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Topological Simplifcation of Jacobi Sets for Piecewise-Linear Bivariate
2D Scalar Fields with Adjustment of the Underlying Data
Felix Raith ID *
Leipzig University
Gerik Scheuermann ID †
Leipzig University & ScaDS.AI
Christian Heine ID ‡
Leipzig University
Figure 1: Comparison of the calculated Jacobi sets in the Cylinder Flow dataset on the left side of the figure for the original dataset
in the upper figure before simplification and the dataset in the lower figure after simplification with the collapse algorithm variant A
with threshold value
t=0.0001
. Furthermore, the corresponding neighborhood graphs are shown on the right side, with the color
corresponding to the orientation and the range area.
ABSTRACT
Jacobi sets are an important tool to study the relationship between
functions. Defined as the set of all points where the function’s gradi-
ents are linearly dependent, Jacobi sets extend the notion of critical
point to multifields. In practice, Jacobi sets for piecewise-linear
approximations of smooth functions can become very complex and
large due to noise and numerical errors. Existing methods that sim-
plify Jacobi sets exist, but either do not address how the functions’
values have to change in order to have simpler Jacobi sets or remain
purely theoretical. In this paper, we present a method that modifies
2D bivariate scalar fields such that Jacobi set components that are
due to noise are removed, while preserving the essential structures
of the fields. The method uses the Jacobi set to decompose the
domain, stores the and weighs the resulting regions in a neighbor-
hood graph, which is then used to determine which regions to join
by collapsing the image of the region’s cells. We investigate the
influence of different tie-breaks when building the neighborhood
graphs and the treatment of collapsed cells. We apply our algorithm
to a range of data sets, both analytical and real-world and compare
its performance to simple data smoothing.
Index Terms: Topological data analysis, bivariate data, Jacobi set,
topological simplification.
*e-mail: raith@informatik.uni-leipzig.de
†e-mail: scheuer@informatik.uni-leipzig.de
‡e-mail: heine@informatik.uni-leipzig.de
1 INTRODUCTION
The analysis of multivariate data is frequently carried out in science,
and the analysis of bivariate data in particular has established itself
as an important field, for example in climate simulations where
temperature and pressure are jointly investigated. The relationship
between the scalar fields can be examined using Jacobi sets [
18
] as
a tool for analysis. This mathematical concept originates from the
field of topology and represents a generalization of critical points
to multifields. In topological data analysis, Jacobi sets are used to
extract Ridge-Valley graph in computer vision and image processing
[
31
], and to track critical points in time [
4
,
18
]. In addition, they
are used to compare scalar fields [
19
] to study the Reeb space [
39
]
and to drive fiber surface extraction [
34
]. They are not only used
in visualization but also in other scientific disciplines [
1
,
2
,
25
,
30
].
This multifaceted application shows the importance of Jacobi sets
in scientific analysis, which is why these structures must be easy to
understand.
However, noise can be a major problem that makes analysis
difficult for domain experts. For this reason, some work has already
been done on the simplification of Jacobi sets [
37
,
3
]. For example,
Reeb graphs are created and the Jacobi sets are simplified using
the comparison measure
κ
.[
37
]. This involves removing loops or
components of the Jacobi set, which occur in particular in the case
of noise and numerical errors. However, only the representation
of the Jacobi sets is simplified, the underlying data itself is not
adjusted, which is unfavorable for further processing. In contrast,
smoothing filters adjust the data globally, but the extent to which
important structures are lost in the process has not been sufficiently
investigated. Important structures are regions in the dataset that lie
above a threshold value of a comparison metric and are also visually
1
arXiv:2408.08097v1 [cs.CG] 15 Aug 2024
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separated from the surrounding area , see Fig. 1 in the upper left
part in the brown region of interest. Subdivision algorithms can also
simplify Jacobi sets and, in particular, remove zig-zag, but only lead
to a better representation of this [
27
]. However, the visual evaluation
indicates that the number of Jacobi set components of the dataset
increases. Therefore, we will compare these approaches and our
approach to determine how suitable they are for simplifying the
Jacobi sets. Our goal is to modify the input functions in such a way
that the extracted Jacobi sets are simplified, while the input functions
are only minimally modified. Our approach to simplifying Jacobi
sets aims to clarify important structures and make them usable for
subsequent analyses by changing the underlying data. This is done
by iteratively collapsing cells, where all scalar values in a cell are set
to the same value, similar to what is shown in the 1D case in Fig. 3.
To identify the components to be collapsed, we use the neighborhood
graph of the components of the Jacobi set.
The contributions in this paper are:
•
We present an algorithm for simplifying Jacobi sets based on
iteratively collapsing cells while changing the underlying data
and identifying this with the help of a neighborhood graph.
•
We investigate the influence of different neighborhood graphs
on this algorithm.
•
We present an adapted Jacobi sets visualization that assigns
degenerate cells according to their point neighborhood.
•
We demonstrate the effect of smoothing and loop subdivision
on the simplification of Jacobi sets using analytical and real
data sets and compare them with our algorithm.
2 RE LATE D WORK
An overview of topology-based methods in visualization can be
found in the survey by Heine et al. [
21
]. He et al. [
20
] focuses more
on the visualization of multivariate data. In scalar field topology,
the essential features of scalar fields can be described by contour
trees [
8
,
10
,
42
], Reeb graphs [
33
] or the Morse-Smale complex [
16
].
FRWith their method, scalar fields can be topologically simplified
and the critical points and their relationships can be reduced. Carr
et al. [
9
] describe a method for simplifying the contour tree by
suppressing smaller topological features. Bremer et al. [
5
] create the
Morse-Smale complex of a 2D function and simplify the topology
by canceling pairs of critical points. Luo et al. [
29
] simplify critical
points based on a point cloud. Edelsbrunner et al. [
17
] introduce
the idea of persistent homology for the topological simplification of
a point cloud, which was introduced by Cohen-Steiner et al. [
14
].
Edelsbrunner et al. [
17
] introduce the idea of persistent homology
for the topological simplification of a point cloud, which extended
Cohen-Steiner et al. [14] to persistence diagrams.
In order to apply the methods mentioned to multifield data, an
adaptation is necessary. Methods such as Jacobi sets and Reeb
graphs cannot extended so easily. In addition, Carlsson et al. [
6
]
showed that a generalization of persistent homology is difficult.
Singh et al. [
35
] use partial clustering of high-dimensional data and
introduce the idea of the mapper based on this. The Joint Contour
Net by Carr et al. [
7
] is partly based on this idea, but use joint
contour plates and their topological connectivity. Reeb graphs were
introduced in the works [
13
] and [
36
] parallel for multifields. Chat-
topadhyay et al. [
13
] introduce the Jacobi structure for subdividing
the Reeb space, which creates a Reeb skeleton that corresponds to
the Reeb graph. An algorithm for calculating the Reeb space of
a bivariate, piecewise-linear scalar function on a tetrahedral grid
is presented by Tierny et al. [
38
]. Chattopadhyay et al. [
12
] use
the Joint Contour Net to further simplify multivariate data and to
simplify the Reeb skeleton.
