## Abstract

Electro-optical implementation of a three-dimensional (3-D) spatial correlation is proposed. A 3-D scene of objects, seen from the paraxial zone, is correlated with a reference object. As an example of application, we describe a 3-D joint transform correlator that is capable of recognizing targets in the 3-D space.

© 1998 Optical Society of America

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### Equations (11)

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(1)
$${x}_{i}=\frac{M({D}_{x}+{x}_{s}^{\prime})}{1-{z}_{s}^{\prime}/L}$$
(2)
$${y}_{i}=\frac{M({D}_{y}+{y}_{s}^{\prime})}{1-{z}_{s}^{\prime}/L}.$$
(3)
$${x}_{i}\cong M({D}_{x}+{x}_{s}^{\prime}+{z}_{s}^{\prime}{D}_{x}/L+{z}_{s}^{\prime}{x}_{s}^{\prime}/L),$$
(4)
$${y}_{i}\cong M({D}_{y}+{y}_{s}^{\prime}+{z}_{s}^{\prime}{D}_{y}/L+{z}_{s}^{\prime}{y}_{s}^{\prime}/L).$$
(5)
$$O(u,v,{D}_{x},{D}_{y})=\iiint o({x}_{s},{y}_{s},{z}_{s})\times exp[i2\pi \kappa ({x}_{i}u+{y}_{i}v)]\mathrm{d}{x}_{s}\mathrm{d}{y}_{s}\mathrm{d}{z}_{s}.$$
(6)
$${x}_{i}\cong M[{D}_{x}+{x}_{s}^{\prime}+{z}_{s}^{\prime}{D}_{x}/L],$$
(7)
$${y}_{i}\cong M[{D}_{y}+{y}_{s}^{\prime}+{z}_{s}^{\prime}{D}_{y}/L].$$
(8)
$$O(u,v,{D}_{x},{D}_{y})=exp[i2\pi M\kappa ({D}_{x}u+{D}_{y}v)]\times \iiint o({x}_{s},{y}_{s},{z}_{s})\times exp\{i2\pi M\kappa [{x}_{s}u+{y}_{s}v+({z}_{s}/L)({D}_{x}u+{D}_{y}v)]\}\mathrm{d}{x}_{s}\mathrm{d}{y}_{s}\mathrm{d}{z}_{s}.$$
(9)
$$o({x}_{s},{y}_{s},{z}_{s})=r({x}_{s},{y}_{s},{z}_{s})+g({x}_{s}+a,{y}_{s}+b,{z}_{s}+c).$$
(10)
$${I}_{4}(u,v,{D}_{x},{D}_{y})={\left|R(u,v,{D}_{x},{D}_{y})+G(u,v,{D}_{x},{D}_{y})\times exp\left\{(i2\pi M/\mathrm{\lambda}f)\left[\mathit{au}+\mathit{bv}+\frac{c}{L}({D}_{x}u+{D}_{y}v)\right]\right\}\right|}^{2}=|R(u,v,{D}_{x},{D}_{y}){|}^{2}+|G(u,v,{D}_{x},{D}_{y}){|}^{2}+G(u,v,{D}_{x},{D}_{y}){R}^{*}(u,v,{D}_{x},{D}_{y})\times exp\left\{(i2\pi M/\mathrm{\lambda}f)\left[\mathit{au}+\mathit{bv}+\frac{c}{L}({D}_{x}u+{D}_{y}v)\right]\right\}+{G}^{*}(u,v,{D}_{x},{D}_{y})R(u,v,{D}_{x},{D}_{y})\times exp\left\{-(i2\pi M/\mathrm{\lambda}f)\left[\mathit{au}+\mathit{bv}+\frac{c}{L}({D}_{x}u+{D}_{y}v)\right]\right\},$$
(11)
$$c({x}_{o},{y}_{o},{z}_{o})=\int {\tilde{I}}_{4}({\omega}_{x},{\omega}_{y},{\omega}_{z})exp[-i2\pi ({x}_{o}{\omega}_{x}+{y}_{o}{\omega}_{y}+{z}_{o}{\omega}_{z})]\mathrm{d}{\omega}_{x}\mathrm{d}{\omega}_{y}\mathrm{d}{\omega}_{z}=r\otimes r+g\otimes g+(r\otimes g)*\delta ({x}_{o}-a,{y}_{o}-b,{z}_{o}-c)+(g\otimes r)*\delta ({x}_{o}+a,{y}_{o}+b,{z}_{o}+c),$$