Ultrahigh-fidelity full-color holographic display via color-aware optimization

Holographic display offers the capability to generate high‑quality images with a wide color gamut since it is laser‑driven. However, many existing holographic display techniques fail to fully exploit this potential, primarily due to the system’s imperfec‑ tions. Such flaws often result in inaccurate color representation, and there is a lack of an efficient way to address this color accuracy issue. In this study, we develop a color‑aware hologram optimization approach for color‑accurate holographic displays. Our approach integrates both laser and camera into the hologram optimization loop, enabling dynamic optimization of the laser’s output color and the acquisition of physi‑ cally captured feedback. Moreover, we improve the efficiency of the color‑aware opti‑ mization process for holographic video displays. We introduce a cascade optimization strategy, which leverages the redundant neighbor hologram information to accelerate the iterative process. We evaluate our method through both simulation and optical experiments, demonstrating the superiority in terms of image quality, color accuracy, and hologram optimization speed compared to previous algorithms. Our approach verifies a promising way to realize a high‑fidelity image in the holographic display, which provides a new direction toward the practical holographic display

Despite the capability of holographic displays to generate arbitrary 3D image content, the generated image inevitably suffers from quality degradation issues.These issues are mainly caused by the interference of coherent random phases [17][18][19], constraints of the spatial light modulator (SLM) [20,21], and imperfections of the optical system [22][23][24].These detrimental factors become an obstacle in making holographic displays a practical technique.Over the past few decades, significant research efforts have been directed toward addressing these challenges.The speckle noise issue [25][26][27], caused by the interference of coherent random phases, has been tackled through techniques such as temporal or spatial multiplexing [7,28,29], and complex-amplitude encoding methods [1,30,31].Additionally, the deep learning-based method provides an efficient approach for reducing speckles in holographic displays by generating and encoding the complex wavefront [32][33][34].The imperfections of the spatial light modulators, such as non-linear voltage-to-phase responses, zeroth diffraction order, and issues related to phase-only or amplitude-only modulation, can heavily degrade the quality of holographic images [21,35].Solutions, including phase calibration [36,37], off-axis encoding [38,39], and phase-only or amplitude-only hologram generation methods [30,40], have been proposed to alleviate these problems.When considering the optical system, lens distortion, dust and scratches on optical element surfaces, and poor illumination conditions [23,41] can also significantly contaminate the holographic display results.Recently, the camera-in-the-loop (CITL) method [17,[42][43][44] has been introduced to address these issues, using the physically captured image as feedback during the hologram optimization process.This camera-integrated optimization accounts for the system's imperfections and can effectively suppress the noises.With these advantages, the CITL method can perform a noiseless full-color holographic display.However, the optimization process is not efficient enough, especially for video hologram generation, since it optimizes the hologram for frame individually.More importantly, there is a lack of consideration for the accuracy of the displayed color, which results in a color-shift problem in the reconstructed images.
As one of the crucial display quality evaluation standards, color accuracy has been widely recognized by commercial display manufacturers [45,46].Color accuracy has a significant influence on the audience's emotions [47].Inaccurate color representations can diverge from our expectations and give us an impression of discrepancy.These inaccurate colors can divert our attention from the ongoing scenario on the screen, thereby deteriorating the our visual experience [48,49].Color accuracy also plays an important role in delivering an immersive visual experience for the audience in holographic displays.Current research mainly focuses on achieving full-color holographic displays through methods such as time multiplexing [7,50], spatial multiplexing [51,52], and frequency multiplexing [53,54], rather than addressing the issue of color accuracy.These methods typically generate holograms for the R, G, and B channels separately and manually adjust the power ratio for each channel to achieve color-corrected full-color displays.However, this approach is less efficient and lacks effective feedback to improve color accuracy.Until now, the problem of color accuracy in full-color holographic displays remains inadequately investigated.
In this study, we propose an effective color-aware hologram optimization strategy to address the color accuracy and hologram optimization speed issues in the full-color holographic display.In our approach, we utilize the gradient descent optimization technique and integrate color-dependent learnable s parameters into illumination.The full-color loss function is employed to simultaneously update the s parameters and SLM phase patterns for the R, G, B channels, facilitating color-accurate optimization.Moreover, we develop an efficient cascade color-aware hologram generation strategy to speed up the iterative process for holographic video displays while maintaining high image quality.In the cascade color-aware hologram optimization, we adopt the previous hologram and s parameter as the initial optimization conditions for the following frame to accelerate the hologram optimization speed.Our method can reproduce high-quality images with superior color accuracy in an efficient way, realizing a ultrahigh-fidelity full-color holographic display.

