On the order of analysis of speckle images in optical astronomy

Abstract

We show in this paper that the statistical properties of the speckle image formed at the focus of a large telescope can be fully described by a joint statistical analysis at N different spatial positions, where N is the number of resolution cells in the object's support. To obtain this result, the statistical properties are defined using multifold moment-generating functions (MGFs). Simplifying assumptions (discrete one-dimensional geometry, stationarity) are used to make the mathematical formalism simpler; they make the imaging process similar to a moving average process. General expressions are given for the twofold MGF and for MGFs of higher order. These relations are then used to show that an analysis of order N is exhaustive. It is shown that an MGF of order N + 1 can be written as the product of two MGFs of order N divided by an MGF of order N - 1. Alternatively, it is also shown that the cumulant of order N + 1 is equal to zero. A particular comment is made for the case of the double-star speckle pattern.