VISUALIZATION OF UNCERTAIN CONTOUR TREES

Martin Kraus

Department of Architecture, Design, and Media Technology, Aalborg University

Niels Jernes Vej 14, DK-9220 Aalborg East, Denmark

Keywords:

Uncertainty visualization, Graph drawing, Information visualization, Volume visualization, Contour tree.

Abstract:

Contour trees can represent the topology of large volume data sets in a relatively compact, discrete data struc-

ture. However, the resulting trees often contain many thousands of nodes; thus, many graph drawing tech-

niques fail to produce satisfactory results. Therefore, several visualization methods were proposed recently

for the visualization of contour trees. Unfortunately, none of these techniques is able to handle uncertain con-

tour trees although any uncertainty of the volume data inevitably results in partially uncertain contour trees. In

this work, we visualize uncertain contour trees by combining the contour trees of two morphologically ﬁltered

versions of a volume data set, which represent the range of uncertainty. These two contour trees are combined

and visualized within a single image such that a range of potential contour trees is represented by the resulting

visualization. Thus, potentially erroneous topological structures are visually distinguished from more certain

structures. Moreover, topological structures can be revealed that are otherwise obscured by data errors. We

present and discuss results obtained with a prototypical implementation using well-known volume data sets.

1 INTRODUCTION

Most of the topological structure of scalar volume

data sets can be efﬁciently represented by its contour

tree (Freeman and Morse, 1967), which is also a use-

ful data structure for several algorithms in volume vi-

sualization.

Unfortunately, contour trees of real volume data

sets tend to be extremely large even for small vol-

ume data sets (thousands or millions of nodes and

edges) since noise in the data results in many local

extrema and, therefore, in many nodes of the con-

tour tree. Thus, it is challenging to visualize contour

trees computed from real data. Moreover, most re-

searchers agree that one coordinate of the visualized

nodes should correspond to an isovalue in the volume

data set. With this requirement, however, it is impos-

sible to avoid crossing edges in the graph layout if

straight edges are employed.

In recent years, various techniques have been pro-

posed to visualize large contour trees of real volume

data sets using dot-and-line diagrams in two and three

dimensions with and without colors and/or icons.

Moreover, functions, surfaces, and terrains have been

proposed as well as combinations of stacked graphs

and Sankey diagrams.

However, the problem of uncertain volume data

has not been addressed by these approaches although

some uncertainty is usually inevitable, for example

because of measurements or simulations with ﬁnite

precision. Moreover, contour trees are susceptible

to arbitrarily small changes of the data of individ-

ual voxels, which can determine whether isosurface

components are separated or form a single compo-

nent, and therefore result in discrete changes of the

contour tree. The lack of visualization techniques for

uncertain contour trees is particularly remarkable as

uncertainty visualization for many other visualization

methods have been suggested in recent years and their

importance is widely accepted (Griethe and Schu-

mann, 2006; Johnson and Sanderson, 2003).

While some parts of contour trees are uncertain,

other parts may be stable against certain kinds of er-

rors and noise. One objective of this work is therefore

to propose a visualization of contour trees that con-

veys the difference between the uncertain parts of a

contour tree and its stable structures. Additionally,

our technique allows us to reveal further topological

structures, which might have been concealed by noise

or other data errors.

The data ﬂow of the proposed technique is illus-

trated in Figure 1. Our approach is based on the obser-

vationthat the uncertainty of volume data corresponds

to the uncertain existence of components of isosur-

132

Kraus M. (2010).

VISUALIZATION OF UNCERTAIN CONTOUR TREES.

