Intro to theory of incoherent matrices, sparse signals, etc:

Stable Recovery of Sparse Overcomplete Representations in the Presence of Noise, Donoho et al

Introduction to Compressed Sensing, Davenport et al

Getting set up with CVX:

Once you are set up with Matlab, download CVX here. Installation instructions here. Test that it's working here.

Applications of CS / structured signal recovery:

Removing outliers from microarray data: Robust Singular Value Decomposition Analysis of Microarray Data, Liu et al

Low-rank approach to localizing sensors: Semi-Definite Programming Relaxation for Non-Line-of-Sight Localization, Ekambaram et al
(background on the data: Locating the Nodes, Patwari et al)

Undersampling for MRI & angiography: Sparse MRI: The Application of Compressed Sensing for Rapid MR Imaging, Lustig et al

Deblurring & depth estimation using coded aperture in photography: Image and Depth from a Conventional Camera with a Coded Aperture , Levin et al

Separating foreground & background in video: Perceptual computing

Images & the wavelet basis: A Wavelet Tour of Signal Processing, Mallat