Some theoretical results on discrete contour trees

TL;DR

This paper introduces a formal framework for discrete contour trees called iso-trees, applicable across all data dimensions, and establishes their equivalence with augmented contour trees, enabling cross-utilization of algorithms.

Contribution

It defines the iso-tree model for discrete data, proves its properties, and shows its isomorphism with augmented contour trees, bridging discrete and continuous contour analysis.

Findings

Iso-trees are applicable to all data dimensions.

Iso-trees are isomorphic to augmented contour trees.

Contour tree algorithms can be used for discrete contour trees.

Abstract

Contour trees have been developed to visualize or encode scalar data in imaging technologies and scientific simulations. Contours are defined on a continuous scalar field. For discrete data, a continuous function is first interpolated, where contours are then defined. In this paper we define a discrete contour tree, called the iso-tree, on a scalar graph, and discuss its properties. We show that the iso-tree model works for data of all dimensions, and develop an axiomatic system formalizing the discrete contour structures. We also report an isomorphism between iso-trees and augmented contour trees, showing that contour tree algorithms can be used to compute discrete contour trees, and vice versa.