Computer Generated Hologram Optimization for Lens Aberration
Kazushi Nakamura

Kenta Yamamoto

University of Tsukuba
University of Tsukuba
Tsukuba, Japan
Tsukuba, Japankazushi@digitalnature.slis.tsukuba.ac.jp kenta.yam@digitalnature.slis.tsukuba.ac.jp

Hologram Plane
u0 (φ)

Aberrated Lensu- u+Propagation

Aberration PSNR=6.15

Image Planeu1

Yoichi Ochiai
University of Tsukuba
Tsukuba, Japanwizard@slis.tsukuba.ac.jp

Correction PSNR=32.05

PropagationzyxfLens Featuref

Calculate
Lossvs.
TargetOptimize (Update Phase Distribution φ)

Figure 1: Left: optical system and optimization model. Right: result of our method.

ABSTRACT
We propose a lens aberration correction method for holographic
displays via a light wave propagation simulation and optimization
algorithm. Aberration correction is an important technology toobtain noise-less hologram images in holographic displays. We
optimized phase holograms with an automatic differentiation technique and Adam optimizer, and aberration corrected images wereachieved. Given an aberrated lens with a focal length of 20mm, theoptimized holographic image has a PSNR value of 32.

CCS CONCEPTS• Human-centered computing → Displays and imagers.

KEYWORDS
hologram, optimization, lens aberration
ACM Reference Format:
Kazushi Nakamura, Kenta Yamamoto, and Yoichi Ochiai. 2022. Computer
Generated Hologram Optimization for Lens Aberration. In SIGGRAPH Asia
2022 Posters (SA ’22 Posters), December 06-09, 2022. ACM, New York, NY,USA, 2 pages. https://doi.org/10.1145/3550082.3564191

1INTRODUCTION

Aberration correction is an essential technology for holographic
displays to generate accurate hologram images. In holographic
display, image distortion can be caused by wavefront aberrationon the lens in the human eye or in the optical hardware, and manymethods have been proposed to correct these lens aberrations.
SA ’22 Posters, December 06-09, 2022, Daegu, Republic of Korea© 2022 Copyright held by the owner/author(s).
This is the author’s version of the work. It is posted here for your personal use. Not forredistribution. The definitive Version of Record was published in SIGGRAPH Asia 2022
Posters (SA ’22 Posters), December 06-09, 2022, https://doi.org/10.1145/3550082.3564191.

Takaki et al. proposed a Maxwellian-view display that corrects
eye aberrations using reversed aberration [Takaki and Fujimoto
2018]. Yamamoto et al. proposed an interactive system with a graphical user interface for eye aberration correction by generating hologram images with reversed aberration [Yamamoto et al. 2021].
To correct aberration more accurately, we adopted an optimization technique. Peng et al. described that noise-less hologram images can be achieved by optimizing phase distribution on a spatial
light modulator (SLM) with an automatic differentiation techniqueand stochastic gradient descent [Peng et al. 2020]. However, Peng
et al. did not cover the aberration correction with the optimization.
By applying this hologram optimization method for optical systems with an aberrated lens, we validated the capability of theaberration correction with optimization. We defined the opticalsystem as shown in Figure 1 (left). Light wave propagation calculations and aberrated lens features were included in the loss function,and the phase on the hologram plane of the SLM was optimized
to minimize the loss between the intensity distribution of the observed image and that of the target image. We obtained aberrationcorrected images by this method as shown in Figure 1 (right).

2METHOD

Computational hologram calculation is based on wave optics [Goodman 2005]. In this study, we calculated light propagation with theangular spectrum method. Propagation from a plane 𝑢𝑎 to anotherplane 𝑢𝑏 is represented as follows:
𝑧 exp(𝑖𝑘𝑟 )−1𝑢𝑏 = FF [𝑢𝑎 ]F(1)𝑖𝜆𝑟2where F [·] is the Fourier transform, F −1 [·] is the inverse Fouriertransform, x, y, z are expressed in the coordinate system shown inthe Figure 1 (left), 𝑖 is the imaginaryunit, 𝜆 is the wavelength, 𝑘 is√︁
the wave number, and 𝑟 = 𝑥 2 + 𝑦 2 + 𝑧 2 . The conversion feature ofPSNRPSNR

SA ’22 Posters, December 06-09, 2022, Daegu, Republic of Korea
35 2Z230Correction25
f = 50 mm
2035 mm
1575 mm
105Aberration 35 mm 50 mm 75 mm
035Z 22f = 20 mm
3025Z-22Z 02
Correction
2015105Aberration Z 22 Z-22 Z 02
050100150200
Iteration

Nakamura et al.

