Non-Iterative Sparse Image Reconstruction from a Few 2-D DFT Frequency Values

We consider the problem of reconstructing a sparse image from a few of its 2-D DFT frequency values. A sparse image has pixel values that are mostly zero, with a few non-zero values at unknown locations. The number of known 2-D DFT values must exceed four times the number of nonzero pixel values. We unwrap the 2-D problem to a 1-D problem using the Good-Thomas FFT, and apply Prony’s method to compute the non-zero pixel value locations. Thus we reformulate the problem as a dual 2-D harmonic retrieval problem. Our solution has three advantages over direct application of 2-D ESPRIT: (1) Instead of solving a huge generalized eigenvalue problem, we compute the roots on the unit circle of a huge polynomial; (2) the locations of the known 2-D DFT values need not form a centrosymmetric region; and (3) there are no matching issues. Our algorithm is also applicable to 2-D beamforming.