10 research outputs found
Local, Smooth, and Consistent Jacobi Set Simplification
The relation between two Morse functions defined on a common domain can be
studied in terms of their Jacobi set. The Jacobi set contains points in the
domain where the gradients of the functions are aligned. Both the Jacobi set
itself as well as the segmentation of the domain it induces have shown to be
useful in various applications. Unfortunately, in practice functions often
contain noise and discretization artifacts causing their Jacobi set to become
unmanageably large and complex. While there exist techniques to simplify Jacobi
sets, these are unsuitable for most applications as they lack fine-grained
control over the process and heavily restrict the type of simplifications
possible.
In this paper, we introduce a new framework that generalizes critical point
cancellations in scalar functions to Jacobi sets in two dimensions. We focus on
simplifications that can be realized by smooth approximations of the
corresponding functions and show how this implies simultaneously simplifying
contiguous subsets of the Jacobi set. These extended cancellations form the
atomic operations in our framework, and we introduce an algorithm to
successively cancel subsets of the Jacobi set with minimal modifications
according to some user-defined metric. We prove that the algorithm is correct
and terminates only once no more local, smooth and consistent simplifications
are possible. We disprove a previous claim on the minimal Jacobi set for
manifolds with arbitrary genus and show that for simply connected domains, our
algorithm reduces a given Jacobi set to its simplest configuration.Comment: 24 pages, 19 figure
Interactive Visualization for Singular Fibers of Functions f : R3 → R2
Scalar topology in the form of Morse theory has provided computational tools that analyze and visualize data from scientific and engineering tasks. Contracting isocontours to single points encapsulates variations in isocontour connectivity in the Reeb graph. For multivariate data, isocontours generalize to fibers—inverse images of points in the range, and this area is therefore known as fiber topology. However, fiber topology is less fully developed than Morse theory, and current efforts rely on manual visualizations.
This paper presents how to accelerate and semi-automate this task through an interface for visualizing fiber singularities of multivariate functions R3 → R2. This interface exploits existing conventions of fiber topology, but also introduces a 3D view based on the extension of Reeb graphs to Reeb spaces. Using the Joint Contour Net, a quantized approximation of the Reeb space, this accelerates topological visualization and permits online perturbation to reduce or remove degeneracies in functions under study. Validation of the interface is performed by assessing whether the interface supports the mathematical workflow both of experts and of less experienced mathematicians
Multivariate Topology Simplification
Topological simplification of scalar and vector fields is well-established as an effective method for analysing and visualising complex data sets. For multivariate (alternatively, multi-field) data, topological analysis requires simultaneous advances both mathematically and computationally. We propose a robust multivariate topology simplification method based on “lip”-pruning from the Reeb space. Mathematically, we show that the projection of the Jacobi set of multivariate data into the Reeb space produces a Jacobi structure that separates the Reeb space into simple components. We also show that the dual graph of these components gives rise to a Reeb skeleton that has properties similar to the scalar contour tree and Reeb graph, for topologically simple domains. We then introduce a range measure to give a scaling-invariant total ordering of the components or features that can be used for simplification. Computationally, we show how to compute Jacobi structure, Reeb skeleton, range and geometric measures in the Joint Contour Net (an approximation of the Reeb space) and that these can be used for visualisation similar to the contour tree or Reeb graph
Computational and Theoretical Issues of Multiparameter Persistent Homology for Data Analysis
The basic goal of topological data analysis is to apply topology-based descriptors
to understand and describe the shape of data. In this context, homology is one of
the most relevant topological descriptors, well-appreciated for its discrete nature,
computability and dimension independence. A further development is provided
by persistent homology, which allows to track homological features along a oneparameter
increasing sequence of spaces. Multiparameter persistent homology, also
called multipersistent homology, is an extension of the theory of persistent homology
motivated by the need of analyzing data naturally described by several parameters,
such as vector-valued functions. Multipersistent homology presents several issues in
terms of feasibility of computations over real-sized data and theoretical challenges
in the evaluation of possible descriptors. The focus of this thesis is in the interplay
between persistent homology theory and discrete Morse Theory. Discrete Morse
theory provides methods for reducing the computational cost of homology and persistent
homology by considering the discrete Morse complex generated by the discrete
Morse gradient in place of the original complex. The work of this thesis addresses
the problem of computing multipersistent homology, to make such tool usable in real
application domains. This requires both computational optimizations towards the
applications to real-world data, and theoretical insights for finding and interpreting
suitable descriptors. Our computational contribution consists in proposing a new
Morse-inspired and fully discrete preprocessing algorithm. We show the feasibility
of our preprocessing over real datasets, and evaluate the impact of the proposed
algorithm as a preprocessing for computing multipersistent homology. A theoretical
contribution of this thesis consists in proposing a new notion of optimality for such
a preprocessing in the multiparameter context. We show that the proposed notion
generalizes an already known optimality notion from the one-parameter case. Under
this definition, we show that the algorithm we propose as a preprocessing is optimal
in low dimensional domains. In the last part of the thesis, we consider preliminary
applications of the proposed algorithm in the context of topology-based multivariate
visualization by tracking critical features generated by a discrete gradient field compatible
with the multiple scalar fields under study. We discuss (dis)similarities of such
critical features with the state-of-the-art techniques in topology-based multivariate
data visualization
Hypersweeps, Convective Clouds and Reeb Spaces
Isosurfaces are one of the most prominent tools in scientific data visualisation. An isosurface is a surface that defines the boundary of a feature of interest in space for a given threshold. This is integral in analysing data from the physical sciences which observe and simulate three or four dimensional phenomena. However it is time consuming and impractical to discover surfaces of interest by manually selecting different thresholds.
