## Abstract

The use of nonlinear techniques in the Fourier plane of pattern-recognition correlators can improve the correlators' performance in terms of discrimination against objects similar to the target object, correlation-peak sharpness, and correlation noise robustness. Additionally, filter designs have been proposed that provide the linear correlator with invariance properties with respect to input-signal distortions and rotations. We propose simple modifications to presently known distortion-invariant correlator filters that enable these filter designs to be used in a nonlinear correlator architecture. These Fourier-plane nonlinear filters can be implemented electronically, or they may be implemented optically with a nonlinear joint transform correlator. Extensive simulation results are presented that illustrate the performance enhancements that are gained by the unification of nonlinear techniques with these filter designs.

© 1996 Optical Society of America

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### Equations (23)

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(1)
$${\mathbf{\text{h}}}_{\text{ECP}}=\mathbf{\text{S}}{({\mathbf{\text{S}}}^{+}\mathbf{\text{S}})}^{-1}\mathbf{\text{c}}*,$$
(2)
$${\mathit{E}}_{\text{ave}}=\frac{1}{N}\sum _{i=1}^{N}\phantom{\rule{0.5em}{0ex}}\sum _{n=1}^{d}\phantom{\rule{0.5em}{0ex}}{\left|\mathbf{\text{\u0125}}\right(n\left)\phantom{\rule{0.2em}{0ex}}\right|}^{2}\phantom{\rule{0.2em}{0ex}}{\left|{\mathbf{\text{\u015d}}}_{i}\right(n\left)\phantom{\rule{0.2em}{0ex}}\right|}^{2},$$
(3)
$${\mathbf{\text{\u0125}}}_{\text{MACE}}={\widehat{\mathbf{\text{D}}}}^{-1}\mathbf{\text{\u015c}}{({\mathbf{\text{\u015c}}}^{+}{\widehat{\mathbf{\text{D}}}}^{-1}\mathbf{\text{\u015c}})}^{-1}\mathbf{\text{c}}*,$$
(4)
$$\widehat{\mathbf{\text{D}}}(r,r)=\frac{1}{N}\sum _{i=1}^{N}{\mathbf{\text{\u015d}}}_{i}(r){\mathbf{\text{\u015d}}}_{i}^{*}(r),r=1,\dots ,d.$$
(5)
$$\begin{array}{ll}g(E)=sgn(E)\phantom{\rule{0.2em}{0ex}}{\left|E\right|}^{k},& k\le 1,\end{array}$$
(6)
$${g}_{1}(E)\propto {[\phantom{\rule{0.1em}{0ex}}|H({f}_{x},{f}_{y})\Vert S({f}_{x},{f}_{y})\phantom{\rule{0.1em}{0ex}}|\phantom{\rule{0.1em}{0ex}}]}^{k}\times exp\{j\phantom{\rule{0.2em}{0ex}}[{\varphi}_{H}({f}_{x},{f}_{y})-{\varphi}_{S}({f}_{x},{f}_{y})]\},$$
(7)
$${\mathbf{\text{\u0125}}}_{\text{ECP}}=\mathbf{\text{\u015c}}{(\mathbf{\text{\u015c}}+\mathbf{\text{\u015c}})}^{-1}\mathbf{\text{c}}*,$$
(8)
$${\mathbf{\text{v}}}^{k}\triangleq \left[\begin{array}{l}{\left|\mathbf{\text{v}}\left[1\right]\phantom{\rule{0.2em}{0ex}}\right|}^{k}exp(j{\varphi}_{\mathbf{\text{v}}}\left[1\right])\\ {\left|\mathbf{\text{v}}\left[2\right]\phantom{\rule{0.2em}{0ex}}\right|}^{k}exp(j{\varphi}_{\mathbf{\text{v}}}\left[2\right])\\ \phantom{\rule{6em}{0ex}}\vdots \\ {\left|\mathbf{\text{v}}\left[d\right]\phantom{\rule{0.2em}{0ex}}\right|}^{k}exp(j{\varphi}_{\mathbf{\text{v}}}\left[d\right])\end{array}\right].$$
(9)
