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Abstract
Joint photographic experts group (JPEG) compression standard is widely adopted for digital images. However, as JPEG encoding is not designed for holograms, applying it typically leads to severe distortions in holographic projections. In this work, we overcome this problem by taking into account the influence of JPEG compression on hologram generation in an end-to-end fashion. To this end, we introduce a novel approach to merge the process of hologram generation and JPEG compression with one differentiable model, enabling joint optimization via efficient first-order solvers. Our JPEG-aware end-to-end optimized holograms show significant improvements compared to conventional holograms compressed using JPEG standard both in simulation and on experimental display prototype. Specifically, the proposed algorithm shows improvements of 4 dB in peak signal-to-noise ratio (PSNR) and 0.27 in structural similarity (SSIM) metrics, under the same compression rate. When maintained with the same reconstruction quality, our method reduces the size of compressed holograms by about $35{\%}$ compared to conventional JPEG-compressed holograms. Consistent with simulations, the experimental results further demonstrate that our method is robust to JPEG compression loss. Moreover, our method generates holograms compatible with the JPEG standard, making it friendly to a wide range of commercial software and edge devices.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Computer-generated holography (CGH) is considered to be the ultimate display technology due to its ability to provide all the cues needed for human vision [1]. To achieve realistic and dynamic display of holographic imagery, current CGH algorithms are required to generate and process massive amounts of data, which forms a major bottleneck for communication channels and edge devices [2,3]. While compressing such heavy holographic data is an option, current compression techniques, including Joint photographic experts group (JPEG) and high-efficiency video coding (HEVC), are designed for traditional images and are not readily compatible with phase hologram data. Additionally, the signal statistics of holograms significantly differ from traditional natural images and videos [4], making conventional compression standards like JPEG sub-optimal for holographic displays [4,5].
Holographic phase retrieval has been a major research focus in the field of computer-generated holography, which includes a wide range of algorithms [6–14]. Two typical examples of iterative algorithms are Gerchberg-Saxton (GS) [15–17] and stochastic gradient descent (SGD) [18–21] algorithms. The GS algorithm iteratively propagates the field between the spatial light modulator (SLM) plane and the target plane while enforcing constraints on each plane. The SGD algorithm takes the phase values on the hologram as the parameters to be optimized. The phase values are updated according to the loss between the target image and the reconstructed image through light diffraction model, as well as the corresponding gradient. SGD algorithms typically have a higher upper performance bound compared to GS algorithms. However, most of the previous hologram generation methods only focus on the quality of optically reconstructed images and ignore the storage and transmission cost. As massive data storage and transmission is essential for industrializing holographic display, more research efforts should be invested into hologram compression field to reduce data size.
The compression coding of holograms is important for the practical application of holography. However, direct lossy encoding and decoding of holograms with compression standards for regular images and videos may produce serious errors in reconstructing the images, resulting in the loss of crucial visual cues. In the field of hologram compression, different transform kernels are used, such as discrete cosine transform (DCT), Gabor wavelet [22] and wavelet-bandelet [23]. Recently, deep learning has been applied to holographic compression. Jiao et al. proposed a "JPEG + deep learning" scheme in 2018, which applies a deep-learning model to the decompressed hologram to recover the degradation due to standard JPEG compression and decompression [24]. Ko et al. proposed an end-to-end deep-learning based image compression network for phase-only holograms in 2021 [25]. Shi et al. proposed deep-learning models for hologram images and videos compression in 2022 [26]. In these papers, the generation and compression of holograms are independent. In 2022, Wang et al. proposed a joint optimization scheme that generates and compresses phase-only holograms with deep-learning models [27]. These methods effectively improved the hologram compression performance. However, the decoders rely on deep-learning models that require large amounts of training data and have limited applicability due to their lack of compatibility with current industrial edge codec devices. To achieve good reconstruction quality and applicability at the same time, we are exploring the generation of holograms optimized for existing widely supported compression standards.
To make generated holograms compatible with the JPEG standard, we propose a novel JPEG-aware SGD hologram generation algorithm that enables high-quality reconstruction while maintaining the ability to resist JPEG compression loss. Our innovation includes two points. First, we perform joint optimization of hologram generation and compression by incorporating differentiable JPEG to the SGD hologram generation framework. Therefore, our holograms are robust to standard JPEG compression loss. Second, we design a new loss function for the optimization task. This loss function includes a reconstruction loss, a total variation (TV) regularization, and an adaptive weight for the TV regularization. The TV regularization smooths the hologram to reduce the noise of the optically reconstructed image. The adaptive weight function finds a suitable weight for the TV regularization at different compression quality levels.