In Proceedings of the International Conference on Imaging Theory and Applications and International Conference on Information Visualization Theor y

and Applications, pages 132-139

DOI: 10.5220/0002817201320139

 SciTePress

volume data





see Section 3

opened version

closed version

? ?

see (Kraus, 2010)

contour tree contour tree





see Section 4

merged contour tree

with classiﬁed components

see Section 5

visualization

Figure 1: Data ﬂow of the proposed technique.

faces and the uncertain separation between these com-

ponents. In order to detect uncertain components and

uncertain separations, we compute two versions of the

volume data set by means of grayscale morphology as

discussed in Section 3. To this end, we propose a new

variant of the opening operator that is capable of pre-

serving small components of isosurfaces.

While one of the two versions of the data set in-

cludes the stable components of isosurfaces, the sec-

ond version includes also uncertain and suspected

components. By computing contour trees for both

versions and matching their nodes, we can there-

fore determine which components and separations be-

tween components should be considered uncertain as

explained in Section 4. In order to visualize the result-

ing classiﬁcation as described in Section 5, we extend

a recently proposed visualization of contour trees for

error-free data (Kraus, 2010).

The main contribution of this work is therefore the

use of grayscale morphology to determine the uncer-

tainty of the elements of a contour tree. In particular,

a new variant of the opening operator is suggested for

this purpose. Moreover, we show how to combine

multiple contour trees of different versions of a data

set in one visualization and how to visually convey

the uncertainty of their elements.

Results of a prototypical implementation of our al-

gorithm are presented in Section 6, while our conclu-

sions and plans for future work are discussed in Sec-

tion 7.

2 RELATED WORK

One of the ﬁrst discussions of the contour tree

was published by Freeman and Morse (Freeman and

Morse, 1967). Efﬁcient algorithms for the computa-

2 18 2845 84101 122 137 179 228

logarithmic surface area

isovalue

Figure 2: Visualization of the contour tree of the fuel data

set with markers for the 10 isosurfaces shown in Figure 3.

tion of contour trees were published, for example, by

Carr et al. (Carr et al., 2004) and Pascucci et al. (Pas-

cucci and Cole-McLaughlin, 2002; Pascucci et al.,

2004).

Visualizations of contour trees are often based on

dot-and-line diagrams in two dimensions (Bajaj et al.,

1997; Pascucci and Cole-McLaughlin, 2002; Carr

et al., 2004) or three dimensions (Takahashi et al.,

2004a; Takahashi et al., 2004b; Pascucci et al., 2004).

Furthermore, the use of icons has been proposed by

Shinagawa et al. (Shinagawa et al., 1991). As it is

difﬁcult to match the resulting graph drawing to the

scalar ﬁeld, the use of colors has been proposed to

identify connected components(Carr et al., 2004; We-

ber et al., 2007b; Weber et al., 2007a). Furthermore,

Weber et al. suggested the metaphor of a topological

landscape (Weber et al., 2007b). Note that all these

visualization techniques are limited to the visualiza-

tion of one error-free contour tree at a time.

In our work, contour trees are computed by the

method published by Kraus (Kraus, 2010), who also

presented a visualization technique for contour trees.

As depicted in Figure 2, contour trees are visualized

within a logarithmic plot of the area of isosurfaces

as a function of the corresponding isovalues. Alter-

natively, a histogram plot could also be employed.

This visualization can be considered a combination of

stacked graphs (see the works by Havre et al. (Havre

et al., 2002) and Byron and Wattenberg (Byron and

Wattenberg, 2008) and references therein) and Sankey

diagrams (Riehmann et al., 2005) or ﬂow maps (Phan

et al., 2005). Related combinations of stacked bar

charts and Sankey diagrams were published by Fry

(Fry, 2004, section 4.6) and by Rosvall and Bergstrom

(Rosvall and Bergstrom, 2008). The approach is also

related to the representation of “levelset trees” as one-

dimensional functions by Klemel¨a (Klemel¨a, 2004),

although level set trees are not identical to contour

trees.

VISUALIZATION OF UNCERTAIN CONTOUR TREES

133

2 18 28 45 84 101 122 137 179 228

Figure 3: Isosurfaces of the fuel data set for the isovalues indicated in Figure 2.

