Temporally Coherent Topological Landscapes

for Time-varying Scalar Fields

Maria Herick, Vladimir Molchanov and Lars Linsen

Department of Mathematics and Computer Science,

Westf

alische Wilhelms-Universit

at M

unster,

Einsteinstr. 62, 48149 M

unster, Germany

Keywords:

Topological Landscapes, Time-varying Scalar Field Visualization.

Abstract:

Topological structures capture the main features of scalar ﬁelds. Topological landscapes have been proposed

for an intuitive depiction of n-dimensional scalar ﬁeld topology using 2D landscapes with matching topology.

For time-varying scalar ﬁelds, each time step could be visualized by a 2D landscape, but there would be

no temporal coherence among the landscapes. We propose the concept of a time-varying contour tree that

is obtained by merging contour trees of all time steps into a meta data structure. The time-varying contour

tree can be exploited to generate temporally coherent topological landscapes. Visual analysis of time-varying

scalar ﬁeld topology is, then, supported by animating landscapes over time or by volume rendering a stack of

temporal slices that represent color-coded landscapes.

1 INTRODUCTION

Scientiﬁc visualization is commonly concerned with

the analysis of time-varying multi-dimensional ﬁelds

representing natural phenomena. Topological repre-

sentations are an effective abstraction of a ﬁeld cap-

turing its fundamental structure. A popular method

to encode a scalar ﬁeld’s topology is the contour

tree (Carr et al., 2003). Since their visual represen-

tation in the form of a graph may be difﬁcult to un-

derstand for untrained users, the metaphor of topo-

logical landscapes has been proposed for a more in-

tuitive depiction (Weber et al., 2007). A topological

landscape is an algorithmically constructed 2D scalar

ﬁeld that is topologically equivalent to the original

scalar ﬁeld, i.e., both ﬁelds have the same contour

tree. However, when generating a topological land-

scape for each time step individually, the landscapes

of a time-varying data set would not expose matching

structures over time.

In this paper, we propose an approach to visualize

the topological changes in time-varying scalar ﬁelds

based on the generation of temporally coherent topo-

logical landscapes. We use contour trees to represent

the structure of the scalar ﬁeld of each time step. The

changes in the topology result in an evolution of con-

tour trees. We identify similarities between timesteps

and merge the contour trees to create a time-varying

contour tree, i.e., a meta data structure that stores the

temporal development of the topology. Using this

data structure, one can generate a sequence of tempo-

rally coherent topological landscapes. This sequence

can then be visualized by animating the landscapes

over time or by converting each landscape into a 2D

color image, stacking these images, and using a vol-

ume rendering approach. Our main contributions are:

• Analyzing similarities between contour trees by

deﬁning a distance metric based on the associated

scalar ﬁeld.

• Deﬁning a meta data structure that represents a

time series of matched contour trees.

• Visualizing the temporal development of the

topology via temporally coherent topological

landscapes.

2 RELATED WORK

Isosurface and direct volume rendering are the most

prominent scientiﬁc visualization methods for 3D

scalar ﬁelds. They both rely on choosing appropri-

ate settings, i.e., isovalue or transfer function selec-

tion. Different approaches such as isosurface sim-

ilarity maps (Bruckner and M

oller, 2010; Fofonov

and Linsen, 2016), stochastic distributions (Pfaffel-

moser and Westermann, 2013), or isosurface statis-

tics (Scheidegger et al., 2008) can be used to support

Herick, M., Molchanov, V. and Linsen, L.

Temporally Coherent Topological Landscapes for Time-varying Scalar Fields.

DOI: 10.5220/0008956300540061

In Proceedings of the 15th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2020) - Volume 3: IVAPP, pages 54-61

ISBN: 978-989-758-402-2; ISSN: 2184-4321

making such choices. Topological descriptions, in-

stead, describe all features in the scalar ﬁeld.

The topological structure of a scalar ﬁeld can be

extracted by representing the evolution of its isosur-

faces in the form of a Reeb graph (Reeb, 1946), which

in case of a simply connected domain (Doraiswamy

and Natarajan, 2013) becomes a contour tree (Carr

et al., 2003). Tierny et al. (Tierny et al., 2018) pre-

sented the topology toolkit to calculate topological

structures like the contour tree, which we used in

our implementation to construct merge and split trees,

i.e., the two trees that are merged to form a con-

tour tree. A common visual representation of the

extracted topology is to draw the respective graph.

