The Phase Diverse Speckle (PDS) problem is formulated mathematically as Multi Frame Blind Deconvolution (MFBD) together with a set of Linear Equality Constraints (LECs) on the wavefront expansion parameters. This MFBD--LEC formulation is quite general and, in addition to PDS, it allows the same code to handle a variety of different data collection schemes specified as data, the LECs, rather than in the code. It also relieves us from having to derive new expressions for the gradient of the wavefront parameter vector for each type of data set. The idea is first presented with a simple formulation that accommodates Phase Diversity, Phase Diverse Speckle, and Shack--Hartmann wavefront sensing. Then various generalizations are discussed, that allows many other types of data sets to be handled.
Background: Unless auxiliary information is used, the Blind Deconvolution problem for a single frame is not well posed because the object and PSF information in a data frame cannot be separated. There are different ways of bringing auxiliary information to bear on the problem. MFBD uses several frames which helps somewhat, because the solutions are constrained by a requirement that the object be the same, but is often not enough to get useful results without further constraints. One class of MFBD methods constrain the solutions by requiring that the PSFs correspond to wavefronts over a certain pupil geometry, expanded in a finite basis. This is an effective approach but there is still a problem of uniqueness in that different phases can give the same PSF. Phase Diversity and the more general PDS methods are special cases of this class of MFBD, where the observations are usually arranged so that in-focus data is collected together with intentionally defocused data, where information on the object is sacrificed for more information on the aberrations. The known differences and similarities between the phases are used to get better estimates.