[Paper Review] Maintaining Contour Trees of Dynamic Terrains

Pankaj K. Agarwal, Lars Arge+3 more|arXiv (Cornell University)|Jun 16, 2014

Topological and Geometric Data Analysis5 references4 citations

TL;DR

This paper presents a kinetic data structure that efficiently maintains the contour tree of a dynamic terrain represented as a piecewise-linear triangulation under time-varying height functions. It processes certificate failures—caused only by adjacent vertices with equal heights or two saddle vertices on the same contour—in O(log n) time, enabling efficient incremental updates to topological analysis of evolving terrains.

ABSTRACT

We consider maintaining the contour tree $\mathbb{T}$ of a piecewise-linear triangulation $\mathbb{M}$ that is the graph of a time varying height function $h: \mathbb{R}^2 ightarrow \mathbb{R}$. We carefully describe the combinatorial change in $\mathbb{T}$ that happen as $h$ varies over time and how these changes relate to topological changes in $\mathbb{M}$. We present a kinetic data structure that maintains the contour tree of $h$ over time. Our data structure maintains certificates that fail only when $h(v)=h(u)$ for two adjacent vertices $v$ and $u$ in $\mathbb{M}$, or when two saddle vertices lie on the same contour of $\mathbb{M}$. A certificate failure is handled in $O(\log(n))$ time. We also show how our data structure can be extended to handle a set of general update operations on $\mathbb{M}$ and how it can be applied to maintain topological persistence pairs of time varying functions.

Motivation & Objective

• To enable efficient incremental maintenance of contour trees as terrain height functions evolve over time.
• To identify and handle only the minimal set of combinatorial changes in the contour tree due to topological events in the terrain.
• To support dynamic updates such as height changes and mesh deformations while preserving topological fidelity.
• To extend the data structure to maintain topological persistence pairs using join and split trees.
• To achieve low update cost per event by restricting certificate failures to critical topological transitions only.

Proposed method

• Maintains kinetic certificates that fail only when adjacent vertices have equal heights or two saddle vertices lie on the same contour.
• Uses a dynamic data structure to represent the contour tree and supports local updates via edge flips, vertex splits, and merges.
• Applies a modified version of the Delete(v) operation that merges edges instead of removing vertices, preserving height consistency.
• Extends the structure to handle global mesh deformations by maintaining certificates on edges and saddle pairs.
• Leverages join and split trees to implicitly represent topological persistence pairs, with pointers from negative saddles to the lowest minimums in their subtrees.
• Uses dynamic tree data structures to maintain and update persistence pair information in O(log n) time per operation.

Experimental results

Research questions

• RQ1 What are the minimal combinatorial changes in the contour tree that occur as a terrain’s height function evolves over time?
• RQ2 Which geometric events trigger changes in the contour tree’s structure during dynamic terrain evolution?
• RQ3 How can kinetic data structures be designed to handle only relevant topological events with minimal certificate failures?
• RQ4 Can topological persistence pairs be maintained efficiently during dynamic terrain updates using the contour tree?
• RQ5 What is the computational cost of maintaining contour trees and persistence pairs under continuous height changes and mesh modifications?

Key findings

• Certificate failures occur only when adjacent vertices have equal heights or two saddle vertices lie on the same contour, minimizing unnecessary updates.
• Each certificate failure is resolved in O(log n) time, significantly improving upon prior O(n) approaches.
• The data structure supports dynamic mesh updates, including height changes and global deformations, with efficient local reconfiguration.
• The join and split trees can be maintained with the same efficiency, enabling incremental computation of topological persistence.
• Topological persistence pairs are implicitly represented via pointers from negative saddles to the lowest minimums in their subtrees, with updates handled in O(log n) time.
• The approach enables efficient, incremental maintenance of contour trees and persistence pairs for large-scale, time-varying terrains in GIS and computational geometry applications.