By Sangwine Horn
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Additional info for Colour Image Processing Handboook
6-12) we can see that only when W(p) > 0 are we assured that a small bias ensures that W(p) will be concentrated near the origin. A. The Minimum-Bias Window Pair The solution to the bias minimization theorem is obtained with the help of the Fejer-Riesz theorem [8, p. 152]. We note that the one-dimensioned form of this problem has been considered in the literature by Papoulis  and that the solution to the two-dimensioned problem has been treated by Stark and Dimitriadis . Hence we shall give the solution here and refer the reader to the listed references for details.
Coherent-Optical Processor As discussed in Chapter 1, the ease with which a lens can perform the Fourier transformation of a two-dimensional distribution is certainly one of the most important assets of coherent-optical processing. Perhaps even more important is the fact that the spectrum of the input is physically accessible and therefore can be manipulated simply by placing masks or optical filters in the Fourier transform plane. The optical processor of Fig. 2-1 is some times called, for obvious reason, a 4-F system.
Smoothing in the frequency plane can be done by analog or digital techniques. Analog smoothing with a uniform window is demonstrated in Figs. 7-2. 7-1 is an estimate of the spectrum offilmgrain 6 X 10 E S D 3h cr \- o UJ Q. CO 32 64 96 128 160 CYCLES / m m FIG. 7-1. Inadequate smoothing of the spectral estimate with a uniform frequency plane window that is too narrow. The window diameter here is 50 μιη. 1. 25 THEORY OF THE OPTICAL FOURIER TRANSFORM 6 X 10 E D I- o ÜJ £L 160 CYCLES / m m FIG. 7-2.
Colour Image Processing Handboook by Sangwine Horn