## Abstract

A multiple-hypothesis method is used to detect a target or a reference signal in the presence of additive noise with unknown statistics. The receiver is designed to detect the target and to be tolerant of the variations in rotation and illumination of the target. A multiple-hypothesis test with unknown-noise parameters is used to locate the target position. The proposed method does not use any specific distortion-invariant-filtering technique, but it relies on a multiple-hypothesis approach. Maximum-likelihood estimates of the illumination constant and the unknown noise parameters are obtained. Computer simulations are presented to evaluate the performance of the receiver for various distorted noisy true-class targets with varying illumination and false-class objects.

© 2002 Optical Society of America

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### Equations (16)

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(1)
$$s\left(t\right)=\mathit{ar}\left(t-{t}_{j}\right)+n\left(t\right).$$
(2)
$${H}_{j}:s\left(t\right)=\mathit{ar}\left(t-{t}_{j}\right)+n\left(t\right).$$
(3)
$${H}_{j}:s\left({t}_{m}\right)=\mathit{ar}\left({t}_{m}-{t}_{j}\right)+n\left({t}_{m}\right),$$
(4)
$${H}_{j}\left(i\right):s\left({t}_{m}\right)={\mathit{ar}}_{i}\left({t}_{m}-{t}_{j}\right)+n\left({t}_{m}\right),m\in \left[1,M\right],$$
(5)
$$P\left[s\left({t}_{m}\right)/{H}_{j}\right]=\frac{1}{N}\sum _{i=1}^{N}{\left(\frac{1}{2\mathrm{\pi}{\mathrm{\sigma}}^{2}}\right)}^{1/2}exp\left\{-{\left[s\left({t}_{m}\right)-\mathrm{\mu}-{\mathit{ar}}_{i}\left({t}_{m}-{t}_{j}\right)\right]}^{2}/2{\mathrm{\sigma}}^{2}\right\},$$
(6)
$$P\left(s/{H}_{j}\right)=\prod _{m:w\left({t}_{m}-{t}_{j}\right)=1}\left(\frac{1}{N}\sum _{i=1}^{N}{\left(\frac{1}{2\mathrm{\pi}{\mathrm{\sigma}}^{2}}\right)}^{1/2}\times exp\left\{-{\left[s\left({t}_{m}\right)-\mathrm{\mu}-{\mathit{ar}}_{i}\left({t}_{m}-{t}_{j}\right)\right]}^{2}/2{\mathrm{\sigma}}^{2}\right\}\right)\times \prod _{m:w\left({t}_{m}-{t}_{j}\right)=0}\left({\left(\frac{1}{2\mathrm{\pi}{\mathrm{\sigma}}^{2}}\right)}^{1/2}exp\left\{-{\left[s\left({t}_{m}\right)-\mathrm{\mu}\right]}^{2}/2{\mathrm{\sigma}}^{2}\right\}\right),$$
(7)
$$log\left[P\left(s/{H}_{j}\right)\right]=\sum _{m:w\left({t}_{m}-{t}_{j}\right)=1}\left[log\left(\sum _{i=1}^{N}\left(\frac{1}{2\mathrm{\pi}{\mathrm{\sigma}}^{2}}\right)\times exp\left\{-{\left[s\left({t}_{m}\right)-\mathrm{\mu}-{\mathit{ar}}_{i}\left({t}_{m}-{t}_{j}\right)\right]}^{2}/2{\mathrm{\sigma}}^{2}\right\}\right)\right]+\sum _{m:w\left({t}_{m}-{t}_{j}\right)=0}\left[log\left(\left(\frac{1}{2\mathrm{\pi}{\mathrm{\sigma}}^{2}}\right)exp\left\{-{\left[s\left({t}_{m}\right)-\mathrm{\mu}\right]}^{2}/2{\mathrm{\sigma}}^{2}\right\}\right)\right].$$
(8)
$$\frac{\partial P\left(s/{H}_{j}\right)}{\partial a}=0,$$
(9)
$$\frac{\partial P\left(s/{H}_{j}\right)}{\partial \mathrm{\mu}}=0,$$
(10)
$$\frac{\partial P\left(s/{H}_{j}\right)}{\partial {\mathrm{\sigma}}^{2}}=0.$$
(11)
$${\mathit{\xe2}}_{j}=\frac{{\displaystyle \sum _{m=1}^{M}}{\displaystyle \sum _{i=1}^{N}}\left[s\left({t}_{m}\right)-\mathrm{\mu}\right]{r}_{i}\left({t}_{m}-{t}_{j}\right)w\left({t}_{m}-{t}_{j}\right)}{{\displaystyle \sum _{i=1}^{N}}{E}_{i}},$$
(12)
$${E}_{i}=\sum _{m=1}^{M}{\left[{r}_{i}\left({t}_{m}\right){w}_{i}\left({t}_{m}\right)\right]}^{2},$$
(13)
$$P{\left(s/{H}_{j}\right)}_{\mathit{wO}}=\sum _{m:w\left({t}_{m}-{t}_{j}\right)=0}log\left(\left(\frac{1}{2\mathrm{\pi}{\mathrm{\sigma}}^{2}}\right)\times exp\left\{-{\left[s\left({t}_{m}\right)-\mathrm{\mu}\right]}^{2}/2{\mathrm{\sigma}}^{2}\right\}\right).$$
(14)
$${\stackrel{\u02c6}{\mathrm{\mu}}}_{j}=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}}\frac{{\displaystyle \sum _{m=1}^{M}}\left[s\left({t}_{m}\right)\right]\left[1-w\left({t}_{m}-{t}_{j}\right)\right]}{{M}_{\mathit{io}}},$$
(15)
$$\stackrel{\u02c6}{\mathrm{\sigma}}_{j}{}^{2}=\frac{1}{N}\sum _{i=1}^{N}\frac{{\left[s\left({t}_{m}\right)-{\stackrel{\u02c6}{\mathrm{\mu}}}_{j}\right]}^{2}}{{M}_{\mathit{io}}}.$$
(16)
$${\mathit{\xe2}}_{j}=\frac{{\displaystyle \sum _{m=1}^{M}}{\displaystyle \sum _{i=1}^{N}}\left\{s\left({t}_{m}\right)-\frac{1}{N}{\displaystyle \sum _{l=1}^{N}}\frac{{\displaystyle \sum _{n=1}^{M}}\left[s\left({t}_{n}\right)\right]\left[1-w\left({t}_{n}-{t}_{j}\right)\right]}{{M}_{\mathit{lo}}}\right\}{r}_{i}\left({t}_{m}-{t}_{j}\right)w\left({t}_{m}-{t}_{j}\right)}{{\displaystyle \sum _{i=1}^{N}}{E}_{i}}.$$