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

The two projections of interest can be written as follows:

number of papers, the relative directions given by fðf Þ and cðf Þ are not considered, and P2 is expressed as

It is possible that fðf Þ is not known for all f. Suppose that fðf Þ is known for f in a set S. Then P2 can be expressed as

where Uðf Þ is the FT of u(t). It is observed that P1 is a linear operator. Then, the optimal l1k is 1. A lower bound for l2k can be derived as [Stark ]

lL can be used as an estimate of l2k. However, lL may be greater than 2. In later iterations, it is necessary to restrict l2k to be less than 2 to guarantee convergence. For example, minðlL; 1:99Þ can be chosen.

Since l1k equals 1, one cycle of iterations can be written as

232 APODIZATION, SUPERRESOLUTION, AND RECOVERY OF MISSING INFORMATION

14.12 RECONSTRUCTION FROM A DISCRETIZED PHASE FUNCTION BY USING THE DFT

The discrete signal of length N will be denoted by u½n&. The phase of the DFT of u½n& as well as the support of u½n& are assumed to be known. The support of u½n& will be defined as follows:

u½n& ¼ 0 outside 0 n < M and u½0& ¼6 0

It can be shown that N 2M for the reconstruction algorithm to be satisfactory [Hayes et al., 1980]. The DFT of u½n& is given by

j½k& is assumed to be known.

The procedure of the algorithm is as follows:

1. Initialize A½k& with an initial guess, A1½k&. Then, the initial U1½k& is given by

2.Compute the inverse DFT of U1½k&, yielding the first estimate u1½n&. Let i equal 1.

3.Define a new sequence vi½n& as

4. Compute the DFT Vi½k& of vi½n&. Vi½k& can be written as

The rest of the procedure can be written in two ways with respect to two different algorithms as follows, respectively.

Algorithm 1 [Levi and Stark]

5. Project Vi½k& as follows:

This operation corresponds to P2. This equation can be modified so that T2 is used by

l2i is computed by using the relation (14.11-6).

RECONSTRUCTION FROM A DISCRETIZED PHASE FUNCTION BY USING THE DFT 233

6.Generate a new estimate of uiþ1½n& by computing the inverse DFT of

Uiþ1½k&.

7.Repeat steps 2–6 until convergence.

Algorithm 2 [Hayes et al.]

The only different step is the following: 8. Project Vi½k& as follows:

All the other steps are the same as in Algorithm 1.

EXAMPLE 14.5 Let the error Ei at the ði þ 1Þth iteration be defined as

where u½n& is the desired signal to be reconstructed. Prove that Ei is nonincreasing as i increases.

Solution: By Parseval’s theorem, the following is true:

Since Z(k) and Zi(k) have the same phase, the last equation can be written as

We also define

The triangle equation for vector differences can be used to note the following:

Eiþ1 Ei0

with equality if Uiþ1½k& ¼ Vi½k&. The inequality above is due to the fact that Uiþ1½k& has the same phase as U½k& whereas Vi½k& has some other phase.

234 APODIZATION, SUPERRESOLUTION, AND RECOVERY OF MISSING INFORMATION

Again using the Parseval’s theorem on this inequality results in

n¼0

Since vi½n& is before the projection to create uiþ1½n&, the above equation shows that the projection reduces the error.

14.13GENERALIZED PROJECTIONS

There are many signal recovery problems involving nonconvex constraints. For example, digitizing a signal and constraining a signal or its Fourier transform to have a specified magnitude are nonconvex projections.

When Si is a nonconvex set, there may be more than one point satisfying the definition of a projector Pi. This discrepancy can be removed by specifying additional constraint(s). Another problem with nonconvex sets is that a proof for the existence of a projection has not been found.

In the discussion below, the number of projections M will be limited to 2. In this case, the following error measure is useful:

The following theorem determines the convergence properties of generalized projections with m ¼ 2 [Levi–Stark]:

Theorem: The recursion given by