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 JPEG compression algorithm, the algorithm utilizing a set of discrete DC coefficients and AC 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 predicting at least one of the AC coefficients using at least the DC coefficients from four neighboring blocks by multiplying those DC coefficients by a Hadamard transform matrix.