Color-aware hologram optimization
To realize the color-accurate holographic display, we introduce the color-aware lasercamera-in-the-loop (LCITL) optimization.The principle of the proposed LCITL optimization is depicted in Fig. 1.Initially, three randomly initialized phase-only holograms (POH) are separately uploaded to the SLM.Following light propagation, the camera captures the reconstructed full-color results.Subsequently, we compute the full-color loss.Then, based on back-propagation, the three POHs and the laser's power can be updated.p r , p g , and p b represent the initial voltage of the laser's RGB channels, while s r , s g , and s b denote the learnable parameters, which can be utilized to adjust the laser's intensity ratio and strength in the R, G, B channels.
We employ the angular-spectrum method to simulate light propagation.The expression is defined as follows: Fig. 1 Principle of color-aware LCITL optimization.Initially, the color of the light source is not well balanced.After loading the hologram onto the SLM, the light is modulated by the SLM and passes through the 4-f filter system.The display results are then captured by the camera.The captured image can be used to calculate the actual loss value.Based on the back-propagation and gradient descent process, the holograms and s factors can be updated.Meanwhile, the laser's voltages can be dynamically adjusted based on the s factors to achieve optimized color output.Images Credits: Unsplash License [55] where ϕ r/g/b and g(s r/g/b , ϕ r/g/b ) represent the input phase and the output complex amplitude distribution, respectively.s r/g/b is a learnable variable with an initial value of 1.0.F and F −1 denote the Fourier transform and its inverse, respectively.H(f x , f y , ) indicate the transfer function, and corresponds to the wavelength of light.k denotes the wavanumber 2π/ .z signifies the propagation distance, and f x , f y are the spatial frequencies.
In the LCITL optimization, the objective is to identify the optimized ϕ r/g/b and s r/g/b , which can be reformulated as the following minimization problem: where I target represents the intensity of the target image, and L m represent the loss func- tion, such as the mean squared error loss.Different from the vanilla CITL method, we do not use any scale factor between the |g(s, φ)| 2 and the I target in the loss function, since we need an absolute value instead a relative value to achieve accurate color reproduction.The relative value could potentially cause overexposure or underexposure issues, thereby exacerbating the color-shift problem in the full-color holographic display.Based on the loss function, we can update our RGB POHs ϕ r , ϕ g , ϕ b , and s factors s r , s g , s b through the gradient descent process.The basic update rules are provided as follows: where gr/g/b and g r/g/b represent the amplitude of the simulated and the captured reconstruction results, respectively.α ϕ and α s denote the learning rates.In the gradi- ent descent process, we replace the value of gr/g/b with that of the g r/g/b , but retain the gradient.In our experiment, we employ the Nadam optimizer [56] instead of the Adam optimizer [57] as optimization update rules.Compared to the Adam optimizer, the Nadam optimizer introduces additional Nesterov momentum [58], which can accelerate the convergence speed and find a superior solution.
Moreover, we compensate for the non-linear relationship between the laser's voltage and the output intensity to improve the accuracy of the voltage update.In our case, the relationship between the voltage and intensity of the red laser exhibits obvious non-linearity, as depicted in Fig. 2a.To compensate this non-linearity, we employ 12 polynomials interpolation create a continues look-up-table (LUT) to make the voltage to intensity for the red channel more linear.Figure 2b represents the new relationship between the original laser's voltage and the new laser's voltage with the LUT, and Fig. 2c shows the relationship between the laser's new voltage and the output intensity, which demonstrates (1) significantly improved linearity after the compensation.More details can be found in the supplementary material.Through proposed the color-aware optimization process, we can dynamically adjust the laser's intensity ratio and strength in the R, G, and B channels and update holograms to achieve accurate color representation.