The simplification of Jacobi sets of multifield data has not seen
much attention so far. The work by Bremer et al. [
4
] describes
a method that removes noise from the Jacobi sets of time-varying
data. In the work of H
¨
uttenberger et al. [
23
,
24
], a method for
extending topological structures from single scalar fields to multi-
fields using Pareto sets is proposed and the approach for simplifying
Jacobi sets [
37
] is extended to multivariate data. Suthambhara and
Natarajan [
37
] use reeb graphs to simplify the calculated Jacobi sets.
Bhatia et al. [
3
] presented a theoretical approach to simplify Jacobi
sets. In contrast to these works, our method also aims to simplify
the underlying data with a simple approach.
Another approach to simplify the data is to use non-topological
methods based on smoothing. This allows noise to be removed
from the entire data, but this is heavily dependent on the filters used.
Well-known filters are the binomial and Gaussian filters. Tong et al.
[
40
] decompose the field into 3 parts, smooth them individually, and
then sum them up again. In contrast, our method only processes the
data in areas where noise occurs, which prevents other structures
from being altered.
A completely different approach is presented by Kl
¨
otzel et al.
[
27
], they introduce a new method for calculating the Jacobi sets
based on local bilinear interpolation, which implements a gener-
alization of the definition by Edelsbrunner and Harer [
18
]. This
method smoothes the Jacobi sets by reducing zig-zag patterns and
better-resolving structures. This effect shows better results, espe-
cially at high resolutions. However, this leads to many small Jacobi
set components in very noisy areas of the dataset.
3 BACKG ROUND
Ascalar field is a function
f:D→C
, smoothly mapping from a
domain
D
, which is a compact
d
-manifold, to a range
C⊂R
. In this
paper, we only consider domains that are subsets of
R2
. A critical
point xof
f
is a point x
∈D
, where the gradient, i.e. the vector
of partial derivatives, of
f
vanishes at
x
:
∇f(
x
) =
0. A bivariate
scalar field can be viewed as two scalar fields
f,g
mapping from the
same domain. Edelsbrunner and Harer [
18
] introduced the Jacobi
set
J(f,g)
for a pair of functions
f,g
as the set of points x
∈D
,
where the gradients ∇f(x)and ∇g(y)are linearly dependent:
J(f,g):={x∈D| ∃λ∈R:∇f(x) + λ∇g(x) = 0
or λ∇f(x) + ∇g(x) = 0}(1)
Note that the critical points of
f
and
g
are trivially part of their Jacobi
set. Edelsbrunner and Harer [
18
] showed that, if
D
is a 2-manifold,
J(g,f)
is generically a collection of pairwise disjoint smooth curves,
free of any self-intersections. We will call the connected components
of a Jacobi set the Jacobi set components. Eq. 1has many equivalent
formulations, for example, Edelsbrunner et al. [
19
] mention that for
two 2D functions, the Jacobi set is the set of all points where the rank
of the Jacobian, i.e., the matrix comprised of the function’s gradients,
is smaller than 2. They also present a more general criterion: the set
of all points
x
where the local
κ
measure is
0
, where
κ(x)
is defined
as the length of the wedge product of the functions gradients. They
argue that the average of this measure over an area is related to the
amount of effort needed to change the topology of the functions in
that area.
A
k
-simplex
σ
is the convex hull of
k+1
affinely independent
points
P
σ={
x
0,...,
x
k}
. 2-simplices are triangles, 1-simplices are
line segments and 0-simplices are points. A simplex
τ
is a face of a
simplex
σ
if
P
τ⊆P
σ
. A simplicial complex is a set of simplices that
are closed under the face relation and any two simplices’ intersection
is either empty or a common face. A piecewise-linear function
assigns a function value to each 0-simplex and extends this to the
other simplices using barycentric interpolation. Edelsbrunner and
Harer [
18
] presented an algorithm to compute the Jacobi set for
piecewise-linear functions approximating smooth functions, which
we will not repeat here because our algorithm works slightly simpler,
due to a more restricted setting.
2
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4 SIMPLIFICATION OF BIVARI ATE 2D SCALAR FIELDS
Our algorithm assumes that the data are given as two piecewise linear
functions
f,g
on a compact domain
D⊆R2
, which we will treat as
a vector-valued function f
:D→R2
. It first computes a measure for
each triangle and uses it to determine the Jacobi set. It then uses the
Jacobi set to decompose the domain into a set of regions separated
by Jacobi set components and determines a neighborhood graph
where the nodes represent the regions and are connected by an edge
if separated by the same Jacobi set. It then assigns a weight to each
region by suitably aggregating the measures from the first step over
all triangles of this region. The weights are used to determine which
region’s functions values are to be manipulated in order to remove a
Jacobi set component.
4.1 Jacobi Set Computation
If
f,g
were smooth, then due to fundamental theorems in linear alge-
bra that relate the determinant of a matrix to the linear dependency
of its rows and columns, Eq. 1is equivalent to:
J(f) = {x∈D|det∇f(x) = 0}.(2)
Note that,
∇
f
(
x
)
is called the Jacobian of fin calculus, and is a
matrix composed of the gradients of
f
and
g
. The absolute value
of the Jacobian’s determinant would equal the value of
κ
by Edels-
brunner et al. [
19
], but by not taking the absolute value, we get an
alternative method to compute the Jacobi set for piecewise-linear
representations of bivariate 2D fields. For piecewise-linear data the
function restricted to the interior of each triangle
σ
can be given as:
f|σ(x) = Aσx+bσ.(3)
If one uses a Taylor expansion of f
|σ
at x
0=0
, one will find that
A
σ=∇
f
|σ(
x
)
, and hence that the determinant of the Jacobian is
constant across the triangle, but is typically discontinuous between
neighboring triangles. Nethertheless, some cells will have a positive
and some a negative determinant, and we determine the Jacobi set
as the list of edges which are faces to triangles with different signs
of the determinant.
Eq. 3also gives a more geometrical intuition useful to understand
the mechanics of our simplification algorithm. Essentially, Eq. 3can
be interpreted as a linear transformation of a space, i.e., how fmaps
the triangle from the domain to the range. Due to this transformation
triangles might get translated, rotated, scaled, and mirrored. If the
determinant of A
σ
, i.e. the Jacobian, is negative, the triangle’s image
is mirrored, i.e., the order of the triangle’s vertices switches between
clockwise and counter-clockwise. If the determinant is
0
then the
transformation will collapse the triangle, i.e., its image under fis a
line segment or a point. We will refer to the sign of the determinant
of the Jacobian as the orientation of a triangle. The magnitude of
the determinant furthermore gives the factor by which the area of
the triangle grows or shrinks, indicating how much effort it takes to
change the function values at a triangle’s vertices to collapse it.
4.2 Neighborhood graph
A neighborhood graph represents the relationship between objects
through nodes and edges. A node represents a Jacobi set compo-
nent, and an edge represents the geometric spatial proximity in the
domain. We examine four approaches for creating neighborhood
graphs labeled A to D, which differ in the grouping of the cells to the
components. An example of the different of neighborhood graphs
is shown in Fig. 2. This figure shows a section of a domain that is
examined in more detail in Fig. 2a and illustrates all four approaches.
The coloring of the cells represents two properties in the range: the
orientation, where positive oriented triangles are colored red and
negative oriented triangles are colored blue, and the area, where high
saturation means a large area. The Jacobi sets are shown in black.
