MetricGrids: Arbitrary Nonlinear Approximation with Elementary Metric Grids based Implicit Neural Representation

TL;DR

MetricGrids tackles the limited nonlinear expressiveness of traditional grid-based implicit neural representations by introducing multiple elementary metric grids that capture higher-order Taylor terms. By combining curves in diverse metric spaces, hashed compression, and a high-order extrapolation decoder, the approach achieves superior fitting across 2D, 3D, and radiance-field tasks while keeping storage and parameter counts modest. The key contributions include the Taylor-inspired grid formulation, a hash-based compact encoding scheme, and a hierarchical decoder that estimates higher-order terms from lower-order grids. This method enhances nonlinear fidelity in grid-based INRs and demonstrates robust generalization across diverse signals, with practical impact for scalable, high-fidelity implicit representations of complex scenes.

Abstract

This paper presents MetricGrids, a novel grid-based neural representation that combines elementary metric grids in various metric spaces to approximate complex nonlinear signals. While grid-based representations are widely adopted for their efficiency and scalability, the existing feature grids with linear indexing for continuous-space points can only provide degenerate linear latent space representations, and such representations cannot be adequately compensated to represent complex nonlinear signals by the following compact decoder. To address this problem while keeping the simplicity of a regular grid structure, our approach builds upon the standard grid-based paradigm by constructing multiple elementary metric grids as high-order terms to approximate complex nonlinearities, following the Taylor expansion principle. Furthermore, we enhance model compactness with hash encoding based on different sparsities of the grids to prevent detrimental hash collisions, and a high-order extrapolation decoder to reduce explicit grid storage requirements. experimental results on both 2D and 3D reconstructions demonstrate the superior fitting and rendering accuracy of the proposed method across diverse signal types, validating its robustness and generalizability. Code is available at https://github.com/wangshu31/MetricGrids}{https://github.com/wangshu31/MetricGrids.

Figures (7)

• Figure 1: Illustration of the nonlinear fitting capabilities for various type of signals using the existing INR method and our MetricGrids.
• Figure 2: Illustration of the proposed MetricGrids. Given an input coordinate x, features are indexed in multiple metric grids, representing the different-order approximations as Taylor expansion. Hash encoding is used based on the sparsity of each metric grid to form a compact representation. A high-order extrapolation decoder is used to further estimate the remaining high-order terms not stored in the metric grids, thereby enhancing the nonlinear fitting capability.
• Figure 3: Illustration of the high-order extrapolation decoder. The ‘Mul’ represents the Hadamard product.
• Figure 4: Qualitative Comparison on Kodak dataset. Zoomed-in views of windows and roofs illustrate the ability of the baseline model and our proposed MetricGrids in fitting complex details.
• Figure 5: Qualitative Comparison on Gigapixel Image. The upper-right shows the ground truth image, containing numerous intricate details. The upper right, lower-left and lower-right images show L2 error maps for the baselines and our method, respectively. All experiments use consistent hyperparameter settings.