Cascade hologram optimization strategy
In holographic video displays, when employing the proposed LCITL method to generate holograms, it requires optimization for each frame separately.However, this strategy is not effective for generating holograms for holographic video display.Inspired by the video compression methods [59], we propose a color-aware cascade laser-camerain-the-loop (CLCITL) optimization strategy that leverages redundant information from neighboring frames to expedite the color-aware optimization process for holographic video display.Besides, this cascade optimization strategy also provide an effective way to mitigate the impact of the system's strong noise and further improve the image quality of the LCITL method.The schematic of the cascade hologram optimization is depicted in Fig. 3. Firstly, the randomly initialized holograms and s factors are used in the LCITL optimization of "Frame1", and the optimized holograms and s factors serve as the initialization conditions for the subsequent frame.We then obtain the newly optimized holograms and s factors, and repeat the process until the final frame.The pseudo-code of CLCLTL optimization can be found in the supplementary material.In this optimization strategy, the previously optimized holograms and s factors are not discarded, but rather used to expedite the optimization process for the next frame.This strategy can significantly reduce the number of optimization steps.Moreover, this strategy can suppress the system's noises at a very early stage, as the previous hologram already contains the noise suppression phase pattern.
In the CLCITL optimization, we propose a soft-mask-based loss function to achieve different learning rates for the background and foreground.The soft-mask is crucial for the background area which undergoes minor change.A large learning speed in such regions might force the optimized hologram to deviate from the optimum value, subsequently generating noise in the background.The soft mask, used to distinguish between the background and foreground, is generated by comparing the intensity difference between frames.Figure 4 shows an example of the soft-mask generation.The generation of the soft mask M s is defined by the following equation: The final loss function in the cascade hologram optimization with the soft mask is defined as follows: The first and second term in Eq. ( 6) represent the loss functions for the foreground and background, respectively.α is weight factor used to balance the background and foreground loss functions.
Figure 5 shows the necessity of the soft-mask in the cascade hologram optimization.In the hologram optimization of "Frame2", the background and foreground will have the same learning speed when there is no soft mask.This will lead to optimization issue in the background and need more iterations to fix it, as shown in Fig. 5a.When with the soft-mask, the learning speed of the background becomes slow, thereby can avoid this background noise issue, as shown in Fig. 5b. (5) Fig. 4 The soft-mask generation process.We let the intensity of the two frames be subtracted and then take the absolute value to obtain the difference mask, the soft-mask is then generated by apply the Gaussian blur to the difference mask.Images Credits: Big Buck Bunny, Blender Foundation [60]

