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

240 APODIZATION, SUPERRESOLUTION, AND RECOVERY OF MISSING INFORMATION

The least-squares solution is given by

" #

0:6

^u ¼ Hþy ¼

0:628

EXAMPLE 14.8 Suppose that the solution is constrained in some sense in the signal domain and the constraint is applied as a matrix acting on ^u. For example, bounding the image to be nonzero in a certain region would be such an operator. Determine the least squares solution in this case.

Solution: The error measure to be minimized can be written as

E ¼ jv HTuj2

where T is the constraint operator in matrix form. In the case of boundedness, T is a diagonal matrix whose diagonal elements are 1s and 0s. We have

E ¼ ðv HTuÞtðv HTuÞ

¼ vtv 2utTtHtv þ utTtHtHTu

The gradient of E is given by

g ¼ 2TtHtv þ 2TtHtHTu

Setting g equal to 0 yields

^u ¼ ½TtHtHT& 1TtHtv

provided that the matrix inverse exists.

14.17THE STEEPEST DESCENT ALGORITHM

When the goal is to obtain ^u rather than Hþ, the iterative gradient method can be used. This method is based on the fact that the steepest descent of E ¼ jv Huj2 is in the direction of the negative gradient of E with respect to u.

Suppose that uk is the vector u at the kth iteration. The gradient of E with respect to uk is given by

where ak is a scalar. Using Eq. (14.17-2) in Eq. (14.17-1), gk can be written as

where lmax is the maximum eigenvalue of A ¼ HtH.

If ak is chosen constant, the method is called the one-step gradient algorithm. The optimum value of a for fastest convergence is given by

where lmin is the minimum eigenvalue of A.

If A is highly ill-conditioned, the condition number of A equal to lmax=lmin is large, and convergence may be very slow.

If ak is optimized at each iteration, the method is called the method of steepest descent. The optimal value of ak is given by

When lmax=lmin is large, convergence may still be slow when ak is used.

EXAMPLE 14.9 Determine the approximate least squares solution of Example 14.8 by (a) the one-step gradient algorithm in 12 iterations, (b) the steepest descent algorithm in three iterations.

Solution: (a) In the one-step gradient algorithm, sopt is found as

2

aopt ¼ l1 þ l2 ¼ 0:1

242 APODIZATION, SUPERRESOLUTION, AND RECOVERY OF MISSING INFORMATION

(b) With the steepest descent algorithm, ak is optimized at each iteration according to Eq. (14.17-6). The result after three iterations is

u3 ¼ 0:5592 0:629

where the error E3 ¼ jv Hu3j2 ¼ 1:0285.

14.18THE CONJUGATE GRADIENT METHOD

In the conjugate gradient method, the scalar ak as well as the correction vector is optimized at each iteration. This allows faster convergence. For this purpose, Eq. (14.17-2) is replaced by

where N is the rank of A.

Due to these properties, the algorithm converges in N iterations. The iterative equations in addition to Eq. (14.18-1) are as follows:

m¼0

When N is a large number, the algorithm can be stopped earlier than N iterations. If the rank of H is M < N, it is satisfactory to run the algorithm with M iterations.

EXAMPLE 14.10 Determine the conjugate gradient method for the least squares problem of Example 14.8.

Solution: The least-squares solution satisfies

TtHtHTu ¼ TtHtv

This is in the same form as before with the replacement of H by H0 ¼ HT. The only iterative equation which needs to be modified is the equation for the gradient, which becomes

gk ¼ gk 1 ak 1TtHtHrk 1

The initial image u0 is chosen as satisfying the constraint represented by T:

Tu0 ¼ u0

g0 rm and r0 are given by

g0 ¼ r0 ¼ TtHtðv Hu0Þ

Since DOEs are based on diffraction, they are highly dispersive, namely, their properties depend on wavelength. Hence they are more commonly used with monochromatic light. However, the dispersive properties can also be used to advantage. For example, light at one wavelength can be focused at one point, and light at another wavelength can be focused at another point in space. In devices such as optical phased arrays used in optical communications and networking, dispersion is used to separate different wavelengths at different focal points. In addition, refractive and diffractive optical elements can be combined in such a way that wavelength dispersion can be removed, and goals such as removal of spherical aberration can be achieved.

There are several steps in the creation of a DOE. These are sampling of waves, computation of wave propagation, usually with the FFT, coding of complex wave information on the hologram, for example, as a real and positive image on the hologram plane, and recording of the resulting DOE.

In 3-D wave coding, phase is usually much more important to be represented correctly than amplitude. Hence, phase is given much more attention. In applications in which the output image intensity is used, output amplitude variations can be significantly reduced by multiplying the output image with a random phase factor (diffuser). This does not change the intensity of the image [Burckhardt, 1970].

This chapter consists of ten sections. The first few methods discussed utilizes the Fourier transformation property of a lens and generate a DOE design for a Fourier transform hologram. The first such method is the Lohmann method discussed in Section 15.2. In this method, amplitude modulation is achieved by controlling the sizes of the apertures, and phase modulation is achieved by controlling the positions of the apertures generated. Section 15.3 discusses the approximations involved in implementing DOEs based on the Lohmann method. These approximations are compensated for in a method called the Lohmann-ODIFIIT method in Section 16.8. Section 15.4 simplifies the Lohmann method by choosing constant aperture size, implying constant amplitude. In order to compensate for the error generated, iterative optimization discussed in Chapter 14 is utilized. Section 15.5 describes computer experiments with the Lohmann methods. Another Fourier method based on hard-clipping of the hologram transmittance function resulting in a binary hologram is discussed in Section 15.6.

The methods discussed so far assume that the reconstructed image is on an output plane. The method discussed in Section 15.7 is capable of generating output image points at arbitrary locations in 3-D. It is a simple method to use. It was the method used in demonstrating for the first time how to make DOEs with a scanning electron microscope system. Section 15.8 extends the method of Section 15.7 and makes it capable of generating many object points in 3-D space.

Essentially all DOE methods involve nonlinear coding of amplitude and phase. As a result, they generate harmonic images. When the FFT is used, the implicit assumption is that the object is periodic. This also causes the images to repeat in a periodic fashion on the reconstruction plane. The one-image-only holography method discussed in Section 15.9 generates a single image by using semi-irregular sampling and a spherical reference wave whose origin is located near the hologram