The systematic way to discover significant isosurfaces in data is with a topological data structure called the contour tree. The contour tree encodes the connectivity and shape of each isosurface at all possible thresholds. The first part of this work has been devoted to developing algorithms that use the contour tree to discover significant features in data using high performance computing systems. Those algorithms provided a clear speedup over previous methods and were used to visualise physical plasma simulations.
A major limitation of isosurfaces and contour trees is that they are only applicable when a single property is associated with data points. However scientific data sets often take multiple properties into account. A recent breakthrough generalised isosurfaces to fiber surfaces. Fiber surfaces define the boundary of a feature where the threshold is defined in terms of multiple parameters, instead of just one. In this work we used fiber surfaces together with isosurfaces and the contour tree to create a novel application that helps atmosphere scientists visualise convective cloud formation. Using this application, they were able to, for the first time, visualise the physical properties of certain structures that trigger cloud formation.
Contour trees can also be generalised to handle multiple parameters. The natural extension of the contour tree is called the Reeb space and it comes from the pure mathematical field of fiber topology. The Reeb space is not yet fully understood mathematically and algorithms for computing it have significant practical limitations. A key difficulty is that while the contour tree is a traditional one dimensional data structure made up of points and lines between them, the Reeb space is far more complex. The Reeb space is made up of two dimensional sheets, attached to each other in intricate ways. The last part of this work focuses on understanding the structure of Reeb spaces and the rules that are followed when sheets are combined. This theory builds towards developing robust combinatorial algorithms to compute and use Reeb spaces for practical data analysis
Visual Analysis of Second and Third Order Tensor Fields in Structural Mechanics
This work presents four new methods for the analysis and visualization of tensor fields. The focus is on tensor fields which arise in the context of structural mechanics simulations.
The first method deals with the design of components made of short fiber reinforced polymers using injection molding. The stability of such components depends on the fiber orientations, which are affected by the production process. For this reason, the stresses under load as well as the fiber orientations are analyzed. The stresses and fiber orientations are each given as tensor fields. For the analysis four features are defined. The features indicate if the component will resist the load or not, and if the respective behavior depends on the fiber orientation or not. For an in depth analysis a glyph was developed, which shows the admissible fiber orientations as well as the given fiber orientation. With these visualizations the engineer can rate a given fiber orientation and gets hints for improving the fiber orientation.
The second method depicts gradients of stress tensors using glyphs. A thorough understanding of the stress gradient is desirable, since there is some evidence that not only the stress but also its gradient influences the stability of a material. Gradients of stress tensors are third order tensors, the visualization is therefore a great challenge and there is very little research on this subject so far.
The objective of the third method is to analyse the complete invariant part of the tensor field. Scalar invariants play an important role in many applications, but proper selection of such invariants is often difficult. For the analysis of the complete invariant part the notion of 'extremal point' is introduced. An extremal point is characterized by the fact that there is a scalar invariant which has a critical point at this position. Moreover it will be shown that the extrema of several common invariants are contained in the set of critical points.
The fourth method presented in this work uses the Heat Kernel Signature (HKS) for the visualization of tensor fields. The HKS is computed from the heat kernel and was originally developed for surfaces. It characterizes the metric of the surface under weak assumptions. i.e. the shape of the surfaces is determined up to isometric deformations. The fact that every positive definite tensor field can be considered as the metric of a Riemannian manifold allows to apply the HKS on tensor fields
Simplification of Jacobi Sets
Abstract. The Jacobi set of two Morse functions defined on a 2-manifold is the collection of points where the gradients of the functions align with each other or where one of the gradients vanish. It describes the relationship between functions defined on the same domain, and hence plays an important role in multi-field visualization. The Jacobi set of two piecewise linear functions may contain several components indicative of noisy or a feature-rich dataset. We pose the problem of simplification as the extraction of level sets and offset contours and describe an algorithm to compute and simplify Jacobi sets in a robust manner.