RaDe-GS: Augmented Gaussian Splatting

• RaDe-GS is a method that augments 3D Gaussian Splatting by analytically computing per-pixel depth and normal maps to enhance shape reconstruction.
• It integrates a closed-form rasterization pipeline with GPU acceleration to deliver high rendering efficiency and precise geometric consistency.
• RaDe-GS significantly improves geometric sharpness and segmentation accuracy in both static scenes and dynamic, radar-augmented driving environments.

RaDe-GS refers to a family of methods and a specific architectural innovation for augmenting 3D Gaussian Splatting (GS) representations with explicit, efficient depth and geometry reasoning. The conceptual foundation and technical realization of RaDe-GS appear across two principal lines: (1) as "Rasterizing Depth in Gaussian Splatting," where analytic per-pixel depth and normal maps are derived from the Gaussian primitives for enhanced shape reconstruction (Zhang et al., 2024); and (2) as a designation for "4DRadar-GS," a dynamic driving-scene pipeline fusing 4D radar with GS, enabling state-of-the-art self-supervised scene flow and segmentation (Tang et al., 16 Sep 2025). The following overview is primarily focused on the core contributions of RaDe-GS (Zhang et al., 2024), while situating their impact within the broader landscape of geometric and radar-augmented Gaussian splatting.

1. Motivation and Conceptual Overview

Classic 3D Gaussian Splatting achieves efficient, photorealistic novel view synthesis by describing a scene as a set of anisotropic Gaussian primitives $\{\mu_i,\,\Sigma_i,\,c_i,\,\alpha_i\}$ (mean, covariance, color, opacity) projected and composited along camera rays. While this approach excels in color fidelity and rendering efficiency, standard GS suffers from blurred and aliased geometry, attributable to its neglect of direct geometric constraints—namely, depth and surface normal information. Shape reconstruction is consequently imprecise, lacking sharp edges and failing to recover thin or intricate structures.

RaDe-GS addresses this by proposing a rasterized, closed-form approach to output analytically derived depth and normal maps alongside each rendered image. Leveraging the full parameterization of the underlying 3D Gaussians, RaDe-GS imposes pixelwise geometric consistency within the splatting framework, thus representing both photometric and geometric scene attributes with high fidelity and computational efficiency (Zhang et al., 2024).

2. Mathematical Formulation

Each 3D Gaussian primitive in RaDe-GS is defined by its spatial mean $\mu_i\in\mathbb{R}^3$, positive definite covariance $\Sigma_i\in\mathbb{R}^{3\times3}$, RGB color $c_i$, and scalar opacity $\alpha_i$. The density at any spatial point $x$ is

$$g_i(x) = \exp\left(-\frac{1}{2}(x-\mu_i)^\top \Sigma_i^{-1} (x-\mu_i)\right).$$

A pixel $(u,v)$ is mapped to a camera ray $x(u,v,t) = C + t\,d(u,v)$, given camera center $C$ and direction $d(u,v)$. To enable analytic depth and normal rasterization, the Gaussians are projected into this "ray space," yielding transformed mean and covariance (denoted $u_{i,c}$, $v_{i,c}$, $t_{i,c}$, and $\Sigma_i'$).

For depth, RaDe-GS derives a closed-form per-pixel composite depth by computing, per Gaussian, the local offset from the Gaussian center along the ray. Specifically, for $\Delta u = u_{i,c} - u$, $\Delta v = v_{i,c} - v$, the estimated depth is

$$d(u,v) = z_{i,c} + p_i\begin{bmatrix} \Delta u & \Delta v \end{bmatrix}^\top,$$

where $p_i$ and intermediate quantities are precomputed via analytic forms tied to the covariance structure [Eq. 17, (Zhang et al., 2024)].

Normals are similarly derived by computing the tangent vectors of the intersection patch on the ray and constructing normals via the cross product, yielding explicit per-pixel orientation. Both per-Gaussian depth and normal contributions are composited for final output, preserving GS's differentiability and compositional structure.

3. Algorithmic Pipeline

The RaDe-GS rasterization procedure is implemented as a fused, GPU-accelerated pass:

1. Precompute per-Gaussian camera-projected means, covariances, and all analytic coefficients required for depth/normal computation.
2. Project each Gaussian into an image-aligned bounding quad.
3. For each covered pixel within the quad:
   • Compute the Gaussian weight in 2D screen space.
   • Analytically determine its per-pixel depth and normal via the closed-form expressions.
   • Accumulate weighted contributions to color, depth, and normal buffers.
   • Composite opacities in correct order to preserve occlusions and front-to-back consistency.
4. Normalize outputs by total accumulated weight to yield final color, depth, and normal outputs per pixel.

