being presented is that the problem of probabilistic output can be addressed by use of the Relevance Vector Machine (RVM). This is sparse Bayesian regression model, for which fast optimisation methods have already been developed. • It is also highlighted that for a probabilistic output to be meaningful, it is better to use data-driven, as opposed to model-based dictionaries, as this captures the covariance structure of original data sets better. • The motivation for this development lies in the compression of ultrasound signals, for engineering applications. Thus, the method is demonstrated, and illustrations are drawn from an ultrasound Non-Destructive-Test (NDT) data set. Abstract The field of Compressive Sensing (CS) has provided algorithms to reconstruct signals from a much lower number of measurements than specified by the Nyquist-Shannon theorem. There are two fundamental concepts underpinning the field of CS. The first is the use of random transformations to project high-dimensional measurements onto a much lower-dimensional domain. The second is the use of sparse regression to reconstruct the original signal. This assumes that a sparse representation exists for this signal in some known domain, manifested by a dictionary. The original formulation for CS specifies the use of an l 1 penalised regression method, the Lasso. Whilst this has worked well in literature, it suffers from two main drawbacks. First, the level of sparsity must be specified by the user, or tuned using sub-optimal approaches. Secondly, and most importantly, the Lasso is not probabilistic; it cannot quantify uncertainty in the signal reconstruction. This paper aims to address these two issues; it presents a framework for performing compressive sensing based on sparse Bayesian learning. Specifically, the proposed framework introduces the use of the Relevance Vector Machine (RVM), an established sparse kernel regression method, as the signal reconstruction step within the standard CS methodology. This framework is developed within the context of ultrasound signal processing in mind, and so examples and results of compression and reconstruction of ultrasound pulses are presented. The dictionary learning strategy is key to the successful application of any CS framework and even more so in the probabilistic setting used here. Therefore, a detailed discussion of this step is also included in the paper. The key contributions of this paper are a framework for a Bayesian approach to compressive sensing which is computationally efficient, alongside a discussion of uncertainty quantification in CS and different strategies for dictionary learning. The methods are demonstrated on an example dataset from collected from an aerospace composite panel. Being able to quantify uncertainty on signal reconstruction reveals that this grows as the level of compression increases. This is key when deciding appropriate compression levels, or whether to trust a reconstructed signal in applications of engineering and scientific interest. [ABSTRACT FROM AUTHOR]