(a) Cutout coordinates (-1.27, -0.52) – (-0.67, -0.28)
(b) Neighborhood graph A (c) Neighborhood graph B (d) Neighborhood graph C (e) Neighborhood graph D
Figure 2: Section of the cylinder flow dataset (a) is an example of
applying the different neighborhood graphs. (b) classical neighbor-
hood graph A, (c) neighborhood graph B with connected negative
cells, (d) neighborhood graph C with connected positive cells, and
(e) neighborhood graph A with connected cells corresponding to the
neighborhood.
For the neighborhood graph A, all cells are combined into one
node if they are not separated by a Jacobi set. This results in a
graph with
69
nodes and
108
edges for the domain section under
consideration. In neighborhood graph B, as shown in Fig. 2c, cells
are merged into one node if they fulfill the conditions of graph A. In
addition, two cells are combined into one node if they are connected
via a point and both have a negative orientation. The neighborhood
graph C, shown in Fig. 2d, follows the same approach as graph B,
but here two cells are combined into one node if they are connected
by a point and both have a positive orientation. These adaptations in
graphs B and C are based on the goal of converting the neighborhood
graph into a neighborhood tree. Such a tree can be simplified by
starting to combine nodes at the leaves as long as the range area
of these nodes are smaller than a threshold according to a suitable
metric. A side effect of this adaptation is the reduction of nodes
in the graph. The effects of these adaptations can be seen in the
resulting graphs. The neighborhood graph B has
39
nodes and
38
edges, while the graph C has
33
nodes and
32
edges. The different
orientations lead to significantly different graphs.
Finally, we consider the neighborhood graph D. Here, all cells
are combined into one node if they are not separated by a Jacobi
set. In addition, two cells are connected if they are connected by a
point, have the same orientation and this orientation prevails in the
neighborhood of the point. The goal of this approach is to minimize
the imbalance that arises in the neighborhood graphs B and C and
produce a more analyzable neighborhood tree. The resulting graph
for this approach is shown in Fig. 2e and has
23
nodes and
22
edges.
This graph shows a compact structure that may be well-suited for
further analysis.
To investigate the effect of the different neighborhood graphs
on the simplification of the Jacobi sets, all four approaches are
combined with the method developed, and the results are compared
visually.
4.3 Metrics to select Jacobi set components
For the reduction of the Jacobi set components a suitable metric
has to be chosen. There are geometric and topological metrics,
whereby we have chosen a topological metric in this work. Based
on the neighborhood graph, we now decide which of the Jacobi set
components should be collapsed. Different measures can be used to
choose the components:
3
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x
f(x)
x
f’(x)
A
B
C
D
A
BC
D
B’ C’
Figure 3: Example of collapsing a cell in 1D. Left before collapsing
the cell BC, right after collapsing. The dashed line shows when the
neighborhood is not taken into account. The solid line shows when it
is taken into account. The red color shows a positive and the blue a
negative orientation.
• The Jacobi set component’s domain area can be used:
AD=
n
∑
i=0
Ai(4)
•
The Jacobi set components range area can also be taken into
account:
A′
C=
n
∑
i=0
A′
i(5)
•
The hypervolume, which is the product of the Jacobi set com-
ponents domain and range area:
HV =
n
∑
i=0
Ai·A′
i(6)
We chose hypervolume, which takes into account both the area
of Jacobi set components in the domain and the range. Furthermore,
the other two metrics pose problems, especially if the Jacobi set
components areas are very different. This could lead to interesting
structures being lost. Another possibility would be the automatic
determination of this variable depending on the neighborhood. How-
ever, this method was not implemented in this work.
4.4 Collapse of Regions
To collapse a region, we use a greedy algorithm that shrinks the
regions starting from the Jacobi set components by collapsing in-
dividual cells until the entire region is collapsed. The collapse a
cell in the 2D case is a complex problem due to its influence on the
neighborhood. In the 1D case, all values at the vertices of a cell
can simply be set to the same value in the range, see Fig. 3. This
collapses the line cell to a point in the 1D case. Equivalently, in the
2D case, variant
V1
mapped all values at the vertices of a triangle
cell to one value and thus collapse the cell in the range, as shown
in Fig. 4a. Variant
V2
collapsed the cell to a line in the range, as
shown in Fig. 4a.
The difference between these two variants is that collapsing to
a point affects more neighboring cells compared to collapsing to a
line. Therefore, we prefer collapsing to a line in this work. Since
only one edge of the triangle cells needs to collapse to collapse the
entire cell, we decide which of the edges to select depending on the
neighborhood. We can therefore divide this into four cases:
1. No cell to be collapsed in the neighborhood Fig. 4a,
2. One cell to be collapsed in the neighborhood Fig. 4b,
3. Two cells to be collapsed in the neighborhood Fig. 4c,
4. Three cells to be collapsed in the neighborhood Fig. 4d.
V1
V2
A
C
BA’
C’
B’
A”
C”
B”
(a) Case 1
V2
A
C
B
A”
C”
B”
(b) Case 2
V2
A
C
B
A”
C”
B”
(c) Case 3
A
C
B
A
C
B
(d) Case 4
Figure 4: Example for the collapse of a triangle cell in the range for
all 4 cases to be considered: V1 shows the collapse to a point, and
V2 shows the collapse to a line. The green dashed edges connect
the cell with a cell to be collapsed. The orange dashed lines are the
edges to be scrambled.
In the first case, all edges of the triangle can be selected for
collapsing. In the second case, the edges that connect the cell to
a cell that is not to be collapsed should be retained. In the third
case, only the edges that connect the cell to a cell to be collapsed
can be collapsed. We skip the fourth case, as the collapsing of the
neighboring cells means that after several iterations they are also
on the edge of the Jacobi set component and can then be treated
according to one of the other cases.
4.5 Method
Motivated by the theoretical approach of Bhatia et al. [
3
], we have
developed a method that aims to change the functions by collapsing
a cell to reduce the complexity of the Jacobi sets while retaining
their essential structures. The collapse of a cell means that the area
that this cell sets up in the range is reduced to the value
0
. This
approach simplifies the Jacobi sets and thus also reduces the Jacobi
set components. The idea of collapsing comes from the 1D case,
in which 2 critical points can be reduced by mapping the function
values
f(x)
of the two points of a cell to the same value. In Fig. 3
this is illustrated for the cell BC.
Here you can see that this change affects at least one neighboring
cell. Therefore, the choice of a suitable value is also crucial to keep
the side effects as low as possible, as otherwise not all critical points
can be removed. In the 2D case, this concept is more difficult, as the
number of affected neighboring cells increases. Therefore, a suitable
metric must be used to keep the side effects as low as possible.
The entire method is presented as pseudocode in Algorithm 1.
Our algorithm uses a 2D bivariate scalar field defined on an un-
structured grid. For these, a neighborhood graph is created in
ComputeNeighborhoodGraph(F)
. In this graph, the nodes repre-
sent individual Jacobi set components, while the edges represent the
relationships between the different components. There are various
ways in which the Jacobi set components can be composed in the
neighborhood graph. These ways and their specifics are described
in detail in Sec. 4.2.
This neighborhood graph is used in our method to select Jacobi
set components. This is done in
FindCollapsibleCells(NG,t)
,
which uses the hypervolume from Sec. 4.3 and an appropriate thresh-
old to identify nodes that are considered irrelevant. The algorithm is
cell-based and therefore returns all cells in a list (
CL
) that belong to
the irrelevant nodes in a list.