f = 35 mm / Z22Aberrationf = 35 mm / Z22
Correction

f = 50 mm / Z22Correction

f = 75 mm / Z 22Correction

f = 20 mm / Z-22
Correction

f = 20 mm / Z-22
Correction

1 iteration

10 iterations

20 iterations

50 iterations

100 iterations

300 iterations

250 300

Figure 2: Row 1: relations between iteration and PSNR, and results for lenses with different focal lengths. Row 2: relations
between iteration and PSNR, and optimization process.
the lens with aberration can be expressed with the following phase
distribution 𝑡 (𝑥, 𝑦) and generalized pupil function P (𝑥, 𝑦):𝑢 + (𝑥, 𝑦) = 𝑢 − (𝑥, 𝑦) ∗ 𝑡 (𝑥, 𝑦) ∗ P (𝑥, 𝑦)(2)

where 𝑢 − and 𝑢 + are the complex amplitude distribution
at the front𝜋 𝑥 2 + 𝑦2
and behind the lens, respectively, 𝑡 (𝑥, 𝑦) = exp −𝑖𝜆𝑓and P (𝑥, 𝑦) = 𝑃 (𝑥, 𝑦)exp(𝑖𝑘𝑊 (𝑥, 𝑦)). 𝑓 is the focal length of thelens, 𝑃 (𝑥, 𝑦) is the pupil function and 𝑊 (𝑥, 𝑦) is the wavefront
aberration which is generally expressed as a linear combination of
the Zernike polynomials [Goodman 2005].
Utilizing Eq.1 and Eq. 2, we formulated our optical system asshown in Figure 1 (left). The image of the object on the image planeis given by the following equation:
𝑢 1 = 𝑃𝑟𝑜𝑝 𝑓 𝑃𝑟𝑜𝑝 𝑓 [𝑢 0 ] ∗ 𝑡 (𝑥, 𝑦) ∗ P (𝑥, 𝑦)(3)
where 𝑃𝑟𝑜𝑝𝑧 [·] is the propagation computation in Eq. 1, 𝑢 0 =exp(𝑖𝜙 (𝑥, 𝑦)) is the hologram plane, and 𝑢 1 ∈ C is the image plane.
In this system, we calculate the gradient by the automatic differentiation technique and optimized with the Adam optimizer [Kingmaand Ba 2014], which is one of the stochastic gradient descent algorithms used to achieve lens aberration correction. We define theloss function as the difference between intensity distribution of theimage plane and that of the target image. The loss function in thissystem is as follows:
21
L (|𝑢 1 | 2,𝑇 ) =|𝑢 1 | 2 − 𝑇(4)2where 𝑇 is the intensity distribution of the target image. The optimization process is as follows:
minimize𝜙 (𝑥,𝑦)L (|𝑢 1 | 2,𝑇 )(5)where 𝜙 (𝑥, 𝑦) is the phase distribution of the hologram plane. The
image quality is gradually improved while repeating the optimization processes as described above.

3RESULTWe conducted 300 steps of optimization and propagation simulations on a group of lenses with different focal lengths at constantaberration and a group of lenses with different aberrations with afixed focal length of 20 mm. Results for these simulations are shownin Figure 2 (row1). In this optical system, aberration makes the resulting images noisy, but applying this optimization technique, weobtained images with reduced noise caused by aberrations.

4CONCLUSION

This paper proposed an aberration correction method for holographic displays with aberrated lens using automatic differentiation and optimization. By correcting aberration with optimization,
Images with less noise were obtained. Application of this method
to actual optical system and optimizations for more complicatedoptical system (e.g. multiple lenses) are planned for future work.

REFERENCES
Joseph W Goodman. 2005. Introduction to Fourier optics. Introduction to Fourier optics,
3rd ed., by JW Goodman. Englewood, CO: Roberts & Co. Publishers, 2005 1 (2005).
Diederik P Kingma and Jimmy Ba. 2014. Adam: A method for stochastic optimization.
arXiv preprint arXiv:1412.6980 (2014).
Yifan Peng, Suyeon Choi, Nitish Padmanaban, and Gordon Wetzstein. 2020. NeuralHolography with Camera-in-the-Loop Training. ACM Trans. Graph. 39, 6, Article
185 (nov 2020), 14 pages. https://doi.org/10.1145/3414685.3417802
Yasuhiro Takaki and Naohiro Fujimoto. 2018. Flexible retinal image formation byholographic Maxwellian-view display. Opt. Express 26, 18 (Sep 2018), 22985–22999.
https://doi.org/10.1364/OE.26.022985
Kenta Yamamoto, Ippei Suzuki, Kosaku Namikawa, Kaisei Sato, and Yoichi Ochiai.
2021. Interactive Eye Aberration Correction for Holographic Near-Eye Display. In
Augmented Humans Conference 2021 (Rovaniemi, Finland) (AHs’21). Association
for Computing Machinery, New York, NY, USA, 204–214. https://doi.org/10.1145/
3458709.3458955