## Abstract

Higher-order correlations are well known for their use in noise removal, image enhancement, and signal identification. They are generalizations of the well-known second-order correlation. The fractionalization of the second-order correlation provides some interesting features that are related to the shift-variance property of the fractional-Fourier-transform operation. This project proposes the fractionalization of the triple-correlation operation (as well as other higher-order correlations). A suggested definition as well as some applications are given. Computer simulations demonstrate some of the features this operation offers.

© 1998 Optical Society of America

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### Equations (14)

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(1)
$${\tilde{l}}^{p}(u)={F}^{p}[l(x)]=C{\int}_{-\infty}^{\infty}l(x)exp\left[\frac{i\pi ({x}^{2}+{u}^{2})}{tan(\pi p/2)}-\frac{2\pi \mathit{ixu}}{sin(\pi p/2)}\right]\mathrm{d}x,$$
(2)
$$C=\frac{exp\left\{-i\left[\frac{\pi \mathrm{sgn}(sin\varphi )}{4}-\frac{\varphi}{2}\right]\right\}}{(|sin\varphi |{)}^{1/2}},$$
(3)
$$l_{3}{}^{1}({x}_{1},{x}_{2})={\int}_{-\infty}^{\infty}l(x)l(x+{x}_{1})l(x+{x}_{2})\mathrm{d}x.$$
(4)
$$\tilde{l}_{3}{}^{1}({u}_{1},{u}_{2})={\int}_{-\infty}^{\infty}l_{3}{}^{1}({x}_{1},{x}_{2})\times exp[-2\pi i({x}_{1}{u}_{1}+{x}_{2}{u}_{2})]\mathrm{d}{x}_{1}\mathrm{d}{x}_{2}={\tilde{l}}^{1}({u}_{1})\xb7{\tilde{l}}^{1}({u}_{2})\xb7{\tilde{l}}^{1}(-{u}_{1}-{u}_{2}).$$
(5)
$$\tilde{l}_{3}{}^{P}({u}_{1},{u}_{2})={\tilde{l}}^{p}({u}_{1})\xb7{\tilde{l}}^{p}({u}_{2})\xb7{\tilde{l}}^{p}(-{u}_{1}-{u}_{2}),$$
(6)
$$l_{3}{}^{P}({x}_{1},{x}_{2})={F}^{-p}[{\tilde{l}}^{p}({u}_{1})\xb7{\tilde{l}}^{p}({u}_{2})\xb7{\tilde{l}}^{p}(-{u}_{1}-{u}_{2})].$$
(7)
$$\tilde{l}_{3}{}^{P}({u}_{1},{u}_{2})={F}^{p}[{l}_{3}({x}_{1},{x}_{2})],$$
(8)
$$tan\left(\frac{\pi}{2}p\right)=\frac{\pi}{b},$$
(9)
$$l_{n}{}^{1}({x}_{1},{x}_{2},\dots ,{x}_{n-1})=\int l(x)\prod _{k=1}^{n-1}l(x+{x}_{k})\mathrm{d}x,$$
(10)
$${\tilde{l}}^{1}(u)=\int l(x)exp(-i2\pi \mathit{xu})\mathrm{d}x,$$
(11)
$$\tilde{l}_{n}{}^{1}({u}_{1},{u}_{2},\dots ,{u}_{n-1})$$
(12)
$$={\tilde{l}}^{1}(-{u}_{1}-{u}_{2}-\cdots -{u}_{n-1})\prod _{k=1}^{n-1}{\tilde{l}}^{1}({u}_{k}).$$
(13)
$$l_{n}{}^{p}({x}_{1},{x}_{2},\dots ,{x}_{n-1})$$
(14)
$$={F}^{-p}\left[{\tilde{l}}^{p}(-{u}_{1}-{u}_{2}-\cdots -{u}_{n-1})\prod _{k=1}^{n-1}{\tilde{l}}^{p}({u}_{k})\right].$$