The algorithm now begins to process the list
CL
. For each cell, it
calculates whether it lies on the border of a Jacobi set component to
be collapsed. To do this,
CalculateCellNeighborhood(CL,c)
returns for a given cell how many neighboring cells are connected
to it and are not contained in
CL
. This number is used to sort cells
and skip cells that are not on the border. For the remaining cells, the
list
(CV)
is returned that contains all possibility variants of how the
cell can be collapsed. These variant are described in more detail in
Sec. 4.4.
4
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Algorithm 1: Collapse Algorithm
Input :2D Bivariate Scalar Field F,
Threshold t
Output : Updated 2D Bivariate Scalar Field F,
Neighborhood Graph NG
1begin
2NG := Com puteNeighborhoodGra ph(F)
3CL := FindCollapsibl eCells(NG,t)
4while CL =/0 do
5for c∈CL do
6nc := Calcul ateCellN eighborhood(CL,c)
7if nc =0then
8continue
9CV := PossibleColl apseVariants(nc)
10 bcv := FindBestColl apseVariant(F,CV,c)
11 Ap plyCollapseVariant(F,c,bcv)
12
13 CL := CL \ {c}
14 CL := CL ∪F lip pedCel lNeighbors(N G,c)
15 if Cell sOscill ated(CL)then
16 break
This list is now used to select the best variant for collapsing the
cell. For this purpose, a metric is used that decides which variant has
the least side effects based on the direct point neighborhood of the
cell. This metric counts how many neighboring cells are negatively
affected when the different options are applied. A negative influence
occurs when cells change their orientation in the range, as shown by
the dashed line in Fig. 3 for the 1D case. In addition, the area in the
range in the neighborhood is also taken into account, and changes
are preferred if the total area in the range becomes smaller.
After selecting the best variant, the bivariate scalar field is ad-
justed accordingly in
ApplyCollapseVariant(F,c,bcv)
accord-
ing to the Sec. 4.4. The cell is then removed from the list
CL
. Since
the algorithm cannot always select a variant in which no side effects
occur,
FlippedCellNeighbors(NG,c)
recognizes negatively in-
fluenced cells and then adds them to the list CL.
To prevent the algorithm from oscillating, the
CellsOscillated(CL)
function checks this and terminates
the algorithm in this case. The result of this method is a modified
bivariate scalar field whose neighborhood graph is greatly reduced
and the Jacobi sets are also simplified.
4.6 Jacobi set Visualization
The calculation of Jacobi sets, which separate cells according to
their range, poses a challenge when a cell is degenerate or collapsed,
because a degenerate cell cannot be assigned to one of the two area
orientations to be separated. A possible solution to this problem is
to assign degenerate cells based on their neighborhood.
The Jacobi sets are calculated as described in Sec. 4.1. In addition,
degenerate cells are assigned based on their point neighborhood of
the Jacobi set components that predominate in the neighborhood.
This means that the cell is assigned to the orientation that is larger
when the neighboring cell orientations are summed up. If there are
only degenerated cells in the direct neighborhood, the neighbors
of the degenerated cells are considered recursively until a clear
assignment is possible.
When applying this method to the four possible neighborhood
graphs from Sec. 4.2, this leads to slightly different results when
dealing with degenerated cells. For the neighborhood graph A and
the neighborhood graph D, which are mapped in Fig. 2b,2e, col-
lapsed cells are assigned to the Jacobi set component as described.
In neighborhood graph B, shown in Fig. 2c, degenerated cells are
assigned to a negative component as long as a negative cell exists in
their neighborhood regardless of what the summed neighbor cell ori-
entation is. For neighborhood graph C, which is mapped in Fig. 2d,
the procedure is identical to that for the neighborhood graph B,
except that it applies to positive cells.
These adjustments ensure a clear assignment of degenerated cells,
which is important for further analysis and interpretation of the
Jacobi sets. However, it cannot be guaranteed that the length of the
Jacobi set is always the shortest.
5 RE SU LTS
To evaluate how much smoothing filter and loop subdivision as
well as the developed collapse algorithm (CA) in all
4
variants
simplify the Jacobi sets, we apply these algorithms to three datasets
and compare them with the original data. To do this, we compare
visually and with two measures whose results for all datasets in
Tab. 1. These measures are the number of Jacobi set components, and
the length of all Jacobi sets in the domain. The number of Jacobi set
components is a good measure to investigate a simplification since
this quantity can be derived directly from the neighborhood graph
and can be considered both visually and as a quantity. However, this
quantity alone could lead to misinterpretations if many Jacobi sets
are connected to a large closed edge, which reduces the components
but has a negative effect on the length of the Jacobi sets. Therefore,
we also consider the length of all Jacobi sets as a second parameter
to evaluate to what extent the structure could also be simplified.
In all figures, the color in the domain shows the orientation of the
cell in the range, while at the same time, the range area is visualized
via saturation. The demonstrations were run on a MacBook Pro (14-
inch, 2021) with a Apple M1 Max, and 64 GB Ram. Implemented
is the algorithm inside our framework with C++.
5.1 Simple Smoothing Approaches
For our evaluations, we use known smoothing methods on the one
hand and loop subdivision on the other. These are classic smoothing
filters to reduce noise and remove irregularities. More precisely, we
use the Gaussian filter and the binomial filter. Loop subdivision
is a method from computer graphics for smooth subdivision of
triangular grids. This method, developed by Loop [
28
], has become
a fundamental tool in computer graphics. For all comparisons we
use
σ=1000
for the Gaussian filter,
r=1
for the binomial filter,
and
4
subdivision steps in the loop subdivision. In the appendix,
there are further comparisons with other
σ
settings for the Gaussian
filter.
5.2 Cylinder Flow (Synthetic)
First, we consider the analytical Cylinder Flow [
26
] dataset, which
is freely available on the ETH Z
¨
urich website [
11
]. The dataset is a
regular 2D grid with a resolution of
450 ×200
and
500
time steps,
on which a synthetic vector field is defined. This vector field was
used as a co-gradient to a stream function of Jung et al. [
26
] and
represents a simplified model of a K
´
arm
´
an vortex street. We use
time step 1 for the comparisons and triangulate the grid.
The aim of the analysis of this dataset is to show the effect of the
CA in all variants, the smoothing filters and the loop subdivision
on the simplification of the Jacobi sets and to find out which CA
variant has the greatest effect and to use this for the other datasets.
The original data serves as a reference. The resulting datasets and
the corresponding Jacobi sets are shown in Fig. 1 and Fig. 5.
When observing the datasets, three regions of interest (ROI) can
be identified that are relevant for the comparison. The first ROI
is marked green in the datasets and contains the two right corners
where many small Jacobi set components can be recognized in Fig. 1
of the original data. These are due to noise, as the range area is very
small, which can also be seen from the saturation of the colors. In the
5
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conference. The final version of this record is available at: xx.xxxx/TVCG.201x.xxxxxxx/
(a) Loop Subdivision (b) Binomial filter (c) Gaussian filter
(d) Collapse Algorithm Variant B (e) Collapse Algorithm Variant C (f) Collapse Algorithm Variant D
Figure 5: Comparison of the calculated Jacobi sets in the Cylinder Flow dataset for the Loop Subdivision (a), the Binomial filter (b), the Gaussian
filter (c), the Collapse Algorithm variant B with
t=0.0001
(d), Collapse Algorithm variant C with
t=0.0001
(e), and Collapse Algorithm variant D
with t=0.0001 (f). Three regions of interest (ROI) are highlighted in color for visual comparison.
loop subdivision, the Jacobi set components can still be recognized
very well; a simplification of the Jacobi sets is therefore not possible.
