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Abstract
A sampled phase-only hologram (SPOH) is the phase component of the hologram of an object image with pixels being sampled with a periodic grid-cross pattern. The reconstructed image of a SPOH is a sparse image with abundant empty voids and degradation in sharpness and contrast. In this paper we proposed a method based on a new sampling scheme, together with stochastic binary search (SBS), to obtain an optimal sampling lattice that can be applied to generate phase-only holograms with enhanced reconstructed image. Experimental results show that with our proposed method, the fidelity and quality of the reconstructed image are increased.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
A three-dimensional (3-D) image can be reconstructed with high fidelity from its hologram, which is a two-dimensional, complex-valued image. Subject to certain degree of distortion, the reconstructed image can also be obtained from either the phase, or magnitude component of its hologram. These two kinds of holograms are known as phase-only hologram, and magnitude (amplitude)-only hologram. The optical setup for displaying a hologram with either one of these 2 components is more simple as compared with reconstructing a complex-valued hologram with a pair of spatial light modulators [1–3]. In comparison, phase-only hologram is preferred in practice as the reconstructed image is brighter than that of the magnitude-only hologram. However, due to the absence of the magnitude component in a phase-only hologram, the quality of the reconstructed image is degraded. To overcome this problem, different research attempts have been conducted in the past 2 decades to reduce the adverse impact caused by the loss of the magnitude information. A popular approach is the Iterative Fourier/Fresnel Transform algorithm (IFTA) [4,5], which is based on the Gerchberg Saxton algorithm (GSA) [6]. The concept of IFTA is given as follows. Starting from an initial phase-only hologram with random pixel phase values, the hologram pixels are adjusted through a series of iterations until the reconstructed image of the hologram is similar to that of a target image. The computation time of IFTA is lengthy as multiple iterations are required, and the reconstructed images of phase-only holograms generated with IFTA are generally heavily contaminated with noise. The noise can be partially reduced by rapidly displaying multiple IFTA generated phase-only holograms, each derived from a different initial phase-only hologram. Despite the success of this approach, a spatial light modulator of high frame-rate is required to display the multiple holograms. Alternatively, the noise level can also be reduced by reallocating the unwanted signals to a different noise region [7,8], at the expense of restricting the reconstructed image to a smaller area that is not overlapping with the noise region. Phase-only hologram can also be generated with the noise addition method [9]. An object image is added with random phase noise, and converted into a hologram, with the phase component extracted as a phase-only hologram. Reconstructed images of phase-only hologram generated with the noise addition method is heavily masked with random noise, and the line patterns are fragmented. In [9], the problem on broken lines can be partially alleviated by restricting the addition of random phase noise to the smooth regions of the image. However, the noisy appearance of the shaded regions remains the same. Another solution to address the noise problem, known as one-step-phase-retrieval (OSPR), is attempted in [10,11]. OSPR is similar in principle to the use of multiple holograms for lowering the noise level in IFTA. A set of sub-holograms, each representing identical source image but added with different random phase noise, are generated and displayed at high speed. The noises are then averaged out, and reduced visually. A major disadvantage of OSPR is the requirement of a high speed SLM for displaying the sub-holograms.
Phase-only hologram can also be generated with the patterned phase-only hologram (PPOH) [12] method. PPOH is similar to random phase noise addition (RPNA), but instead of random phase noise, periodic blocks of a 2-D phase pattern are added to the source image. Reconstructed images of PPOHs are overlaid with texture of the periodic blocks patterns, and rather blocky in appearance. In addition, the quality of the reconstructed images of PPOHs is poorer for larger and high frequency images. Insofar, there is no effective method for further improving the PPOH method.
The direct binary search (DBS) [13,14] is another method for generating binary phase-only hologram by flipping the binary state of each hologram pixel, keeping the flipped value if it increases the fidelity of the reconstructed image. The method is only used to generate binary holograms and the computation time is extremely long.
