## Abstract

The goal of discrimination of one color from many other similar-appearing colors even when the colored objects show substantial variation or noise is of obvious import. We show how to accomplish that using a technique called Margin Setting. It is possible not only to have very low error rates but also to have some control over the types of errors that do occur. Robust spectral filtering prior to spatial pattern recognition allows subsequent filtering processes to be based on conventional coherent optical correlation that can be done monochromatically.

© 2007 Optical Society of America

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### Equations (14)

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(1)
$${N}^{\left(j+1\right)}{\le N}^{\left(j\right)};1\le j\le M$$
(2)
$${X}^{\left(j\right)}=\bigcup _{i=1}^{{N}^{\left(1\right)}}\left\{{X}_{\left(i\right)}^{\left(j\right)}\right\}\phantom{\rule{2em}{0ex}};\phantom{\rule{2em}{0ex}}{X}_{\left(i\right)}^{\left(j\right)}=\{{X}_{\left(i\right)k}^{\left(j\right)};{1\le k\mathrm{\le M}}_{\left(i\right)}^{\left(j\right)}\}$$
(3)
$${R}_{\left(i\right)k}^{\left(j\right)}=\{\underset{\forall l\ne i}{min}\parallel {x}_{\left(i\right)k}^{\left(j\right)}-{x}_{\left(l\right)m}^{\left(j\right)}\parallel ;1\le m\le {M}_{\left(l\right)}^{\left(j\right)}\};1\le k\le {M}_{\left(i\right)}^{\left(j\right)}$$
(4)
$${F}_{\left(i\right)k}^{\left(j\right)}=n\left({X\xb4}_{\left(i\right)k}^{\left(j\right)}\right);1\le k\le {M}_{\left(i\right)}^{\left(j\right)}$$
(5)
$${X\u0301}_{\left(i\right)k}^{\left(j\right)}=\{{x}_{\left(i\right)l}^{\left(j\right)}\mid \parallel {x}_{\left(i\right)l}^{\left(j\right)}-{x}_{\left(i\right)k}^{\left(j\right)}\parallel <{R}_{\left(i\right)k}^{\left(j\right)};1\le l\le {M}_{\left(i\right)}^{\left(j\right)}\}$$
(6)
$${X\u0301}_{\left(i\right)k}^{\left(j\right)}\subseteq {X}_{\left(i\right)}^{\left(j\right)}$$
(7)
$${F}_{\left(i\right)}^{\left(j\right)}=\underset{k}{max}\left\{{F}_{\left(i\right)k}^{\left(j\right)}\right\}$$
(8)
$${X}_{\left(i\right)}^{\left(j\right)}\to {\stackrel{=}{X}}_{\left(i\right)}^{\left(j\right)}$$
(9)
$${\stackrel{=}{X}}_{\left(i\right)}^{\left(j\right)}=\bigcup _{l=1}^{n\left({X}_{\left(i\right)}^{\left(j\right)}\right)}{\overline{X}}_{\left(i\right)l}^{\left(j\right)}$$
(10)
$${\stackrel{}{\overline{X}}}_{\left(i\right)l}^{\left(j\right)}=\left\{{\stackrel{}{\overline{x}}}_{\left(i\right)l}^{\left(j\right)}\mid {\stackrel{}{\overline{x}}}_{\left(i\right)l}^{\left(j\right)}\sim N({\mathbf{\mu}}_{\left(i\right)l}^{\left(j\right)},{\mathbf{\Sigma}}_{\left(i\right)})\right\}$$
(11)
$${\mathbf{\mu}}_{\left(i\right)l}^{\left(j\right)}={x}_{\left(i\right)l}^{\left(j\right)}$$
(12)
$${\Sigma}_{\left(i\right)}=\left[\begin{array}{ccc}{\sigma}_{\left(i\right)\mathrm{1,1}}^{2}& \cdots & {\sigma}_{\left(i\right)1,{N}^{d}}^{2}\\ .& & .\\ .& & .\\ .& & .\\ {\sigma}_{\left(i\right){N}^{d},1}^{2}& \cdots & {\sigma}_{\left(i\right){N}^{d},{N}^{d}}^{2}\end{array}\right]$$
(13)
$${f}_{{\stackrel{\xaf}{\stackrel{}{x}}}_{\left(i\right)l}^{\left(j\right)}}\left({\stackrel{}{\stackrel{\xaf}{x}}}_{\left(i\right)l,1}^{\left(j\right)},\dots ,{\stackrel{}{\overline{x}}}_{\left(i\right)l,{N}^{d}}^{\left(j\right)}\right)=\frac{1}{{\left(2\pi \right)}^{\frac{{N}^{d}}{2}}{\mid {\mathbf{\Sigma}}_{\left(i\right)}\mid}^{\frac{1}{2}}}\mathrm{exp}\left(-\frac{1}{2}{\left({\stackrel{}{\stackrel{}{\overline{x}}}}_{\left(i\right)l}^{\left(j\right)}-{\mathbf{\mu}}_{\left(i\right)l}^{\left(j\right)}\right)}^{T}{\mathbf{\Sigma}}_{\left(i\right)}^{-1}\left({\overline{x}}_{\left(i\right)l}^{\left(j\right)}-{\mathbf{\mu}}_{\left(i\right)l}^{\left(j\right)}\right)\right)$$
(14)
$${\stackrel{\u2322}{X}}_{\left(i\right)}^{\left(j\right)}\subset {\stackrel{=}{X}}_{\left(i\right)}^{\left(j\right)}$$