## Abstract

Two types of filter are proposed to detect a noisy target embedded in nonoverlapping background noise by optimization of two proposed criteria that are used in the assessment of filter design and performance. Criterion 1 is defined as the ratio of the square of the expected value of the correlation-peak amplitude to the expected value of the output-signal energy. Criterion 2 is defined as the ratio of the square of the expected value of the correlation-peak amplitude to the average output-signal variance. It is shown that, for the nonoverlapping target and scene noise models, the target window and the scene noise window affect the filter functions significantly. Computer-simulation tests of the generalized optimum filter for various kinds of noisy input image are provided to investigate filter performance in terms of peak-to-output-energy ratio, discrimination against undesired objects, and tolerance to target distortion (for example, target rotation and scaling). We compare the results with those of other filters to verify the performance of the optimum filters.

© 1994 Optical Society of America

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### Equations (33)

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(1)
$$\text{POE}=\hspace{0.17em}\mid E[y(\tau ,\tau )]{\mid}^{2}/E\{\overline{{[y(t,\tau )]}^{2}}\},$$
(2)
$$\text{S}\xd1\text{R}=\hspace{0.17em}\mid E[y(\tau ,\tau )]{\mid}^{2}/\overline{\text{Var}[y(t,\tau )]},$$
(3)
$$s(t,\tau )=r(t-\tau )+\xf1(t).$$
(4)
$$y(t,\tau )=\frac{1}{2\pi}\int H(\omega )S(\omega ,\tau )\text{exp}(j\omega t)\text{d}\omega ,$$
(5)
$$E[y(\tau ,\tau )]=\frac{1}{2\pi}\int H(\omega )E[S(\omega ,\tau )\text{exp}(j\omega \tau )]\text{d}\omega .$$
(6)
$$E\{\overline{{[y(t,\tau )]}^{2}}\}=\frac{1}{2\pi L}\int \mid H(\omega ){\mid}^{2}E[\mid S(\omega ,\tau ){\mid}^{2}]\text{d}\omega .$$
(7)
$$\overline{\text{Var}[y(t,\tau )]}=\frac{1}{2\pi L}\int \mid H(\omega ){\mid}^{2}\text{Var}[S(\omega ,\tau )]\text{d}\omega .$$
(8)
$$\text{POE}=\frac{{\left|\int H(\omega )E[S(\omega ,\tau )\text{exp}(j\omega \tau )]\text{d}\omega \right|}^{2}}{2\pi /L\hspace{0.17em}\int \mid H(\omega ){\mid}^{2}E[\mid S(\omega ,\tau ){\mid}^{2}]\text{d}\omega}.$$
(9)
$$\text{POE}\le \frac{L}{2\pi}\int {\left|\frac{E[S(\omega ,\tau )\text{exp}(j\omega \tau )]}{{\{E[\mid S(\omega ,\tau ){\mid}^{2}]\}}^{1/2}}\right|}^{2}\text{d}\omega .$$
(10)
$${H}_{\text{opt}}^{*}(\omega )=\frac{E[S(\omega ,\tau )\text{exp}(j\omega \tau )]}{E[\mid S(\omega ,\tau ){\mid}^{2}]},$$
(11)
$$\text{S}\xd1\text{R}=\frac{{\left|\int H(\omega )E[S(\omega ,\tau )\text{exp}(j\omega \tau )]\text{d}\omega \right|}^{2}}{2\pi /L\hspace{0.17em}\int \mid H(\omega ){\mid}^{2}\text{Var}[S(\omega ,\tau )]\text{d}\omega}.$$
(12)
$$\text{S}\xd1\text{R}\le \int {\left|\frac{E[S(\omega ,\tau )\text{exp}(j\omega \tau )]}{{\{\text{Var}[S(\omega ,\tau )]\}}^{1/2}}\right|}^{2}\text{d}\omega .$$
(13)
$${H}_{\text{gmf}}^{*}(\omega )=\frac{E[S(\omega ,\tau )\text{exp}(j\omega \tau )]}{\text{Var}[S(\omega ,\tau )]}.$$
(14)
$$\xf1(t)={\xf1}_{b}(t)+{\xf1}_{r}(t),$$
(15)
$${\xf1}_{b}(t)={n}_{b}(t)[{w}_{0}(t)-{w}_{r}(t-\tau )],$$
(16)
$${\xf1}_{r}(t)={n}_{r}(t){w}_{r}(t-\tau ),$$
(17)
$$f(\tau )=\{\begin{array}{ll}1/d\hfill & \tau \in \text{scene}\hspace{0.17em}\text{area}\hfill \\ 0\hfill & \text{elsewhere}\hfill \end{array},$$
(18)