Results and discussion
We conduct both simulations and optical experiments to demonstrate the superiority of our method.We simulate a SLM with a resolution of 3840×2160 and a pixel pitch of 3.74 µ m, and the propagation distance between the SLM plane and target plane is 80 mm.The wavelengths of the RGB light source are 678 nm, 520 nm, and 450 nm.The original resolution of the input image is 2560×1440, which we zero-pad to 3840×2160.In the simulation, we add intensity variations to the light propagation model to simulate the imperfect illumination condition in the optical experimental system.Figure 6 presents the numerical simulation results of the stochastic gradient descent (SGD), CITL, and the proposed LCITL methods.To better evaluate the color accuracy of the reconstructed results, we introduced the color difference metric E [61] for color accuracy evaluation.This metric is developed based on the human vision system, which offers a more accurate reflection of color differences as perceived by the human eye.A lower value of E indicates better color representation.More details can be found in the supplementary material.In the result of the SGD method, the image quality is the lowest among these methods, and the color tone of the image is shifted to blue.The CITL method successfully suppresses the noises, but the color representation is still unsatisfactory.In the results of the proposed LCITL method, it is evident that the noises are almost eliminated and the color tone aligns well with the target image.In comparison to the CITL method, the LCITL method demonstrates a significant improvement, yielding a 10.08 dB increase in peak signal-to-noise ratio (PSNR) and an 82.30% decrease in E.
The simulation results of the CLCITL optimization are presented in Fig. 7.The first row illustrates the optimization process of "Frame1", and we can observe that the color tone of the image is progressively corrected by the color-aware optimization.The second row displays the optimization results of "Frame2".It is evident that the background noise is effectively suppressed within a few iterations, and the reconstructed image fully transitions from "Frame1" to "Frame2" at approximately 15 iterations in this case.The convergence speed of "Frame2" is much faster than that of "Frame1".A larger discrepancy between frames may require additional iterations in the CLCITL method to obtain comparable image quality.Techniques such as early stopping in CLCITL optimization can be employed to adaptively reduce hologram generation time for different frames.
In our holographic display prototype, we utilize an RGB fiber-coupled module with three optically aligned laser diodes as the light source.The wavelengths of the laser in R, G, and B channels are same as the light source in the simulation.The SLM is HOLO-EYE MEGA phase-only liquid crystal on silicon, which has the same resolution and pixel pitch as the SLM in numerical simulation.The bit depth and diffraction efficiency of the Fig. 5 Cascade optimization results without and with soft-mask.a When the soft-mask is not applied, the noises are appear in the background due to the over-large learning rate.b The introduced soft-mask can efficiently eliminate these noises.Images Credits: Big Buck Bunny, Blender Foundation [60] SLM are 8 bits and over 90%, respectively.We employ a 4 − f system to filter high-order lights and noises.The initial laser's voltages are same for all methods.A color chargecoupled device camera is used to capture the results.Additional hardware details can be found in the supplementary material.
Figure 8 presents experimentally captured 2D holographic images using the SGD, CITL, and proposed LCITL methods.In the results of the SGD method, it is evident that both system noise and color shift issues are not adequately addressed, and the image quality is the lowest among these methods.In the CITL method, the noises are effectively suppressed while the color shift problem persists.The proposed LCITL method presents the best color reproduction performance and maintains high image quality.Compared to the CITL method, the LCITL method demonstrates a substantial improvement, achieving a 4.35 dB quantitative increase in PSNR and a 37.3% decrease in E when averaged over 10 distinct images.We also demonstrate the color representation capability of our method under different illumination conditions, and the details can be found in the supplementary material.Figure 9a and b indicate the captured 3D results when focusing at the "flower" and "butterfly".The enlarged images provide a comparison among the SGD method, Vanilla CITL method, proposed LCITL method and the target image.It is evident that the proposed LCITL method exhibits superior performance in terms of color accuracy.
Since the E may not intuitively reflect how the reproduced color shifts away from the target color, we project all the pixels' color of Fig. 8a-d into the International Commission on Illumination (CIE) 1931 color space [62].Figure 10 illustrates the color The simulated results of "Frame2", the reconstructed image is approach to the target image faster than that of "Frame1".Images Credits: Big Buck Bunny, Blender Foundation [60] Fig. 8 Experimentally captured holographic images with different methods.a, e In the results of SGD method, the images are contaminated by noises, and the color tone is shifted due to the color of the light source.b, f The CITL method can suppress the noises well while the accuracy of color reproduction is limited.Among these methods, the LCITL method c, g performs the best image quality and color accuracy, and the color tone of the reproduced images are almost the same as that of the target images d, h.Additional captured results can be found in the supplementary material.Images Credits: Unsplash License [55] Fig. 9 Experimentally captured 3D holographic images with different methods.Reconstruction results when focusing at the "flower" a and focusing at the "butterfly" b.Images Credits: Unsplash License [55] Fig. 10 Color distribution comparison among the target image and the reconstructed image from the SGD, CITL, and LCITL methods in the CIE 1931 color space.a The SGD method shows a severe color shift issue, as we can see the blue color shifts away from the red color.b This situation is slightly better in the CITL method, while the blue color still shifts from the center of the red color.c Our color-aware LCITL method shows the best color accuracy, as we can see that most of the blue and red colors overlap each other distributions of Figs.8a-d in the CIE 1931 color space.Note that before we project the images into other color spaces, we applied the same Gaussian blur operation to them to avoid the effect of noises on color.In Fig. 10, the red and blue areas in the color space indicate the distribution of project color.The red and blue edges represent the mum range of the project color in the color space.From Fig. 10a and b, we can see that the color shift issue in the SGD and CITL methods can be clearly observed in the chromatic domain.Our proposed color-aware LCITL optimization can mitigate this issue effectively.The color reconstruction achieved by our method closely aligns with that of the target image, as demonstrated in Fig. 10c.
In the CLCITL optimization, since the second hologram reutilizes the previous frame's hologram, which has already effectively eliminated system noise, the subsequent optimization can focus on the differences between frames.As illustrated in Fig. 11, the first row displays the captured results during the optimization of "Frame 1", where the color is gradually corrected and noise is well suppressed, albeit requiring many iterations.The second row presents the optimization results of "Frame 2".As shown in Fig. 11g, 15 iterations already guarantee image quality in this case, and the image quality is much better than that of Fig. 11c. Figure 12 presents the comparison results of the SGD, vanilla CITL, LCITL, and CLCITL methods.Notably, the CLCITL method outperforms others in achieving superior image quality and color accuracy while also effectively mitigating the system's noises.
To demonstrate the effectiveness of our method for holographic video display, we compute holograms using different methods for continuous frames.Figure 13 illustrates a comparison among the SGD, CITL, LCITL, and CLCITL methods at different frames.It is evident that the CLCITL method requires only 30 iterations while achieving the best image quality.Fig. 12 Experimentally captured holographic images with different methods.a SGD method, b CITL method, c LCITL method, d CLCITL method, e Target image.The CLCITL method out-performs the other methods achieved the best image quality and color accuracy, the heavy system noise indicated in the blue box is also eliminated (see visualization 2 in supplementary material).Additional comparisons can be found in the supplementary material.Images Credits: Big Buck Bunny, Blender Foundation [60] Figure 14 illustrates the averaged PSNR and overall iterations of the SGD, CITL, LCITL, and the CICITL methods of 50 frames.The overall iterations of the SGD, CITL, and LCITL methods are the same, while the LCITL method performed the best image quality, showing a 9.03 dB and 4.46 dB improvement compared to the SGD and CITL methods.The overall iterations for the CLCITL method reduced by 57.2% compared to the LCITL method, while the image quality is still maintained at a high level.Note that each iteration time in the Fig. 13 Experimentally captured results with SGD, CITL, LCITL, and CICITL method at different frames.Compared to the SGD method a-e, the CITL method f-j can eliminate most of the noises, while the color shift issue still exists in both methods.When we adopt the color-aware LCITL optimization k-o, both noise and color accuracy issues are addressed well.p-t The CLCITL method can significantly speed up the color-aware optimization while maintaining the similar image quality (see visualization 3 in supplementary material).Images Credits: Big Buck Bunny, Blender Foundation [60] Fig. 14 Comparison among the SGD, CITL, LCITL and the CICITL methods in terms of the averaged PSNR and overall iterations of 50 frames.The averaged PSNR in the CICITL method shows a 4.76 dB improvement compared to the CITL method.Meanwhile, the overall iterations reduced from 5000 to 2140 CITL, LCITL, and CLCITL are similar, which means the CLCITL method reduced the hologram generation time by approximately 57.2% compared to the CITL and LCITL methods.

