## Abstract

The Fukunaga–Koontz (F–K) transform is a linear transformation
that performs image-feature extraction for a two-class image classification
problem. It has the property that the most important basis functions for
representing one class of image data (in a least-squares sense) are also the
least important for representing a second image class. We present a new method
of calculating the F–K basis functions for large dimensional imagery by
using a small digital computer, when the intraclass variation can be
approximated by correlation matrices of low rank. Having calculated the
F–K basis functions, we use a coherent optical processor to obtain the
coefficients of the F–K transform in parallel. Finally, these
coefficients are detected electronically, and a classification is performed by
the small digital computer.

© 1982 Optical Society of America

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### Equations (21)

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(1)
$$\begin{array}{l}{W}_{1}=\left[{\left(\frac{{P}_{1}}{{M}_{1}}\right)}^{1/2}{{\mathbf{X}}_{1}}^{(1)}\hspace{0.17em}{\left(\frac{{P}_{1}}{{M}_{1}}\right)}^{1/2}{{\mathbf{X}}_{2}}^{(1)}\hspace{0.17em}{\left(\frac{{P}_{1}}{{M}_{1}}\right)}^{1/2}{{\mathbf{X}}_{3}}^{(1)}\cdots {\left(\frac{{P}_{1}}{{M}_{1}}\right)}^{1/2}{{\mathbf{X}}_{{M}_{1}}}^{(1)}\right],\\ {W}_{2}=\left[{\left(\frac{{P}_{2}}{{M}_{2}}\right)}^{1/2}{{\mathbf{X}}_{1}}^{(2)}\hspace{0.17em}\left(\frac{{P}_{2}}{{M}_{2}}\right){{\mathbf{X}}_{2}}^{(2)}\hspace{0.17em}{\left(\frac{{P}_{2}}{{M}_{2}}\right)}^{1/2}{{\mathbf{X}}_{3}}^{(2)}\cdots {\left(\frac{{P}_{2}}{{M}_{2}}\right)}^{1/2}{{\mathbf{X}}_{{M}_{2}}}^{(2)}\right],\end{array}$$
(2)
$${W}_{t}=\left[{\left(\frac{{P}_{1}}{{M}_{1}}\right)}^{1/2}{{\mathbf{X}}_{1}}^{(1)}\hspace{0.17em}{\left(\frac{{P}_{1}}{{M}_{1}}\right)}^{1/2}{{\mathbf{X}}_{2}}^{(1)}\hspace{0.17em}\cdots {\left(\frac{{P}_{1}}{{M}_{1}}\right)}^{1/2}{{\mathbf{X}}_{{M}_{1}}}^{(1)}\hspace{0.17em}{\left(\frac{{P}_{2}}{{M}_{2}}\right)}^{1/2}{{\mathbf{X}}_{1}}^{(2)}\hspace{0.17em}{\left(\frac{{P}_{2}}{{M}_{2}}\right)}^{1/2}{{\mathbf{X}}_{2}}^{(2)}\cdots {\left(\frac{{P}_{2}}{{M}_{2}}\right)}^{1/2}{{\mathbf{X}}_{{M}_{2}}}^{(2)}\right],$$
(3)
$${W}_{t}{{W}_{t}}^{+}={W}_{1}{{W}_{1}}^{+}+{W}_{2}{{W}_{2}}^{+},$$
(4)
$${W}_{i}{{W}_{i}}^{+}=\frac{{p}_{i}}{{M}_{i}}\sum _{j=1}^{{M}_{i}}{{\mathbf{X}}_{j}}^{(i)}\hspace{0.17em}{{\mathbf{X}}_{j}}^{(i)+},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}i=1,2$$
(5)
$$({{W}_{t}}^{+}{W}_{t})E=E\mathrm{\Lambda}$$
(6)