After smoothing with the binomial filter, a reduction of the Jacobi
set components can already be recognized. The smoothing is further
intensified with the Gaussian filter, which even leads to the removal
of all Jacobi set components in this ROI. A similar result is obtained
after applying the CA variants. Here, all Jacobi set components can
also be removed, which can be seen in Fig. 5d, e, f. Only in variant
A do individual Jacobi set components remain. The second ROI
is marked in brown in the top center of the original data. A larger
reddish Jacobi set component can be seen here, with several smaller
components adjacent to it , which should be removed. After applying
the loop subdivision, the large Jacobi set component is visible and
the small ones are no longer recognizable. When smoothing with the
binomial filter, there is hardly any effect in Fig. 5b and the large and
small Jacobi set components are still present. In contrast, the effect
of smoothing by the Gaussian filter is too large and all components
are removed, which is visible in Fig. 5c. When using the CA variant
D, the result is similar to the binomial filter, and all components are
still present. CA variants B and C do better here, where the large
Jacobi set component remains and almost all small components
disappear. In CA variant A, only the large component remains, as
can be seen in Fig. 1 at the bottom left. The third ROI is marked
in yellow in the center of the original data. In this ROI, several
small Jacobi set components can be recognized in the original data.
In the loop subdivision, several large Jacobi set components are
separated from each other as can be seen in Fig. 5a. This indicates
that the small components are caused by noise or numerical errors
and should disappear. With the smoothing filters, the result is similar
to before. The binomial filter smoothes too little and the Gaussian
filter too much, whereby in this ROI the Gaussian filter does not
remove all the small Jacobi set components, but even creates many
new ones. Using the CA variants produces very different results
here. In variant C, for example, the Jacobi set components are not
separated but connected. In variants B and D, the larger components
are separated from each other, but many small components remain.
In variant A, the large components of the Jacobi set are completely
separated and almost all small components disappear so that the
results of the loop subdivision are very similar.
Overall, the visual comparison of the cylinder flow shows that the
CA variant A is the best variant. This is also shown in the Tab. 1. To
confirm this, the Fig. 1 on the right shows the neighborhood graphs
of the Jacobi set components for the original data and for the dataset
after applying CA variant A. Here we can see that the algorithm
(a) (b)
Figure 6: The datasets from Fig. 1 are mapped into the range. The
border of the domain is shown in green, the Jacobi sets in black,
and the grid in orange for the original data (a) and the dataset after
applying CA variant A (b) in orange.
greatly simplifies the Jacobi set components. In addition, the range
of the original data is compared with the dataset after applying the
CA variant A in Fig. 6 to examine the effects of modifying the data.
Despite the differences in the domain, there is hardly any visual
difference between the two datasets. The simplification from the
Jacobi set components from
679
to
17
using the CA variant A takes
0.2 s.
5.3 Tensile Bar
Next, we look at a collection of real-world datasets, the Tensile bars
[
43
,
44
,
32
]. Various notches or holes are made in these specimens,
as shown in Fig. 7, to create different loading conditions and test
the material properties. The tensile bars were simulated using the
commercial software package Abaqus/Standard CAE [
15
]. The
datasets are unstructured 3D datasets on which 3D symmetric tensor
fields of second order are defined. This datasets have a dimension of
the volume of the data set of
120 ×30 ×3
. Since the tensile forces
are mainly in one plane, we were able to use a single layer from the
data sets and triangulate the grid. Here, the 3D tensor is mapped to a
2D tensor from which we derive an invariant set for the comparison.
For this comparison, we use the principal invariants
I
which are
the coefficients of the characteristic polynomial. They are defined
for 2D tensor fields as
I1=trT =λ1+λ2
, and
I2=detT =λ1·λ2
,
where
trT
denotes the trace and
detT
the determinant of
T
for
the eigenvalues
λ
. A detailed description of these and other stress
tensors can be found in Holzapfel’s textbook [22].
In Fig. 8 we visually compare the extracted Jacobi sets in the
original data as seen in the upper Fig. 8a–h with the datasets after
6
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conference. The final version of this record is available at: xx.xxxx/TVCG.201x.xxxxxxx/
Table 1: The results for the length of the Jacobi sets, and the number
of Jacobi set components can be seen for all datasets and methods
tested, with the bolded values being the best. CA is short for Collapse
Algorithm.
Dataset / Method Length of # of Jacobi
Cells Jacobi sets set components
Cylinder Flow Original Data 92.4614 679
Fig. 1, and Binomial filter 64.9239 448
Fig. 5 Gaussian filter 40.0536 101
178’702 Loop subdivision 82.9961 6’311
CA Variant A 44.056 17
CA Variant B 48.8755 41
CA Variant C 50.2173 42
CA Variant D 49.8907 23
Tensile Bar A Original Data 1382.95 817
Fig. 8a / Loop subdivision 1845.95 17’858
30’960 CA Variant A 615.601 19
Tensile Bar B Original Data 1476.09 800
Fig. 8b / Loop subdivision 2200.9 23’625
22’360 CA Variant A 767.146 50
Tensile Bar C Original Data 1503.17 883
Fig. 8c / Loop subdivision 2068.83 21’529
29’560 CA Variant A 675.679 18
Tensile Bar D Original Data 963.471 519
Fig. 8d / Loop subdivision 1010.44 7’164
26’850 CA Variant A 465.583 40
Tensile Bar E Original Data 1004.65 490
Fig. 8e / Loop subdivision 1238.53 9’442
36’336 CA Variant A 535.383 38
Tensile Bar F Original Data 791.891 329
Fig. 8f / Loop subdivision 1004.82 7’259
27’210 CA Variant A 478.767 24
Tensile Bar G Original Data 815.444 326
Fig. 8g / Loop subdivision 1094.49 6’827
34’012 CA Variant A 514.535 36
Tensile Bar H Original Data 782.771 322
Fig. 8h / Loop subdivision 846.473 4’484
28’226 CA Variant A 447.887 28
Hurricane Original Data 915’064 43’838
Isabel Binomial filter 662’059 27’344
Fig. 9 / Gaussian filter 122’871 4’832
498’002 Loop subdivision 801’508 336’823
CA Variant A 393’343 2’657
applying the CA variant A as seen in the Fig. 8i–p. The range area
is displayed using color coding, with the saturation indicating how
high the stresses in the dataset are. The first thing to recognize is that
the Jacobi sets in all datasets are visually simplified after applying
the CA variant A and parts with small Jacobi set components could
be greatly reduced. Differences can be seen in the datasets with
notches and the datasets with holes. For example, Fig. 8a, i shows
that the Jacobi sets are simplified at the notches and the structure
becomes more recognizable for domain experts. In Fig. 8c, k, it
can be seen that in parts where a high density of small Jacobi set
components occurs, the Jacobi sets are simplified, but the symmetry
of the dataset is partially broken. For example, in Fig. 8e, m, it can
be seen that the Jacobi sets are simplified in the part around the hole.
In Fig. 8f, n it can also be seen that the Jacobi sets can be simplified
at the upper and lower end of the tie rod where low stress values also
appear in the dataset, which can also be seen from the white color.
