10 research outputs found
A Family of Metrics from the Truncated Smoothing of Reeb Graphs
In this paper, we introduce an extension of smoothing on Reeb graphs, which we call truncated smoothing; this in turn allows us to define a new family of metrics which generalize the interleaving distance for Reeb graphs. Intuitively, we "chop off" parts near local minima and maxima during the course of smoothing, where the amount cut is controlled by a parameter ?. After formalizing truncation as a functor, we show that when applied after the smoothing functor, this prevents extensive expansion of the range of the function, and yields particularly nice properties (such as maintaining connectivity) when combined with smoothing for 0 ? ? ? 2?, where ? is the smoothing parameter. Then, for the restriction of ? ? [0,?], we have additional structure which we can take advantage of to construct a categorical flow for any choice of slope m ? [0,1]. Using the infrastructure built for a category with a flow, this then gives an interleaving distance for every m ? [0,1], which is a generalization of the original interleaving distance, which is the case m = 0. While the resulting metrics are not stable, we show that any pair of these for m, m\u27 ? [0,1) are strongly equivalent metrics, which in turn gives stability of each metric up to a multiplicative constant. We conclude by discussing implications of this metric within the broader family of metrics for Reeb graphs
A Comparative Study of the Perceptual Sensitivity of Topological Visualizations to Feature Variations
Color maps are a commonly used visualization technique in which data are
mapped to optical properties, e.g., color or opacity. Color maps, however, do
not explicitly convey structures (e.g., positions and scale of features) within
data. Topology-based visualizations reveal and explicitly communicate
structures underlying data. Although we have a good understanding of what types
of features are captured by topological visualizations, our understanding of
people's perception of those features is not. This paper evaluates the
sensitivity of topology-based isocontour, Reeb graph, and persistence diagram
visualizations compared to a reference color map visualization for
synthetically generated scalar fields on 2-manifold triangular meshes embedded
in 3D. In particular, we built and ran a human-subject study that evaluated the
perception of data features characterized by Gaussian signals and measured how
effectively each visualization technique portrays variations of data features
arising from the position and amplitude variation of a mixture of Gaussians.
For positional feature variations, the results showed that only the Reeb graph
visualization had high sensitivity. For amplitude feature variations,
persistence diagrams and color maps demonstrated the highest sensitivity,
whereas isocontours showed only weak sensitivity. These results take an
important step toward understanding which topology-based tools are best for
various data and task scenarios and their effectiveness in conveying
topological variations as compared to conventional color mapping
Labeled Interleaving Distance for Reeb Graphs
Merge trees, contour trees, and Reeb graphs are graph-based topological
descriptors that capture topological changes of (sub)level sets of scalar
fields. Comparing scalar fields using their topological descriptors has many
applications in topological data analysis and visualization of scientific data.
Recently, Munch and Stefanou introduced a labeled interleaving distance for
comparing two labeled merge trees, which enjoys a number of theoretical and
algorithmic properties. In particular, the labeled interleaving distance
between merge trees can be computed in polynomial time. In this work, we define
the labeled interleaving distance for labeled Reeb graphs. We then prove that
the (ordinary) interleaving distance between Reeb graphs equals the minimum of
the labeled interleaving distance over all labelings. We also provide an
efficient algorithm for computing the labeled interleaving distance between two
labeled contour trees (which are special types of Reeb graphs that arise from
simply-connected domains). In the case of merge trees, the notion of the
labeled interleaving distance was used by Gasparovic et al. to prove that the
(ordinary) interleaving distance on the set of (unlabeled) merge trees is
intrinsic. As our final contribution, we present counterexamples showing that,
on the contrary, the (ordinary) interleaving distance on (unlabeled) Reeb
graphs (and contour trees) is not intrinsic. It turns out that, under mild
conditions on the labelings, the labeled interleaving distance is a metric on
isomorphism classes of Reeb graphs, analogous to the ordinary interleaving
distance. This provides new metrics on large classes of Reeb graphs
Data analysis with merge trees
Today’s data are increasingly complex and classical statistical techniques need growingly more refined mathematical tools to be able to model and investigate them. Paradigmatic situations are represented by data which need to be considered up to some kind of trans- formation and all those circumstances in which the analyst finds himself in the need of defining a general concept of shape. Topological Data Analysis (TDA) is a field which is fundamentally contributing to such challenges by extracting topological information from data with a plethora of interpretable and computationally accessible pipelines. We con- tribute to this field by developing a series of novel tools, techniques and applications to work with a particular topological summary called merge tree. To analyze sets of merge trees we introduce a novel metric structure along with an algorithm to compute it, define a framework to compare different functions defined on merge trees and investigate the metric space obtained with the aforementioned metric. Different geometric and topolog- ical properties of the space of merge trees are established, with the aim of obtaining a deeper understanding of such trees. To showcase the effectiveness of the proposed metric, we develop an application in the field of Functional Data Analysis, working with functions up to homeomorphic reparametrization, and in the field of radiomics, where each patient is represented via a clustering dendrogram
Topological Analysis of the 2D von Kármán Street
International audienceTopology based analysis and feature tracking is a well studied area. In this work, we focus exclusively on a dataset called the von Kármán street, and apply topology-based methods to understand its vortices. For this analysis, we adapt the recently proposed edit distance between merge trees. We discern several interesting results. One, we observe spatial periodicity between the vortices, alternating every half-cycle. Two, we observe a distinct difference in spatial probability of vortex regions during a half-cycle. Further, we compare the accuracy of our spatial probability with an off-the-shelf machine learning approach