$$({W}_{t}{{W}_{t}}^{+})\hspace{0.17em}({W}_{t}E)=({W}_{t}E)\mathrm{\Lambda}.$$
(7)
$$({W}_{t}{{W}_{t}}^{+})\hspace{0.17em}({W}_{t}E{\mathrm{\Lambda}}^{-1/2})=({W}_{t}E{\mathrm{\Lambda}}^{-1/2})\mathrm{\Lambda}.$$
(8)
$${({W}_{t}E{\mathrm{\Lambda}}^{-1/2})}^{+}\hspace{0.17em}({W}_{t}E{\mathrm{\Lambda}}^{-1/2})={\mathrm{\Lambda}}^{-1/2}{E}^{+}\hspace{0.17em}({{W}_{t}}^{+}{W}_{t}E){\mathrm{\Lambda}}^{-1/2}={\mathrm{\Lambda}}^{-1/2}{E}^{+}E\mathrm{\Lambda}{\mathrm{\Lambda}}^{-1/2}={\mathrm{\Lambda}}^{-1/2}\mathrm{\Lambda}{\mathrm{\Lambda}}^{-1/2}=I.$$
(9)
$$({\mathrm{\Lambda}}^{-1/2}{E}^{+}{{W}_{t}}^{+})\hspace{0.17em}({W}_{t}{{W}_{t}}^{+})\hspace{0.17em}({W}_{t}E{\mathrm{\Lambda}}^{-1/2})=\mathrm{\Lambda}.$$
(10)
$${\mathrm{\Lambda}}^{-1}{E}^{+}{{W}_{t}}^{+}({W}_{t}{{W}_{t}}^{+}){W}_{t}E{\mathrm{\Lambda}}^{-1}=I.$$
(11)
$$[{\mathrm{\Lambda}}^{-1}{E}^{+}{{W}_{t}}^{+}({W}_{1}{{W}_{1}}^{+}){W}_{t}E{\mathrm{\Lambda}}^{-1}]\mathrm{\Psi}=\mathrm{\Psi}\mathrm{\Gamma},$$
(12)
$$[{\mathrm{\Lambda}}^{-1}{E}^{+}{{W}_{t}}^{+}({W}_{2}{{W}_{2}}^{+}){W}_{t}E{\mathrm{\Lambda}}^{-1}]\mathrm{\Theta}=\mathrm{\Theta}M.$$
(13)
$${\mathrm{\Psi}}^{+}{\mathrm{\Lambda}}^{-1}{E}^{+}{{W}_{t}}^{+}\hspace{0.17em}({W}_{1}{{W}_{1}}^{+})\hspace{0.17em}{W}_{t}E{\mathrm{\Lambda}}^{-1}\mathrm{\Psi}=\mathrm{\Gamma}.$$
(14)
$$({{\psi}_{i}}^{+}{\mathrm{\Lambda}}^{-1}{E}^{+}{{W}_{t}}^{+}),$$
(15)
$${\mathrm{\Lambda}}^{-1}{E}^{+}{{W}_{t}}^{+}({W}_{t}{{W}_{t}}^{+}-{W}_{2}{{W}_{2}}^{+}){W}_{t}E{\mathrm{\Lambda}}^{-1}\mathrm{\Psi}=\mathrm{\Psi}\mathrm{\Gamma}$$
(16)
$${\mathrm{\Lambda}}^{-1}{E}^{+}{{W}_{t}}^{+}\hspace{0.17em}({W}_{2}{{W}_{2}}^{+}){W}_{t}E{\mathrm{\Lambda}}^{-1}\mathrm{\Psi}=\mathrm{\Psi}(I-\mathrm{\Gamma}).$$
(17)
$$\mathrm{\Theta}=\mathrm{\Psi}$$
(18)
$$M=I-\mathrm{\Gamma}.$$
(19)
$${\mu}_{i}=1-{\gamma}_{i},$$
(20)
$$h*(-x,-y)=\sum _{pq}^{b}{f}_{pq}*(x+p\mathrm{\Delta},y+q\mathrm{\Delta})\times \text{exp}\hspace{0.17em}[-i{\varphi}_{r}(x+p\mathrm{\Delta},y+q\mathrm{\Delta})],$$
(21)
$$\begin{array}{l}C\hspace{0.17em}({x}^{\prime},{y}^{\prime})=\iint g(x,y)\hspace{0.17em}\text{exp}\hspace{0.17em}[i{\varphi}_{r}(x,y)]\\ \times \hspace{0.17em}\sum _{p,q}^{M}{f}_{pq}*(x-{x}^{\prime}+p\mathrm{\Delta},y-{y}^{\prime}+q\mathrm{\Delta})\\ \times \hspace{0.17em}\text{exp}\hspace{0.17em}[-i{\varphi}_{r}(x-{x}^{\prime}+p\mathrm{\Delta},y-{y}^{\prime}+q\mathrm{\Delta})]\text{d}x\text{d}y\\ =\sum _{p,q}^{M}\left[\iint g(x,y){f}_{pq}*(x,y)\text{d}x\text{d}y\right]\\ \times \hspace{0.17em}\delta ({x}^{\prime}-p\mathrm{\Delta},{y}^{\prime}-q\mathrm{\Delta}).\end{array}$$