For example, Heine et al. (Heine et al., 2011) dis-

cussed planar visualizations of contour trees, while

Pascucci et al. (Pascucci et al., 2004) suggested 3D

visualizations of contour trees. As the graph draw-

ings of contour trees are often hard to understand, We-

ber et al. (Weber et al., 2007) proposed topological

landscapes, which visualize the topology of a mul-

tidimensional data set by constructing a height ﬁeld

with an identical contour tree. They extended the con-

cept by taking into account geometrical properties of

topological features (Beketayev et al., 2012). Harvey

et al. (Harvey and Wang, 2010) presented an ensem-

ble of volume-preserving topological landscapes from

higher-dimensional scalar ﬁelds. However, all these

approaches focus on a single scalar ﬁeld, while scien-

tiﬁc data sets are commonly time-varying. Of course,

one can extract the topological landscape of each time

step individually, but the location of extracted features

of consecutive time steps would not match, which ren-

ders this approach useless for visualizing topological

changes over time.

A topological approach for time-varying data was

presented by Edelsbrunner et al. (Edelsbrunner et al.,

2004) who suggested a mathematical approach to

track changes in Reeb graphs. Such space-time topo-

logical structures can be used to track and visualize

topological features over time (Bremer et al., 2010;

Weber et al., 2011). However, such a visualization fo-

cuses on the selected topological structure only. Sim-

ilarly, Sohn et al. (Sohn and Bajaj, 2006) proposed to

compute time-varying contour topology by selecting

an isovalue and visualizing the topological changes in

the corresponding isosurfaces over time. Again, this

method is restrained to a single isovalue. Oesterling

et al. (Oesterling et al., 2017) computed time-varying

merge trees and visualize their evolution over time in

a bottom-up layout. Our time-varying contour trees

follow a similar idea, but use both merge and split

trees, which allows for a direct application to topo-

logical landscape generation.

In other contexts than contour trees, there exist

approaches that try to merge trees or graphs. For

example, Lukasczyk et al. (Lukasczyk et al., 2017)

present an approach to merge tracking graphs into

a static visualization. Also, static visualizations

when generating temporal treemaps have been pre-

sented recently by K

opp and Weinkauf (K

opp and

Weinkauf, 2019), which extends earlier work on sta-

ble treemaps via local moves (Sondag et al., 2018),

using Voronoi treemaps (Hahn et al., 2014), or for

evolving treemaps (Scheibel. et al., 2018). The data

structure introduced by K

opp and Weinkauf is related

to our data structure presented below.

3 BACKGROUND

Let f (x,t) be a time-varying d-dimensional scalar

ﬁeld sampled at spatial points {p

∈ R

|i = 1, . . . , n}

and time points t

, j = 1, . . . , k , i.e., the given ﬁeld

values are f (p

) = x

i j

. We assume that f (x, t) is a

Morse function at any time to exclude the existence of

degenerate critical points (Milnor, 2016) and that it is

deﬁned on a simply connected set. Then, the contour

tree describes the development of isocontours in f . At

any time t

, the level set L

for an isovalue z ∈ R is de-

ﬁned as the set of all points x ∈ R

with f (x, t

) = z.

An isocontour for isovalue z at time t

is then deﬁned

as a simply connected component of L

Let T

be the contour tree of f (x, t

) for a ﬁxed

time. T

is constructed by contracting each isocon-

tour for a speciﬁc isovalue of f (x,t

) into a point in

, which is then placed at the height indicated by

the isovalue. Thus, the edges of the contour tree rep-

resent the life span of topological features, while its

nodes symbolize critical points in the function. Crit-

ical points are points where a new isocontour occurs

(minima in f ), an existing isocontour vanishes (max-

ima in f ), multiple existing isocontours merge (saddle

points in f , referred to as merge nodes), or an exist-

ing isocontour splits (saddle points in f , referred to

as split nodes). To generate a contour tree, an inter-

mediate step is to compute a merge tree (or join tree)

and a split tree, which are directed trees that contain

only merge nodes or split nodes, respectively (Carr

et al., 2003). Subsequently, merge and split tree are

combined to form the contour tree.

