One of the key applications is in coherent x-ray imaging – coherent diffraction patterns are fourier transforms of real-space density of an object, but due to phase problem (loss of phases during the measurements of x-ray intensities) one can’t simply perform inverse fourier transform to get back to real-space image. But oversampling – measuring intensities at at least twice the spatial frequency of the object – can solve the phase problem by bringing the number of equations equal to the number of unknown variables again. One can solve for phases by considering restraints, or rules, imposed by such experiments – for example, intensities are known (while phases are not), real space densities are real positive numbers, typically contained within a finite volume (support). There may be more elaborate restraints.
By using an approach based on differential map algorithm one can alternate performing simple projections in real and reciprocal space, using the restraints mentioned above, alternated with Fourier transforms to go from real space to reciprocal space and back.
This approach can be applied to other problems – from finding the local minimum in energy landscape for protein folding, packing and tiling probelms to finding a unique solution for sudoku problems, jumble and other puzzles.