2. Method
2.1 JPEG and differentiable JPEG simulator
JPEG is a widely adopted image compression standard that is compatible with most software and devices [28]. The encoding and decoding procedures of JPEG are illustrated in Fig. 1, including color space conversion (optional), block splitting, 2D-DCT, quantization, zigzag encoding, and entropy encoding. The JPEG encoder $E(x, q)$ compresses each image $x$ using quality level $q$ (between 0 and 100), and the decoder $D(\cdot )$ inverts these steps, recovering the image. Of these steps, note that the downsampling process during color space conversion and quantization are responsible for lossy coding and decoding.
Fig. 1. Diagram of JPEG codec, including color space conversion (optional), block splitting, 2D-DCT, quantization, zigzag encoding, and entropy encoding. The decoder contains the inverse of these steps.
In JPEG, the 2D DCT of the blocks $C$ are quantized with $8 \times 8$ quantization matrices $Q$ to generate $D_{i, j} = \lfloor \frac {C_{i, j}}{Q_{i, j}} \rceil$, where $\lfloor \cdot \rceil$ represents the rounding operation. Since the derivative of the rounding operation is $0$ almost everywhere, JPEG can not be directly embedded in the SGD algorithm. This makes it challenging to apply any iterative algorithms and therefore we employ a differentiable JPEG simulator. Specifically, to this end, the rounding operation $\lfloor x \rceil$ is replaced by Eq. (1), which has non-zero derivatives nearly everywhere [29].
2.2 JPEG-aware SGD algorithm
SGD algorithm is a commonly used iterative CGH algorithm. It aims to solve the optimization problem described below in Eq. (2).
where $\mathcal {L}(\cdot )$ is a loss function, $f(\cdot )$ is a differentiable propagation model, $a_\text {target}$ is the square root of target image, $\phi$ is the phase-only hologram to be optimized, $\phi ^*$ is the optimized phase-only hologram.Specifically, the phase values on the SLM are the parameters to be optimized, and the field is propagated to the target plane using differentiable operations like the angular spectrum method (ASM),
The hologram optimized via the described SGD algorithm is displayed on the SLM to achieve high-quality holographic reconstructions. However, in cloud-edge case where the hologram is computed on the cloud and transmitted to the edge devices [27], the conventional SGD algorithm would fail since it ignores the post-processing for the storage and transmission of the holograms. Considering the JPEG compression for holograms, the optimization becomes
Instead of designing a new hologram compression algorithm, we incorporate the encoding and decoding procedures into CGH algorithm to make the generated hologram compatible with JPEG standard. With the differentiable JPEG simulator, the optimization problem in Eq. (4) can be solved with first-order iterative SGD methods. The pipeline of the proposed JPEG-aware SGD algorithm is shown in Fig. 2.
Fig. 2. (a) JPEG-aware SGD algorithm pipeline. The hologram passes through a differentiable JPEG codec followed by numerical diffraction to obtain the reconstructed image. The reconstruction loss between the target and the reconstructed image ensures per-pixel reconstruction accuracy. The total variation of the hologram is used to suppress the speckle noise due to the compression. (b) The weight for total variation regularization at different quality levels.
For an SLM with a resolution of $N\times M$, the mean square error $\mathcal {L}_{\mathscr {l}_{2}}$ is used for per-pixel accuracy of the reconstruction, and the weight $W_{\mathscr {l}_{2}}$ for $\mathcal {L}_{{\mathscr {l}_{2}}}$ is set as a constant.
A low quality level $q$ implies that less information is retained during compression, leading to a higher compression rate. JPEG compression with a low $q$ introduces artifacts consisting of discontinuities at the blocks’ borders and oscillations or ringing artifacts next to strong edges [30,31]. When applied to holograms, which inherently contain numerous details and value jumps, low-quality JPEG compression can result in complex artifacts and discontinuities. The compressed phase distribution will cause severe speckle noise which heavily affects the quality of the reconstructed image [32,33]. Meanwhile, recent works [34,35] suggest that smooth phase holograms with natural-image-like portfolios may induce less speckle noise. Thus, the TV regularization term $\mathcal {L}_{TV}$ is added into the loss function $\mathcal {L}$, which is a measure of how much the solution changes between adjacent elements. This encourages the algorithm to find phase-only holograms that are smooth.
3. Results and discussion
We perform extensive simulations and experiments to evaluate the performance of the JPEG-aware SGD algorithm. The algorithms are implemented in Python 3.8.5 and PyTorch 1.12, and we use the Adam optimizer with a learning rate of 0.01. All the algorithms are run on a computer with Windows 11, an AMD Ryzen 9 5900X 12-core processor, 128 GB RAM, and one 24 GB NVIDIA GeForce RTX 3090 GPU. All tested images are from the validation set of DIV2K [36] and resized into $1920\times 1072$. The pixel pitch is 6.4 $\mu m$, and the wavelengths are 638 $nm$, 520 $nm$ and 450 $nm$. For the hyper-parameters in the weight function of the JPEG-aware SGD algorithm, the $w_{max}$ and $w_{min}$ are set as 0.001 and 0, the $k$ is set as 0.6, and $q_{1}$ is set as 60.
