## Abstract

We describe an approach to compute filters that automatically performs a spatial frequency selection to improve interclass discrimination and to reduce intraclass sensitivity. This approach is achieved by using as input to the filter synthesis a set of reference images to compute the filters and a set of distorted images to introduce the distortion or noise model of the reference images. Simulation results of correlation examples are provided for two pattern-recognition problems and are compared with the ones obtained with the standard minimum average correlation energy filters.

© 1993 Optical Society of America

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### Equations (17)

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(1)
$${r}_{kqm}=\sum _{s,t=0}^{n-1}{h}_{m}(s,t){x}_{kq}(s,t).$$
(2)
$${r}_{kqm}=\sum _{U,V=-n/2}^{(n/2)-1}{H}_{m}(U,V){{X}_{kq}}^{*}(U,V).$$
(3)
$${r}_{kqm}={\mathbf{x}}_{kq}\xb7{\mathbf{h}}_{m},$$
(4)
$${r}_{kqm}={{\mathbf{X}}_{kq}}^{*}\xb7{\mathbf{H}}_{m},$$
(5)
$${F}_{m}=\sum _{s,t=-n/2}^{(n/2)-1}\sum _{kqi}\mid {H}_{m}(U,V){[{X}_{kq}(U,V)-{Y}_{kqi}(U,V)]}^{*}{\mid}^{2}.$$
(6)
$${X}_{kq}(U,V)=0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{if}\frac{1}{2LK}\sum _{k,{k}^{\prime},q,{q}^{\prime}}\frac{{X}_{kq}(U,V){{X}_{{k}^{\prime}{q}^{\prime}}}^{*}(U,V)}{\mid {X}_{kq}(U,V){X}_{{k}^{\prime}{q}^{\prime}}(U,V)\mid}\ge \mathrm{\eta},$$
(7)
$$E(Z,Z)=\sum _{k,q,i}\mid {X}_{kq}(Z)-{Y}_{kqi}(Z){\mid}^{2},$$
(8)
$${F}_{m}={{\mathbf{H}}_{m}}^{+}E{\mathbf{H}}_{m},$$
(9)
$${\mathbf{H}}_{m}={E}^{-1}X({X}^{+}{E}^{-1}X){\mathbf{r}}_{m}.$$
(10)
$${Y}_{kqi}(U,V)={X}_{kq}(U,V)+{N}_{i}(U,V),$$
(11)
$${F}_{m}=\sum _{s,t=-n/2}^{(n/2)-1}\sum _{kqi}\mid {H}_{m}(U,V){{N}_{i}}^{*}(u,V){\mid}^{2}.$$
(12)
$$\text{FR}=\frac{\mid \text{E}[{y}_{1}(0,0)]-\text{E}[{y}_{2}(0,0)]{\mid}^{2}}{\{\text{VAR}[{y}_{1}(0,0)]+\text{VAR}[{y}_{2}(0,0)]\}/2},$$
(13)
$$G=\frac{\text{E}[\mid {y}_{1}(0,0){\mid}^{2}]-\text{E}[\text{max}(\mid {y}_{2}{\mid}^{2})]}{{\{\text{VAR}[\mid {y}_{1}(0,0){\mid}^{2}]\}}^{1/2}+{\{\text{VAR}[\text{max}(\mid {y}_{2}{\mid}^{2})]\}}^{1/2}},$$
(14)
$$G\approx \frac{\text{I}1-\text{I}0}{\text{SD}1+\text{SD}0},$$
(15)
$$\text{PCE}=\frac{\mid {y}_{1}(0,0){\mid}^{2}}{{\displaystyle \sum _{s,t=0}^{n-1}}\mid {y}_{1}(s,t){\mid}^{2}}.$$
(16)
$$\text{SNR}=\frac{\mid \text{E}[{y}_{1}(0,0)]{\mid}^{2}}{\text{VAR}[{y}_{1}(0,0)]}.$$
(17)
$$\text{HE}=\frac{\mid {y}_{1}(0,0){\mid}^{2}}{{\displaystyle \sum _{U,V}}\mid X(U,V){\mid}^{2}},$$