We develop new discrete uncertainty principles in terms of numerical sparsity, which is a continuous proxy for the 0-norm. Unlike traditional sparsity, the continuity of numerical sparsity naturally accommodates functions which are nearly sparse. After studying these principles and the functions that achieve exact or near equality in them, we identify certain consequences in a number of sparse signal processing applications.

Hyperspectral data is commonly used by astronomers to discern the chemical composition of stars. Unfortunately, conventional hyperspectral platforms require long exposure times, which can hamper their use in applications like celestial navigation. We propose a compressed sensing platform that exploits the spatial sparsity of stars to quickly sample the hyperspectral data. We leverage certain combinatorial designs to devise coded apertures, and then we apply block orthogonal matching pursuit to quickly reconstruct the desired imagery.