In the Tab. 1 it can be seen that CA variant A can greatly reduce
Figure 7: Example of the 3D tensile bar F geometry.
all two measures. The results of the loop subdivision can also be
seen here. It is noticeable that the length of all Jacobi sets increases
significantly compared to the original data. For the smoothing filters,
we have no results for these datasets, as the used filters only work
on structured grids. Overall, you can see in the figures that the noise
in the datasets can be greatly reduced. The simplification from the
Jacobi set components for the most complex tensile bar D from
519
to
40
using the CA variant A takes
0.085
s. All the other tensile bars
need less time.
5.4 Hurricane Isabel
Another real-world dataset from the field of meteorology is Hur-
ricane Isabel[
41
,
27
], which was published as a freely available
dataset as part of the 2004 SciVis contest. This modeled hurricane
is based on Hurricane Isabel which caused severe destruction in
September 2003. The dataset is a 3D dataset with a resolution of
500 ×500 ×100
and
48
time steps and contains
13
scalar variables,
such as precipitation, at each data point. For our comparison, we use
a layer from the dataset at height
50
at time step
30
and triangulate
the grid.
To do this, we look at the scalar fields for pressure and temperature
for the corresponding visual analysis in Fig. 9. This figure shows the
algorithms to be compared, and the original data. An overall view
of the dataset can be seen in Fig. 9c,9d,9i, and Fig. 9j. Visually, it
can be seen that all
3
algorithms simplify the Jacobi sets compared
to the original data. In Fig. 9a,9e,9g and Fig. 9k a cutout of the
center of the hurricane is shown. It is easy to see that CA variant
A and the loop subdivision simplify the Jacobi sets but still retain
the structures. In contrast, the Gaussian filter simplifies the center
so much that it looks completely different compared to the original
data. A visual comparison in Fig. 9j shows that the Gaussian filter
greatly simplifies many large structures. A second cutout of the
edge of the hurricane can be seen in Fig. 9b,9f,9h, and Fig. 9l.
Here it can be seen in the original data that in the parts where the
difference between the range areas is small, many small Jacobi set
components occur here. Visually, these small Jacobi set components
have decreased in the loop subdivision as well as in the CA variant
A. This also shows that the Gaussian filter greases the data too much.
The Tab. 1 clearly shows that CA variant A reduces the Jacobi set
components the most. It can also be seen that the loop subdivision
leads to more Jacobi set components despite a visual simplification.
For the Gaussian filter, it can be seen that it reduces the components
of the Jacobian sets and even reduces the length of the Jacobian sets
the most, but this is at the expense of important structures.
Simplifying the Jacobi set components from
43′344
to
2′657
using the CA variant A takes 5.5 s.
6 DISCUSSION
In this study, we compared the variants of the collapse algorithm,
smoothing filters, and a loop subdivision with the original data
using different data sets to investigate the effect on the Jacobi set
simplification. In addition, we visually examined the influence of
the different neighborhood graphs on the result using two measures
and compared the best variant with the smoothing filters and the
loop subdivision.
7
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conference. The final version of this record is available at: xx.xxxx/TVCG.201x.xxxxxxx/
(a) Tensile Bar A (b)Tensile Bar B (c) Tensile Bar C (d) Tensile Bar D (e) Tensile Bar E (f) Tensile Bar F (g) Tensile Bar G (h) Tensile Bar H
(i) t=0.001 (j) t=0.001 (k) t=0.001 (l) t=0.01 (m) t=0.001 (n) t=0.001 (o) t=0.0001 (p) t=0.001
Figure 8: Comparison of the calculated Jacobi sets in the Tensile Bar datasets for the original data in the upper figures and the Collapse Algorithm
Variant A with the corresponding tvalues shown in the lower figures.
Our investigations have shown that the structure of the neigh-
borhood graph is of great importance, as the results differed from
one another. The idea of converting the neighborhood graph into a
neighborhood tree had certain advantages, as this made it possible to
assign an orientation direction in particularly noisy parts. However,
the decision as to whether the negative (variant B) or the positive
orientation (variant C) was preferred led to very different results.
Therefore, the choice of variant varies depending on the case. The
assignment of the cells according to their local neighborhood (vari-
ant D) could already solve this problem better but led to the fact that
many small Jacobi set components were still not simplified because
they were assigned according to their neighborhood. For this reason,
the neighborhood graph was used in collapse algorithm variant A
without further adjustments. This has the advantage that the struc-
ture is simpler and in the examples considered, the results were not
worse, but even better. However, there were small noisy parts in
Fig. 1 at the bottom left where not all the noise could be removed.
Compared to the original data, the results of the collapse algorithm
variant A are very good. The neighborhood graphs can be greatly
simplified, which is particularly beneficial for domain experts, as
the analysis becomes more difficult as the data gets larger and larger.
This can also be seen when looking at all datasets together, whereby
collapse algorithm variant A was able to reduce the number of Ja-
cobi set components by more than an order of magnitude on average
compared to the original data. The length of the Jacobi sets and
the number of Jacobi set components were reduced by more than
half on average with collapse algorithm variant A. The visual results
also show that the dataset provides a significantly better overview.
The comparison of the collapse algorithm variant A with the loop
subdivision showed that the collapse algorithm does not affect the
essential structures in the dataset and can simplify the datasets at the
same time. However, there were still deficits, particularly in the case
of data with high symmetry and very noisy parts. Here, the collapse
algorithm should be further optimized to deliver even better results.
Although the loop subdivision can visually provide very good re-
sults, the actual problem of noisy small Jacobi set components is
not solved, but the triangles where this occurs become smaller, as
already shown in the work of Kl
¨
otzel et al. [
27
] can be seen. The
comparison with the smoothing filters, in particular the Gaussian
filter, showed that although it removes common noise in the green
regions of interest very well, the essential structures are also changed
and thus the content is lost. This can also be seen in Fig. 9k where
the Gaussian filter can simplify very noisy parts, leaving many small
Jacobi set components. Further comparative tests with different
σ
’s
for the Gaussian filter can be found in the appendix.
Overall, it can be seen that the collapse algorithm variant A can
greatly simplify the Jacobi sets without removing essential structures
and achieves better results compared to the other algorithms. Also,
the time analysis shows that this depends on the number of Jacobi set
components to be reduced. On average, the CA variant A requires
13.80 ms for all datasets to reduce 100 Jacobi set components.
