BiWGS: Bidirectional Gaussian Splatting

• BiWGS is a framework that employs explicit Gaussian primitives, bidirectional scattering models, and coupled warping techniques for high-dimensional spatial mapping.
• The method integrates mathematical tools like bidirectional spherical harmonics and adaptive Gaussian optimization to boost performance in wireless channel and relightable rendering tasks.
• Its bidirectional consistency constraints and forward-backward warping enhance both novel view synthesis and channel prediction accuracy across diverse applications.

Bidirectional Gaussian Splatting (BiWGS) is a family of approaches that advance the representation and synthesis of high-dimensional spatial, physical, and visual fields using explicit Gaussian primitives, bidirectional scattering models, and coupled forward–backward warping. By generalizing classic 3D Gaussian Splatting (3DGS) to multidimensional, bidirectionally aware settings, BiWGS supports accurate modeling and prediction in applications ranging from wireless channel mapping to relightable novel view synthesis and physically plausible rendering. This article details the core mathematical frameworks, algorithmic strategies, applications, and performance characteristics of BiWGS methods, emphasizing documented advances in wireless environment modeling and computer graphics.

1. Mathematical and Computational Foundations

1.1. High-Dimensional Mapping Problems

BiWGS addresses problems where the goal is to learn a compact, high-dimensional function mapping between configuration parameters and measurements. For example, in wireless channel knowledge mapping (CKM), the function $q_E(\mathbf{p}_t, \mathbf{p}_r)$ outputs the single-input-multi-output (SIMO) channel vector $h \in \mathbb{C}^{N\times1}$ corresponding to a transmitter (Tx) position $\mathbf{p}_t\in\mathbb{R}^3$ and receiver (Rx) array position $\mathbf{p}_r\in\mathbb{R}^3$, given an environment $E$ (Zhou et al., 30 Oct 2025). Similarly, in novel view synthesis and relightable rendering, the mapping can relate object geometry and viewing/light direction to image or radiance content (Liu et al., 2024).

1.2. Gaussian Primitives

The core geometric representation uses explicit Gaussian primitives. Each primitive or ellipsoid is parameterized by a center $\boldsymbol{\mu}_m \in \mathbb{R}^3$, covariance structure $\Sigma_m = R_m S_m S_m^T R_m^T$ with rotation $R_m$ and scale $S_m$ (diagonal), and optional per-primitive features (opacity, color, or scattering coefficients). The spatial weight is

$$G_m(\mathbf{x}) = \exp\left(-\frac{1}{2} (\mathbf{x} - \boldsymbol{\mu}_m)^T \Sigma_m^{-1} (\mathbf{x} - \boldsymbol{\mu}_m) \right).$$

Projection and warping of Gaussians into lower-dimensional hyperplanes or camera views is accomplished by affine transforms and Jacobians, propagating parameterizations exactly (Ma et al., 29 Sep 2025, Liu et al., 2024).

1.3. Bidirectional Scattering and Spherical Harmonic Expansions

Bidirectional scattering in BiWGS is encoded by parameterizing the dependence of observed measurement or radiance on both incident and outgoing directions, using bidirectional spherical harmonics (BSH). For a scattering or transfer function $h \in \mathbb{C}^{N\times1}$0 on the product of two spheres, the BSH representation is

$h \in \mathbb{C}^{N\times1}$1

Symmetry and reciprocity constraints are imposed by parameter tying (Zhou et al., 30 Oct 2025, Liu et al., 2024). This enables modeling of view- and light-dependent effects, or, in channel modeling, the correspondence between incident and scattered electromagnetic directions.

2. BiWGS for 6D Wireless Channel Knowledge Mapping

2.1. Problem Setting and BiWGS Model

The BiWGS framework for 6D CKM fundamentally extends 3D Gaussian splatting by providing a mapping

$h \in \mathbb{C}^{N\times1}$2

for arbitrary transmitter–receiver spatial configurations. Major steps include:

• Representing virtual scatterers and obstacles as anisotropic 3D Gaussian ellipsoids.
• Expressing the angle-dependent complex scattering coefficient $h \in \mathbb{C}^{N\times1}$3 per ellipsoid by a real scale $h \in \mathbb{C}^{N\times1}$4 and BSH expansion $h \in \mathbb{C}^{N\times1}$5.
• Modeling attenuation and phase along Tx and Rx paths using projected 2D Gaussians.
• Single-bounce channel synthesis via summing contributions of all intersected ellipsoids along discretized angle-of-arrival (AoA) bins.

2.2. Optimization and Adaptive Model Control

Learning proceeds by minimizing a combined spectrum MSE and channel-power MAE loss:

$h \in \mathbb{C}^{N\times1}$6

where $h \in \mathbb{C}^{N\times1}$7 is the $h \in \mathbb{C}^{N\times1}$8-scale spatial-spectrum MSE and $h \in \mathbb{C}^{N\times1}$9 the channel-power MAE, with parameters including geometry, attenuation, and BSH coefficients. Adaptive density control is applied by splitting, cloning, or pruning Gaussians in response to underfitting or low opacity (Zhou et al., 30 Oct 2025).