$${\mathbf{\text{\u0125}}}_{\text{ECP},k}^{k}={\mathbf{\text{\u015c}}}^{k}{[({\mathbf{\text{\u015c}}}^{k})+{\mathbf{\text{\u015c}}}^{k}]}^{-1}\mathbf{\text{c}}*.$$
(10)
$${\mathbf{\text{v}}}^{1/k}\triangleq \left[\begin{array}{l}{\left|\mathbf{\text{v}}\left[1\right]\phantom{\rule{0.2em}{0ex}}\right|}^{1/k}exp(j{\varphi}_{\mathbf{\text{v}}}\left[1\right])\\ {\left|\mathbf{\text{v}}\left[2\right]\phantom{\rule{0.2em}{0ex}}\right|}^{1/k}exp(j{\varphi}_{\mathbf{\text{v}}}\left[2\right])\\ \phantom{\rule{5em}{0ex}}\vdots \\ {\left|\mathbf{\text{v}}\left[d\right]\right|}^{1/k}exp(j{\varphi}_{\mathbf{\text{v}}}\left[d\right])\end{array}\right],$$
(11)
$${\mathbf{\text{\u0125}}}_{\text{ECP},k}={\{{\mathbf{\text{\u015c}}}^{k}{[({\mathbf{\text{\u015c}}}^{k})+{\mathbf{\text{\u015c}}}^{k}]}^{-1}\mathbf{\text{c}}*\}}^{1/k},$$
(12)
$${\mathbf{\text{h}}}_{\text{ECP},k}={F}^{-1}\{{\{{\mathbf{\text{\u015c}}}^{k}{[({\mathbf{\text{\u015c}}}^{k})+{\mathbf{\text{\u015c}}}^{k}]}^{-1}\mathbf{\text{c}}*\}}^{1/k}\}.$$
(13)
$${\mathbf{\text{h}}}_{\text{MACE},k}={F}^{-1}\{{\{{({\widehat{\mathbf{\text{D}}}}^{k})}^{-1}{\mathbf{\text{\u015c}}}^{k}{[({\mathbf{\text{\u015c}}}^{k})+{({\widehat{\mathbf{\text{D}}}}^{k})}^{-1}{\mathbf{\text{\u015c}}}^{k}]}^{-1}\mathbf{\text{c}}*\}}^{1/k}\},$$
(14)
$${\widehat{\mathbf{\text{D}}}}^{k}(j,k)=\{\begin{array}{ll}\text{\u2211}_{i=1}^{N}{\left|{\mathbf{\text{\u015d}}}_{i}^{k}(k)\phantom{\rule{0.2em}{0ex}}\right|}^{2}& j=k=1,\dots ,d\\ 0& j\ne k\end{array}.$$
(15)
$${R}_{0}^{2}=\underset{x,y}{max}[\phantom{\rule{0.2em}{0ex}}{|C(x,y)\phantom{\rule{0.2em}{0ex}}|}^{2}],$$
(16)
$$\text{SL}=\underset{\begin{array}{l}x,y\phantom{\rule{0.2em}{0ex}}>\phantom{\rule{0.2em}{0ex}}\text{circular region}\\ \text{centered at position}\\ \text{of}\phantom{\rule{0.2em}{0ex}}{R}_{0}\phantom{\rule{0.2em}{0ex}}(x,y)\phantom{\rule{0.2em}{0ex}}\text{and with}\\ \text{diameter given by}\\ {R}_{0}^{2}(x,y)/2\end{array}}{max}[\phantom{\rule{0.2em}{0ex}}{|C(x,y)\phantom{\rule{0.2em}{0ex}}|}^{2}].$$
(17)
$${R}_{0}\text{SL}=\frac{{R}_{0}^{2}}{\text{SL}}$$
(18)
$$\text{PCE}=\frac{[{R}_{0}^{2}-\overline{{\left|C(x,y)\right|}^{2}}]}{{\left\{\text{\u2211}_{\begin{array}{l}\text{all}\phantom{\rule{0.2em}{0ex}}x,y\\ x,y\ne 0\end{array}}\frac{{[{\left|C(x,y)\right|}^{2}-\overline{{\left|C(x,y)\right|}^{2}}]}^{2}}{({N}_{x}{N}_{y}-1)}\right\}}^{1/2}},$$
(19)
$$\overline{{\left|C(x,y)\phantom{\rule{0.2em}{0ex}}\right|}^{2}}=\text{\u2211}_{\text{all}\phantom{\rule{0.2em}{0ex}}x,y}\frac{{\left|C(x,y)\phantom{\rule{0.2em}{0ex}}\right|}^{2}}{{N}_{x}{N}_{y}}$$
(20)
$$\text{DR}=\frac{E\{{R}_{0,\text{target}}\}}{E\{{R}_{0,\text{nontarget}}\}},$$
(21)
$$\text{SNR}=\frac{E\left\{{R}_{0}^{2}\right\}}{\sqrt{E\{{({R}_{0}^{2}-E\{{R}_{0}^{2}\})}^{2}\}}},$$
(22)
$$E\{{R}_{0}^{2}\}=\text{\u2211}_{s}\frac{{R}_{0,s}^{2}}{{N}_{s}}$$
(23)
$$E\{{({R}_{0}^{2}-E\{{R}_{0}^{2}\})}^{2}\}=\text{\u2211}_{s}\frac{{({R}_{0,s}^{2}-E\{{R}_{0}^{2}\})}^{2}}{{N}_{s}}$$