Exact Computation of Trait-induced Merge Trees for Bivariate Fields

2026-08-07

AI Summary

This work addresses the limitations of existing methods for computing Trait-Induced Merge Trees (TIMTs), which rely on vertex sampling and piecewise linear interpolation and often fail to accurately capture the true topology of Euclidean distance functions, leading to feature loss or extremum displacement. Focusing on piecewise linear bivariate fields, the authors analyze the convexity of sublevel sets within tetrahedra to construct a weighted graph that enables exact TIMT reconstruction and reveals its intrinsic connection to the Jacobi set. The study presents the first exact algorithm for TIMT computation in bivariate fields, provides theoretical proof of the triviality of local merge structures, demonstrates that global topological changes arise solely from simplex boundary gluing, and derives an upper bound on interpolation error. Implemented using CGAL and VTK, the method robustly preserves zero-level features, accurately locates minima and merge events across diverse trait types, and validates its accuracy and practicality on both synthetic and real-world data.

Abstract

Trait-induced merge trees (TIMTs) provide a robust topology-based method for selecting and browsing feature level sets in multivariate data by analyzing the distance field induced by a user-specified trait in attribute space. Existing TIMT computations typically sample this distance field at mesh vertices and assume piecewise-linear interpolation, although the Euclidean distance-to-trait function is generally not piecewise linear on the original mesh. As a result, the resulting merge tree may miss zero-valued features and may perturb the locations and values of minima and merge events. We study the exact computation of TIMTs for piecewise-linear bivariate fields, focusing first on point traits. We show that the restricted sublevel sets inside each tetrahedron are convex and therefore have trivial local merge-tree structure, implying that global topological changes arise only through gluing across simplex boundaries. Based on this observation, we construct a weighted graph whose merge tree is isomorphic to the exact merge tree of the induced distance field. We further relate TIMTs to Jacobi sets, showing how nonzero edge events of the TIMT are localized by the singular structure of the underlying bivariate map. We establish a theoretical upper bound on the error of the vertex-sampled linear interpolation, expressed in terms of the maximum length of projected mesh edges in the range. We discuss extensions to line, line-segment, and finite point-set traits, and implement the method robustly using CGAL and VTK, demonstrating results on both synthetic and real-world datasets.

Problem

• merge trees
• trait-induced
• bivariate fields
• distance field
• topological errors

Innovation

• exact merge tree
• trait-induced topology
• bivariate fields
• Jacobi sets
• convex sublevel sets