1. A method implemented at least in part by a computer, the method comprising: obtaining data that has been created through an algorithm that compresses a first image, the algorithm comprising a lossy compression algorithm, the algorithm utilizing a set of discrete coefficients during compression; and constructing a second image from the data and recovering spectral information by performing actions, comprising: performing nonlinear partial differential equation regularization on the data; and enforcing constraints during the partial differential equation regularization.