A single fused GPU kernel performs color, depth, and normal computation, minimizing memory bandwidth and maximizing throughput. Data structures are organized in structure-of-arrays format, and transformed covariances are cached to avoid per-frame recomputation. Level-of-detail control is applied to maintain real-time rates for large scenes (Zhang et al., 2024).

4. Quantitative and Qualitative Evaluation

RaDe-GS was evaluated on standard multiview datasets, including DTU, LLFF, and Tanks & Temples:

• Chamfer distance (DTU): RaDe-GS achieves 0.38 mm, a 16% improvement over standard GS (0.52 mm) and outperforming NeuraLangelo (0.45 mm).
• Depth RMSE (DTU): 1.8 mm for RaDe-GS, compared to 2.3 mm for GS alone.
• Runtime: On RTX 3090, GS color-only runs at 28 fps; RaDe-GS, including color, depth, and normal rasterization, runs at 26 fps (7% additional overhead).

Qualitative results demonstrate significant improvements in geometric sharpness and surface recovery, notably in thin structures (chair legs, wires), where previous GS methods produced blurred or merged forms. Normal maps recovered via RaDe-GS show crisp shading transitions consistent with ground-truth scans [Figs. 5,6, (Zhang et al., 2024)].

5. Comparative Analysis with Related Techniques

A direct comparison with other representations and approaches reveals several differentiating strengths:

• 2D Gaussian Splatting: RaDe-GS surpasses non-geometric 2D GS in shape quality with negligible computational overhead.
• Mesh-based reconstruction (PSR, marching cubes): RaDe-GS requires no mesh extraction or topological heuristics, supporting real-time rendering without post-processing.
• NeRF-style volumetric rendering: RaDe-GS trains orders of magnitude faster and retains differentiability, with near-matching novel-view synthesis quality.
• Point-based graphics (NPBG): Gaussian splatting naturally interpolates color and geometry, while points require either neighborhood-normal estimation or learnable features.

A plausible implication is that RaDe-GS bridges the gap between explicit mesh-based geometry and neural implicit fields, offering the tractability of GS with enhanced geometric integrity (Zhang et al., 2024).

6. Practical Extensions and Applications

RaDe-GS extends the utility of GS-based representations in several emerging areas:

• Hybrid rasterization and ray tracing: Per-pixel depth and normal maps can drive mesh extraction or hybrid pipelines for interactive scene editing, relighting, and physically based rendering.
• Online SLAM and AR/VR tracking: The geometry-aware rasterization enables robust real-time pose tracking and dynamic scene understanding.
• Scene editing and mesh optimization: Joint optimization of GS and meshes, or use of RaDe-GS losses to supervise differentiable mesh extraction, becomes feasible.

Within 4DRadar-GS, the "RaDe-GS" principle is adopted implicitly for radar-driven segmentation, depth-priming, and canonical shape recovery in dynamic scenes, demonstrating its applicability in multi-modal, self-supervised pipelines (Tang et al., 16 Sep 2025).

7. Significance in Dynamic and Radar-Vision Contexts

RaDe-GS concepts have been adapted in the broader context of self-supervised, radar-augmented scene reconstruction, as in 4DRadar-GS (Tang et al., 16 Sep 2025). Here, accurate per-pixel geometry (depth, normals) derived via closed-form splatting facilitates:

• Radar-vision fusion for dynamic/static segmentation,
• Velocity-supervised deformation tracking (VGPT),
• Improved scale recovery and canonicalization across challenging dynamic driving scenes.

Quantitatively, systems leveraging RaDe-GS show compelling improvements in PSNR, SSIM, and LPIPS on the OmniHD-Scenes dataset, with qualitative results indicating superior artifact resistance in dynamic scenes compared to both self-supervised and LiDAR-supervised baselines. This illustrates RaDe-GS as a foundational geometric module not only for shape-enhanced splatting but also as a robust backbone for multi-sensor, multi-task scene modeling in mobile robotics and autonomous driving (Tang et al., 16 Sep 2025, Zhang et al., 2024).