Past research has demonstrated that a digital, complex Fresnel hologram can be converted into a phase-only hologram with the use of the bi-direction error diffusion (BERD) algorithm. However, the recursive nature error diffusion process is lengthy and increases monotonically with hologram size. In this paper, we propose a method to overcome this problem. Briefly, each row of a hologram is partitioned into short non-overlapping segments, and a localized error diffusion algorithm is applied to convert the pixels in each segment into phase only values. Subsequently, the error signal is redistributed with low-pass filtering. As the operation on each segment is independent of others, the conversion process can be conducted at high speed with the graphic processing unit. The hologram obtained with the proposed method, known as the Localized Error Diffusion and Redistribution (LERDR) hologram, is over two orders of magnitude faster than that obtained by BERD for a hologram, exceeding the capability of generating quality phase-only holograms in video rate.
© 2014 Optical Society of America
Displaying a complex Fresnel hologram is a difficult issue in practice, as a holographic image is composing of a pair of orthogonal (e.g. the real and the imaginary) components that cannot be displayed directly with a single device. A straightforward solution is to employ a pair of spatial light modulators (SLM), each displaying a component of the complex hologram, and combining the reconstructed images of each component with certain optical means [1–4]. Alternatively, the orthogonal components can be displayed on a single SLM, and integrated optically via a high resolution grating [5–8]. Either approaches require rather complicated optical setup with precise alignment of the building blocks. Another solution is to convert the complex hologram into an amplitude image by retaining only the real component of the hologram. However, the amplitude hologram is comprising of semi-transparent fringes that lower the optical efficiency. In addition, the removal of the imaginary part of the hologram also leads to the generation of a twin image. Although the latter can be suppressed with optical filtering, or by adding an inclined reference beam signal to the complex hologram prior to removing the imaginary component, the twin image consumes about half of the frequency spectrum. A more promising solution is to convert the complex hologram into a phase only hologram (POH) hologram. A POH has a number of attractive features when it compares with the complex or the amplitude hologram. First, the reconstructed image of a POH is inherently free from the zero order diffraction and the twin image. Second, a POH can be directly displayed with a single phase only spatial SLM. Third, a POH exhibits higher optical efficiency and a fuller utilization of the frequency spectrum. Despite these favorable features, generating a POH is a difficult problem. It was found that deriving a POH through eliminating the magnitude component of a complex hologram will lead to substantial degradation to the reconstructed image. An effective and popular approach is to employ the Gerberg-Saxton , or the iterative Fresnel transform  algorithms to compute the phase hologram so that the reconstructed image will match with a target planar image. On the downside, these methods are computationally intensive especially if the object scene is composed of many planar images each positioned at different axial distance from the hologram. A faster solution known as the “One-Step Phase Retrieval” (OSPR) has been developed and reported in [11–13]. In this method, a complex hologram is generated from the points in an object space that has been perpetuated with a random phase signal. Subsequently, a POH is obtained by retaining only the phase component of the complex hologram. The phase hologram can be converted into a binary hologram with error diffusion . The reconstructed images of the phase holograms generated with such means are generally noisy in appearance, albeit processing some of the advantages of a phase-only hologram. Lately, a fast and effective method for converting a complex Fresnel hologram into a phase-only hologram, based on the bidirectional error diffusion (BERD), has been presented in . Experimental evaluation reveals that high fidelity is preserved on the reconstructed image of a POH obtained with this approach, as compare from one derived directly from the original complex hologram. However, due to the recursive nature of the error diffusion process, the speed of conversion is rather slow and not suitable for speed demanding applications (e.g. conversion of complex to phase holograms at video rate). In this paper we shall present an enhanced method known as the “localized error diffusion and redistribution” (LERDR) in that the conversion of a complex Fresnel hologram to a POH can be conducted in a parallel manner. The POH obtained with this approach is referred to as the LERDR hologram. Organization of this paper is given as follows. In section 2, we shall provide a brief outline of the BERD hologram. Our proposed LERDR method is reported in section 3. Experimental evaluation on the proposed method, and a conclusion summarizing the essential findings, are presented in sections 4 and 5, respectively.
