Compressed Sensing: “When sparsity meets sampling”∗

The recent theory of Compressed Sensing (Candès, Tao & Romberg, 2006, and Donoho, 2006) states that a signal, e.g. a sound record or an astronomical image, can be sampled at a rate much smaller than what is commonly prescribed by Shannon-Nyquist. The sampling of a signal can indeed be performed as a function of its “intrinsic dimension” rather than according to its cutoff frequency. This chapter sketches the main theoretical concepts surrounding this revolution in sampling theory. We emphasize also its deep affiliation with the concept of “sparsity”, now ubiquitous in modern signal processing. The end of this chapter explains what interesting effects this theory may have on some Compressive Imaging applications.