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Typography and Annotations: A Survey of Topology-Based
Methods

NAME
INSTITUTION

Abstract
The aim of this paper is presents state of the art in topology-based
visualization methods.
The paper describes process and results of annotations for generating
definition of terminologies in this area.
Terminologies enables a topology for models used to organize research
results and state of the art.
Focus is on the relationship between different topology based
visualization methods and then identifying themes that are common in
this research area.

Introduction
Data visualization is a critical part of research in the last few decades
Topological methods have become a solid foundation in providng insight into
complicated data set and the phenomenon that is behind it
Topology based methods have been around since the start of visualization
The concept has gained interests precisely when it comes to expressing user interests.
Focus on scalar and vector fields and topology based methods in each field

Literature Research and Annotation
Topology based methods are created on the idea of:
◦ abstracting characteristic structure like the skeleton from data and to construct visualization thereafter accordingly.
◦ utilize typological research to organize, describe and classify data to be used in visualization.

Typography:
◦ refers to the arrangement of types that involves pint-size, line length, typefaces,
leading, tracking and kerning

Regular topological concept includes
◦ homotopic equivalence
◦ connectedness
◦ and quotient space.

Conti…
In this project, we took keywords from these uses or annotation then grouped them.
This helps discuss and establish a clear definition for each group.

Since the primary focus of the project is topological methods with the primary application
being visualization, computational geometry articles or conference proceedings were not the
primary consideration of the paper.
Topology and annotation is based on certain and uncertain classification.
For scalar field certain classification consist persistent homology, level-based approach,
critical points and gradient-based approaches.
Then, for vector, this analysis has classified the certain and uncertain data visualization The
certain includes critical points, invariant sets, saddles, separatrices and connectors.
For uncertain category, this include field topology
(See table 1 next slide)

Literature Research and
Annotations

Topology-based Methods in Flow
visualization
Topography based method can be well analyzed through
◦ scalar field
◦ Vector fields.

1) Scalar field.
Two classes of feature definition exist in this field;
◦ threshold based feature (burning regions in the combustion simulation)
◦ gradient-based features (vector fields obtained by taking gradient in each location of the scalar field).

2 ) Vector Field

The State of the Art Scalar Fields
Certain data are time-dependent or time-independent.
Time independent: for scalar includes critical pints, merge tree, contour tree, merge
tree, and extremum graph.
The time-dependent: encompasses the critical point tracking as well as the dynamic
contour tree. The uncertain data include the mandatory critical points and confidence
region.

Critical point:
A point p at which gradient f is zero or when it vanishes. Δf (p)=0. The rest of the points
are d-manifolds called ordinary. The index of the critical path is essentially the negative
eigenvalues of H and separate Maxima, minima, and saddle.

Conti..

Certain Time-dependent scalar
fields.
Persistent Homology
Most topology-based methods of visualization for certain time-dependent are built on Morse theory
◦ Edelsbrunner and Harer (2008) uses this method to drive simplification
◦ EdlesonBrunner et al., (2008), started at a set point method, which is turned into a parameter
filtration of a cell through a mathematical algebra method.
Level set methods
A level set can be essentially be defined as a preimage of a function f: Ώ – R. (Farine et al., 2012).
◦

Carr et al, (2010), who computed the merge tree for a piecewise linear function

Exact time-dependent scalar field.
The classification in this includes critical point tracking, level set tracking and dynamic level set
graph.

Critical point tracking
◦ Edelbrunner et al. (2008) outlined an idea to track critical points of the time-dependent fields
on a structured grid.
◦ Level set tracking.
◦ Samtaney et al., ( 2012) track the sublevels grown from minima until the levels exceeded the
required geometric measures.
◦ Similarly, Szymczak and Brunhart‐Lupo (2012) proposed subdomain contour tree, where
he computed t it for adjacent time step and annotate each part with the domain of interest

Uncertain Time Independent Scalar
Field
Different topology methods have been presented to compute probabilities for critical
points and confident region.
.Bubeni (2004) constructed a function from a persistence diagram and used the linear
combination to provide a visualization aggregate of different scalar functions and then defined
measures such as standard deviation.
.Gunter et al proposed a characterization of critical point and the spatial relation for uncertain
piecewise in 2D.
.Zheng and Pang (2004, )and Kraus (2010) utilized the Monto Carlo method to query 2D
terrains in an uncertain time dependence classification.

State of the Art in Vector fields
Initially introduced by Helman and Hesselink as the first generation of topology-based method
visualization (Farine et al., 2012).
the topology methods were designed to locate classifies and display critical points in given pints in
a given vector and assign an icon.
In vector fiels critical points can be easily be identified as isolate points where the field velocity is
zero.Such a point can be classified in different types based on the approximation of the Jacobian
Matrix.

In practice, this can only be achieved through numerical analysis of the matrix

Conti..
Typographical analysis can be a critical step followed by the application of other
visualization methods.

Using topology to segment the vector field into regions can generate a good flow
especially in 2D.
In 3D, segmentation has to changes to more free topological properties.
The invariant sets represent basic topology-based visualization methods.

By far, the largest body of work on this method in vector field is based on the model of
the limit set as well as their associated connection

Exact time-independent vector fields
Critical points extraction from tetrahedral discretion was first described by Gunther and
Theisel (2016).

Mann and Rockwood (2002) used a mathematical model to identify cells containing
critical points.
In this research, Ren et al., (2020) extract and visualized higher-order critical points and
used other techniques for 3D

Mahrous et al., (2004) derive segmentation of the 3D vector field domain, explicitly
observing behaviors of integral curves. Their research shows significant segmentation
boundaries.
Weinkauf et al., (2004) also points putouts that the saddle connector is a good solution
to reducing separatrix

Conti..
Topography tracking: Topography tracking
interprets time and examines the alterations in
topological structure when there is a parameter
change or when the parameters changes.
One of the most important features of topography
tracking is that the parameters can vary.
◦ For 2D flows, Vejdemo-Johansson & Skraba (2016),
used a cell wise continuation topology method to and
to calculate the critical points evolution and its
bifurcations assuming that the linear interpolation
between time and vector field is discreet.
◦ Other researchers have extended this approach using
3D cases. In recent years, researchers who have
tried to use the approach using 4D trajectories have
found it hard to depict the critical point.

Uncertainty in Vector Field
The use of topological concepts to the vector field when it is under uncertainty is critical
in research.
◦ Lavine et al. (2012) based on the special characterization of critical points into two given fields,
uses a quantitative comparison method also termed as a distant measure.
◦ Based on their analysis, it is apparent the topology-based methods for stationary vectors could be applied

◦ Petz et al., (2012) ad...