2. Bidirectional error diffusion (BERD) hologram
For the sake of clarity of explanation, we would like to outline the BERD algorithm in . To start with, consider a complex Fresnel hologram with each pixel represented by (where and are the vertical and horizontal axes of the co-ordinate system, respectively as shown in Fig. 1(a)). To convert the hologram into a POH with the BERD, the hologram is processed sequentially in a row by row manner. Along the odd and the even rows, the pixels are scanned from left to right direction, and from the right to the left direction, respectively. The value of the current pixel under evaluation is converted to a phase-only quantity by forcing the magnitude to '1' (i.e. a transparent pixel which only changes the phase angle of the light passing through it) as given by Eq. (1).Eq. (2) to a pixel, its error is diffused with pre-defined weightings to the neighborhood members. The error diffusion of a pixel in the odd and the even rows, together with the weighting factors, to , are shown in Figs. 1(b) and 1(c), respectively.
Although the BERD hologram results in less amount of noise in the reconstructed image, the conversion of each hologram pixel involves the accumulated errors of all the previous visited pixels in the scan path. In another words, the BERD algorithm is a strictly recursive process that cannot be realized with parallel computation. For example, the time taken to convert a medium size complex to a POH requires about 0.72sec based on a commodity personal computer (PC). In addition, the computation time will also escalate monotonically with the size of the hologram. As such, the method is not suitable for high speed operation, especially for large size hologram.
3. Proposed localized error diffusion and redistribution (LERDR) hologram
Our proposed method, which is referred to as localized error diffusion and redistribution (LERDR), can be divided into 2 stages and outlined as follows.
3.1 First stage: The localized error diffusion (LERD) process
In the 1st stage, a complex hologram is processed sequentially in a row by row manner. Along each row, the hologram pixels are uniformly partitioned into contiguous segments of M pixels as depicted in Fig. 2(a). In each segment, the pixels are scanned from the left to right, and each visited pixel is converted into a phase only quantity with Eq. (1). For each of the first pixels, its error is diffused to the one on the right, and the 3 pixels directly beneath it as shown in Fig. 2(b). For the last pixel in the segment, its error is only diffused to the pixels beneath it (Fig. 2(c)). Mathematically, the diffusion of error from the first M-1 pixels, and that from the last pixel in a segment, is described in Eqs. (3)-(6), and Eqs. (7)-(9), respectively.14], with , , , , , , . From Figs. 2(b) and 2(c), it can be seen the distribution of error from each pixel is always localized within its own segment, and not propagated to other segment. As such, the error diffusion process of the segments can be conducted in a parallel fashion.
3.2 Stage 2: Error redistribution
As we shall show later, the reconstructed image of a POH obtained with the localized error diffusion (LERD) process in stage 1 will be heavily degraded. The reason is that the error of the last pixel in each segment is not compensated, as in the case of the rest of the members, by the pixel on its right (which belongs to another segment). To alleviate this problem, we apply low-pass-filtering to redistribute the error signal in the POH, so that the error in the last pixel of each segment can be diffused to its adjacent segment. In the realization of low-pass-filtering, the hologram is convolved with a low-pass function asEq. (10) is realized in the frequency spectrum using Fast Fourier Transform, with the transfer function of being a simple box function given by
4. Experimental results
A binary image “CTU” in Fig. 3(a), and the grey level images “Lena” and “Peppers” in Figs. 3(b) and 3(c), are taken to evaluate our proposed method. All the test images have a size of 10241024 pixels. A digital complex Fresnel hologram comprising of 20482048 pixels, each having a square pixel size of 7μm7μm, is generated for each image based on Fresnel diffraction . The wavelength of the optical beam is , and all the images are parallel to, and located at 0.3m from the hologram plane. As the numerical reconstructed images of the 3 complex holograms are visually identical to the original images, they are not shown in here. Next, we convert the holograms into phase only hologram with LERD (1st stage of our proposed method). The length of the segment is 8, i.e., M = 8, and the numerical reconstructed images are shown in Figs. 4(a) to 4(c). We observe that the reconstructed images are very noisy, especially at the dark background regions. Next, we applied the error redistribution process (2nd stage of our proposed method) to the phase only holograms obtained in stage 1, and the reconstructed images of the LERDR holograms are shown in Figs. 5(a) to 5(c). It can be seen that the noise signals are no longer present, and the reconstructed images are very similar to the original ones. For clarity of illustration, the reconstructed images of the 3 complex holograms, together with the images in Figs. 4(a) to 4(c), and Figs. 5(a) to 5(c), are shown in their original size in the media files Media 1, Media 2, and Media 3, respectively. Quantitative evaluation on the quality of the reconstructed images of the abovementioned phase only holograms are listed in Table 1. The evaluation is based on the peak-signal-to-noise ratio (PSNR), a fidelity measurement which is computed with reference to the reconstructed image of the original complex hologram. The PSNR values for the phase-only holograms derived simply from the use of the LERD, and the LERDR process are given in the first and second rows of the table, respectively. The results are in line with the visual quality of the corresponding images, reflecting very low fidelity (small PSNR) for the reconstructed images of the LERD holograms, and high fidelity for the ones derived from the LERDR method. Slight blurring is imposed on the reconstructed images of the LERDR holograms due to low pass filtering, but the effect is not prominent. For the sake of comparison, the results obtained from the BERD method are shown in the last row of Table 1. Regarding the computation efficiency, the time taken to generate a LERDR hologram is less than 6ms based on a personal computer (with a Nvidia GTX 590 GPU), which is about 0.72 sec/6ms = 120 times, over 2 orders of magnitude faster than the BERD method.
