numerical experiments on kinoform synthesis with DSRT method

Set of 10 test images was used in numerical experiments on kinoforms synthesis with DSRT method. Images have 128×128 pixels and 256 halftones. They are shown in Fig. 2. Small image size was chosen to minimize calculation time.

Figure 2. Set of 10 test images used in numerical experiments on kinoforms synthesis with DSRT method. Test images are numbered 1-10 from top left to down right

Synthesized kinoforms had 128×128 elements and 16 phase levels. Typical dependency of NSTD of reconstructed image from desired one during iterative process for image number 1 (“Lenna”) is presented in Fig. 3. Iterative algorithm was designed so that it continues to work until NSTD decrease between two last iterations is higher than certain fixed value. For presented dependency this value equals to 0.00005. Initial NSTD value 0.14 corresponds to kinoform generated with Gerchberg-Saxton method (“Gerchberg-Saxton” kinoform). Achieved NSTD value is 0.10. One iteration calculation time is approximately 100 seconds. PC that was used in calculations has AMD FX-8350 eight cores CPU with operating frequency 4.0 GHz and 8 Gb RAM. It should be noted though those 23 iterations were excessive, they merely demonstrate stagnation process. First iteration provides 65% of total NSTD decrease, second iteration – 82%, third – 88%, fourth – 92%, fifth – 94%. Considering each iteration calculation cost, there is not much point to go on after second or third iteration. However, at numerical experiments minimum NSTD decrease between iterations was set to 0.0002 which led to 10÷14 iterations depending on test image.

Figure 3. Typical dependency of NSTD of reconstructed image from desired one during iterative process

Images reconstructed from initial “Gerchberg-Saxton” kinoform and from “DSRT” kinoform for test image number 1 are presented in Fig. 4.

Figure 4. Images reconstructed from initial “Gerchberg-Saxton” kinoform (left) and from “DSRT” kinoform (right) for test image number 1

It can be seen that image reconstructed from “DSRT” kinoform is significantly less noisy. Results for other test images are similar. They are presented in Table 1. Dependencies of reconstructed images NSTD on test image number for Gerchberg-Saxton and DSRT methods are shown in Fig. 5. Average NSTD of reconstructed images for “Gerchberg-Saxton” and “DSRT” kinoforms equal to 0.152 and 0.115 respectively. NSTD decrease after application of DSRT method varies in range 24÷31%, average NSTD decrease value equals to 26%.

Table 1. NSTD of reconstructed images for Gerchberg-Saxton and DSRT methods

Test image number	Method	NSTD decrease
Gerchberg-Saxton	DSRT
	0.140	0.101	27.9%
	0.157	0.120	23.6%
	0.152	0.115	24.3%
	0.150	0.114	24.0%
	0.134	0.093	30.6%
	0.144	0.106	26.4%
	0.143	0.104	27.3%
	0.149	0.110	26.2%
	0.152	0.115	24.3%
	0.153	0.115	24.8%

Figure 5. Dependencies of reconstructed images NSTD on test image number for Gerchberg-Saxton and DSRT methods

Conclusion

Method of direct search with random trajectory (DSRT) and its application to task of kinoform synthesis are presented. DSRT method allows for easy change of objective criterion to obtain matrix with desired properties. Calculation costs for arrays with different sizes and quantities of gradations were assessed. DSRT method can work with virtually any initial phase distribution, but it was determined that better and faster results are obtained if in place of initial phase distribution kinoform generated with conventional method is used. We used it in conjunction with well-known Gerchberg-Saxton method. Numerical experiments on synthesis of kinoforms with 128×128 elements and 16 phase levels were performed using set of 10 test images. Results are similar for all used test images. Average normalized standard deviation (NSTD) of reconstructed images for initial “Gerchberg-Saxton” kinoforms and final “DSRT” kinoforms equal to 0.152 and 0.115 respectively. NSTD decrease after application of DSRT method varies in range 24÷31%, average NSTD decrease value equals to 26%.

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