7 CONCLUSION & FUTURE WORK
In this study, an algorithm for simplifying Jacobi sets for bivariate
2D scalar fields in unstructured triangulated grids was presented
by adjusting the underlying data. This algorithm is based on the
collapsing of cells which changes the underlying data, whereby the
collapsing cells are identified using a neighborhood graph. This
8
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(a) original Cutout 1
(b) original Cutout 2 (c) original (d) CA Variant A
(e) CA Variant A Cutout 1
(f) CA Variant A Cutout 2
(g) Loop Subdivision Cutout 1
(h) Loop Subdivision Cutout 2 (i) Loop Subdivision (j) Gaussian filter
(k) Gaussian filter Cutout 1
(l) Gaussian filter Cutout 2
Figure 9: Comparison of the calculated Jacobi sets in the Hurricane Isabel dataset for the original dataset (c), the Collapse Algorithm (CA) Variant
A with
t=200
(d), the Loop Subdivision (i), and the Gaussian filter (j). Additional comparison of the four methods in detail for cutout 1 (coordinates
(772, 683) – (1278, 1122)) (a) (e) (g) (k) and cutout 2 (coordinates (1572, 1382) – (2077, 1822)) (b) (f) (h) (l).
resulted in four algorithm variants. How well the algorithm variants,
smoothing filters, and loop subdivision can simplify the Jacobi sets
was then studied with the original data using three datasets from the
literature. The best of the four variants was determined, which is the
collapse algorithm variant A, especially with regard to the number
of components of the Jacobi set, which was an order of magnitude
lower than in the original data. The Gaussian filter, on the other
hand, was able to reduce the length of the Jacobi sets the most, but
changed the data set too much. However, the collapse algorithm was
also able to reduce the length of the Jacobi sets by half. The results
of the collapse algorithm are promising, although some adjustments
are still needed for symmetric data such as the tension rods to better
preserve symmetry. In addition, an adapted Jacobi set visualization
was presented to account for degenerate cells and assign them to a
Jacobi set component according to their neighborhood.
As indicated at various points in the paper, there is still a lot of
potential for future work: Currently, a threshold is used to select
Jacobi set components to be collapsed, here an automatic selection
that works over the neighborhood could be a possible extension.
Furthermore, the presented collapse algorithm has the potential for
accelerating the extraction of fiber surfaces or lines, especially if
the extraction is done using Jacobi sets, as in the work of Sharma
et al. [
34
]. This further development would not only increase the
extraction speed but also simplify the visual analysis of the range.
Symmetrical datasets show optimization possibilities for the algo-
rithm at various points, as the assignment of collapsed cells cannot
always be unambiguous. One approach would be to adapt the algo-
rithm so that cells do not collapse directly, but collapse gradually.
This could achieve a smooth transition between the cells of the re-
gions to be collapsed. An obvious next step is to adapt the algorithm
to work with 3D bivariate datasets, as the domain still consists of
triangles. However, this requires further investigation and evaluation
of different datasets to investigate the influence on the domain. Espe-
cially the selection of the Jacobi set components to be collapsed can
be difficult. Furthermore, the focus is on extending the algorithm
for 3D case. Here, it would first have to be investigated how the
collapsing of tetrahedra can be realized and what effects this has on
the dataset. This would introduce a new dimension of complexity
and require careful investigation and testing with different datasets.
ACKNOWLEDGMENTS
This work was funded by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) - SCHE 663/17-1. The au-
thors would like to thank Markus Stommel from Leibniz Institute of
Polymer Research Dresden, Germany for providing the tensile bar
datasets. The authors will like to thank Bill Kuo, Wei Wang, Cindy
Bruyere, Tim Scheitlin, and Don Middleton of the U.S. National
Center for Atmospheric Research (NCAR), and the U.S. National
Science Foundation (NSF) for providing the Weather Research and
Forecasting (WRF) Model simulation data of Hurricane Isabel. The
authors acknowledge the financial support by the Federal Ministry of
Education and Research of Germany and by S
¨
achsische Staatsminis-
terium f
¨
ur Wissenschaft, Kultur und Tourismus in the programme
Center of Excellence for AI-research “Center for Scalable Data
Analytics and Artificial Intelligence Dresden/Leipzig”, project iden-
tification number: SCADS24B.
9
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Topological Simplifcation of Jacobi Sets for Piecewise-Linear Bivariate
2D Scalar Fields with Adjustment of the Underlying Data (Supplement)
Felix Raith ID *
Leipzig University
Gerik Scheuermann ID †
Leipzig University & ScaDS.AI
Christian Heine ID ‡
Leipzig University
A EXTENDED RE SU LTS
In this section, we present detailed results of our study. We
show additional comparison for the Gaussian filter with a
σ∈
{10,50,100,500}
and give a detailed overview of the comparative
tests in the extended Tab. 2. We consider another metric in the ex-
tended table, the number of Jacobi set segments. This metric can
be relevant, especially when Jacobi sets are used in acceleration
structures to enable a faster selection of the areas to be considered.
In the Tab. 1 a temporal analysis is given for the collapse algo-
rithm variant A. This shows that the time depends on the Jacobi set
components to be reduced.
Table 1: Time analysis of the Collapse Algorithm Variant A for all
datasets. All times are given in ms.
Reduced jacobi Time to reduce
Dataset Overall time set components 100 jacobi
set components
Cylinder Flow 211.07 662 31.88
Tensile Bar A 58.74 798 7.36
Tensile Bar B 60.81 750 8.11
Tensile Bar C 54.38 865 6.29
Tensile Bar D 85.82 479 17.92
Tensile Bar E 52.47 452 11.61
Tensile Bar F 36.44 305 11.95
Tensile Bar G 38.17 290 13.16
Tensile Bar H 48.42 294 16.47
Hurricane 5’461.22 41181 13.26
Isabel
Overall, the application of the Gaussian filter with different
σ
values shows that important structures, as can be seen in the center
of the hurricane (see Fig. 6a), are reduced.
A.1 Cylinder Flow (Synthetic)
The section contains corresponding figures for the different Gaussian
filters. The visual analysis shows that the Gaussian filters with
σ∈ {10,50,100}
already have problems reducing all Jacobi set
components from the first green region of interest (ROI). In the other
two ROIs, the reduction is also not sufficient to remove the small
Jacobi set components. For the Gaussian filter with
σ=500
, all
Jacobi set components can be removed in the first green ROI. In the
second brown ROI, the large Jacobi set component is already greatly
*e-mail: raith@informatik.uni-leipzig.de
†e-mail: scheuer@informatik.uni-leipzig.de
‡e-mail: heine@informatik.uni-leipzig.de
reduced in size, and the small ones cannot be removed. A similar
picture emerges in the third yellow ROI in which the larger Jacobi
set components have already been removed, and only many small
Jacobi set components can be seen in their place.
A.2 Tensile Bar
To fully analyze the effectiveness of the loop subdivision to simplify
the Jacobi sets, we give the visual results of applying the loop
subdivision to the tensile bars. Here can see that the Jacobi sets
can be simplified visually for the different tensile bars. In detail,
however, it can be seen that there are some problems. For example,
in tensile bar G in Fig. 2g it is possible to find regions in which more
new Jacobi set components have formed. Such areas can also be seen
in the other tensile bars, especially where Jacobi sets intersect in the
original data. These visual results confirm the results from the Tab. 2,
in which the length of the Jacobi sets in the tensile bars is greater
than the length of the original data, in contrast to the cylinder flow
and Hurricane Isabel data sets. It can also be seen in the Tab. 2 that
the number of Jacobi set components increases much more than in
the two other data sets before. This could lead to a misinterpretation
of the visual analysis.