$$\begin{array}{l}E[S(\omega ,\tau )\text{exp}(j\omega \tau )]=\mathcal{F}\{E[s(t+\tau ,t)]\}\\ =\mathcal{F}\{E[r(t)+{n}_{b}(t+\tau )\times [{w}_{0}(t+\tau )-{w}_{r}(t)]+{n}_{r}(t+\tau ){w}_{r}(t)]\}\\ =R(\omega )+{m}_{b}{W}_{1}(\omega )+{m}_{r}{W}_{r}(\omega ),\end{array}$$
(19)
$${W}_{1}(\omega )=\hspace{0.17em}\mid {W}_{0}(\omega ){\mid}^{2}/d-{W}_{r}(\omega ).$$
(20)
$$E[\mid S(\omega ,\tau ){\mid}^{2}]=\hspace{0.17em}\mid R(\omega )+{m}_{b}{W}_{1}(\omega )+{m}_{r}{W}_{r}(\omega ){\mid}^{2}+\frac{1}{2\pi}{W}_{2}(\omega )*{{N}_{b}}^{0}(\omega )+\frac{1}{2\pi}\mid {W}_{r}(\omega ){\mid}^{2}*{{N}_{r}}^{0}(\omega )+{{m}_{b}}^{2}[{W}_{2}(\omega )\hspace{0.17em}-\mid {W}_{1}(\omega ){\mid}^{2}],$$
(21)
$${W}_{2}(\omega )=\hspace{0.17em}\mid {W}_{0}(\omega ){\mid}^{2}+\mid {W}_{r}(\omega ){\mid}^{2}-\hspace{0.17em}2\mid {W}_{0}(\omega ){\mid}^{2}\text{real}[{W}_{r}(\omega )]/d.$$
(22)
$${H}_{\text{opt}}^{*}(\omega )=\frac{R(\omega )+{m}_{b}{W}_{1}(\omega )+{m}_{r}{W}_{r}(\omega )}{\mid R(\omega )+{m}_{b}{W}_{1}(\omega )+{m}_{r}{W}_{r}(\omega ){\mid}^{2}+\frac{1}{2\pi}{W}_{2}(\omega )*{{N}_{b}}^{0}(\omega )+\frac{1}{2\pi}\mid {W}_{r}(\omega ){\mid}^{2}*{{N}_{r}}^{0}(\omega )+{{m}_{b}}^{2}[{W}_{2}(\omega )\hspace{0.17em}-\mid {W}_{1}(\omega ){\mid}^{2}}.$$
(23)
$$E[S(\omega ,\tau )]=R(\omega ){W}_{0}(\omega )/d+{m}_{b}[{W}_{0}(\omega )-{W}_{r}(\omega ){W}_{0}(\omega )/d]+{m}_{r}{W}_{r}(\omega ){W}_{0}(\omega )/d.$$
(24)
$$\begin{array}{l}{H}_{\text{gmf}}^{*}(\omega )=\frac{E[S(\omega ,\tau )\text{exp}(j\omega \tau )]}{E[\mid S(\omega ,\tau ){\mid}^{2}]\hspace{0.17em}-\mid E[S(\omega ,\tau )]{\mid}^{2}}\\ =\{R(\omega )+{m}_{b}{W}_{1}(\omega )+{m}_{r}{W}_{r}(\omega )\}/\\ \{\mid R(\omega )+{m}_{b}{W}_{1}(\omega )+{m}_{r}{W}_{r}(\omega ){\mid}^{2}\\ +\hspace{0.17em}\frac{1}{2\pi}{W}_{2}(\omega )*{{N}_{b}}^{0}(\omega )+\frac{1}{2\pi}\mid {W}_{r}(\omega ){\mid}^{2}\\ *\hspace{0.17em}{{N}_{r}}^{0}(\omega )+{{m}_{b}}^{2}[{W}_{2}(\omega )\hspace{0.17em}-\mid {W}_{1}(\omega ){\mid}^{2}]\\ -\hspace{0.17em}\mid R(\omega ){W}_{0}(\omega )/d\\ +\hspace{0.17em}{m}_{b}[{W}_{0}(\omega )-{W}_{r}(\omega ){W}_{0}(\omega )/d]\\ +\hspace{0.17em}{m}_{r}{W}_{r}(\omega ){W}_{0}(\omega )/d{\mid}^{2}\}.\end{array}$$
(25)
$$w(t,\tau )=1-{w}_{r}(t-\tau ).$$
(26)
$${W}_{1}(\omega )=\delta (\omega )-{W}_{r}(\omega ),$$
(27)
$${W}_{2}(\omega )=\hspace{0.17em}\mid {W}_{1}(\omega ){\mid}^{2}.$$
(28)
$${H}_{\text{opt}}^{*}(\omega )=\frac{R(\omega )+{m}_{b}{W}_{1}(\omega )+{m}_{r}{W}_{r}(\omega )}{\mid R(\omega )+{m}_{b}{W}_{1}(\omega )+{m}_{r}{W}_{r}(\omega ){\mid}^{2}+\frac{1}{2\pi}\mid {W}_{1}(\omega ){\mid}^{2}*{{N}_{b}}^{0}(\omega )+\frac{1}{2\pi}\mid {W}_{r}(\omega ){\mid}^{2}*{{N}_{r}}^{0}(\omega )}.$$
(29)
$${H}_{\text{opt}}(\omega )=\frac{1}{R(\omega )+{m}_{b}{W}_{1}(\omega )+{m}_{r}{W}_{r}(\omega )}.$$
(30)
$${H}_{\text{opt}}^{*}(\omega )=\frac{R(\omega )+{m}_{b}\delta (\omega )}{\mid R(\omega )+{m}_{b}\delta (\omega ){\mid}^{2}+\frac{1}{2\pi}{{N}_{b}}^{0}(\omega )}.$$
(31)
$${H}_{\text{opt}}^{*}(\omega )=\frac{R(\omega )}{\mid R(\omega ){\mid}^{2}+\frac{1}{2\pi}{{N}_{b}}^{0}(\omega )}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{for}\hspace{0.17em}\omega \ne 0.$$
(32)
$${H}_{\text{gmf}}(\omega )=\frac{{R}^{*}(\omega )}{\frac{1}{2\pi}\mid {W}_{1}(\omega ){\mid}^{2}*{{N}_{b}}^{0}(\omega )+\frac{1}{2\pi}\mid {W}_{r}(\omega ){\mid}^{2}*{{N}_{r}}^{0}(\omega )}.$$
(33)
$${H}_{\text{gmf}}(\omega )={R}^{*}(\omega )/\frac{1}{2\pi}{{N}_{b}}^{0}(\omega ).$$