In this paper, we report a novel and very fast method for converting a digital, complex Fresnel hologram into a phase-only hologram called the LERDR holograms. Being different from existing approaches that employ a recursive error diffusion mechanism in deriving the POH, our proposed method adopts a localized error diffusion operation that can be executed in parallel on multiple, non-overlapping segments on each row of hologram pixels. As such, the process can be realized in less than 6ms, faster than video rate, for a hologram using a commodity PC that is equipped with a typical graphical processing unit. Experimental evaluation reveals that the reconstructed images of the proposed LERDR holograms are very similar to those obtained with the original complex holograms.
This work is supported by the Chinese Academy of Sciences Visiting Professorships for Senior International Scientists Program under Grant Number 2010T2G17 and the High-End Foreign Experts Recruitment Program, China, under Grant Number GDJ20130491009.
References and links
1. M. Makowski, A. Siemion, I. Ducin, K. Kakarenko, M. Sypek, A. Siemion, J. Suszek, D. Wojnowski, Z. Jaroszewicz, and A. Kolodziejczyk, “Complex light modulation for lensless image projection,” Chin. Opt. Lett. 9, 120008 (2011).
2. M.-L. Hsieh, M.-L. Chen, and C.-J. Cheng, “Improvement of the complex modulated characteristic of cascaded liquid crystal spatial light modulators by using a novel amplitude compensated technique,” Opt. Eng. 46(7), 070501 (2007). [CrossRef]
3. R. Tudela, E. Martín-Badosa, I. Labastida, S. Vallmitjana, I. Juvells, and A. Carnicer, “Full complex Fresnel holograms displayed on liquid crystal devices,” J. Opt. A5, S189–S194 (2003).
5. X. Li, Y. Wang, J. Liu, J. Jia, Y. Pan, and J. Xie, “Color holographic display using a phase-only spatial light modulator” in Dig. Holo. and 3D Img., OSA Technical Digest, OSA, paper DTh2A.3 (2013).
6. S. Reichelt, R. Häussler, G. Fütterer, N. Leister, H. Kato, N. Usukura, and Y. Kanbayashi, “Full-range, complex spatial light modulator for real-time holography,” Opt. Lett. 37(11), 1955–1957 (2012). [CrossRef] [PubMed]
7. H. Song, G. Sung, S. Choi, K. Won, H. S. Lee, and H. Kim, “Optimal synthesis of double-phase computer generated holograms using a phase-only spatial light modulator with grating filter,” Opt. Express 20(28), 29844–29853 (2012). [CrossRef] [PubMed]
9. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).
10. J. Yeom, J. Hong, J.-H. Jung, K. Hong, J.-H. Park, and B. Lee, “Phase-only hologram generation based on integral imaging and its enhancement in depth resolution,” Chin. Opt. Lett. 9(12), 120009 (2011).
11. E. Buckley, “Holographic laser projection technology” Proc. SID Symp., 1074–1078 (2008). [CrossRef]
12. A. J. Cable, E. Buckley, P. Marsh, N. A. Lawrence, T. D. Wilkinson, and W. A. Crossland, “Real-time binary hologram generation for high-quality video projection applications” SID International Symposium Digest of Technical Papers, 35,1431–1433 (2004). [CrossRef]
13. E. Buckley, “Real-time error diffusion for signal-to-noise ratio improvement in a holographic projection system,” J. Disp. Tech. 7(2), 70–76 (2011). [CrossRef]
14. R. W. Floyd and L. Steinberg, “An adaptive algorithm for spatial grey scale,” Proc Soc. Info. Disp. 17, 75–77 (1976).
15. P. W. Tsang and T.-C. Poon, “Novel method for converting digital Fresnel hologram to phase-only hologram based on bidirectional error diffusion,” Opt. Express 21(20), 23680–23686 (2013). [CrossRef] [PubMed]