As contour trees can be rather complex and their

depiction in the form of graph drawings hard to in-

terpret, Weber et al. (Weber et al., 2007) proposed to

visualize them using the metaphor of a topological

landscape, which is a 2D heightﬁeld with a contour

tree that is identical to the contour tree of f . The

location of peaks and valleys and their area covered

Temporally Coherent Topological Landscapes for Time-varying Scalar Fields

provide a degree of freedom that can be used to en-

code further properties of f such as having the area

represent the volume of the topological feature.

4 TEMPORALLY COHERENT

TOPOLOGICAL LANDSCAPES

Our goal is to visualize topological changes in time-

varying scalar ﬁelds via temporally coherent topolog-

ical landscapes. Our approach starts off by comput-

ing merge and split trees for each of the given time

steps individually. We combine the merge and split

trees separately by iteratively matching the nodes of

the trees from consecutive time steps. The matching

of the nodes is based on a proper distance metric us-

ing the spatial information associated with the nodes

(Section 4.1). The aggregated merge and split trees

are combined to create the time-varying contour tree,

a meta data structure that embeds the matched con-

tour trees of all time steps (Section 4.2). The time-

varying contour tree can be used to lay out temporally

coherent topological landscapes and to visualize their

temporal development (Section 4.3).

4.1 Distance Metrics

To identify matching nodes in two (either merge or

split) trees, we recursively deﬁne a distance met-

ric for two nodes of different trees. First, we de-

ﬁne distances between two leaf nodes. Leaves cor-

respond to isolated critical points (minima or max-

ima). Critical points are considered similar, if they

have a similar spatial location and a similar function

value. Hence, we deﬁne the distance of two leaves

i and j as the weighted sum of the Euclidean dis-

tance of the spatial locations of the respective criti-

cal points kp

− p

and the difference in their func-

tion values | f (p

) − f (p

)| (after normalizing the two

terms). Next, we deﬁne distances of two inner nodes.

Each inner node i corresponds to a connected com-

ponent R

in the scalar ﬁeld’s domain surrounding

the point p

associated with the inner node. This re-

gion R

contains all spatial points connected to p

that

have a function value lower than the value assigned

to p

’s parent. Two inner nodes are considered sim-

ilar, if their regions match in size and location and

if the leaves of the subtrees rooted at the inner nodes

have many matches. To compute how well the regions

match, we ﬁrst calculate a one-sided distance of i to a

node j by

(i, j) =

∑

p∈R

min

q∈R

kp − qk

where | · | denotes the cardinality and k.k

the Eu-

clidean distance. Computations are sped up by only

considering the margins of the regions. Then, a two-

sided distance of inner nodes is deﬁned by

δ(i, j) =

(i, j) · |R

| + δ

( j, i) · |R

∪ R

To compute how well the leaves of the subtrees match,

the average similarity of all possible leaf node pairs is

calculated as well as the percentage of matched leaf

nodes in the subtrees (if the two inner nodes were

matched). The distance measures are combined in a

weighted fashion to deﬁne the overall distance mea-

sure for inner nodes, where the weights can be ad-

justed to give one or the other aspect more impact.

We do not need to deﬁne distances for the root nodes,

as the root nodes match by deﬁnition.

4.2 Time-varying Contour Tree

Combining Two Trees. Let T

and T

with t

< t

be (either merge or split) trees, which we want to

combine to a meta tree T . Two nodes a

∈ T

and

∈ T

match, if they correspond to a topological fea-

ture existing in both time steps. If we detect a match,

and a

are combined in a single node a ∈ T . If

∈ T

or b

∈ T

have no match, they correspond to

a disappearing (b

) or appearing (b

) topological fea-

ture and should persist in T . Every node in T stores

information about its life span in temporal dimension

and its function values within the life span.

