Reduction of speckle in an image by a moving aperture - second order statistics

In an earlier paper the second-order statistics of time-averaged speckle were investigated for the case of speckle formed in the Fraunhofer plane when an aperture was moved over the surface of a uniformly illuminated diffuse object. In this paper it is the intention to substantiate a claim that was made in that paper that the same treatment will also apply to time-averaged speckle in an image plane when an aperture is moved in the pupil plane of the imaging lens. The optical system is allowed to have aberrations of any degree of severity, and the object is allowed to have non-uniform brightness, subject to the condition that the brightness variations are slow in comparison to the size of the (aberrated) point spread function. The treatment makes use of the usual assumption that a very large number of independent scattering points in the object contributes to any one point in the image: for this assumption to be realistic, however, it will be shown that the point spread function of the lens must be of a size that restricts us either to systems where the numerical aperture in object space is low or, alternatively, to systems where the aberrations are severe.