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

Figure 16.11. The binary hologram generated with the IIT method for the cat brain image.

Figure 16.11 shows the binary hologram generated with the IIT method for the cat brain image. Figure 16.12 shows how the error is reduced as a function of the number of iterations. Figure 16.13 shows the corresponding He–Ne laser beam reconstruction.

Iteration number

Figure 16.12. Error reduction as a function of iteration number in IIT design.

Figure 16.15. Interlacing of subholograms in ODIFIIT with m ¼ n ¼ 2.

conjugate of the reconstructed image exists in the region Rþ due to the real-valued CGH transmittance. Since the binary CGH has cell magnitude equal to unity, it is important that the desired image is scaled so that its DFT is normalized to allow a direct comparison to the reconstructed image hðm; nÞ.

The total CGH is divided into m n subholograms, or blocks, where m ¼ M=A and n ¼ N=B: m and n are guaranteed to be integers if M, N, A, and B are all powers of two. Utilizing decimation-in-frequency [Brigham, 1974], the

Defining Hðk; lÞ as the sum of all the subholograms, the expression for the reconstructed image becomes

where 0 m < M, 0 n < N.

The reconstructed image in the region R is computed by replacing m and n by m þ M1 and n þ N1, respectively, and letting m and n span just the image

region:

hðm þ M1; n þ N1Þ

where 0 m A 1, 0 n B 1.

Let ha;bðm; nÞ be the size A B inverse discrete Fourier transform of the ða; bÞth subhologram:

ha;bðm; nÞ ¼ IDFTAB½Hðmk þ a; nl þ bÞ&m;n

where 0 a m 1, 0 b n 1, 0 m A 1, 0 n B 1.

Using the IDFT of size A B, the reconstructed image inside the region R becomes

the size A B IDFTs of all the subholograms. From this equation, it can be seen that the reconstructed image in the region R due to the ða; bÞth subhologram is given by

which is the IDFT of the ða; bÞth block times the appropriate phase factor, divided by mn.

An array, which will be useful later on, is defined as follows:

~ ð þ þ Þ ¼ ð þ þ Þ 0 ð þ þ Þ ð Þ ha;b m M1; n N1 h m M1; n N1 ha;b m M1; n N1 : 16:5-6

This is the reconstructed image in the region R due to all the subholograms except the ða; bÞth subhologram.

Conversely, given the desired image in the region R, the transmittance values can be obtained. From Eq. (16.5-2)

where 0 k A 1, 0 l B 1, 0 a m 1, 0 b n 1:

Therefore, the transmittance values of the subhologram ða; bÞ that create the image hðm þ M1; n þ N1Þ in the region R are given by

h i

Hðmk þ a; nl þ bÞ ¼ WA M1kWB N1lDFTAB hðm þ M1; n þ N1ÞWMðmþM1ÞaWN ðnþN1Þb

k;l

ð16:5-9Þ

where 0 k A 1, 0 l B 1:

Using Eqs. (16.5-5) and (16.5-9), we can compute the reconstructed image in the region R due to each individual subhologram, or, given a desired image in the region R, we can determine the transmittance values needed to reconstruct that desired image. Therefore, we can now utilize the IIT to design a CGH.

Letting f0ðm þ M1; n þ N1Þ, 0 m A 1, 0 n B 1, be the the desired image of size A B, the ODIFIIT algorithm can be summarized as follows:

1.Define the parameters M, N, A, B, M1, and N1, and determine m and n. Then, divide the total CGH into m n interlaced subholograms.

2.Create an initial M N hologram with random transmittance values of 0 and 1.

3.Compute the M N IDFT of the total hologram. The reconstructed image in

the region R is the points inside the region R, namely, hðm þ M1; n þ N1Þ, 0 m A 1, 0 n B 1.

4.The desired image f ðm þ M1; n þ N1Þ is obtained by applying the phase of each point hðm þ M1; n þ N1Þ to the amplitude f0ðm þ M1; n þ N1Þ as in the POCS method. So,

where fmþM1;nþN1 ¼ argfhðm þ M1;n þ N1Þg.

is the reconstructed image in the region R due to all the subholograms except the ða; bÞth subhologram.

7.Determine the error image that the ða; bÞth subhologram uses to reconstruct (i.e., the error image) as

which is equivalent to the error image in the IIT method.

8.Using Eq. (16.5-9), find the transmittance values Eðmk þ a; nl þ bÞ for the current block that reconstructs the error image.

9.Design the binary transmittance values of the current block as

10. Find the new reconstructed image h0a;bðm þ M1; n þ N1Þ in the region R due to the current block.

11. Determine the new total reconstructed image hðm þ M1; n þ N1Þ by adding

0 ð þ þ Þ ~ ð þ þ Þ the new ha;b m M1; n N1 to ha;b m M1; n N1 .

12. With the new hðm þ M1; n þ N1Þ, use Eq. (16.5-10) to update f ðm þ M1;

n þ N1Þ.

13.Repeat steps 7–12 until the transmittance value at each point in the current block converges.

14.Update the total hologram with the newly designed transmittance values.

15.Keeping l the same, repeat steps 3–14 (except step 5) for all the subholograms.

16.After all the blocks are designed, compute the MSE from Eq. (16.3-7).

17.Repeat steps 3–16 until the MSE converges. Convergence indicates that the optimal CGH has been designed for the current l.

16.5.1Experiments with ODIFIIT

The ODIFIIT method was used to design the DOEs of the same binary E and girl images that were used in testing Lohmann’s method. There two images are shown in Figure 15.3 and Figure 15.8, respectively. A higher resolution 256 256 grayscale image shown in Figure 16.16 was also used.

The computer reconstructions from the ODIFIIT holograms are shown in Figures 16.17–16.19.

All the holograms designed using the ODIFIIT used the interlacing pattern as shown in Figure 16.15. There are many different ways in which the subholograms