Smooth digital terrain modelling in irregular domain using finite element thin plate splines and adaptive refinement

TL;DR

This work develops and evaluates the finite element thin plate spline (TPSFEM) for digital terrain modeling in irregular domains, coupling FE-based smoothing with adaptive refinement to efficiently interpolate large, irregularly distributed elevation data. The TPSFEM minimizes a data-fit plus smoothing functional on a FE mesh, yielding a system whose size scales with the mesh nodes $m$ and remains independent of the data count $n$, and uses Dirichlet boundary conditions approximated by a TPS on a small data subset. An iterative, error-indicator–driven refinement scheme is used, with careful handling of boundary initialization to avoid over-refinement near complex boundaries; nodal averaging of new boundary values improves stability. Across three real-world datasets (Mountain, Canyon, River), adaptive TPSFEM substantially improves RMSE vs. node count in both square and irregular domains and compares favorably to TPS and CSRBFs in accuracy and cost, supporting its use for irregular-domain terrain modeling and motivating extensions to 3D.

Abstract

Digital terrain models (DTMs) are created using elevation data collected in geological surveys using varied sampling techniques like airborne lidar and depth soundings. This often leads to large data sets with different distribution patterns, which may require smooth data approximations in irregular domains with complex boundaries. The thin plate spline (TPS) interpolates scattered data and produces visually pleasing surfaces, but it is too computationally expensive for large data sizes. The finite element thin plate spline (TPSFEM) possesses smoothing properties similar to those of the TPS and interpolates large data sets efficiently. This article investigates the performance of the TPSFEM and adaptive mesh refinement in irregular domains. Boundary conditions are critical for the accuracy of the solution in domains with arbitrary-shaped boundaries and are approximated using the TPS with a subset of sampled points. Numerical experiments are conducted on aerial, terrestrial and bathymetric surveys. It is shown that the TPSFEM works well in square and irregular domains for modelling terrain surfaces and adaptive refinement significantly improves its efficiency. A comparison of the TPSFEM, TPS and compactly supported radial basis functions indicates its competitiveness in terms of accuracy and costs.

Figures (21)

• Figure 1: (a) Observed data of a bathymetric survey; and (b) corresponding interpolant.
• Figure 2: (a) U-shaped domain; and (b) square domain with an interior hole.
• Figure 3: Adaptively refined mesh with over-refinement near $[0.6,1.0]$.
• Figure 4: FE meshes built in (a) square domain; and (b) irregular domain; with triangular elements. The data points are represented as blue dots.
• Figure 5: Distribution of 500 data points within $[-2.4,2.4]^2$ region sampled (a) randomly; (b) using quadtree; and (c) using quadtree outside $[-1.9,1.9]^2$ region.