In this work, we extend the approach of Kraus

to the case of uncertain contour trees by combining

the visualization of multiple contour trees in a single

image. Uncertainty visualization has been an estab-

lished topic in scientiﬁc visualization for several years

(Johnson and Sanderson, 2003). A survey of this re-

search was published by Griethe and Schumann (Gri-

ethe and Schumann, 2006). One of the most common

techniques to visualize uncertainty is the utilization of

free graphical variables, in particular color. However,

the simultaneous use of color to visualize data and its

uncertainty at the same time is problematic. For ex-

ample, Kardos et al. (Kardos et al., 2006) found that

the utilization of saturation to visualize uncertainty

is rather ineffective if hue is employed to visualize

the actual data. Differences between graphs are usu-

ally determined by graph matching techniques and are

also often visualized by color coding as discussed, for

example, by Delugach and de Moor (Delugach and

de Moor, 2005).

Grayscale morphology was published by Stern-

berg (Sternberg, 1986) and is a well-established im-

age processing technique, in particular for segmenta-

tion of medical data. In this context, grayscale mor-

phology is therefore more suitable to determine the

uncertainty of parts of the contour tree than simpli-

ﬁcation methods for contour trees, which were sug-

gested, for example, by Carr et al. (Carr et al., 2004)

and Pascucci et al. (Pascucci et al., 2004). Moreover,

grayscale morphology allows us to reveal additional

topological structures that might have been hidden by

data errors.

3 MORPHOLOGICAL IMAGE

PROCESSING

We employ opening and closing operators of

grayscale morphology (with a ﬂat structuring func-

tion) to compute two versions of the data set, which

contain uncertain and certain topological structures,

respectively. The degree of uncertainty can be con-

trolled by the number of applications of these mor-

phological operators. In this work, however, we only

show results for at most one application of each oper-

ator.

original data

standard erosion

dilation

opened version

Figure 4: Illustration of a standard opening operation (ero-

sion followed by dilation).

3.1 Modiﬁed Opening

To reveal uncertain topological structures that are ob-

scured by noise, we apply a new variant of the open-

ing operator.

The standard opening is illustrated in Figure 4 and

starts with an erosion, which is applied to each voxel

← min

w∈N(v)

{ f

}

where f

is the data value of voxel v and N(v) is the

set of the 26 neighbors of v and v itself. This erosion

is followed by a dilation:

← max

w∈N(v)

{ f

}

IVAPP 2010 - International Conference on Information Visualization Theory and Applications

134

original data

modified erosion dilation

dilation erosion

opened version

with preserved maxima

closed version

Figure 5: Opening with preservation of maxima (left) and

closing (right).

Almost separated peaks and plateaus in the data set

will be separated by the opening operator. Addition-

ally, small separations tend to be widened. Thus, iso-

surface components are often split into multiple sepa-

rated components by the opening operator. In terms of

the contour tree, this corresponds to additional edges.

Therefore, topological structures that were obscured

by noise or other errors can be revealed by the open-

ing operator.

However, the standard opening operator also re-

moves peaks from the data as shown in Figure 4.

Since these peaks correspond to small isosurface

components, the opening can also prune edges from

the contour tree. There are several reasons why this

removal of edges is undesirable in the context of this

work. First of all, small isosurface components might

be removed by the standard opening operator and

joined with a nearby component by the closing op-

erator; thus, they vanish in both cases even though

they exist in the original data. This contradicts the

idea of representing the range of uncertainty by only

two versions of the data. Furthermore, the merging

and rendering of the two contour trees computed from

the opened and closed volumes can be signiﬁcantly

simpliﬁed if the “closed tree” (the contour tree com-

puted from the closed data set) is a pruned version

of the “opened tree” (the contour tree computed from

the opened data set); i.e., if no edges of the latter are

removed.

