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Abstract
We report a novel and non-iterative method for the generation of phase-only Fourier hologram for image projection. Briefly, target image is first added with a special quadratic phase and then padded with zeros. A complex Fourier hologram is generated via the simple fast Fourier transform. Subsequently, the error diffusion algorithm is applied to convert the complex hologram into a phase-only hologram. The numerical, as well as the optical reconstructed images with the proposed method are of higher visual quality and contain less speckle noise compared to the original random phase method, which add the random phase to the target image and then preserve the phase component of the complex hologram. The influences of quadratic phase and zero-padding on the image quality are also discussed in detail.
© 2017 Optical Society of America
1. Introduction
Holographic projection is an attractive technique, because it possesses not only the merit of laser display but also the character of aberration-free, very simple optical system and potentials of three-dimensional display [1–4]. Nowadays, spatial light modulators (SLMs) are adopted to display the hologram. Yet, available SLM are only capable of displaying either the magnitude or the phase component of a complex hologram. Compared to amplitude hologram, phase-only hologram has theoretically 100% diffraction efficiency and no conjugate image. Therefore, how to generate a phase-only hologram has been a topic of immense interest [5].
Random phase method is most widely used. Briefly, random phase is superimposed on the target image to diffuse the light and then the magnitude component of the complex hologram is discarded and only the phase component is retained as phase-only hologram [6–8]. Due to the random phase is imposed on the object points, strong intensity fluctuations occur in the reconstructed image, which severely degrades the image quality. Furthermore, discarding the magnitude component directly will also introduce the noise. To improve the speckle noise, Gerchberg-Saxton algorithm [9] and various iterative phase retrieval algorithms [10–12], have been proposed. These methods improve the speckle noise. However, they are time-consuming because repeated forward and inverse diffraction calculations between image plane and hologram plane are needed.
Subsequently, random-phase free and no time-consuming methods have been proposed, such as the down-sampling method [13]. The image is first down-sampled into a sparse form uniformly and then retained the phase component of the Fresnel hologram as the phase-only hologram. The reconstructed image is clear and bright, but degraded with numerous empty holes. Then two phase-only holograms are generated for an image, each based on a different down-sampling lattice. And then the holograms are displayed alternately to achieve a densely sampled image [14]. Another representative method is the error diffusion method [15], which is used to convert a complex Fresnel hologram to phase-only hologram. For each pixel in the complex hologram, the magnitude is forced to be unity and the phase is preserved. Subsequently, the resulting error is diffused to the neighboring pixels [16]. The reconstructed image is almost identical to the source image. But the size cannot exceed the hologram because the object light does not widely spread, since no random phase is added in the hologram generation.
Then a simple and computationally inexpensive method was proposed by multiplying the object light with the virtual convergence light, which drastically improve the image quality and reduce the speckle noise [17–19]. The essence of the method lies in that the object light should be widely distributed on the hologram. Then the real part or the argument of the complex amplitude on the hologram is enough to reconstruct the image with high image quality. Based on this idea, a new non-iterative method to generate the phase-only Fourier hologram is proposed in this paper. High quality reconstruction from Fourier holograms has significant applications in holographic display, beam shaping and so on. The reconstructed images with the proposed method have high quality and needs no other optimization.
2. Proposed method
The proposed non-iterative method for the generation of phase-only Fourier hologram is shown in Fig. 1. Let I(m,n) represent the intensity image of an object. Then an quadratic phase φ(m,n) is added to the target image, which can be expressed as
where a and b are constant and lies between 0 and 1, m and n are pixel coordinates. If the target image has M × N pixels, then x and y take the values of m = -M/2, -M/2 + 1,…,M/2-1, y = -N/2, -N/2 + 1,…,N/2-1. In result, the complex amplitude distribution of the object can be written as followsThe collimated plane wave is always used to illuminate the hologram for image projection. And the Fourier holograms record the spectrum of the object wave. Consequently, the complex hologram can be derived via the simple fast Fourier transform operation of the complex amplitude distribution of the object.