In our simulations and experiments, holograms are computed and subsequently processed through the JPEG codec to simulate a cloud-edge scenario. The numerical reconstructions are obtained via ASM, and the optical reconstructions are captured at a predetermined location after loading the hologram onto the SLM. By manipulating the quality level $q$ of JPEG, we obtain simulated and captured reconstructions at varying bpps.
Peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) are used as the metrics of the quality of the reconstructions as shown in Eq. (9) and Eq. (10).
3.1 JPEG compression of holograms
This section aims to validate our hypothesis that the direct application of JPEG compression on holograms will degrade the quality of the reconstruction, particularly at high compression rates. To demonstrate this, we run two classical CGH algorithms, SGD and angular spectrum method plus the double phase method (ASM+DPM), followed by JPEG compression at varying quality levels. ASM+DPM is a non-iterative method for generating phase-only holograms [34]. It initializes the phase component on the target plane and combines it with the target amplitude to produce a complex field. This field is then back-propagated to the hologram plane, where the complex field is encoded into a phase-only hologram using the double phase method.
The simulation results are shown in Fig. 3. Bpp denotes bits per pixel used to encode the compressed hologram. Both ASM+DPM and SGD show the same trend, i.e., the quality of the reconstructed image dramatically decreases as the $q$ and bpp decrease. The simulation results in Fig. 3 indicate our motivation to design a JPEG-resistant algorithm.
Fig. 3. The simulation results of the angular spectrum method plus the double phase method (ASM+DPM) and SGD algorithms at various quality levels, without considering any coding or decoding process. These results support our hypothesis that the direct application of JPEG on holograms leads to severe distortions in the reconstructed images.
3.2 Simulation results
The numerical reconstructions using the SGD and JPEG-aware SGD algorithms at different bpps with a wavelength of 520 $nm$ are presented in Fig. 4. When the bpp is high, the results of the SGD algorithm and JPEG-aware SGD algorithm are similar and close to the ground truth. When the bpp starts to decrease, the results of the SGD algorithm start to degrade. Specifically, severe speckle noise is observed in the reconstructed images, and the PSNR and SSIM of the reconstructions drop significantly. Instead, the JPEG-aware SGD algorithm is robust to the lossy compression and produces reconstructed images with less speckle. Especially when the compression rate is so high that the reconstructions of the SGD algorithm are almost indistinguishable, the JPEG-aware SGD still exhibits resistance to JPEG lossy compression and generates meaningful reconstructed images. The simulation results of full-color reconstruction are shown in Fig. 5, and the results are similar to the results in Fig. 4.
Fig. 4. The simulated reconstructions of the SGD and JPEG-aware SGD hologram generation algorithms at different bpps with a wavelength of 520 $nm$.
Fig. 5. The simulated reconstructions of the SGD and JPEG-aware SGD hologram generation algorithms at different bpps with wavelengths of 638 $nm$, 520 $nm$, and 450 $nm$.
The average reconstruction performance of different CGH algorithms at different bpps is shown in Fig. 6. One hundred high-resolution images from the DIV2K validation set are used for testing, and the average PSNR and SSIM are calculated. The curves of SGD and ASM+DPM algorithms show severe degradation in the quality of reconstructed images as the bpp decreases, further validating the need for designing JPEG-aware CGH algorithms. Instead, the proposed JPEG-aware SGD algorithms show high reconstruction performance against JPEG compression in SSIM and PSNR metrics. When the bpp is high, the holograms are not significantly compressed, so the reconstruction performance difference between the algorithms is not significant. When bpp starts to decrease, the advantages of the proposed method begin to appear. Specifically, the reconstruction performance improvement reaches its maximum at a bpp of about 2.7, with a PSNR improvement of about 4 dB and an SSIM improvement of about 0.27. In addition, the Fig. 6 shows that the TV regularization term $\mathcal {L}_{TV}$ and its weighting function $W_{TV}(q)$ are beneficial to the reconstruction results of the JPEG-aware SGD algorithm, with significant improvements in SSIM and PSNR metrics. This implies that the smooth phase pattern helps remove speckle noise due to hologram compression when the bpp is low.
Fig. 6. The reconstruction performance of different hologram generation algorithms at different bpps. One hundred high-resolution images from the DIV2K validation set are used for testing, and the average PSNR and SSIM are calculated.
