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

226 APODIZATION, SUPERRESOLUTION, AND RECOVERY OF MISSING INFORMATION

Figure 14.5. Deconvolution of blurred 2-D signal: (a) Gaussian blurring function, (b) blurred signal,

(c) recovered signal by deconvolution [Courtesy of Schafer, Mersereau, Richards].

Figure 14.6. Examples of a convex set and a nonconvex set.

EXAMPLE 14.3 Consider the set S3 of all real nonnegative functions uðx; yÞ (such as optical field amplitude) that satisfy the energy constraint

ðð

juj2 ¼ juðx; yÞj2dxdy E

Show that S3 is convex.

Solution: Consider u1; u2 2 S3. For 0 l 1. The Cauchy–Schwarz inequality to will be used to write

jlu1 þ ð1 lÞu2j lju1j þ ð1 lÞju2j lE1=2

Hence, the set is convex.

EXAMPLE 14.4 Consider the set S4 of all real functions in S whose amplitudes must lie in the closed interval [a,b], a < b, and a 0, b > 0. Show that S4 is convex. Solution: For 0 l 1, we write

u3 ¼ lu1 þ ð1 lÞu2 la þ ð1 lÞa ¼ a

for the lower bound, and

u3 ¼ lu1 þ ð1 lÞu2 lb þ ð1 lÞb ¼ b

for the upper bound. Hence, u3 2 S0, and S4 is convex.

14.8METHOD OF PROJECTIONS ONTO CONVEX SETS

The method of POCS has been successfully used in signal and image recovery. The viewpoint is that every known property of an unknown signal u is considered as a constraint that restricts the signal to be a member of a closed convex set Si. For M properties, there are M such sets. The signal belongs to S0, the intersection of these sets:

\M

The signal restoration problem is defined as projecting v, the corrupted version of u, onto S0.

Let Pi denote the projector for the set Si. Let us also define the operator Ti by

If ui is the fixed-point of Pi in Si, we get

There are two possible ways to estimate the optimal values of l1k and l2k for fast convergence. In the first case, l1k is first optimized, and T1 is applied. This is followed by the optimization of l2k and the application of T2. In general, it can be shown that for a linear projector lik ¼ 1, and lik 0 otherwise. In the second case, l1k and l2k can be optimized simultaneously so that T ¼ T2T1 results in minimum error [Stark].

14.9GERCHBERG–PAPOULIS (GP) ALGORITHM

The original GP algorithm is a special case of the POCS algorithm, and uses projectors P1 and P2 in the signal and the Fourier domains, respectively [Gerchberg]. Hence

where P1 and P2 are defined as follows:

Let A be a region in R2, and S1 be the set of all functions in S that vanish almost everywhere (except, possibly, over a point set of measure zero) outside A. P1 is

Let S2 be the set of all functions whose Fourier transforms assume a prescribed function Vðfx; fyÞ over a closed region B in the Fourier plane. P2 is defined by

It can be shown that both S1 and S2 are convex sets.

14.10OTHER POCS ALGORITHMS

In the GP algorithm, the two projection operators P1 and P2 are with respect to the convex sets S1 and S2 discussed above, respectively. Now, we introduce three more projection operators utilizing convex sets.

Consider the set S3 introduced in Example 14.3. Assume u is complex. It can be written as

230 APODIZATION, SUPERRESOLUTION, AND RECOVERY OF MISSING INFORMATION

where uR; uI denote the real and complex parts of u, respectively. xþR is also defined as

and E is the estimated or known energy of the desired image. ERþ is bounded. Consider the set S4 defined in Example 14.4. The projection operator P4 is

defined by

Finally, consider the subset S5 of all real nonnegative functions in S. S5 can be

where u ¼ uR þ juI as before.

Some examples of the applications of the theory presented above is given in the following sections.

14.11RESTORATION FROM PHASE

For the sake of simplicity, the 1-D case will be discussed below. The signal uðxÞ will be assumed to have a compact support ½ a; a&. The known phase of the FT of uðxÞ is fðf Þ. The amplitude of the signal is to be estimated. It can be shown that the necessary condition for uðxÞ to be completely specified by fðf Þ within a scale factor