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.
Highlights
The autocorrelation of a real-valued function f is another real function af created by integrating the product of f and a shifted copy of itself: a function of the general form af(s) = f f(x)f(x + s)dx
In recent years there has been growing interest in the possibility of recovering phase information from higher-order autocorrelation function (ACF)'s created by integrating the product of a function and multiple shifted copies of itself.' 9 Generalizing the concept of autocorrelation, one can construct a sequence {ak, f: k = 1,2 ... } of ACF's of a real function f where the kth-order ACF a,f(S1,..-Sk) is created by an integraol ftheformf f(x)f(x + l)... f(x + s)dx
In this sequence a is the ordinary ACF and a 2,f is the triple-correlation function, which has been widely applied in optics' and is beginning to find uses in vision research4.'1 0 (The Fourier transform of the triple correlation is commonly known as the bispectrum, so our numbering agrees with standard terminology in the spectral domain.)
Summary
For establishing triple-correlation uniqueness in cases in which the transform of a band-limited image vanishes over an interval belowthe frequency cutoff It shows that the triple correlation determines the functions sinc 2 (x) (1 + cos 57rx) and sinc 2(x) (1 + sin 5irx), which are similar to the counterexamples of Fig. 1 except for the size of the gaps in their spectra. We sketch the uniqueness properties of the higherorder ACF's of infinitely extended periodic images, drawing on the work of Klein and Tyler.[4] Here again two images f and g have the same triple-correlation function if and only if their transforms satisfy Eq (1), but the discreteness of the spectrum in this case weakens the force of that constraint, making the uniqueness problem more difficult. It is noted explicitly that, while the followinganalysis often relies onprobabilistic arguments (exploiting the formal similarity between images and probability distributions), the images that concern us are always deterministic: the paper does not deal with higher-order ACF's of stochastic processes
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