A.3 Hurricane Isabel
In Fig. 3a,Fig. 4a,Fig. 5a,Fig. 6a, and Fig. 7a an overall view of
the hurricane Isabelle dataset is shown after applying the binominal
filter and the gaussian filter with
σ∈ {10,50,100,500}
. These
smoothing filters also show visually that the Jacobi set components
can be simplified visually. In detail, the cutout of Fig. 3b,Fig. 4b,
and Fig. 5b shows that the reduction is present in the center of
the hurricane and that the structures remain largely intact when
compared with the original data. In the Fig. 6b, and Fig. 7b, the
filter effect is already too strong, and important structures are no
longer fully recognizable. The cutout in Fig. 3c,Fig. 4c,Fig. 5c,
Fig. 6c, and Fig. 7c provides a slightly different picture, whereby
a real simplification and thus the recognition of structures has not
been improved. This is particularly evident in the comparison with
the loop subdivision and the CA variant A. The image of the visual
analysis is confirmed in the values from the Tab. 2, whereby a
reduction of the Jacobi set components can be recognized and thus
also a reduction of the length of the Jacobi sets themselves. It
turns out that these smoothing filters are also not better suited for
simplifying the Jacobi set components.
©
2024 IEEE. This is the author’s version of the article that has been published in the proceedings of IEEE
Visualization conference. The final version of this record is available at: xx.xxxx/TVCG.201x.xxxxxxx/
12
(a) Gaussian filter σ=10
(b) Gaussian filter σ=50
(c) Gaussian filter σ=100
(d) Gaussian filter σ=500
Figure 1: Comparison of the calculated Jacobi sets in the Cylinder
Flow dataset for the Gaussian filter
σ=10
(a),
σ=50
(b),
σ=100
(c),
andσ=500 (d).
(a) Tensile Bar A (b) Tensile Bar B (c) TensileBar C (d) Tensile Bar D
(e) Tensile Bar E (f) Tensile Bar F (g) Tensile Bar G (h) Tensile Bar H
Figure 2: Calculated Jacobi sets in the Tensile Bar datasets for the
Loop subdivision Algorithm with 4 subdivision steps.
(a)
(b)
(c)
Figure 3: The calculated Jacobi sets in the Hurricane Isabel dataset for
the Binomial filter (a) and additional in detail for cutout 1 (coordinates
(772, 683) – (1278, 1122)) (b) and cutout 2 (coordinates (1572, 1382)
– (2077, 1822)) (c).
©
2024 IEEE. This is the author’s version of the article that has been published in the proceedings of IEEE
Visualization conference. The final version of this record is available at: xx.xxxx/TVCG.201x.xxxxxxx/
13
(a)
(b)
(c)
Figure 4: The calculated Jacobi sets in the Hurricane Isabel dataset for
the Gaussian filter with
σ=10
(a) and additional in detail for cutout 1
(coordinates (772, 683) – (1278, 1122)) (b) and cutout 2 (coordinates
(1572, 1382) – (2077, 1822)) (c).
(a)
(b)
(c)
Figure 5: The calculated Jacobi sets in the Hurricane Isabel dataset for
the Gaussian filter with
σ=50
(a) and additional in detail for cutout 1
(coordinates (772, 683) – (1278, 1122)) (b) and cutout 2 (coordinates
(1572, 1382) – (2077, 1822)) (c).
(a)
(b)
(c)
Figure 6: The calculated Jacobi sets in the Hurricane Isabel dataset
for the Gaussian filter with
σ=100
(a) and additional in detail for
cutout 1 (coordinates (772, 683) – (1278, 1122)) (b) and cutout 2
(coordinates (1572, 1382) – (2077, 1822)) (c).
(a)
(b)
(c)
Figure 7: The calculated Jacobi sets in the Hurricane Isabel dataset
for the Gaussian filter with
σ=500
(a) and additional in detail for
cutout 1 (coordinates (772, 683) – (1278, 1122)) (b) and cutout 2
(coordinates (1572, 1382) – (2077, 1822)) (c).
©
2024 IEEE. This is the author’s version of the article that has been published in the proceedings of IEEE
Visualization conference. The final version of this record is available at: xx.xxxx/TVCG.201x.xxxxxxx/
14
Table 2: The results for the length of the Jacobi sets, the number of separated Jacobi set segments, and the number of Jacobi set components in
detail can be seen for all datasets and methods tested, with the bolded values being the best.
Dataset Method Cells Length of all # of # of separated
Jacobi sets Jacobi set segments Jacobi sets components
Cylinder Flow Original Data 178’702 92.4614 3839 679
(jungtelziemniak2d) Binominal filter 178’702 64.9239 2746 448
Gauss filter with σ=10 178’702 79.4928 3’329 564
Gauss filter with σ=50 178’702 61.1041 2’393 405
Gauss filter with σ=100 178’702 56.3566 2’393 353
Gauss filter with σ=500 178’702 43.8264 1’872 143
Gauss filter with σ=1000 178’702 40.0536 1’724 101
Loop subdivision Algorithm 45’747’712 82.9961 55’192 6’311
Collapse Algorithm Variant A 178’702 44.056 1’879 17
Collapse Algorithm Variant B 178’702 48.8755 2’073 41
Collapse Algorithm Variant C 178’702 50.2173 2’126 42
Collapse Algorithm Variant D 178’702 49.8907 2’121 23
Tensile Bar A Original Data 30’960 1’382.95 2’349 817
Loop subdivision Algorithm 79’25’760 1’845.95 50’018 17’858
Collapse Algorithm Variant A 30’960 615.601 1’067 19
Tensile Bar B Original Data 22’360 1’476.09 2’289 800
Loop subdivision Algorithm 5’724’160 2’200.9 55’415 23’625
Collapse Algorithm Variant A 22’360 767.146 1’182 50
Tensile Bar C Original Data 29’560 1’503.17 2’439 883
Loop subdivision Algorithm 7’567’360 2’068.83 53’992 21’529
Collapse Algorithm Variant A 29’560 675.679 1’103 18
Tensile Bar D Original Data 26’850 963.471 1’681 519
Loop subdivision Algorithm 6’873’600 1’010.44 28’336 7’164
Collapse Algorithm Variant A 26’850 465.583 829 40
Tensile Bar E Original Data 36’336 1’004.65 1’776 490
Loop subdivision Algorithm 9’302’016 1’238.53 35’312 9’442
Collapse Algorithm Variant A 36’336 535.383 980 38
Tensile Bar F Original Data 27’210 791.891 1’392 329
Loop subdivision Algorithm 6’965’760 1’004.82 28’972 7’259
Collapse Algorithm Variant A 27’210 478.767 870 24
Tensile Bar G Original Data 34’012 815.444 1’426 326
Loop subdivision Algorithm 8’707’072 1’094.49 30’530 6’827
Collapse Algorithm Variant A 34’012 514.535 926 36
Tensile Bar H Original Data 28’226 782.771 1’399 322
Loop subdivision Algorithm 7’225’856 846.473 24’196 4’484
Collapse Algorithm Variant A 28’226 447.887 820 28
Hurricane Isabel Original Data 498’002 915’064 194’923 43’838
Binominal filter 498002 662’059 141’847 27’344
Gauss filter with σ=10 498’002 815’635 174’267 36’244
Gauss filter with σ=50 498’002 572’704 122’935 22’068
Gauss filter with σ=100 498’002 427’678 91’946 15’905
Gauss filter with σ=500 498’002 181’582 39’165 6’979
Gauss filterwith σ=1000 498’002 122’871 26’524 4’832
Loop subdivision Algorithm 127’488’512 801’508 2’737’567 336’823
Collapse Algorithm Variant A 498’002 393’343 84’089 2’657
©
2024 IEEE. This is the author’s version of the article that has been published in the proceedings of IEEE
Visualization conference. The final version of this record is available at: xx.xxxx/TVCG.201x.xxxxxxx/
15
