ImplicitTerrain: a Continuous Surface Model for Terrain Data Analysis

TL;DR

ImplicitTerrain tackles the challenge of scalable, topology-aware terrain modeling by adopting an implicit neural representation (INR) framework. A Surface-plus-Geometry (SPG) cascade—comprising a smooth surface model $\Psi_s$ and a geometry residual $\Psi_g$—is trained progressively on Gaussian pyramids to yield a differentiable, high-fidelity terrain surface from which derivatives enable topological and topographical analyses. Topological features are extracted via Morse theory, constructing a Morse-Smale complex and a Morse Incidence Graph (MIG), with alignment to discrete Forman-gradient baselines validated on synthetic and real terrain data and robustness to noise. The results show accurate surface fitting, coherent topology extraction, and practical terrain feature computations (slope, aspect, curvature), indicating a scalable pathway for continuous-terrain analysis with potential impact on hydrology, geomorphology, and land management.

Abstract

Digital terrain models (DTMs) are pivotal in remote sensing, cartography, and landscape management, requiring accurate surface representation and topological information restoration. While topology analysis traditionally relies on smooth manifolds, the absence of an easy-to-use continuous surface model for a large terrain results in a preference for discrete meshes. Structural representation based on topology provides a succinct surface description, laying the foundation for many terrain analysis applications. However, on discrete meshes, numerical issues emerge, and complex algorithms are designed to handle them. This paper brings the context of terrain data analysis back to the continuous world and introduces ImplicitTerrain (Project homepage available at https://fengyee.github.io/implicit-terrain/), an implicit neural representation (INR) approach for modeling high-resolution terrain continuously and differentiably. Our comprehensive experiments demonstrate superior surface fitting accuracy, effective topological feature retrieval, and various topographical feature extraction that are implemented over this compact representation in parallel. To our knowledge, ImplicitTerrain pioneers a feasible continuous terrain surface modeling pipeline that provides a new research avenue for our community.

Figures (8)

• Figure 1: A 3D view of topological features derived from ImplicitTerrain's smooth surface model (left) and discrete mesh model (right) over a synthetic terrain dataset. Different critical point types are highlighted in different colors. Separatrix lines from our ImplicitTerrain are smoothly aligned with the terrain surface and color-coded by the critical point pair they connected.
• Figure 2: The pipeline of the ImplicitTerrain. Firstly, terrain data is preprocessed as a Gaussian pyramid for progressive fitting. Then, the cascaded Surface-plus-Geometry (SPG) model is trained to fit the smoothed terrain surface and the residual/displacement map in order. Finally, various terrain data analyses are supported by the smooth surface model. Model weights can be serialized for storage and inference to reconstruct the terrain surface with flexible structures (grids or TINs) and resolutions. Better viewed in the digital version.
• Figure 3: Gaussian pyramid and frequency view of $\text{Swiss}_1$ dataset.
• Figure 4: Comparison of topological analysis results of the synthetic terrain. Node colors and shapes represent the critical point types and the edge colors represent the separatrix lines as in the legend of (c) and (d). Better viewed in the digital version.
• Figure 5: ImplicitTerrain fitting results and topological features extracted from $\text{Swiss}_1$. (b) shows the fitting error with red color mapping to 0.25% error. Better viewed in the digital version.