The algorithm for combining trees T

and T

to T

traverses the trees T

and T

top-down to build tree T

step by step. We start by matching the root nodes of

and T

and generating the respective root node in

T . Then, we iteratively proceed to the next depth level

of nodes in both trees. We use a greedy algorithm to

calculate the nodes’ best matching using the metrics

in Section 4.1. To ﬁnd the best match for a node of

, we take into account that the best match may not

be at the same depth level in T

. Thus, for each node

of T

we consider the nodes of T

at the current and

the subsequent depth level and vice versa. The neces-

sity of taking into account the subsequent depth level

is illustrated with examples of topological changes in

Figure 1. Assuming that T

is shown in red and T

in blue, the case of an emerging topological feature

(isocontour 4) is shown in Figure 1a. Figure 1b illus-

trates that the correct match is found by shifting the

red subtree one depth level down. Figure 1c shows a

second example, where the hierarchical order of topo-

logical features changes, i.e., red isocontour 0 is a par-

ent of red isocontours 2 and 3, while blue isocontour

0 is a parent of blue isocontours 1 and 2. A correct

IVAPP 2020 - 11th International Conference on Information Visualization Theory and Applications

combination of T

and T

needs to have two parent

nodes of isocontour 2 in tree T , see Figure 1d. This

case destroys the tree property of T , which we have to

handle appropriately in subsequent steps. For match-

ing nodes of two trees we only consider nodes of the

current and subsequent depth level. Due to temporal

coherence, we can assume trees of consecutive time

steps to be similar to each other. Thus, we do not

proceed to further levels, which prevents exploding

computational costs.

Combining Tree Sequences. The combination of

multiple trees to one meta tree processes the trees it-

eratively in chronological order. Thus, we start by

combining T

and T

to a meta tree, then the result is

combined with T

, etc. Combining a tree with a meta

tree requires further considerations: As we have seen

in Figure 1, there are nodes and edges that only ex-

ist in some of the combined trees. Thus, we have to

decide which nodes and edges are to be considered

valid for the matching step. Because of temporal co-

herence of subsequent time steps, we can assume that,

e.g., T

is closer to T

than to T

. Hence, when adding

a tree to the meta tree of preceding time steps, only

the nodes and edges of the last time step within the

meta tree are considered during the matching, e.g., T

would be matched with T

while neglecting T

. This is

necessary, because the algorithm we used to calculate

the combination of the two structures relies on them

being trees. We thus need to ensure this property by

temporarily ignoring all edges that destroy it, namely

those which were described in Figure 1d. Of course,

all already existing nodes and edges in the meta tree

persist during the extension.

Generating Contour Tree. Given a sequence of

time steps, we combine the respective sequence of

merge trees and split trees separately as described

above. Then, we can join the combined merge tree

and the combined split tree to a time-varying contour

tree using the original algorithm (Carr et al., 2003).

However, this algorithm requires that merge and split

trees are actual trees, which we violated when com-

bining them, cf. Figure 1d. The requirement can be

met by duplicating the subtree whose root has two

parents (node 2 in Figure 1d). We perform such du-

plications where necessary and store a respective link

between the duplicates. Note that at any time point at

most one of the duplicates is active. Having created

the duplicates, the combined merge and split trees can

be joined using the original algorithm. The result-

ing time-varying contour tree stores all nodes of the

merge and split trees of all time steps, the respective

contour tree edges, the life span for each node and

edge, the function values of the nodes during their life

span, and links between duplicates.

4.3 Visualizing Time-varying

Landscapes

Having created the time-varying contour tree, we can

generate the topological landscape of all time steps

together using the original algorithm (Weber et al.,

2007). Of course, it is not meaningful to visualize

this landscape. Instead, we can generate topologi-

cal landscapes for each time step individually. These

landscapes are temporally coherent, as they follow the

same layout.

We can visualize the topological changes by ani-

mating the landscapes over time. For each time step,

we have stored the nodes’ height (i.e., the respective

function values), which we interpolate linearly be-

tween consecutive time steps. The animation leads to

smoothly changing renderings. The only discontinu-

ity occurs when switching between duplicates, as du-

plicates are positions at different locations in the land-

scape. However, duplicates occur at time steps when

there are topological changes. Hence, having these

steps emphasized by quickly disappearing/emerging

structures is arguably a desired effect. We color-code

duplicates in the same color to visualize the matching

structures.

