2Université de Toulouse, IRIT

Abstract — One of the fundamental theorem in information theory is the so-called sampling theorem also known as Shannon-Nyquist theorem. This theorem aims at giving the minimal frequency needed to sample and reconstruct perfectly an analog band-limited signal. Compressive sensing (or compressed sensing, compressive sampling) or CS in short is a recent theory that allows, if the signal to be reconstructed satisfies a number of conditions, to decrease the amount of data needed to reconstruct the signal. As a result this theory can be used for at least two purposes: i) accelerate the acquisition rate without decreasing the reconstructed signal quality (e.g. in terms of resolution, SNR, contrast …) ii) improve the image quality without increasing the quantity of needed data. Even if medical ultrasound is a domain where several potential applications can be highlighted, the use of this theory in this domain is extremely recent. In this paper we review the basic theory of compressive sensing. Then, a review of the existing CS studies in the field of medical ultrasound is given: reconstruction of sparse scattering maps, pre-beamforming channel data, post-beamforming signals and slow time Doppler data. Finally the open problems and challenges to be tackled in order to make the application of CS to medical US a reality will be given