Conclusion
In this paper, we have proposed a color-aware hologram optimization strategy, which could reproduce color-accurate high-quality holographic images.In our approach, the laser's output color and phase pattern were dynamically adjusted to achieve optimal color reproduction.Compared to the SGD method and the original CITL method, our method demonstrated a clear advantage in terms of color accuracy and image quality.The proposed CLCITL method also exhibited significant improvement in hologram generation time.With the proposed color-aware hologram generation strategy, we could produce a vivid full-color holographic video display.We believe our method has the potential to be used in the area of AR/VR display and holographic projection since our approach is capable of providing high-quality holographic images, which may give audiences a more immersive visual experience.
However, our method has room for improvement.The current optimization model is based on the camera's color feedback, which may differ from the human visual system.In the future, we will aim to improve the optimization process from the viewer's perspective to realize a more realistic color holographic display.Another improvement can focus on the differentiable color-difference loss functions (such as E ).Currently, the loss functions for the full-color image are based on the RGB color space.Nevertheless, this color space cannot accurately reflect the human eye's perception of color difference, and thereby, we may lose some accuracy in color reproduction.A possible solution is to calculate the differentiable color-difference loss function in other color spaces (such as CIELab) to further enhance our control of color accuracy.

Fig. 2 Fig. 3
Fig. 2 Compensation process for the laser's voltage to intensity response.a Original laser's voltage to intensity response of the red laser.b The calculate continues LUT map for the laser's original voltage and the new voltage.c The laser's new voltage to intensity response

Fig. 6 Fig. 7
Fig. 6 Numerical simulation results.a Due to the non-uniform intensity distribution of the light source, the noises appear in the SGD method.bThe CITL method shows a better noise reduction while the color tone is shifted.c The LCITL method demonstrates better performance in terms of color accuracy and image quality compared to the CITL and SGD methods.d Target images.Additional numerical simulation results can be found in the supplementary material.Images Credits: Unsplash License[55]

Fig. 11
Fig. 11 Experimentally captured holographic images of the CLCITL optimization.a-d The reconstruction results in optimization of the "Frame1".e-hThereconstruction results in the optimization of "Frame2".The optical experimental results are matched well with the simulation results, the convergence speed and the noises are improved in the second frame optimization (see visualization 1 in supplementary material).Images Credits: Big Buck Bunny, Blender Foundation[60]