Animations are suitable to show topological

changes of consecutive time steps, i.e., for temporally

local analyses. For temporally global analyses, ani-

mations induce a high cognitive load. Thus, we also

propose an alternative visualization for global analy-

ses. We transform each topological landscape into a

2D image, where for each pixel we store the heights

of the landscape and a node ID for each node of the

time-varying contour tree. We stack the 2D images in

chronological order to form a 3D image. This 3D im-

age can be rendered using a direct volume rendering

approach. By providing the node ID, we can assign to

each node a unique color in all time steps it exists. For

example, we can color-code all peaks of the topolog-

ical landscape using a categorical color map (similar

to (Weber et al., 2007)). Then, the transfer function

of the height values only maps to opacities without

changing the colors. The transfer function can be in-

teractively adjusted, e.g., to show all peaks or only

highest peaks.

5 RESULTS

To validate our approach we generated a synthetic

time-varying 2D scalar ﬁeld (resolution 64 × 64, 20

Temporally Coherent Topological Landscapes for Time-varying Scalar Fields

(a)

-1

2 3

(b)

2 3

(c)

1 2

2 3

(d)

Figure 1: Examples for combining tree T

(red) and tree T

(blue) with occurring topological changes: Emerging isocontour

4 (a) requires us to shift to next depth level for subtree matching (b). Changing hierarchical order of isocontour 2 (c) requires

us to store two parents for the respective node in the combined tree (d).

time steps). Initially, it has two maxima, of which

one persists over time, while the other vanishes af-

ter some time steps and a third maximum emerges.

The volume rendering of the stacked topological land-

scapes in Figure 2 (left) correctly visualizes this be-

havior, where the Z-axis represents time: The per-

sisting (red), vanishing (yellow), and emerging (blue)

maxima can be easily observed. The larger areas of

the yellow and blue maxima when compared to the

red one indicate that they correspond to larger regions

in the scalar ﬁeld. The transfer function has been cho-

sen to only show high values, which is why only max-

ima show up and the yellow and blue regions have

rounded shapes. Figure 2 (right) shows two time steps

of the corresponding landscape animation, namely at

the beginning (bottom) and when the red peak has

emerged while the yellow one has vanished (top).

A second synthetic time-varying 2D scalar ﬁeld

example (resolution 64×64, 20 time steps) illustrates

that the merging of topological structures are also

visualized correctly (Figure 3). Here, two maxima

(green and red) fuse after some time steps and become

one maximum (blue).

Next, we applied our approach to a more com-

plicated example: We generated two 2D scalar ﬁeld

with several critical points over a grid of resolution

128 × 128. Then, we produced a time-varying data

set with 20 time steps by linearly interpolating the

scalar ﬁelds at each grid point. Figure 4 shows 5

of the time steps by displaying the interpolated 2D

scalar ﬁelds on top as a grey-scale image and the cor-

responding topological landscape below. We frame

each of the extrema in the color that was assigned to

it in the topological landscape. We can observe that

the peaks in the landscape only change their heights

from Figure 4(a) to Figure 4(b). Then, in Figure 4(c),

two new maxima arise, while in Figure 4(d), a new

minimum appears. Eventually, in Figure 4(e), some

extrema vanish. We observe that the layout of the

topological landscape is stable throughout the entire

process such that it is easy to observe the changes oc-

curring in the landscapes. The accompanying video

shows the respective animation.

We also tested our algorithm on a climate reanaly-

sis data set (Copernicus Atmosphere Monitoring Ser-

vice, 2018). The data set we used consists of a time-

varying 2D temperature scalar ﬁeld for 48 hours (sam-

pled every hour). Figure 5 shows three time steps of

the topological landscape animation and the respec-

tive color-coded scalar ﬁeld. We observe that the

overall structure remains quite stable over time (yel-

low peak exists in all time steps), while individual

structures (several valleys) disappear.

6 DISCUSSION

Experiments showed that all components of the pro-

posed distance metric were necessary to produce de-

sired results. Adjusting the (non-zero) weights in

the distance metric does affect the result, but did

not change the main features in the overall structure.