where denotes the convolution operation, h(u, v) and p(u, v) are the Fourier transform of and , respectively. Generally, images contain many low spatial frequency components, and thus its spectrum cannot spread widely, as can be seen the h(u,v) in Fig. 1. Thereafter, if the error diffusion method is adopted to convert the h(u,v) into phase-only distribution. The reconstructed image with this phase-only hologram has poor image quality and very low diffraction efficiency, which will be given in section 4. Conversely, the spectrum p(u,v) of the quadratic phase is relative smooth and meanwhile band-limited when a suitable parameter a and b are set. We can see that when the convolution of h(u,v) and p(u,v) is taken, the spectrum H(u,v) will be more smooth than the original spectrum h(u,v).Thereafter, the bidirectional error diffusion method is adopted to convert the complex hologram into a phase-only hologram. In this method, the magnitude of each pixel of the hologram is set to the value of unity and the error is diffused to the neighborhood pixels as follows:
where arg() denotes the argument part of complex amplitude, ← indicates the current value is overwritten. The amplitude distribution of the spectrum H(u,v) should be normalized before the error diffusion operation. The Floyd-Steinberg coefficients w1 = 7/16, w2 = 3/16, w3 = 5/16, w4 = 1/16 are adopted. The pixels are scanned sequentially in a row by row manner. And the odd and even rows are scanned in the opposite direction. For more details, see [15].Now, how to determine the quadratic phase is described. From the mathematical point of view, quadratic phase can be considered as a point light source. Thus, the addition of a quadratic phase to the image is equivalent to illuminating the image with a divergent point light source at a specific distance. As shown the simple Fourier transform optical system in Fig. 2, the image and the hologram are located in the front and back focal planes of the lens, respectively. It can be seen that when the image is illuminated with a point light source located at a different position in the axial direction, different band width of the spectrum can be obtained at the hologram plane, which lies at the back focal plane of the lens. As mentioned earlier, the main function of the quadratic phase is to smooth the spectrum of the object and make the spectrum band-limited. Therefore, the position of the point light source should be chosen carefully so that the band width of the spectrum is as close as possible to the hologram size.
Assuming that the size of the object and hologram are So and Sh respectively, the distance between the point light source and the lens is l1, the distance between the image point and the lens is l2. According to the imaging relationship of the lens and simple geometric relations, we can derive the following equations:
Thereafter the position of the divergent point light source can be expressed as:And the light field distribution of the divergent point light source on the image plane can be written as:where the dx and dy are the sampling interval of the image in the x and y direction, respectively. Compare to Eq. (1), the parameter a is:Combine the Eq. (8) and the Eq. (10), we can obtain:where dxh is the sampling interval of the hologram. Usually, the size of the hologram Sh is much smaller than the size of the object So. Hence the parameter a in the quadratic phase can be approximately determined by π/M, where M is the pixel number of the image in one direction. For example, if the hologram has a sampling interval dxh = 8µm, the pixel number of the image is 256 × 256, the wavelength is 650nm, the focal lens of the lens is 30cm. Then the size of the hologram and image are Sh = M × dxh = 2.048mm and So = λf/dxh = 23.7mm. The ratio Sh/ So is 0.086, which is much smaller than 1. And the calculated parameter is a = 0.011, which is similar to the value predicted by π/M = 0.012.3. Simulation and experiment
Grey images “Orchid” [20], “Cameraman” and “Peppers” as shown in Fig. 3, which are all comprising of 256 × 256 pixels, are employed as the test images to evaluate the proposed method. In the hologram design, all the images are padded with zeros to form 1024 × 1024 pixels. The influences of zero-padding will be discussed in the “discussion” section. The pixel numbers of the phase-only Fourier holograms are 1024 × 1024 and the sampling interval is 8µm × 8µm. The wavelength of the laser beam is 632nm. Computer simulation is firstly conducted and the result is shown in Fig. 4. Quadratic phase φ(m,n) = 0.011(m2 + n2), which is determined according to the Eq. (11), is adopted for all the three images in the design.
Fig. 4 (a) Generated hologram. (b) Reconstructed image with the proposed method. (c) Quadratic phase. (d) Phase distribution of the image in Fig. 4(b). (e) Comparison of the reconstructed phase and the original quadratic phase. (f) Reconstructed image with the original random-phase method.
