Gaussian Entropy Fields: Driving Adaptive Sparsity in 3D Gaussian Optimization

TL;DR

This work reframes 3D surface reconstruction in Gaussian Splatting as an entropy minimization problem, promoting low configurational entropy to drive dominant primitives and suppress redundant ones. It introduces the Surface Neighborhood Redundancy Index (SNRI), image-entropy guided weighting, and multi-scale entropy alignment to adaptively preserve geometric detail while enforcing sparsity. The approach achieves competitive geometric accuracy (DTU, T&T) and superior perceptual rendering (Mip-NeRF 360), demonstrating robust performance across dense and sparse scenes with efficient optimization. By leveraging an information-theoretic perspective, the method enables natural surface emergence and robust reconstruction without relying on hard geometric priors.

Abstract

3D Gaussian Splatting (3DGS) has emerged as a leading technique for novel view synthesis, demonstrating exceptional rendering efficiency. \replaced[]{Well-reconstructed surfaces can be characterized by low configurational entropy, where dominant primitives clearly define surface geometry while redundant components are suppressed.}{The key insight is that well-reconstructed surfaces naturally exhibit low configurational entropy, where dominant primitives clearly define surface geometry while suppressing redundant components.} Three complementary technical contributions are introduced: (1) entropy-driven surface modeling via entropy minimization for low configurational entropy in primitive distributions; (2) adaptive spatial regularization using the Surface Neighborhood Redundancy Index (SNRI) and image entropy-guided weighting; (3) multi-scale geometric preservation through competitive cross-scale entropy alignment. Extensive experiments demonstrate that GEF achieves competitive geometric precision on DTU and T\&T benchmarks, while delivering superior rendering quality compared to existing methods on Mip-NeRF 360. Notably, superior Chamfer Distance (0.64) on DTU and F1 score (0.44) on T\&T are obtained, alongside the best SSIM (0.855) and LPIPS (0.136) among baselines on Mip-NeRF 360, validating the framework's ability to enhance surface reconstruction accuracy without compromising photometric fidelity.

Figures (11)

• Figure 1: Overview of the structured sparse strategy. In contrast to 2DGS huang20242d which exhibits performance degradation in complex scenes, the proposed method enables superior detail reconstruction while maintaining computational efficiency. Representative details illustrate improved thin-structure preservation and reduced aliasing under GEF.
• Figure 2: Overview: Entropy-Driven Surface Reconstruction.[id=] (a) Starting from 3D Gaussian primitives, RGB images are rendered for training. (b) The rendered RGB images are used to compute multi-scale image-entropy maps, while delayed updates of the opacity field produce rendered entropy maps. A multi-scale entropy alignment module encourages the entropy of the rendered opacity to match the image entropy so that geometric complexity follows image complexity. (c) Geometric regularization operates directly on Gaussian primitives: within each local neighborhood, primitives are encouraged to aggregate around the underlying surface and to share consistent normal directions. The rendered depth and normal maps are shown only for illustration and are not used as supervision during training. (d) The entropy-regularized sparsity constraint penalizes high-entropy opacity profiles within local neighborhoods (where $H(\alpha) > \eta$) and drives the configuration towards a low-entropy state with a dominant primitive aligned with the surface. (e) The ideal low-configurational-entropy state, where opacity histograms become sharp and concentrated on the surface, indicating a well-defined surface geometry.The framework transforms surface reconstruction from explicit geometric constraints to entropy minimization. Key components include: (1) geometric regularization via depth and normal consistency, (2) multi-scale entropy alignment with competitive weighting, and (3) entropy-regularized sparsity constraints. This method converts high-entropy primitive aliasing into low-entropy surface-aligned distributions, enabling natural surface emergence through information-theoretic optimization. See Sections \ref{['sec:entropy-sparsity']}, \ref{['sec:multi-scale-alignment']}, and \ref{['subsec:losses-design']} for details.
• Figure 3: Dual perspectives on entropy-driven sparsification.Left: [id=]Ray-based perspective showing how redundant primitive stacking creates multi-peaked opacity profiles along viewing rays.Ray-integrated entropy computation requires dense sampling along viewing rays, leading to high computational overhead. Right: [id=]Neighborhood-based perspective showing the same redundancy as diffuse opacity histograms in 3D spatial neighborhoods. Both perspectives capture primitive aliasing from different angles—the neighborhood formulation enables efficient optimization (1280x speedup) while inherently promoting multi-view consistency.The neighborhood-based method discretizes the optimization space into localized influence regions, achieving equivalent sparsity constraints with reduced computational complexity. The transformation from multi-peak opacity response to structured bi-modal distribution demonstrates effective entropy minimization within local neighborhoods. Curves and color encodings are illustrative and not to scale.
• Figure 4: Geometric illustration of SNRI metric for different primitive arrangements.
• Figure 5: Competitive weight distributions across multiple scales showing spatial complementarity with $\beta=6.0$. Fine scale $W_1$ (k=3) concentrates on textural regions and object boundaries (warm colors), medium scale $W_2$ (k=5) captures intermediate structural features, while coarse scale $W_3$ (k=9) dominates smooth background areas (cool colors). Note the spatial exclusivity where each region is primarily governed by one dominant scale.