Change tensor: Estimating complex topographic changes from point clouds using Riemann manifold surfaces

Abstract

Estimating complex 3D topographic surface changes including rigid spatial movement and non-rigid morphological deformation is an essential task to investigate Earth surface dynamics. However, for current 3D point comparison approaches, it is challenging to separate rigid and non-rigid topographic surface changes from multi-temporal 3D point clouds. Additionally, these methods are affected by challenges including topographic surface roughness and point cloud heterogeneities (i.e., discrete and irregular point distributions). To address these challenges, in this paper, we consider the dynamic evolution of topographic surfaces as the geometric changes of Riemann manifold surfaces. By building Euclidean (straight) and non-Euclidean (curved) coordinate systems on Riemann manifold surfaces that are represented from point clouds, the rigid transformation and non-rigid deformation of the Riemann manifold surfaces are solved to conceptualize rigid and non-rigid change tensors, respectively. On this basis, we design rigid (i.e., translation and rotation) and non-rigid (i.e., stretch and distortion) change features to describe various topographic surface changes and quantify the associated uncertainties to capture significant changes. The proposed method is tested on pairwise point clouds with simulated and real topographic surface changes in mountain regions. Simulation experiments demonstrate that the proposed method performed better than the baseline (i.e., M3C2) and state-of-the-art methods (i.e., LOG), with a higher translation accuracy (more than 50% improvement), a lower translation uncertainty (more than 61% reduction), and strong robustness to varying point densities. These results also show that the proposed method accurately quantifies three additional types of change features (i.e., the mean accuracies of rotation, stretch, and distortion are 1.5°, 0.5%, and 3.5°, respectively). Moreover, the real-scene experimental results demonstrate the effectiveness and superiority of the proposed method in estimating various topographic changes in real environments, the applicability in analyzing geomorphological processes, and the potential contribution for understanding spatiotemporal patterns of Earth surface dynamics.