Applications of Optical Fourier Transforms by Henry Stark

By Henry Stark

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11 CONCLUSION In this chapter we have explored the basic principles associated with obtaining and measuring Fourier spectra using optical systems. We reviewed some basic principles of diffraction theory and showed under what circum­ stances an ideal lens would produce a good Fourier transform of an object. We considered the tradeoff between stability and fidelity in measuring the power spectrum and derived some smoothing windows that had certain optimal properties. We briefly reviewed the problem of measuring the spec­ trum at low spatial frequencies and at the origin.

That \F(u9 v) - Fn(u9 v)\2 du dv = 0. 10-7) n-*ao J Thus not only is the mean square error reduced at every step but, by Eq. 10-7), convergence is ensured. In the case of discrete data, the rms error may still decrease but the convergence of fn(x, y) to /(x, y) generally does not occur, since the discrete signal does not have the analyticity property that the continuous signal has. The dual problem has the same structure. Here we are given that / ( x , y) is spatially bounded to a rectangle |x| < L/2, \y\ < W/2, and we observe its spectrum F(u, v) over a region $ = {(w, I;):|H| < Bx,\v\ < By}.

T-l/006-1-00. Riverside Res. , New York. [7] M. G. Jenkins and D. G. Watts (1968). " HoldenDay, San Francisco, California. [8] N. I. Achieser (1956). " Ungar, New York. [9] A. Papoulis (1973). IEEE Trans. Inf. Theory IT-19, 9-12. [10] H. Stark and B. Dimitriadis (1975). J. Opt. Soc. Am. 65, 425-431. [11] A. Papoulis (1972). J. Opt. Soc. Am. 62, 1423-1429. [12] P. Jacquinot and B. Roizen-Dossier (1964). In "Progress in Optics" (E. ), Vol. Ill, p. 31. North-Holland, Amsterdam. [13] H. Stark, D. Lee, and B.

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