Phase retrieval for a complex-valued object by using a low-resolution image

It is difficult to reconstruct an image of a complex-valued object from the modulus of its Fourier transform (i.e., retrieve the Fourier phase) except in some special cases. By using additionally a low-resolution intensity image from a telescope with a small aperture, a fine-resolution image of a general object can be reconstructed in a two-step approach. First the Fourier phase over the small aperture is retrieved, using the Gerchberg–Saxton algorithm. Then that phase is used, in conjunction with the Fourier modulus data over a large aperture together with a support constraint on the object, to reconstruct a fine-resolution image (retrieve the phase over the large aperture) by the iterative Fourier-transform algorithm. The second step requires a modified algorithm that employs an expanding weighting function on the Fourier modulus.