2.3. Experimental Performance

BiWGS achieves strong performance on synthetic indoor ray-traced 6D datasets, outperforming classic MLPs for generalization on unseen Tx–Rx pairs:

• In channel-power prediction, BiWGS approximately halves the MAE relative to MLP, providing NMAE drops from 0.096→0.080, 0.158→0.101, and 0.237→0.109 (conference room, bedroom, office).
• In 3D spectrum prediction, BiWGS yields best-in-class LPIPS (0.4565) and near-best SSIM (0.6787), confirming perceptual and structural fidelity (Zhou et al., 30 Oct 2025).

3. BiWGS in View Synthesis and Relightable Rendering

3.1. Bidirectional Gaussian Primitives for Relightability

Relightable 3D Gaussian Splatting (BiGS) employs a bidirectional scattering formulation directly encoded in the Gaussian primitives, supporting real-time relighting with complex, volumetric, or subsurface effects (Liu et al., 2024). Light, geometry, and view dependence is captured by intrinsic light decomposition and BSHs, allowing the representation of transport and scattering without requiring surface normals or classical BRDFs.

3.2. Learning and Losses

Training optimizes geometric, opacity, and BSH coefficients using sequences acquired under One-Light-At-a-Time (OLAT), photometric and regularization losses that include energy conservation and nonnegativity constraints. High-quality appearance and geometry are achieved with moderate model size and fast inference (up to 40 fps for 40k Gaussians at 1080p).

3.3. Application Impact

The BiGS approach excels in representing scenes with volumetric or complex materials, supporting plausible rendering of hair, fur, iridescence, and subsurface scattering. Key architectural contributions are direct handling of S² × S² scattering, avoidance of normal-dependent models, and efficient SH evaluation at render time (Liu et al., 2024).

4. Bidirectional Warping and Consistency Constraints in BiWGS

4.1. Bidirectional Warping for Virtual View Synthesis

DWGS extends Gaussian splatting by introducing bidirectional warping between real and virtual views for robust supervision under sparse input conditions (Ma et al., 29 Sep 2025). Warping is performed both forward (real→virtual) and backward (virtual→real), synthesizing depths and colors in each domain and enforcing round-trip consistency in position, shape, and appearance.

4.2. Towards Fully Bidirectional Gaussian Splatting

A generalized BiWGS scheme envisions Gaussians as actively warped and splatted into both forward and backward views. Explicit consistency losses link Gaussians' spatial parameters and features across warping directions, leading to end-to-end, Gaussian-only novel view synthesis pipelines that do not rely on external depth estimation or pixel-based interpolation.

4.3. Empirical Findings

Bidirectional virtual view constraints consistently improve novel view synthesis quality. On LLFF with 3 sparse views, DWGS achieves higher PSNR and lower LPIPS compared to prior Gaussian and neural methods, affirming the advantage of explicit bidirectional coupling for geometry and appearance learning (Ma et al., 29 Sep 2025).

5. Comparison of BiWGS to Related Gaussian Splatting Methods

Framework	Application Domain	Bidirectionality Handling	Key Representation	Performance
BiWGS (Zhou et al., 30 Oct 2025)	6D Channel Knowledge Maps	BSH-based scattering, 6D mapping	3D Gaussian ellipsoids	Outperforms MLP/NeRF² for 6D CKM
BiGS (Liu et al., 2024)	Relightable NVS	BSH-based light/view, transport SH	Volumetric Gaussians	High-fidelity, real-time relighting
DWGS (Ma et al., 29 Sep 2025)	Sparse-view Synthesis	Forward+backward view warping pipeline	Gaussians + warping	Superior under sparse input

The above table highlights that BiWGS uniquely tackles high-dimensional (6D) mappings and electromagnetic phenomena, while BiGS and DWGS target photorealistic rendering, generalizing BSH models and introducing bidirectional learning or warping for appearance and geometry fidelity.

6. Outlook and Open Directions

Documented BiWGS frameworks enable the expansion of classical splatting methods into higher dimensional or physical domains through the explicit incorporation of bidirectionality, adaptive geometric control, and parameter-efficient harmonic expansions. Several open directions have been outlined:

• Computational scalability improvements (e.g., more advanced pruning or merging schedules) (Zhou et al., 30 Oct 2025).
• Extension to wideband or time-varying signals by parameterizing frequency dependence or dynamic splats.
• Integration of dynamic objects and temporal variation for time-varying high-dimensional knowledge maps.
• Extending Gaussian-level bidirectional consistency to more general scene types and measurement modalities (Ma et al., 29 Sep 2025).

A plausible implication is that end-to-end, high-dimensional BiWGS models will continue to supplant neural-only baselines in domains where physical interpretability, compactness, and generalization are critical.