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.

No abstract

No abstract

No abstract

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.