Figure 4(a) shows the phase-only Fourier hologram calculated by the proposed method. And the numerically reconstructed intensity distribution of the hologram is illustrated in Fig. 4(b). We can see that there exist lots of noises at the edge area, which will reduce the diffraction efficiency to some extent. But meanwhile, the image in the middle area is very clear and shows no significant differences to the target image. In order to evaluate the reconstructed image quantitatively. The peak signal noise ratio (PSNR) was introduced. And the calculated PSNR is 66. Another important criterion in measuring the performance of a hologram is the diffraction efficiency. Here, the diffraction efficiency, which represents the ratio of the energy in the middle 256 × 256 area to the whole 1024 × 1024 area, is η = 25%. Although the diffraction efficiency is not as high as the traditional random phase method, it is still acceptable. The reconstructed phase is presented in Fig. 4(d). It can be seen from the enlarged view of the middle region that the phase distribution also exhibits the characteristics of quadratic phase. But there is a slight movement compared to target quadratic phase φ(x,y) = 0.011(m2 + n2), as plotted in Fig. 4(e). This means that the reconstructed quadratic phase has larger parameters a and b. And larger value will widen the spectrum bandwidth of the image and make the edges blurred. So when determine the quadratic phase, a litter smaller value calculated by the Eq. (11) can be chosen. As a comparison, the reconstructed image with the original random phase method is shown in Fig. 4(f), which is contaminated by the speckle noise seriously. The PNSR is just 57.
Numerical reconstruction of the “orchid” and “Cameraman” images with the proposed method and the original random phase method are also taken and the results are presented in Fig. 5. The reconstructed images with the proposed method are very similar to the original images and have very high image quality. Meanwhile, the phase distributions of the images are all quadratic phase, which can eliminate the destructive interference between the sampling points and improve the speckle noise significantly. Conversely, the reconstructed images with the random phase method are very noisy due to discarding the magnitude component of the hologram directly. Furthermore, the phase distribution are all random and erratic, which will cause the destructive interference between the sampling points and the image quality will be further reduced. Quantitative comparison of the image quality is given in Table 1. Evidently, the PSNR of the three reconstructed images with the proposed method are all higher than the ones with the original method.
Fig. 5 Numerical reconstructed images of the phase-only Fourier hologram: (a)-(b) proposed method, (c)-(d) original random phase method.
Table 1. The PSNR of the reconstructed images with two different methods.
Experiments are also taken to evaluate the proposed method. Experimental setup for optical reconstruction of the hologram is shown in Fig. 6. A phase-only SLM of Holoeye PLUTO is used to display the calculated hologram. The pixel number and pixel size are 1920 × 1080 and 8 × 8µm, respectively. A He-Ne laser with wavelength of 632 nm is collimated and expanded to illuminate the reflective SLM. The reconstructed images are reflected from the beam splitter and captured with digital camera at the focal plane of the Fourier Lens. The focal length of the lens is f = 30cm. The HR16000CTLGEC camera was used with resolution of 4896 × 3248 and pixel size of 7.4 × 7.4µm. In order to avoid the influences of the zero-order diffraction of SLM, a linear phase is superimposed on the hologram to perform off-axis reconstructions. All the holograms in the experiments have the same resolution 1024 × 1024. The calculated sampling interval of the reconstructed image is determined by 632nm × 40cm/(1024 × 8μm) = 31μm, which is larger than the pixel size 7.4μm of the camera. Accordingly, the reconstructed image can be recorded correctly.
The optical reconstructed images of the random phase method and the proposed method are displayed in Fig. 7 and Fig. 8, respectively. We observe that the images in Fig. 7 are heavily contaminated with noise and some details are blurred and hardly to be distinguished. In contrast, the images in Fig. 8 are reconstructed with favorable visual quality and the speckle noise is greatly reduced. A lot of details, such as the stones, the camera and the small pepper, as shown in the Fig. 8 marked with a yellow dashed box, all can be clearly discerned. Due to the inevitable speckle noise caused by the rough surface, as well as the non-ideal phase modulation of the SLM, the quality of the reconstructed images is not as good as those obtained with numerical reconstruction. However, we can still draw the conclusion that the reconstructed image with our proposed method is clearer and less contaminated with the speckle noise.
