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Hologram Representation via Quadratic Phase Gaussian Splatting
Haolong Wang, Yicheng Zhan, Kaan Akşit, Simeng Qiu
TL;DR
Hologram representations must preserve dense high-frequency variations, but flat-phase Gaussian primitives have limited wavefront flexibility. CVQPG adds a learnable curvature parameter and shape-adaptive quadratic phase to complex Gaussian primitives, improving reconstruction quality under equal primitive counts. The method’s gains are strongest in grayscale, while shared curvature across wavelengths remains a limitation.
Problem
Efficient hologram representations must preserve dense high-frequency details, while flat-phase Gaussian primitives leave their representational capacity under-explored.
Method
CVQPG adds one learnable curvature parameter per complex Gaussian primitive and uses quadratic phase modulation with sampling-aware curvature control.
Results
+0.19 dB RGB and +0.33 dB grayscale mean PSNR gains are achieved over the planar Gaussian baseline under equal primitive counts.
Takeaways & Limitations
Equal-parameter evaluations indicate that modulating primitive wavefronts is a lightweight enhancement, with CVQPG preserving mid-to-high frequencies in natural images.
Takeaways & Limitations
Shared curvature cannot be simultaneously optimal for all wavelengths, producing larger gains in grayscale than RGB.
Abstract
from arXiv · showhide
We introduce Complex-Valued Quadratic Phase Gaussian (CVQPG), a novel hologram representation method that replaces standard 2D Gaussian representations used in 2D Gaussian Splatting with 2D quadratic phase functions. CVQPG incorporates additional learnable parameters to control the curvature of these bases. We evaluate our approach against state-of-the-art methods, exceeding the visual quality by +0.19 dB (RGB) and +0.33 dB (grayscale) on average in holographic reconstructions. Specifically, our equal parameter count evaluations show that modulating the primitive's wavefront is an effective and lightweight enhancement for hologram representations. In addition, our frequency domain analysis illustrates that CVQPG has successfully preserved the mid-to-high frequency band of natural images.
1 Introduction
CVQPG addresses the limited representational capacity of flat-phase Gaussian primitives by adding a learnable curvature parameter to modulate holographic wavefronts. Experiments report generalized quality improvements and better parameter effectiveness than the baseline.
- 1 Introduction: The method targets dense hologram variations where efficient representations must preserve high-frequency details.The introduction contrasts this challenge with low-frequency preference in implicit neural representations and Gaussian compression approaches.
- 1 Introduction: CVQPG adds one curvature parameter per primitive to explicitly model the field’s wavefront.The curvature parameter is derived with a per-primitive bound from the display sampling limit to avoid high-frequency aliasing.
- 1 Introduction: Experiments evaluate CVQPG across settings, including frequency-domain features and component ablations.The authors report generalized quality improvement and improved parameter effectiveness over the baseline.
2 Methods
CVQPG represents complex holograms with Gaussian primitives whose quadratic phase profiles provide learnable, shape-adaptive curvature. Rendering propagates the accumulated hologram across depth planes, while bounded curvature and staged optimization control aliasing and training stability.
- 2 Methods: CVQPG adds a curvature parameter to complex-valued Gaussian primitives, replacing flat phase with a quadratic phase wavefront.Each primitive retains opacity, color, position, scale, rotation, and phase parameters while adding scalar γ.
- 2 Methods: Mahalanobis QPF aligns elliptical phase fringes with the Gaussian envelope, avoiding the mismatch caused by standard circular wavefronts.The method uses the covariance-based squared Mahalanobis distance as the spatial metric.
- 2 Methods: The curvature parameter represents optical power and supports continuous wavefront transformation between divergence, convergence, and a planar state.The parameter is symmetrically bounded, with γ = 0 representing a planar wavefront.
- 2 Methods: Curvature is bounded using the SLM’s Nyquist limit over each primitive’s 3σ footprint to prevent aliasing artifacts.The bound is scale-adaptive because it depends on the primitive’s equivalent standard deviation, wavelength, and pixel pitch.
- 2 Methods: The hologram is formed by rasterizing primitive contributions, then propagated with BLASM and optimized against synthesized target intensities across depth planes.The loss combines reconstruction and SSIM terms summed over the reconstructed planes.
- 2 Methods: A two-stage curvature warm-up first optimizes planar Gaussians, then unlocks curvature to refine residual high-frequency details.This coarse-to-fine schedule is intended to stabilize early training.
3 Evaluation and Discussion
CVQPG improves holographic reconstruction quality across settings and preserves mid-to-high frequency details, while its ablations expose both the benefits of adaptive curvature and a boundary-related artifact.
- Qualitative Observations: +3.04 dB PSNR on the Siemens star demonstrates CVQPG’s improved high-frequency reconstruction, although abrupt boundary truncation produces ringing artifacts.A cosine-tapered window suppresses the ringing caused by truncating the chirp phase.
- Ablation Study: The component ablation evaluates scale-adaptive curvature control and the Mahalanobis QPF under an RGB 10% budget across five scenes.The comparison is reported in Table 2 for N=15,360 primitives.
- Ablation Study: Replacing the adaptive bound with a constant bound reduces the PSNR gain from +0.185 to +0.048 dB, identifying adaptive curvature control as decisive.
- Frequency Analysis: CVQPG’s gains cover the mid-to-high frequency band containing fine details of natural images across ten scenes.
- Ablation Study: Replacing the Mahalanobis QPF with the standard QPF costs 0.036 dB on average, while the vanilla model retains only a +0.046 dB gain.The ablation indicates that simply appending a standard QPF does not capture the full potential of phase modulation.
- Conclusion and Discussion: CVQPG improves reconstruction quality across metrics and settings, and equal-parameter evaluation favors optimizing curvature over increasing planar Gaussian count.The curvature parameter reaches its full potential only when bounded by the per-primitive sampling limit.
1 Degenration of Complex-Valued Quadratic Phase Gaussian
CVQPG degenerates to simpler phase representations under isotropic scales or zero curvature, linking its general form to standard quadratic-phase and planar Gaussian cases.
- Degeneration Cases: When the scale vector has identical elements, the covariance becomes a scaled identity and the Mahalanobis QPF becomes a standard QPF.
- Degeneration Cases: When the curvature parameter is zero, the phase profile is flattened and the CVQPG primitive degrades to a planar Gaussian with uniform phase.
2 Scale-Adaptive Curvature Control Derivation
The scale-adaptive curvature bound derives from the chirp’s spatial frequency and the SLM’s Nyquist limit, with a relaxation factor providing a safety margin.
- Frequency Derivation: The instantaneous spatial frequency is derived from the Mahalanobis QPF phase, with wavelength and SLM pixel pitch determining the frequency scale.
- Curvature Bound: The maximum chirp frequency is bounded by the SLM Nyquist limit of 0.5 cycles/pixel to avoid aliasing.
- Curvature Bound: The scale-adaptive curvature bound uses this frequency constraint, while η supplies an empirical safety margin.
3 Visual Results
Visual reconstructions compare CVQPG with the baseline across RGB and grayscale settings and multiple compression ratios.
- Visual Results: The visual comparison lists RGB 20%, RGB 10%, grayscale 20%, and grayscale 10% settings for CVQPG and the baseline.