Abstract

The kth order autocorrelation function (ACF) 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.¹ 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 one another but that have identical triple correlations. Further examples show that for every k there are distinct band-limited images that have identical kth order ACFs. 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, the other on image moments.

© 1991 Optical Society of America