Physical phenomena in science and engineering are frequently modeled using scalar fields. In scalar field topology, graph-based topological descriptors such as merge trees, contour trees, and Reeb graphs are commonly used to characterize topological changes in the (sub)level sets of scalar fields. One of the biggest challenges and opportunities to advance topology-based visualization is to understand and incorporate uncertainty into such topological descriptors to effectively reason about their underlying data. In this paper, we study a structural average of a set of labeled merge trees and use it to encode uncertainty in data. Specifically, we compute a 1-center tree that minimizes its maximum distance to any other tree in the set under a well-defined metric called the interleaving distance. We provide heuristic strategies that compute structural averages of merge trees whose labels do not fully agree. We further provide an interactive visualization system that resembles a numerical calculator that takes as input a set of merge trees and outputs a tree as their structural average. We also highlight structural similarities between the input and the average and incorporate uncertainty information for visual exploration. We develop a novel measure of uncertainty, referred to as consistency, via a metric-space view of the input trees. Finally, we demonstrate an application of our framework through merge trees that arise from ensembles of scalar fields. Our work is the first to employ interleaving distances and consistency to study a global, mathematically rigorous, structural average of merge trees in the context of uncertainty visualization.
Mihai and Westermann @cite_55 measure the likelihood of the occurrence of critical points with respect to both the positions and types of the critical points. Specifically, when the data uncertainty is described by a Gaussian distribution, confidence intervals are derived for the gradient and the determinant and trace of the Hessian matrix in scalar field ensembles to infer confidence regions for critical points @cite_55 . @cite_75 characterize critical points and their spatial relation for 2D uncertain scalar fields, where each vertex in a regular grid is assigned a probability density function (PDF) describing its scalar value. They identify so-called mandatory critical points -- spatial regions and function ranges where critical points have to occur in any realization of the input based on the PDF.
@cite_55 @cite_75
1988628849 1604082823
Abstract In scalar fields, critical points (points with vanishing derivatives) are important indicators of the topology of iso-contours. When the data values are affected by uncertainty, the locations and types of critical points vary and can no longer be predicted accurately. In this paper, we derive, from a given uncertain scalar ensemble, measures for the likelihood of the occurrence of critical points, with respect to both the positions and types of the critical points. In an ensemble, every instance is a possible occurrence of the phenomenon represented by the scalar values. We show that, by deriving confidence intervals for the gradient and the determinant and trace of the Hessian matrix in scalar ensembles, domain points can be classified according to whether a critical point can occur at a certain location and a specific type of critical point should be expected there. When the data uncertainty can be described stochastically via Gaussian distributed random variables, we show that even probabilistic measures for these events can be deduced. This paper introduces a novel, non-local characterization of critical points and their global relation in 2D uncertain scalar fields. The characterization is based on the analysis of the support of the probability density functions (PDF) of the input data. Given two scalar fields representing reliable estimations of the bounds of this support, our strategy identifies mandatory critical points: spatial regions and function ranges where critical points have to occur in any realization of the input. The algorithm provides a global pairing scheme for mandatory critical points which is used to construct mandatory join and split trees. These trees enable a visual exploration of the common topological structure of all possible realizations of the uncertain data. To allow multi-scale visualization, we introduce a simplification scheme for mandatory critical point pairs revealing the most dominant features. Our technique is purely combinatorial and handles parametric distribution models and ensemble data. It does not depend on any computational parameter and does not suffer from numerical inaccuracy or global inconsistency. The algorithm exploits ideas of the established join split tree computation. It is therefore simple to implement, and its complexity is output-sensitive. We illustrate, evaluate, and verify our method on synthetic and real-world data.