Another simple technique for generating phase-only hologram is proposed in [15]. The intensity component of an object image is first down-sampled with a lattice, and the phase component of its hologram is extracted as a sampled phase-only hologram (SPOH). The down-sampling lattice is formed by periodic square blocks of a basic sampling pattern having a size of $\tau \times \tau $. The basic sampling pattern is binary with each member being either 1 or 0. In [15], the basic sampling pattern is in the form of a grid-cross pattern. The reconstructed image of a SPOH is similar to the original object image, but filled with high proportion of empty voids, leading to a sparse and dim appearance. Although the problem can be alleviated by displaying 2 or more SPOHs of the same source image [16], each generated based on a different sampling lattices, the computation time will be lengthened, leading to a decrease in frame-rate. It is also possible to use time division spatial filtering [17] to lower the sparsity of the reconstructed image of a SPOH. This approach suffers the same drawback as [16] with reduction in frame-rate. On the bright side, the simplistic structure of the sampling pattern (with each element having value of either 1 or 0), suggested a small search space. Hence it is possible to determine an optimal sampling pattern (through existing optimization techniques) that results in the best reconstructed image. In this paper, we work along this direction and proposed a method to enhance the SPOH method. Briefly, we adopt a new sampling method that assigns positive and negative polarities to pixels at sampled and non-sampled locations, respectively. An optimization process based on stochastic binary search (SBS) is adopted to derive an optimal sampling pattern. Note that the optimization process is conducted only once. The optimized sampling pattern is then repeated along the horizontal and vertical directions to form an optimized sampling lattice. After the optimal sampling lattice is obtained, it can be applied to generate phase-only hologram of any source image with the following steps. First, the optimal sampling lattice is used to sample the source image. Next, a digital Fresnel hologram of the sampled source image is generated. Subsequently, phase component of the hologram is extracted as an optimal sampled phase-only hologram (OSPOH). Experimental results reveals that with our proposed method, the fidelity of the reconstructed image of an OSPOH is enhanced with higher contrast, and prominent reduction of the empty voids as compare with the reconstructed image of a SPOH.
2. Optimal sampled phase-only hologram (OSPOH)
2.1 Sampled phase-only hologram (SPOH)
The principles of generating a sampled phase-only hologram [15] is shown in Fig. 1., and outlined as follows. For the sake of simplicity, we assumed that the object is a planar image, and parallel to the hologram. The separation, sometimes referred to as the depth, between the image and the hologram is ${z_0}$. This can be easily extended to a 3-D object image with multiple depth layers [18]. The Fresnel hologram of the image can be obtained with Fresnel diffraction as
where $f({m,n;{z_0}} )= exp\left( {\frac{{i2\pi }}{\lambda }\sqrt {{m^2}{\delta^2} + {n^2}{\delta^2} + {z_0}^2} } \right)$ is the free space impulse response, with $\lambda $ and $\delta $ being the wavelength of light, and the pixel size, respectively. It can be seen that $H({m,n} )$ is a complex-valued hologram with both magnitude and phase components. To generate a SPOH, the intensity image is first down-sampled with a grid-cross lattice $GD({m,n} )$ of down-sampling factor $\tau $. A grid-cross down-sampling lattice is a binary array of the same size as the intensity image, and which is comprising of repetitive square blocks of down-sampling patterns, denoted by $B({u,v} ){|_{0 \le u,v < \tau }}$. The size of $B({u,v} )$ is $\tau \times \tau $. Each entry in $B({u,v} )$ is either 1 or 0, corresponding to sampled and non-sampled points, respectively. A point in the block $B({u,v} )$ is assigned as a sampled point if either one of the following criteria is satisfied where $mod({R,S} )$ is the modulus operator returning the remainder of $R/S$, with R and S being integers. The down-sampled image is given by Subsequently, the down-sampled image is converted into a digital phase-only Fresnel hologram ${H_P}({m,n} )$. Denoting $arg[\cdot ]$ to be the phase component of a complex quantity, we have2.2 Proposed method for generating the optimal sampled phase-only hologram (OSPOH)
From Eqs. (2a)–(2d), it can be inferred that after down-sampling, non-sample points are all set to zero. As such, the reconstructed image of a sampled-phase only hologram based on the grid-cross lattice is full of empty voids. To overcome this problem, we proposed a method to generate an optimal sampling lattice that is formed by a periodic two dimensional square pattern denoted by $B^{\prime}({u,v} )$. Once the optimal sampling lattice is obtained, it can be generally applied to generate sampled phase-only holograms, and no further modification on the sampling lattice is necessary. The process is conducted in repetitively rounds of iteration, with the operation in each iteration shown in Fig. 2. To begin with, a square block $B^{\prime}({u,v} )$ of size $\tau \times \tau $ is defined, with each element being either 1 or -1, is defined. Initially, the elements in $B^{\prime}({u,v} )$ are assigned random values of either 1 or -1. In each iteration, a randomly selected entry in $B^{\prime}({u,v} )$ is inverted. For example, if a pixel $B^{\prime}({{u_0},{v_0}} )$ is selected, its value will be changed to $- B^{\prime}({{u_0},{v_0}} )$. Equations (3) and (4) are applied to generate the SPOH, with the sampling block $B({u,v} )$ replaced by $B^{\prime}({u,v} )$ in Eq. (3). Next, a reconstructed image is computed from the SPOH as
