Hologram Representation via Quadratic Phase Gaussian Splatting

Haolong Wang ¹, Yicheng Zhan ², Kaan Akşit ², Simeng Qiu ¹

¹ Swansea University ² University College London
SIGGRAPH Asia 2026 Technical Communications

Optimizing CVQPG on the Windmill scene (RGB, 15,360 Gaussians). The curvature γ of every Gaussian is locked at zero for the first 400 iterations (warm-up stage), then optimized jointly, improving the PSNR gain over the planar Gaussian baseline.

Abstract

We introduce Complex-Valued Quadratic Phase Gaussian (CVQPG), a novel hologram representation method that augments each 2D Gaussian primitive with a quadratic phase profile controlled by a learnable curvature parameter. Against the planar Gaussian baseline, CVQPG improves the average PSNR of holographic reconstructions by 0.19 dB (RGB) and 0.33 dB (grayscale) at equal primitive counts, and by 0.05 dB (RGB) and 0.08 dB (grayscale) at equal parameter counts, where it still leads in all visual quality metrics. Our frequency-domain analysis shows that CVQPG better preserves the mid-to-high frequency band of natural images, where the reconstruction MSE drops by up to 11% (RGB) and 22% (grayscale), indicating that modulating primitive wavefronts is an effective and lightweight enhancement.

Method

Pipeline

A set of 2D Gaussian primitives are rasterized into a complex hologram (amplitude and phase), propagated to several depth planes, and compared with synthesized RGB+D targets at each focus.

Pipeline of CVQPG. We only introduce one extra curvature parameter \(\gamma\) for each primitive.

Complex-Valued Quadratic Phase Gaussian (CVQPG)

Learnable curvature. Each primitive carries a quadratic phase controlled by one curvature parameter \(\gamma = 1/f\):

\[ \begin{gathered} \mathcal{G}(\mathbf{p}) = \alpha\,\mathbf{c}\, \exp\!\big(-\tfrac{1}{2} D^2_{mah}(\mathbf{p})\big)\, \exp(j\boldsymbol{\varphi})\, Q_{mah}(\mathbf{p}), \\[6pt] Q_{mah}(\mathbf{p}) = \exp\!\Big(j\,\frac{k\gamma\det(\mathbf{S})}{2}\, D^2_{mah}(\mathbf{p})\Big). \end{gathered} \]

The curvature acts like a virtual lens associated with the planar Gaussian light source: converging for \(\gamma < 0\), planar at \(\gamma = 0\), diverging for \(\gamma > 0\).

Phase of one primitive as \(\gamma\) sweeps.

Side view of the propagated field from a CVQPG primitive.

Mahalanobis QPF. Measuring the phase with the Mahalanobis distance keeps the fringes aligned with the anisotropic envelope.

Standard QPF vs. Mahalanobis QPF as the primitive changes shape.

Curvature Control and Optimization

Scale-adaptive curvature bound. Each primitive's curvature is bounded by the SLM Nyquist limit to avoid aliasing:

\[ \gamma_{\max} = \eta\,\frac{\lambda}{6\sigma\,(\Delta x)^2}, \qquad \eta = 0.9, \]

where \(\sigma\) is the equivalent standard deviation, \(\lambda\) is the wavelength, and \(\Delta x\) is the pixel pitch.

Curvature warm-up strategy. \(\gamma\) stays at 0 for the first 400 of 2,000 iterations, then is optimized jointly.

Optimization on the Flower scene (RGB, 15,360 Gaussians).

Results

Quantitative Comparison

CVQPG improves PSNR and visual quality consistently across all settings and metrics, by 0.19 dB (RGB) and 0.33 dB (grayscale) at equal primitive counts, and keeps its lead over Baseline⁺ at equal parameter counts, showing that wavefront modulation is more effective than adding more planar Gaussians.

Setting	Method	N	PSNR ↑	SSIM ↑	LPIPS ↓	FLIP ↓	CVVDP ↑
RGB 20%	Baseline	30,720	29.265	0.8307	0.2895	0.1144	9.168
	Baseline⁺	33,280	29.386	0.8336	0.2853	0.1134	9.186
	CVQPG	30,720	29.450	0.8364	0.2828	0.1125	9.203
	Δ		+0.185	+0.0057	−0.0066	−0.0019	+0.035
RGB 10%	Baseline	15,360	28.034	0.7992	0.3292	0.1292	8.917
	Baseline⁺	16,640	28.179	0.8034	0.3241	0.1271	8.950
	CVQPG	15,360	28.225	0.8053	0.3217	0.1265	8.966
	Δ		+0.191	+0.0060	−0.0076	−0.0026	+0.049
Grayscale 20%	Baseline	15,360	29.504	0.8340	0.3813	0.0753	8.935
	Baseline⁺	17,280	29.753	0.8394	0.3766	0.0733	8.988
	CVQPG	15,360	29.834	0.8443	0.3694	0.0725	9.019
	Δ		+0.331	+0.0103	−0.0119	−0.0028	+0.085
Grayscale 10%	Baseline	7,680	28.076	0.7964	0.4204	0.0888	8.548
	Baseline⁺	8,640	28.317	0.8030	0.4134	0.0862	8.621
	CVQPG	7,680	28.398	0.8079	0.4069	0.0850	8.658
	Δ		+0.322	+0.0115	−0.0135	−0.0038	+0.109

Averaged over ten scenes and two depth planes. Baseline has the same primitive count N as CVQPG, Baseline⁺ the same parameter count. Δ: CVQPG over Baseline. CVQPG (bold) is best in every column.

Frequency Analysis

The gain concentrates in the mid-to-high frequency band: up to 11% (RGB) and 22% (grayscale) MSE reduction.

Frequency-resolved MSE reduction of CVQPG over the baseline: near zero at low frequency, peaking in the mid-to-high band, decaying near Nyquist.

Frequency-resolved \(\Delta\)MSE of CVQPG over the baseline.

Visual Comparison

Flower scene, RGB 20%: CVQPG vs. baseline at both focal planes.

Flower, RGB 20%

Sushi scene, RGB 20%: CVQPG vs. baseline at both focal planes.

Sushi, RGB 20%

Bento scene, grayscale 20%: CVQPG vs. baseline at both focal planes.

Bento, grayscale 20%

Siemens star, single plane: CVQPG 34.57 dB vs. baseline 31.53 dB PSNR.

Siemens star comparison (grayscale, single plane)

Holographic Display Prototype

Phase-only holograms encoded from CVQPG (3840 × 2160), captured on our display prototype.

Holographic display prototype.

Captured Burger scene at rear and front focus.

Captured result, rear (top) and front (bottom) focus.

BibTeX

@inproceedings{wang2026hologram,
  author = {Wang, Haolong and Zhan, Yicheng and Ak{\c{s}}it, Kaan and Qiu, Simeng},
  title = {Hologram Representation via Quadratic Phase Gaussian Splatting},
  booktitle = {SIGGRAPH Asia 2026 Technical Communications (SA Technical Communications '26)},
  year = {2026},
  month = {December 01--04},
  publisher = {Association for Computing Machinery},
  location = {Kuala Lumpur, Malaysia},
  pages = {4},
  isbn = {979-8-4007-2841-9/2026/12},
  doi = {10.1145/3829339.3847858},
  url = {https://arxiv.org/abs/2609.11434}
}