## Abstract

A distortion-invariant joint transform correlator based on the concepts
of the fringe-adjusted joint transform correlator and the synthetic
discriminant function is presented. Computer-simulation results show that the
proposed joint transform correlator is distortion-invariant for the target
image from the training set and produces sharper correlation peaks and lower
sidelobes compared with the classical joint transform
correlator.

© 1997 Optical Society of America

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### Equations (15)

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(1)
$$f\left(x,y\right)=r\left(x,y+{y}_{0}\right)+s\left(x,y-{y}_{0}\right).$$
(2)
$$F\left(u,v\right)=R\left(u,v\right)exp\left({\mathit{jvy}}_{0}\right)+S\left(u,v\right)exp\left(-{\mathit{jvy}}_{0}\right),$$
(3)
$${\left|F\left(u,v\right)\right|}^{2}={\left|R\left(u,v\right)\right|}^{2}+{\left|S\left(u,v\right)\right|}^{2}+R\left(u,v\right){S}^{*}\left(u,v\right)exp\left(j2{\mathit{vy}}_{0}\right)+{R}^{*}\left(u,v\right)S\left(u,v\right)exp\left(-j2{\mathit{vy}}_{0}\right).$$
(4)
$$H\left(u,v\right)=\frac{B\left(u,v\right)}{A\left(u,v\right)+{\left|R\left(u,v\right)\right|}^{2}},$$
(5)
$$G\left(u,v\right)=H\left(u,v\right)\xb7{\left|F\left(u,v\right)\right|}^{2}.$$
(6)
$$G\left(u,v\right)=2+2cos\left(2{\mathit{vy}}_{0}\right).$$
(7)
$$P\left(u,v\right)={\left|F\left(u,v\right)\right|}^{2}-{\left|R\left(u,v\right)\right|}^{2}-{\left|S\left(u,v\right)\right|}^{2},=R\left(u,v\right)S*\left(u,v\right)exp\left(j2{\mathit{vy}}_{0}\right)+{R}^{*}\left(u,v\right)S\left(u,v\right)exp\left(-j2{\mathit{vy}}_{0}\right).$$
(8)
$$G\left(u,v\right)=H\left(u,v\right)P\left(u,v\right)=2cos\left(2{\mathit{vy}}_{0}\right).$$
(9)
$$r\left(x,y\right)=\sum _{n=1}^{N}{a}_{n}{r}_{n}\left(x,y\right),$$
(10)
$$R\left(u,v\right)=\sum _{n=1}^{N}{a}_{n}{R}_{n}\left(u,v\right),$$
(11)
$$H\left(u,v\right)=\frac{B\left(u,v\right)}{A\left(u,v\right)+{\left|{\displaystyle \sum _{n=1}^{N}}{a}_{n}{R}_{n}\left(u,v\right)\right|}^{2}}.$$
(12)
$$G\left(u,v\right)=\frac{B\left(u,v\right)}{A\left(u,v\right)+{\left|{\displaystyle \sum _{n=1}^{N}}{a}_{n}{R}_{n}\left(u,v\right)\right|}^{2}}\times \sum _{n=1}^{N}{a}_{n}{R}_{n}\left(u,v\right){S}^{*}\left(u,v\right)exp\left(j2{\mathit{vy}}_{0}\right)+\sum _{n=1}^{N}{a}_{n}^{*}{R}_{n}^{*}\left(u,v\right)S\left(u,v\right)exp\left(-j2{\mathit{vy}}_{0}\right).$$
(13)
$$G\left(u,v\right)\approx \frac{S*\left(u,v\right)}{\left|{\displaystyle \sum _{n=1}^{N}}{a}_{n}{R}_{n}\left(u,v\right)\right|}exp\left[j\mathrm{\varphi}\left(u,v\right)\right]exp\left(j2{\mathit{vy}}_{0}\right)+\frac{S\left(u,v\right)}{\left|{\displaystyle \sum _{n=1}^{N}}{a}_{n}{R}_{n}\left(u,v\right)\right|}exp\left[-j\mathrm{\varphi}\left(u,v\right)\right]\times exp\left(-j2{\mathit{vy}}_{0}\right),$$
(14)
$$\iint \frac{{R}_{n}\left(u,v\right)}{{\displaystyle \sum _{m=1}^{N}}{a}_{m}{R}_{m}\left(u,v\right)}exp\left(-j2{\mathit{vy}}_{0}\right)\mathrm{d}u\mathrm{d}v={c}_{n},$$
(15)
$$a_{n}{}^{i+1}=a_{n}{}^{i}+\mathrm{\beta}\left[{c}_{n}-{c}_{0}\left(\frac{p_{n}{}^{i}}{p_{0}{}^{i}}\right)\right],$$