where ${f^\dagger }({m,n;{z_0}} )$ is the complex conjugate of $f({m,n;{z_0}} )$. Note that the sampling in our proposed method is different from the one adopted in [15]. In the SPOH method, the entries in the sampling block $B({u,v} )$ are either 1 or 0, resulting in lots of empty voids when $B({u,v} )$ is used to sample an image. In our proposed method, the entries in $B^{\prime}({u,v} )$ are either 1 or -1. In another words, the non-sampled points are inverted, rather than removed from the original image. Although this will result in a decrease in the emptiness that are caused by complete removal of the non-sample points, the quality of the reconstructed image of the phase-only hologram is still poor if the values of the elements in the sampling block $B^{\prime}({u,v} )$ is not correctly determined. The iterative process in Fig. 2 is used to derive a sampling block that minimizes the difference between the reconstructed image of the SPOH and the source image. Initially, each element in the sampling block is randomly assigned a value of either +1 or -1. Referring back to Fig. 2, peak-signal-to-noise-ratio (PSNR) of the reconstructed image, as compare with the original image $I({m,n} )$ is computed. Suppose the pixels in the image are within the range $[{0,1} ]$, and let M and N be the number of columns and rows in the original image, respectively, the mean square error (MSE) is given by
Fig. 2. Proposed method for generating a down-sampling lattice for generating the optimal sampled phase-only hologram
The peak-signal-to-noise ratio (PSNR) is obtained as
If the PSNR is higher than the one obtained with the previous down-sampling block the new down-sampling is retained. Otherwise it is reverted to the previous one. The above process is repeated until the number of iterations exceed a given threshold. It can be seen that in each round of iteration, the content of $B^{\prime}({u,v} )$ is either unchanged, or modify to result in a better reconstructed image. As we shall illustrate later, this will lead to significant improvement on the reconstructed image after certain number of iterations have been conducted.3. Experimental results
The training image with randomly generated contents in Fig. 3(a) is used to obtained the optimal sampling lattice with our proposed method. The size of the image and the hologram are $1024 \times 1024$, and $2048 \times 2048$, respectively. The training image is comprising of numerous small blocks of size $\tau \times \tau $, that are having diversified intensity distribution. The optimization process determines an optimal sampling pattern that, when applied to generate a SPOH, maximizes (on average) the fidelity of all the image blocks in the reconstructed image as compared with the original set of image blocks. Presumably, real-world images are also largely containing image blocks that are similar to that of the training image. As such, the optimized sampling pattern can also be applied to generate SPOHs of other images, and result in reconstructed images with higher quality as compared with phase-only holograms that are generated with the grid-cross pattern in [15]. Referring to the iterative process described in Section 2.2, Eqs. (3) and (4) are applied to generate the optimal SPOH (OSPOH) of the image with wavelength $\lambda $=520 nm, pixel size $\delta $=6.4um, ${z_0} = 0.16\textrm{m}$ and $\tau = 12$. After conducting 500 iterations, the reconstructed image of the OSPOH is shown in Fig. 3(b). The PSNR of the reconstructed image of the OSPOH, as compared with that of the complex valued hologram $H({m,n} )$, is 18.78db. A plot of the PSNR of the reconstructed image of the OSPOH versus number of iterations is shown in Fig. 4. It can be seen that the reconstructed image is quite close to the optimal value, and exceeds 18db in less than 500 iterations. The computation time for one iteration is around 0.6sec to 0.7sec on a typical commodity PC.
Fig. 3. (a) Source image with randomly generated content, (b) Reconstructed images of the OSPOH of the source image, based on the optimal down-sampling lattice.
Two source images “Mandrill” and “ABC”, as shown in Figs. 5(a) and 5(b), are employed to evaluate our proposed method, and to compare with existing methods for generating phase-only holograms. Both images have identical size of $1024 \times 1024$. The “Mandrill” image has higher frequency contents and the “ABC” image is mainly comprising of smooth shaded regions. The size of the holograms to be generated is $2048 \times 2048$.
The SPOHs of the 2 source images are generated. The numerical reconstructed images of the 2 SPOHs are shown in Figs. 6(a) and 6(b), with PSNR values of 7.66db and 12.17db. It can be seen that the reconstructed images are heavily overlaid with the texture of the sampling lattice.