IVAPP 2020 - 11th International Conference on Information Visualization Theory and Applications

Figure 2: (left) Volume rendering of peaks for chronologically (Z-axis) stacked topological landscapes: Red peak persists

throughout the 20 time steps, while the yellow vanishes and the blue emerges. (right) Two time steps (beginning, after red

peak emerged and yellow vanished) of topological landscape animation.

Figure 3: Volume rendering of peaks for chronologically

(Z-axis) stacked topological landscapes: Two peaks (red

and green) fuse into one (blue).

Hence, the layout is stable against small changes of

the weights. The greedy approach does not guarantee

an optimal solution, but it is deterministic and pro-

duces good results. One topic for future work is to

re-consider the choice of having a discontinuity when

a topological feature switches its parent. It may be

desirable to have this discontinuity to explicitly see

such a change, but one may also argue for avoiding

it, as the feature itself is not changing but only the

parents.

Computation times were moderate for the pre-

sented examples, but optimization would be required

before applying to big data sets. For example, the

merging of time steps is currently processed sequen-

tially in chronological order. Merging consecutive

time steps in a hierarchical fashion should make the

individual computations less complex and would be

amenable to parallel computing.

7 CONCLUSION

We have presented an approach for temporally coher-

ent visualization of topological landscapes to analyze

temporally local or global topological changes. It is

based on a matching strategy of topological features

between consecutive time steps and a respective com-

bination of the contour trees. The combined contour

tree as a meta-data structure is used for the topolog-

ical landscape generation. The landscapes are then

visualized using animations of surface renderings or

using volume rendering after stacking the individual

time steps in temporal dimension. To exploit tempo-

ral coherence, we assumed that consecutive time steps

are indeed sufﬁciently similar.

ACKNOWLEDGEMENTS

This work was supported in part by DFG grant

MO 3050/2-1.

Temporally Coherent Topological Landscapes for Time-varying Scalar Fields

(a) (b) (c)

(d) (e)

Figure 4: Topological landscapes from a time series obtained by linearly interpolating between two 2D scalar ﬁelds with

multiple extrema. The landscapes are consistent throughout the time series such that emerging and vanishing structures as

well as height changes can be easily observed.

(a) 10 hours

(b) 20 hours (c) 34 hours

Figure 5: (top) Three time steps (after 10, 20, and 34 hours) of topological landscape animation of climate reanalysis data set.

(bottom) Color-coded visualization of respective 2D temperature ﬁelds.

IVAPP 2020 - 11th International Conference on Information Visualization Theory and Applications

REFERENCES

Beketayev, K., Weber, G. H., Morozov, D., Abzhanov, A.,

and Hamann, B. (2012). Geometry-preserving topo-

logical landscapes. In Proceedings of the Workshop at

SIGGRAPH Asia, WASA ’12, pages 155–160, New

York, NY, USA. ACM.

Bremer, P., Weber, G., Pascucci, V., Day, M., and Bell, J.

(2010). Analyzing and tracking burning structures in

lean premixed hydrogen ﬂames. IEEE Transactions

on Visualization and Computer Graphics, 16(2):248–

260.

Bruckner, S. and M

oller, T. (2010). Isosurface similar-

ity maps. In Computer Graphics Forum, volume 29,

pages 773–782. Wiley Online Library.

Carr, H., Snoeyink, J., and Axen, U. (2003). Computing

contour trees in all dimensions. Computational Ge-

ometry, 24(2):75–94.

Copernicus Atmosphere Monitoring Service (2018). ERA5

hourly data on pressure levels from 1979 to present.

https://cds.climate.copernicus.eu/.

Doraiswamy, H. and Natarajan, V. (2013). Computing Reeb

graphs as a union of contour trees. IEEE Transactions

on Visualization and Computer Graphics, 19(2):249–

262.

Edelsbrunner, H., Harer, J., Mascarenhas, A., and Pascucci,

V. (2004). Time-varying Reeb graphs for continuous

space-time data. In Proceedings of the twentieth an-

nual symposium on Computational geometry, pages

366–372. ACM.

Fofonov, A. and Linsen, L. (2016). Fast and robust iso-

surface similarity maps extraction using quasi-Monte

Carlo approach. In Wilhelm, A. F. and Kestler, H. A.,

editors, Analysis of Large and Complex Data, pages

497–506. Springer International Publishing.

