Topology-Preserving Terrain Simplification

TL;DR

This paper establishes criteria for terrain modifications that preserve topological features, enabling effective terrain simplification while maintaining essential geometric and topological properties.

Contribution

It provides necessary and sufficient conditions for elementary terrain operations to preserve persistent homology, aiding in topological terrain simplification.

Findings

Criteria for edge flips and vertex removals preserving persistent homology

Method reduces terrain complexity while maintaining topological integrity

Post-processing approach for terrain simplification that preserves key properties

Abstract

We give necessary and sufficient criteria for elementary operations in a two-dimensional terrain to preserve the persistent homology induced by the height function. These operations are edge flips and removals of interior vertices, re-triangulating the link of the removed vertex. This problem is motivated by topological terrain simplification, which means removing as many critical vertices of a terrain as possible while maintaining geometric closeness to the original surface. Existing methods manage to reduce the maximal possible number of critical vertices, but increase thereby the number of regular vertices. Our method can be used to post-process a simplified terrain, drastically reducing its size and preserving its favorable properties.