Compressive sensing generally relies on the 2 norm for data fidelity, whereas in many applications, robust estimators are needed. Among the scenarios in which robust performance is required, applications where the sampling process is performed in the presence of impulsive noise, i.e., measurements are corrupted by outliers, are of particular importance. This article overviews robust nonlinear reconstruction strategies for sparse signals based on replacing the commonly used 2 norm by M-estimators as data fidelity functions. The derived methods outperform existing compressed sensing techniques in impulsive environments, while achieving good performance in light-tailed environments, thus offering a robust framework for CS.

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Ultrasound imaging is used for detecting and characterizing breast lesions. A state of the art imaging method is the contrast source inversion (CSI), which solves the full wave nonlinear inverse problem. However, when the measurements are acquired in noisy environments, CSI can diverge from the correct solution after several iterations. Problems associated with noisy data were originally solved by including total variation (TV) regularization. Unfortunately, for very noisy data, TV regularization alone is not sufficient. In this work, compressed sensing ideas are used to regularize the inversion process by restricting the solution of the CSI method to be sparse in a transformation domain. The proposed method estimates the contrast source and contrast function by minimizing the mean squared error between the measured and modeled data. An extra penalty term is added to measure sparsity in the transformation domain. A second method that combines sparsity of the contrast source and minimal TV in the contrast function is also presented. The proposed methods are tested on noise-free and noisy synthetic data sets representing a scan of a cancerous breast. Numerical experiments show that, for measurements contaminated with 1% noise, the sparsity constrained CSI improves the normalized mean squared error of the reconstructed speed-of-sound profiles up to 36% in comparison with traditional CSI. Also, for measurements contaminated with 5% noise, the proposed methods improve the quality of the reconstruction up to 70% in comparison with the traditional CSI method. Experimental results also show that the methods remain convergent to the correct speed-of-sound profile as the number of iterations increases.

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Abstract-The design of ultrasound scanning systems for applications such as breast cancer detection is a challenging task, especially when the number of sources and receivers increases and they become spread over a large surface. In order to determine the number of transducers and above all their position, several approaches could be followed. A simple and straightforward approach is to compute the energy distribution in the region of interest for several configurations, and treat each receiver as if it is a source. Here the underlying assumption is that the number of A-scans, i.e. number of sources times receivers, determines the image quality. This assumption is mainly based on reciprocity; the observation that the response RAB measured by a receiver located at B and a source at A is identical to the response RBA obtained after interchanging the source and receiver locations. This is valid for linear imaging methods. However, the question arises whether this is also the case for nonlinear inversion methods. In this work, we evaluate the SAFT, CG and CSI imaging and inversion methods for different configurations of sources and receivers located uniformly distributed over a circular array, surrounding an heterogeneous medium. The obtained results show that with the linear SAFT and CG methods, the sources and receivers can be interchanged, and the resulting images are the same. However, for the non-linear CSI method, the results show that it is preferred to have more receivers than sources, despite the fact the measured signals are reciprocal.