Hahn, S., Tr

umper, J., Moritz, D., and D

ollner, J. (2014).

Visualization of varying hierarchies by stable layout

of voronoi treemaps. In 2014 International Confer-

ence on Information Visualization Theory and Appli-

cations (IVAPP), pages 50–58.

Harvey, W. and Wang, Y. (2010). Topological landscape

ensembles for visualization of scalar-valued functions.

In Computer Graphics Forum, volume 29, pages 993–

1002. Wiley Online Library.

Heine, C., Schneider, D., Carr, H., and Scheuermann, G.

(2011). Drawing contour trees in the plane. IEEE

Transactions on Visualization and Computer Graph-

ics, 17(11):1599–1611.

opp, W. and Weinkauf, T. (2019). Temporal treemaps:

Static visualization of evolving trees. IEEE Transac-

tions on Visualization and Computer Graphics (Proc.

IEEE VIS), 25(1):534–543.

Lukasczyk, J., Maciejewski, R., Weber, G. H., Garth,

C., and Leitte, H. (2017). Nested tracking graphs.

Computer Graphics Forum (Special Issue, Proceed-

ings Eurographics/IEEE Symposium on Visualiza-

tion), 36(3):12–22. Best paper award.

Milnor, J. (2016). Morse Theory.(AM-51), volume 51.

Princeton university press.

Oesterling, P., Heine, C., Weber, G., Morozov, D., and

Scheuermann, G. (2017). Computing and visualizing

time-varying merge trees for high-dimensional data.

In Topological Methods in Data Analysis and Visual-

ization IV, pages 87–101.

Pascucci, V., Cole-McLaughlin, K., and Scorzelli, G.

(2004). Multi-resolution computation and presenta-

tion of contour trees. In Proc. IASTED Conference on

Visualization, Imaging, and Image Processing, pages

452–290. Citeseer.

Pfaffelmoser, T. and Westermann, R. (2013). Visualizing

contour distributions in 2D ensemble data. EuroVis-

Short Papers, 71(72):55–59.

Reeb, G. (1946). Sur les points singuliers d’une forme

de Pfaff completement integrable ou d’une fonction

numerique [on the singular points of a completely

integrable Pfaff form or of a numerical function].

Comptes Rendus Acad. Sciences Paris, 222:847–849.

Scheibel., W., Weyand., C., and D

ollner., J. (2018). Evo-

cells - a treemap layout algorithm for evolving tree

data. In Proceedings of the 13th International Joint

Conference on Computer Vision, Imaging and Com-

puter Graphics Theory and Applications - Volume 3:

IVAPP,, pages 273–280. INSTICC, SciTePress.

Scheidegger, C. E., Schreiner, J. M., Duffy, B., Carr, H.,

and Silva, C. T. (2008). Revisiting histograms and iso-

surface statistics. IEEE Transactions on Visualization

and Computer Graphics, 14(6):1659–1666.

Sohn, B.-S. and Bajaj, C. (2006). Time-varying contour

topology. IEEE Transactions on Visualization and

Computer Graphics, 12(1):14–25.

Sondag, M., Speckmann, B., and Verbeek, K. (2018). Stable

treemaps via local moves. IEEE Transactions on Vi-

sualization and Computer Graphics, 24(1):729–738.

Tierny, J., Favelier, G., Levine, J. A., Gueunet, C., and

Michaux, M. (2018). The topology toolkit. IEEE

transactions on visualization and computer graphics,

24(1):832–842.

Weber, G., Bremer, P., Day, M., Bell, J., and Pascucci, V.

(2011). Feature tracking using Reeb graphs. In Topo-

logical Methods in Data Analysis and Visualization.

Mathematics and Visualization. Springer, Berlin, Hei-

delberg, pages 241–253, Berlin, Heidelberg. Springer

Berlin Heidelberg.

Weber, G., Bremer, P.-T., and Pascucci, V. (2007). Topo-

logical landscapes: A terrain metaphor for scientiﬁc

data. IEEE Transactions on Visualization and Com-

puter Graphics, 13(6):1416–1423.

Temporally Coherent Topological Landscapes for Time-varying Scalar Fields