4. Discussion
The selection of quadratic phase is critical to reconstruct high-quality image. In this section, the influence of the quadratic phase on the image quality is discussed. For the sake of simplicity, we assume that the image has the same pixels in the horizontal and vertical direction. Then the parameter a and b in the Eq. (1) is equal. The “Peppers” image with 256 × 256 pixels is used as the test image. In the hologram design, the image is also padded with zeros to form 1024 × 1024 pixels. Further simulation, which is not presented in this paper, shows that zeros-padding has no influences on the bandwidth of the spectrum. Firstly, four different quadratic phase with a = 0, a = 0.00275, a = 0.011 and a = 0.014 are superimposed on the image and the spectrum is calculated and illustrated in Fig. 9(a)-9(d). Thereafter, the phase-only Fourier holograms are generated with the proposed method. And the numerical reconstructed images of the holograms are presented in Fig. 9(e)-9(h). When a = 0, the quadratic phase is equal to the constant phase. The spectrum mainly concentrates on the low spatial frequency and thus cannot spread widely. The reconstructed image in Fig. 0.9(e) is blurred. As the parameter a becomes larger, the bandwidth of the spectrum increases gradually. The image quality also improves significantly. When a = 0.011, the bandwidth is almost the same as the hologram size. The calculated PSNR reaches the maximum value 66dB. When a = 0.014, the bandwidth is larger than the hologram size. Due to the spectrum is not band-limited, the phase distribution of the sampling points at the edge of the image will cause destructive interference and speckle noise will be introduced and the image quality is reduced.
Fig. 9 (a)-(d) Calculated spectrum of the image when different quadratic phases are added. (e)-(h) Numerical reconstructed images with different quadratic phase added in the design.
The parameter a is set from 0 to 0.015, and the calculated PSNR and diffraction efficiency are plotted in Fig. 10. It shows that diffraction efficiency increases with the parameter a. When a = 0.011, the diffraction efficiency reaches the maximum value 25.2%. When a is larger than 0.011, the diffraction efficiency gets lower. Through these simulations, we can draw a conclusion that the phase superposed on the image should make the spectrum band-limited and meanwhile as smooth as possible to reach high image quality and diffraction efficiency. Moreover, we observe that when the same quadratic phase is added, the spectrum band width of the image with 512 × 512 pixels is one times wider than the image with 256 × 256 pixels. And for the different grey image with the same resolution, the band width is almost the same.
Fig. 10 Calculated PSNR and diffraction efficiency of the reconstructed image when different quadratic phase is adopted.
Another factor that affects the quality of the reconstructed image is the number of zeros which are padded to the target image during the hologram calculation. Assume that the target image has m × m pixels. And after zero-padding, the image has M × M pixels. The “Peppers” image with 256 × 256 pixels is used as the target image. Then different number of zeros is padded and the holograms are calculated. The numerical reconstructed intensity distribution of the whole output plane with different M/m are shown in Fig. 11. It can be seen that when the target image is not zero-padding (M/m = 1), the image is reconstructed successfully only in the central area. As the M/m increases, the noise and the higher-order diffraction image gradually move toward the edge area, making the reconstructed image clear. Figure 12 shows that the PSNR of the reconstructed image increases linearly with the M/m. At the same time, the diffraction efficiency will decrease, due to the proportion of the image to the whole output plane drops. However, we find that when the M/m is larger than 4, the rate of decrease of diffraction efficiency is gradually slowing down. And finally, the diffraction efficiency maintain at about 23% when the M/m reaches 10.
Fig. 11 Numerical reconstructed results when the target image is padded with different number of zeros.
Fig. 12 Calculated PSNR and diffraction efficiency of the reconstructed image when different number of zeros is padded to the target image.
Although the zero-padding can improve the imaging quality, it will reduce the image size for a fixed hologram pixel size. The size of the whole output plane is determined by λf/dxh. When the zeros are padded, the size of the target image is limited to λz/dxh × (m/M). Therefore, to balance the PNSR, diffraction efficiency and image size, M/m can be set to 4. This is very similar to the design of diffractive optical elements, which usually pad the target pattern with zeros 4 times to balance the mean square error and the diffraction efficiency.
5. Conclusion
In conclusion, we proposed a fast and non-iterative method for the generation of phase-only Fourier hologram based on quadratic phase addition and bidirectional error diffusion. Compared to the original widely used random phase method, the reconstructed image with the proposed method has very low speckle noise and high peak signal noise ratio. Meanwhile, the proposed method is simple and computationally inexpensive, which has potential in holographic projection and holographic three-dimensional display, and so on.
Funding
The National Natural Science Foundation of China (Nos. 61505214, 61605211); The Applied Basic Research Programs of Department of Science and Technology of Sichuan Province (Nos.2016JY0175, 2016RZ0067); National Defense Foundation of China (No.CXJJ-16M116).
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