In recent years, interest has grown in the study of sparse solutions to underdetermined systems of linear equations because of their many potential applications. In particular, these types of solutions can be used to describe images in a compact form, provided one is willing to accept an imperfect representation. We shall develop this approach in the context of sampling theory, and for problems in image compression. We use various error estimation criteria - PSNR, SSIM, and MSSIM - to conduct a presentation that is phenomenological and computational, as opposed to theoretical. This machinery leads naturally to a compressed sensing problem that can be seen as a non-uniform sampling reconstruction problem with promising applications. Joint work with John J. Benedetto (University of Maryland, College Park)