Файл: Ersoy O.K. Diffraction, Fourier optics, and imaging (Wiley, 2006)(ISBN 0471238163)(427s) PEo .pdf

218 APODIZATION, SUPERRESOLUTION, AND RECOVERY OF MISSING INFORMATION

where d0 is the distance from the circular aperture. d is called the Rayleigh distance and is a measure of the limit by which two point sources can be resolved. Note that d is due to the exit pupil of the optical system, which is a finite aperture.

Finite aperture means the same thing as missing information since the wave field is truncated outside the aperture. Below the wave field is referred to as the signal since the topic discussed is the same as signal recovery in signal processing. In addition, the window function is generalized to be a distortion or transformation operator D, as commonly used in the literature on signal recovery. The coverage will still be in 1-D without loss of generality.

For a space-limited signal, the exact recovery of the signal is theoretically possible since the Fourier transform of such a signal is an analytic function [Stark, 1987]. If an analytic function is exactly known in an arbitrarily small region, the entire function can be found by analytic continuation. Consequently, if only a small part of the spectrum of such a signal is known exactly, the total spectrum and the signal can be recovered.

Unfortunately, especially due to measurement noise, the exact knowledge of part of the spectrum is often impossible. Instead, imperfect measurements and a priori information about the signal, such as positivity, finite extent, etc. are the available information.

The measured signal v can be written in terms of the desired signal u as

where D is a distortion or transformation operator. With known D, the problem is to recover z (in system identification, the problem is to estimate D when v and u are known).

The straightforward approach to solve for u is to find D 1 such that

Unfortunately, D is often a difficult transformation. For example, when more than one u corresponds to the same v, D 1 does not exist. Such is the case, for example, when D corresponds to a low-pass filter. Even if D 1 can be approximated, it is often ill-conditioned such that slight errors cause large errors in the estimation of u. The problem of recovery of u under such conditions is often referred to as the inverse problem. In the following sections, signal recovery by contractions and projections will be studied as potential methods for solving inverse problems. These methods are usually implemented in terms of a priori information in the signal and spectral domains.

220 APODIZATION, SUPERRESOLUTION, AND RECOVERY OF MISSING INFORMATION

Another topic of importance is the fixed point(s) of the linear operator A. Consider

Any vector u* which satisfies this equation is called a fixed point of A.

If the mapping A is strictly nonexpansive, and there are two fixed points u* and v*, then

is a contradiction. It is concluded that there cannot be more than one fixed point in a strictly nonexpansive contraction.

14.4.1Contraction Mapping Theorem

When the mapping by A is strictly nonexpansive so that

where 0 < a < 1, then, A has a unique fixed point in S0.

Proof:

Let u0 be an arbitrary point in S0. The sequence uk ¼ Auk 1, k ¼ 1; 2; . . . is formed. The sequence ½uk& belongs to S0. We have