DARB-Splatting: Generalizing Splatting with Decaying Anisotropic Radial Basis Functions

TL;DR

This work generalizes the splatting kernel in 3D Gaussian Splatting by introducing Decaying Anisotropic Radial Basis Functions (DARBFs) as plug-and-play alternatives to Gaussians. A correction factor $\psi$ is used to approximate the Gaussian projection, enabling efficient CUDA-based backpropagation for multiple kernels (Raised Cosine, Half-Cosine, Sinc, Inverse MQ, Parabolic) while maintaining comparable novel-view quality. Empirical results show substantial training-time speedups (e.g., up to ~34%) and memory reductions (up to ~45%) with certain kernels, along with modest or comparable PSNR/SSIM/LPIPS gains. The paper provides detailed simulations, Monte Carlo analyses, and extensive implementation notes, demonstrating practical viability and outlining directions for future exploration of non-exponential splatting kernels in radiance field reconstruction.

Abstract

Splatting-based 3D reconstruction methods have gained popularity with the advent of 3D Gaussian Splatting, efficiently synthesizing high-quality novel views. These methods commonly resort to using exponential family functions, such as the Gaussian function, as reconstruction kernels due to their anisotropic nature, ease of projection, and differentiability in rasterization. However, the field remains restricted to variations within the exponential family, leaving generalized reconstruction kernels largely underexplored, partly due to the lack of easy integrability in 3D to 2D projections. In this light, we show that a class of decaying anisotropic radial basis functions (DARBFs), which are non-negative functions of the Mahalanobis distance, supports splatting by approximating the Gaussian function's closed-form integration advantage. With this fresh perspective, we demonstrate up to 34% faster convergence during training and a 45% reduction in memory consumption across various DARB reconstruction kernels, while maintaining comparable PSNR, SSIM, and LPIPS results. We will make the code available.

Figures (19)

• Figure 1: This figure compares two different splat functions---Gaussian and cosine (specifically, half-cosine)---initialized for the same 3D covariance matrix, $\mathbf{\Sigma}$. Half-cosine functions, defined as $\cos\left(\frac{d_M^2}{\xi}\right)$ for $d_M^2 < \frac{\xi \pi}{2}$ where $\xi>0$, have finite support and exhibit effective reconstruction performance comparable to Gaussians. Notably, this non-exponential-based approach enhances training speed by 34% and reduces memory usage by 15%, providing a more efficient alternative.
• Figure 2: Overview of decaying anisotropic radial basis functions (DARBFs) shown with their respective 1D curves. These functions decay with distance and vary in their sensitivity to direction, making them effective for capturing anisotropic features in spatial data.
• Figure 3: Simulations of 1D signal reconstruction through backpropagation across different DARBFs. Here, the target, reconstructed, and individual components are represented by red, blue, and green lines, respectively. We achieve competitive reconstruction results, validating our hypothesis that signals can be effectively reconstructed using non-exponential alternative reconstruction kernels.
• Figure 4: Block diagram: We use the standard optimization pipeline from 3DGS kerbl3Dgaussians with modifications, introducing a correction factor ($\psi$) to obtain the projected 2D covariance matrix ($\Sigma'_{2 \times 2}$) compatible with splatting for DARBFs within the existing framework. Our changes within the pipeline are highlighted in red text.
• Figure 5: Training loss and speed curves for the Kitchen (Mip-NeRF 360 dataset) and Train (Tanks and Temples dataset) scenes across different reconstruction kernels. These plots reveal performance differences in terms of training speed while showcasing a similar loss curve. These differences can be attributed to the inherent characteristics of each kernel, which influence the training dynamics of the functions. Notice the significant performance improvement of half-cosines compared to other DARBFs in training speed, while exhibiting a similar loss curve.