1. Introduction

Holographic projection is an attractive technique due to high quality reconstruction, high optical efficiency and reducing the system size [1, 2]. In color holographic projection, there are two approaches: the first approach is the use of three Spatial Light Modulator (SLMs) such as a phase modulated Liquid Crystal Display (LCD). For example, one method [3] uses three LCD panels to display holograms, which are Computer Generated Holograms (CGHs) or kinoforms (kinoform is also referred as to phase-only CGH), upon which we record the red, green and blue components of a color image. The system can reconstruct the color image by compounding the diffracted lights from each LCD.

The other approach is the use of one SLM. In addition, there are several methods in one SLM approach. Time-division methods [4–7] use red, green and blue reference lights, and one LCD panel. We divide a color image into red, green and blue components, and compute three holograms corresponding to the components individually. The LCD panel displays one of the holograms in sequence and outputs synchronized signals, indicating that one of the holograms is currently displayed on the LCD panel. Red, green and blue reference lights are switched by the synchronized signals and due to the afterimage effect on human eyes, we can observe the color reconstructed image.

Unlike the time-division method, other methods using one SLM, which allows us to record RGB color information on one hologram have been proposed [8, 9, 12]. References [8] and [9] record RGB images on a hologram by placing them at various distances from the hologram. The distances are calculated in order to obtain a fine color reconstructed image when the hologram is simultaneously illuminated by three reference lights of RGB colors. This method is hereafter referred as to the depth-division method (DMM). The detail of the algorithm is described in Refs. [10] and [11].

The space-division method (SDM) [12] can also reconstruct a color image by one hologram and avoid superimposing unwanted images on a wanted color image. The method divides an original color image into RGB components and distributes the RGB components of a color image transversely, then records them on a hologram. The reconstruction from the hologram is simultaneously illuminated by three reference lights of RGB colors.

DMM and SDM correspond evidently to multi-exposure hologram, where three R,G,B holograms are calculated and then are added in a final hologram. As was proven in Ref. [9], such hologram has a decreased depth of a phase modulation and in the consequence they lead to worse reconstructions than elements designed by three hologram method [3]; However, the methods need only one SLM.

In this paper, we propose a color holographic projection using the SDM. We calculated three holograms corresponding to red, green and blue, and then generated one hologram to add the three holograms. Three holograms were optimized by the Gerchberg-Saxton (GS) algorithm [13, 14] for improvement of reconstructed color images. We numerically evaluated the quality of reconstructed color images in terms of the color space of YCbCr, and compared the quality of reconstructed color images by the SDM with that of reconstructed color images by the DMM. In Section 2, we describe the details of color holographic projection with the SDM and DMM. In Section 3, we evaluate results from the SDM and DMM numerically. In Section 4, we conclude this work.

2. Color holographic projection with the space-division method

Figures 1(a) and 1(b) show the recording and reconstruction processes of the DMM. In the recording process, the method divides the RGB components of a color image, and distributes each divided component along the depth direction as shown in Fig. 1(a). We calculate kinoforms from each divided component by calculating Fresnel diffraction. The kinoforms are obtained by taking only the phase components of each diffraction result. Finally, we superimpose the phase components in order to obtain a final kinoform. In the reconstruction process, we obtain a color reconstruction by simultaneously illuminating three reference lights of RGB colors to the final kinoform perpendicularly. The DMM causes unwanted images on the axis; however, when we observe the wanted image, the unwanted images are blurred.

 

Fig. 1 Recording and reconstruction processes of the depth-division method (DMM).(a) recording (b) reconstruction.

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Figures 2(a) and 2(b) show the recording and reconstruction processes of the SDM. In the recording process, the method divides the RGB components of a color image, and distributes transversely each divided component as shown in Fig. 2(a). We calculate kinoforms from each divided component by calculating Fresnel diffraction. Calculation of the blue image employs typical Fresnel diffraction because of on-axis propagation, whereas, calculations of the red and green images employ Shifted-Fresnel diffraction [16] because of off-axis propagation. The kinoforms are obtained by taking only the phase components of each diffraction result. Finally, we superimpose the phase components in order to obtain a final kinoform. In the reconstruction process, we obtain a color reconstruction by simultaneously illuminating three reference lights of RGB colors to the final kinoform. The SDM causes unwanted images; however, it is no problem for the observation of the color reconstructed image because the unwanted images are separated from the wanted image.

 

Fig. 2 Recording and reconstruction processes of the space-division method (SDM).(a) recording (b) reconstruction.

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If we can display a diffraction result with a complex number on an appropriate electric device, we will obtain a complete reconstructed image. However, as it is difficult to develop an electric device that is capable of displaying a complex number, we need to select either the amplitude or the phase components of a diffraction result. This will give rise to deterioration of the reconstructed image. We employ the Gerchberg-Saxton (GS) algorithm as an iterative algorithm [13, 14] in order to improve the deterioration of a reconstructed image.

