Uniqueness and Other Aspects of the Optical Phase Problem

The phase problem is examined as it occurs in stellar speckle interferometry, wherein one assumes that the image to be reconstructed is of finite spatial extent. Algorithms for reconstructing the Fourier phase of an image are studied, as well as a detailed investigation of the two-dimensional uniqueness question undertaken. Criteria for unique phase retrieval are derived for two-dimensional discrete objects, with particular emphasis placed on the importance of support constraints. The support is shown to have a profound effect on the uniqueness properties of the function defined upon it. It is established that some, such as Eisenstein's support, ensure a single solution independent of the function defined on that region, excluding mild restrictions necessary to define the region. An alternate method of demonstrating the solution uniqueness associated with Eisenstein's support is presented. Moreover, this approach is generalized to produce a large number of supports which are not described by Eisenstein's criterion, but which nonetheless guarantee solution uniqueness. An algorithm is developed to test an arbitrary discrete support for membership in this special family. A related criterion is also derived which, when satisfied, ensures solution uniqueness when support information is explicitly incorporated into a reconstruction algorithm. A one-dimensional phase retrieval technique for stellar speckle interferometric imaging is presented, which is based on the identification and manipulation of complex zeros. The algorithm is applied to data produced in a laboratory simulation of stellar speckle. It is found that it is possible to recover one-dimensional images in this manner, however, the quality of the reconstruction is generally no better than that produced by the Knox-Thompson algorithm, and the roots method is substantially more difficult to implement. A brief survey of modulus-only reconstruction algorithms is undertaken, with an emphasis on Fienup's iterative approach. An implementation of Fienup's algorithm proves to be capable of reconstructing a limited number of objects from the Fourier modulus, but in general requires stronger constraints than nonnegativity to converge to the correct solution. The algorithm is capable of reconstructing rather complex images defined on Eisenstein's support, in addition to others which are defined on a sufficiently irregular region.