## Abstract

The limitation of the conventional signal-to-noise ratio as a performance measure in matched-filter-based optical pattern recognition for input-scene noise that is disjoint (or nonoverlapping) with the target is investigated.

© 1992 Optical Society of America

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### Equations (9)

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(1)
$$s(x,y)=r(x,y)+n(x,y),$$
(2)
$$S(\mathrm{\alpha},\mathrm{\beta})\text{exp}[j{\mathrm{\varphi}}_{S}(\mathrm{\alpha},\mathrm{\beta})]=R(\mathrm{\alpha},\mathrm{\beta})\text{exp}[j{\mathrm{\varphi}}_{R}(\mathrm{\alpha},\mathrm{\beta})]+N(\mathrm{\alpha},\mathrm{\beta})\text{exp}[j{\mathrm{\varphi}}_{N}(\mathrm{\alpha},\mathrm{\beta})].$$
(3)
$${H}_{k1}(\mathrm{\alpha},\mathrm{\beta})={[R(\mathrm{\alpha},\mathrm{\beta})]}^{k}\text{exp}\{j[{x}_{0}\mathrm{\alpha}-{\mathrm{\varphi}}_{R}(\mathrm{\alpha},\mathrm{\beta})]\}.$$
(4)
$${C}_{1}(\mathrm{\alpha},\mathrm{\beta})={[R(\mathrm{\alpha},\mathrm{\beta})]}^{k}\text{exp}\{j[{x}_{0}\mathrm{\alpha}-{\mathrm{\varphi}}_{R}(\mathrm{\alpha},\mathrm{\beta})]\}\times R(\mathrm{\alpha},\mathrm{\beta})\text{exp}[j{\mathrm{\varphi}}_{R}(\mathrm{\alpha},\mathrm{\beta})]+{[R(\mathrm{\alpha},\mathrm{\beta})]}^{k}\times \text{exp}\{j[{x}_{0}\mathrm{\alpha}-{\mathrm{\varphi}}_{R}(\mathrm{\alpha},\mathrm{\beta})]\}\times N(\mathrm{\alpha},\mathrm{\beta})\text{exp}[J{\mathrm{\varphi}}_{N}(\mathrm{\alpha},\mathrm{\beta})].$$
(5)
$$\text{SNR}=\frac{{E}^{2}\{{s}_{\text{o}}\}}{\text{Var}\{{s}_{\text{o}}\}}.$$
(6)
$${s}_{\text{o}}=\int R{(\mathrm{\alpha},\mathrm{\beta})}^{k+1}\text{d}\mathrm{\alpha}\text{d}\mathrm{\beta}+\int R{(\mathrm{\alpha},\mathrm{\beta})}^{k}N(\mathrm{\alpha}\mathrm{\beta})\text{exp}\{j[{\mathrm{\varphi}}_{N}(\mathrm{\alpha},\mathrm{\beta})-{\mathrm{\varphi}}_{R}(\mathrm{\alpha},\mathrm{\beta})]\}\text{d}\mathrm{\alpha}\text{d}\mathrm{\beta}.$$
(7)
$$\begin{array}{l}{n}_{\text{o}}={\int}_{(x,y)\in {S}_{1}\cup {S}_{2}}n(x,y){r}_{k}(x,y)\text{d}x\text{d}y\\ ={\int}_{(x,y)\in {S}_{2}}n(x,y){r}_{k}(x,y)\text{d}x\text{d}y,\end{array}$$
(8)
$$\text{SNR}=\frac{{\left\{\int R{(\mathrm{\alpha},\mathrm{\beta})}^{k+1}\text{d}\mathrm{\alpha}\text{d}\mathrm{\beta}+{\int}_{(x,y)\in {S}_{2}}E[n(x,y)]{r}_{k}(x,y)\text{d}x\text{d}y\right\}}^{2}}{E\left({\left\{{\int}_{(x,y)\in {S}_{2}}n(x,y){r}_{k}(x,y)\text{d}x\text{d}y-{\int}_{(x,y)\in {S}_{2}}E[n(x,y)]{r}_{k}(x,y)\text{d}x\text{d}y\right\}}^{2}\right)}.$$
(9)
$$\text{PNR}\frac{\mid {s}_{\text{o}}{\mid}^{2}}{\left\{{\displaystyle \sum _{j}}{\displaystyle \sum _{j}}[n({x}_{i},{y}_{j})]-\mid {s}_{\text{o}}{\mid}^{2}\right\}/N},$$