Fig. 6. (a) Reconstructed images of the SPOHs of the source image “Mandrill”, (b) Reconstructed images of the SPOHs of the source image “ABC”,
Next, we applied two existing methods; the random phase noise addition [9] and the patterned phase [12] hologram generation methods to generate the phase-only holograms of the 2 sources images. The numerical reconstructed images of the 2 holograms generated with the random phase noise addition method are shown in Figs. 7(a) to 7(b). It can be seen that they are heavily contaminated with noise, with low PSNR of 13.11 dB for “Mandrill”, and 15.063 dB for “Three”. Numerical reconstructed images of the phase-only holograms obtained with the PPOH method (Figs. 8(a) and 8(b)) are better than the SPOH and the random phase noise addition methods, with PSNR of 17.31 dB for “Mandrill”, and 18.17 dB for “Three”. The difference of the PSNR between the two images are over 0.8 dB, showing that the performance of the PPOH method is inferior to source image with higher frequency content. The reconstructed images in Figs. 8(a) and 8(b) also show rather prominent texture of the periodic block patterns, though to a lesser extent than that of the SPOH method.
Fig. 7. (a) Reconstructed images of the hologram of the source image “Mandrill”, generated with the random phase noise addition method, (b) Reconstructed images of the hologram of the source image “ABC”, generated with the noise addition method.
Fig. 8. (a) Reconstructed images of the hologram of the source image “Mandrill”, generated with the patterned phase method, (b) Reconstructed images of the hologram of the source image “ABC”, generated with the patterned phase.
Finally, we applied our proposed OSPOH method to generate the phase-only holograms of the 2 source images. The reconstructed images are shown in Figs. 9(a) and 9(b). We observed that the visual quality of the 2 reconstructed images are better than those resulted from the SPOH, random phase noise addition, and the PPOH methods. The texture of the sampling lattice is also less obvious than that of the SPOH and the PPOH methods. The PSNR of the reconstructed images are 19.20 dB for “Mandrill” and 19.46 dB for “Three”, both of which are highest as compared with the ones obtained in the other 3 phase-only hologram generation methods. In addition, the PSNR of the two reconstructed images are similar despite the large difference between the source images.
Fig. 9. (a) Reconstructed images of the hologram of the source image “Mandrill”, generated with the our proposed OSPOH method, (b) Reconstructed images of the hologram of the source image “ABC”, generated with our proposed OSPOH method.
Optical reconstruction of the SPOH and the OSPOH is conducted with a reflective spatial light modulator (SLM) manufactured by Jasper Display Corp. (model number JD955B). The resolution, pixel pitch and maximum frame rate are is 1920 × 1080, 6.4 micron, and 60 Hz, respectively. The SLM has 2π phase modulation (256 levels) at 532 nm, and no calibration is made before the experiment. Source of illumination is obtained with a diode-pumped solid state (DPSS) laser with wavelength λ=532 nm, which is expanded and used to illuminate the SLM. A CMOS sensor (Sony IMX183) with 5440×3648 resolution, and pixel pitch 2.4 micron is employed at the reconstruction plane to directly record the reconstructed image. The optical reconstructed images of the SPOH, and the OSPOH are shown in Figs. 10(a) and 10(b), and 11(a) and 11(b), respectively. Due to imperfection of the optical setup, the optical reconstructed images are not as good as the ones obtained with numerical reconstruction. However, we can still observe that the reconstructed images of the OSPOH are superior to that of the SPOH, with less contamination of the sampling lattice, better contrast and sharpness. These findings are in line with the numerical simulation in Figs. 6(a), 6(b), 9(a) and 9(b).
Fig. 10. (a)(b) Optical reconstructed images of the SPOHs of the source images in Figs. 5(a) and 5(b), respectively.
Fig. 11. (a)(b) Optical reconstructed images of the OSPOHs of the source images in Figs. 5(a) and 5(b), respectively.
4. Conclusion
In this paper, we proposed a new method for generating sampled phase-only holograms. First, a new sampling scheme is employed with which sampled, and non-sampled pixels of a source image are assigned positive, and negative polarity, respectively. Second, stochastic binary search is applied to obtain an optimal sampling lattice. Determining the optimal sampling lattice from the training image with binary search is a one-off process. Once obtained, the optimal sampling lattice can be used to generate phase-only holograms of other source images. As such, although computing the optimal sampling lattice may take about 6 to 7 minutes, applying it to generate hologram of an arbitrary source image is only around 0.3 to 0.4 second with a typical commodity PC. Experimental evaluation based on quantitative measurement, numerical simulation, and optical reconstruction demonstrate that our proposed method is effective in improving both the contrast and clarity of the reconstructed image as compare with the existing sampled phase-only hologram generation method. The results are also better than the existing random phase noise addition and the PPOH methods, which are also used to generate phase-only holograms
Funding
Ministry of Science and Technology, Taiwan (109-2221-E-035-076-MY3); Research Grants Council, University Grants Committee (11200319).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
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