Uniqueness properties of higher-order autocorrelation functions

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Abstract
The kth-order autocorrelation function of an image is formed by integrating the product of the image and k independently shifted copies of itself: The case k = 1 is the ordinary autocorrelation; k = 2 is the triple correlation. Bartelt et al. [Appl. Opt. 23, 3121 (1984)] have shown that every image of finite size is uniquely determined up to translation by its triple-correlation function. We point out that this is not true in general for images of infinite size, e.g., frequency-band-limited images. Examples are given of pairs of simple band-limited periodic images and pairs of band-limited aperiodic images that are not translations of each other but that have identical triple correlations. Further examples show that for every k there are distinct band-limited images that have identical kth-order autocorrelation functions. However, certain natural subclasses of infinite images are uniquely determined up to translation by their triple correlations. We develop two general types of criterion for the triple correlation to have an inverse image that is unique up to translation, one based on the zeros of the image spectrum and the other based on image moments. Examples of images satisfying such criteria include diffraction-limited optical images of finite objects and finite images blurred by Gaussian point spreads.