Figure 3 shows the GS algorithm we used. In the GS algorithm for Fourier hologram, Fourier and inverse Fourier transforms correspond to reconstruction from a hologram and hologram generation, respectively. In this paper, instead of Fourier and inverse Fourier transforms, we use Fresnel and inverse Fresnel diffractions because a Fresnel hologram can reconstruct an image at an arbitrary distance from a hologram. During the iteration, each divided component of a color image is individually optimized. We start the iteration by adding a random phase to each divided component.

 

Fig. 3 Gerchberg-Saxton (GS) algorithm in Fresnel region in order to improve the deterioration of a reconstructed image.

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In SDM, the red and green components employ Shifted-Fresnel diffraction, then, we extract only the phase components (“Phase constraint” in Fig. 3) from the diffracted lights, in order to generate kinoforms. Likewise, the blue image employ Fresnel diffraction, then, we extract only the phase component from the diffracted light.

The kinoforms are reconstructed by inverse Fresnel diffraction for the blue and Shifted-Fresnel diffraction for the red and green. We replace the amplitudes of the results with the original divided components (“Amplitude constraint” in Fig. 3). We execute 30 iterations per each divided component. After the iteration, we superimpose the phase components to make a final kinoform. We used a GPU to accelerate the calculation [15].

3. Results

In this section, we compare the quality of the reconstructed color image by SDM with that by DMM. Figure 4 shows original images with 512 × 512 pixels. The sampling interval is 8.0μm × 8.0μm. We use RGB reference lights whose wavelengths are 633 nm for red, 532 nm for green, and 470 nm for blue, respectively. The kinoform size is 2,048 × 2,048 pixels.

 

Fig. 4 Original images. (a) simple color object (b) Lena (c) Mandrill.

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In DMM, the red, green and blue components are set at 0.2 m, 0.168 m and 0.136 m from a kinoform, repectively. Then, each kinoform correponding to each component is generated by using only the red reference light. The distances are decided by z×λ ₁/λ ₂, where z is reconstruction distance of the wanted image, λ ₁ is wavelength for reconstruction and λ ₂ is wavelength for recording. We obtain the wanted color image at 0.2 m by simultaneously illuminating the RGB reference lights to the final kinoform. Figures 5(a), 5(c), and 5(e) show reconstructed wanted images by DMM.

 

Fig. 5 Reconstructed color images. (a), (c), (e) show reconstructed wanted images by DMM. (b), (d), (f) show reconstructed wanted images by SDM.

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In SDM, as shown in Fig. 2, all components are set at z = 0.2 m from a kinoform, and each component is set at d = 0.01 m transversely. We obtain a wanted color image at 0.2 m by simultaneously illuminating the RGB reference lights to the final kinoform. The incident angle θ of the red and green reference lights is 2.93°. Figures 5 (b), 5(d), and 5(f) show reconstructed wanted images by SDM.

As shown in Fig. 5, SDM can reconstruct better image quality than DMM, because reconstructed wanted and unwanted images by DMM overlap each other, whereas, those by SDM do not. Next, we estimate numerically the image quality in terms of luminance and chrominance, so that we use YCbCr as a color space. “Y” of YCbCr means luminance, “Cb” means blue chrominance and “Cr’ means red chrominance.

Figure 6 shows the Peak Signal to Noise Ratio (PSNR) between the original (Fig. 4(b)) and reconstructed images by DMM and SDM (Figs. 5(e) and 5(f)) versus the number of the iteration. The PSNR is defined as $\mathit{\text{PSN}}R(\mathit{\text{dB}})\hspace{0.17em}=\hspace{0.17em}10{\text{log}}_{10}({255}^2/((\frac{1}{\mathit{\text{MN}}}{\Sigma }^{\mathit{\text{MN}}}{(I_o\hspace{0.17em}-\hspace{0.17em}{I}_r)}^2))$, where M and N are the horizontal and vertical number of pixels for a reconstructed image and original image, I _o and I _r are original and reconstructed images, respectively. In the graph, solid lines indicate PSNRs of SDM and dotted lines indicate PSNRs of DMM. The PSNR almost converges at 30 iterations. Both PSNRs of luminance for SDM and DMM are small values, whereas, PSNRs of Cb and Cr for SDM and DMM are over 20 (dB) at 30 iterations. Moreover, all PSNRs of SDM are better than that of DMM.

 

Fig. 6 PSNR between the original (Fig. 4(b)) and reconstructed images by DMM and SDM (Figs. 5(e) and 5(f)) versus the number of the iteration. Solid lines indicate PSNRs of SDM and dotted lines indicate PSNRs of DMM.

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4. Conclusion

We evaluated a color holographic projection with the space-division method, which could record color information on one hologram. We calculated a final kinoform using the GS algorithm in order to optimize the image quality of a reconstructed image from the kinoform. The image quality of the space-division method was better than that of the depth-division method. In our next work, we will show the optical experimental results of the proposed method.