Figure 7 shows the file size ratio between our method (JPEG-aware SGD) and the traditional method (SGD) hologram generation followed by JPEG compression. To compare the file size ratios of the two methods under the same image quality, we plotted the data obtained from Fig. 6 with PSNR and SSIM as the horizontal axis, and the bpp ratio, which is the ratio of the bit per pixel (bpp) of our method divided by the bpp of the images obtained by the traditional method, as the vertical axis. Since both methods generate the same number of hologram pixels, the bpp ratio is equal to the ratio of the file sizes of the two methods. As shown in Fig. 7, at PSNR = 15.7 dB, the bpp ratio achieves a minimum value of 0.65. This means that at PSNR = 15.7 dB, our method has the most significant compression effect, and the hologram file obtained is 35${\%}$ smaller than that of the conventional method.
Fig. 7. The bpp ratio between the results of the JPEG-aware SGD and SGD hologram generation algorithms at different PSNRs and SSIMs. The bpp ratio is equal to the storage space ratio, and a bpp ratio being less than 1 means that the storage space required for a hologram generated by the JPEG-aware SGD algorithm is less than that required for a hologram generated by the SGD algorithm for the same reconstruction quality. The data is extracted from Fig. 6 (the purple and yellow lines).
3.3 Experimental results
Finally, we verify the JPEG-aware SGD algorithm in our experimental setup shown in Fig. 8. The laser is FISBA READYBeam operating at 450 $nm$, 520 $nm$, and 638 $nm$. The polarizer is employed to ensure that the incident beam on the SLM is linearly polarized, and the beam is collimated by a lens. The beam splitter (BS) splits the beam into two paths, and the incident beam is modulated by a HOLOEYE LETO-3-CFS-127 SLM with a pixel pitch of 6.4 $um$. The modulated beam gets reflected, and an aperture is used to filter out the unwanted high diffraction orders. The target plane is 20 $cm$ away from the SLM.
Fig. 8. The experimental results setup. The laser module is not shown in this figure, and BS is the abbreviation for beam splitter.
Figure 9 presents the experimental results obtained at a wavelength of 520 $nm$. At bit rates of 1.60 and 2.45 bpp, where the holograms undergo severe compression, the reconstructed images using the SGD algorithm exhibit substantial speckle noise and significant quality degradation. Conversely, the images reconstructed using the JPEG-aware SGD algorithm display markedly reduced speckle, suggesting that our JPEG-aware SGD algorithm generates holograms resistant to JPEG loss. At higher bit rates, the reconstruction results from the JPEG-aware SGD and SGD algorithms are comparable, which is attributed to less information loss.
Fig. 9. The optical reconstructions of the SGD and JPEG-aware SGD hologram generation algorithms at different bpps with a wavelength of 520 $nm$.
Full-color optical reconstruction results, as shown in Fig. 10, are captured at wavelengths of 450 $nm$, 520 $nm$, and 638 $nm$. These results are consistent with the full-color simulation results in Fig. 5. In low bpp scenarios, the reconstructed images from our JPEG-aware SGD show significantly less speckle noise compared to the SGD results, while in high bpp cases, the performance of the JPEG-aware SGD and SGD algorithms is similar.
Fig. 10. The optical reconstructions of the SGD and JPEG-aware SGD hologram generation algorithms at different bpps with wavelengths of 638 $nm$, 520 $nm$, and 450 $nm$.
4. Conclusion
In the cloud-edge case, compression methods for digital holograms have limited applicability due to their lack of compatibility with edge codec devices. Moreover, current compression methods for regular images are not designed for holograms. When holograms are compressed with the JPEG lossy compression standard, severe noise may be observed in the reconstructed images. In this paper, we proposed a JPEG-aware SGD algorithm that incorporates a differentiable JPEG codec in generating holograms, which is usually ignored when designing hologram generation algorithms. The algorithm jointly optimizes the phase-only holograms for high-quality reconstruction and JPEG-resistant compression. In our simulation results, the proposed algorithm improves up to 4 dB in PSNR and 0.27 in SSIM under the same compression rate. With the same reconstruction quality, our method reduces the size of compressed holograms to about $35{\%}$. Simulations and experiments have confirmed that, in comparison to the SGD hologram generation without optimization for JPEG approach, the proposed algorithm produces reconstructed images with significantly reduced speckle noise.
It is indeed worth mentioning that the proposed scheme is not limited to JPEG compression. As long as the process can be modeled in a differentiable form, the scheme can be extended to other hologram generation methods such as different image and video compression standards and transmission channel characteristics. This makes the generated holograms resistant to attacks from other types of noise. We believe that our method motivates researchers to explore this new and exciting area of compression-aware hologram generation.
Funding
National Natural Science Foundation of China (62305184); Basic and Applied Basic Research Foundation of Guangdong Province (2023A1515012932); Science, Technology and Innovation Commission of Shenzhen Municipality (WDZC20220818100259004).
Acknowledgments
We are very grateful to Prof. Hongtao Li from Tsinghua University for meaningful discussions.
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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