Due to these considerations, we propose a new

variant of the opening operator that tries to preserve

arbitrarily small peaks. While the dilation is un-

changed, the modiﬁed erosion is:

←







if f

= max

w∈N(v)

{ f

}

min

w∈N(v)

{ f

} otherwise

This erosion operator is illustrated in Figure 5. It

makes sure that the data values of local maxima

are preserved by the corresponding opening operator.

While it might still remove some very small isosur-

face components, it cannot remove components that a

larger than about one voxel.

3.2 Closing

The standard closing consists of a dilation, i.e.,

← max

w∈N(v)

{ f

followed by an erosion, i.e.,

← min

w∈N(v)

{ f

The closing joins peaks and plateaus that are close to

each other as illustrated in Figure 5. Thus, isosur-

face components are often joined and their total num-

ber is reduced. The remaining isosurface components

are therefore considered particularly stable or “cer-

tain.” In terms of the contour tree, this corresponds

to a pruning of the tree. In fact, our algorithm as-

sumes that the closed tree is a pruned version of the

opened tree. Therefore, it is unnecessary to preserve

local extrema in the way we proposed for the opening

operator.

3.3 Variants

By modifying the neighborhood N(v) or applying

the dilation and erosion operators multiple times, the

level of uncertainty can be adjusted to speciﬁc re-

quirements.

It should also be noted that the employed visual-

ization of the contour tree (Kraus, 2010) is a conser-

vative approximation in the sense that it never shows a

separation between two isosurface components unless

they are clearly separated in the volume data, i.e., only

rather stable components are visualized. If this kind

VISUALIZATION OF UNCERTAIN CONTOUR TREES

135

of conservative visualization is employed, the closing

might not be necessary but the original data can be

used instead.

4 MERGING MULTIPLE

CONTOUR TREES

After different versions of the original data set have

been compute by morphological image processing

(see Section 3), an approximation to the contour tree

is computed for each version of the data set as de-

scribed in (Kraus, 2010). Then, these contour trees

are merged into one tree and the uncertainty of their

nodes is determined.

While it is possible to merge any number of con-

tour trees, we will focus on the case of just two trees:

the “opened tree,” which is based on the morphologi-

cally opened data set, and the “closed tree,” which is

based on the morphologically closed data set. Fur-

thermore, we will assume that the closed tree is a

pruned version of the opened tree; i.e., it lacks some

(uncertain) nodes of the opened tree but it has no ad-

ditional nodes.

The approximate computation of the contour tree

described in (Kraus, 2010) partitions the whole data

range into uniform intervals. For each interval, it

determines the voxels with data values in the inter-

val and computes connected groups of these voxels,

which are used to approximateconnected components

of isosurfaces. These are the nodes of the computed

contour tree. Moreover, overlaps between connected

components of neighboring intervals are recorded in

order to construct edges of the contour tree.

In this work, we also record the largest overlap of

each connectedcomponent in the opened data set with

the connected components in the closed data set for

the same data interval. If no such overlap exists for a

speciﬁc connected component of the opened data set

then this component is removed from the processing.

The recorded overlaps allow us to match the nodes

of the two trees instead of matching their tree struc-

tures. Speciﬁcally, we distinguish the following cases

for each node of the closed tree:

1. No node of the opened tree is matched to a spe-

ciﬁc node of the closed tree: This contradicts our

assumption that the opened tree contains all nodes

of the closed tree. Nonetheless the node of the

closed tree is included in the merged tree since

this does not result in any ambiguities.

2. Exactly one node of the opened tree is matched

to the node of the closed tree: In this case the

two nodes are merged into one; in particular, all

recorded overlaps with nodes of neighboring data

intervals are inherited from the two nodes.

3. More than one node of the opened tree is matched

to the node of the closed tree: In this case,

the nodes of the opened tree are included in the

merged tree but the node of the closed tree is not.

Recorded overlaps with the latter node are ignored

since they are ambiguous.

Merging nodes consists mainly of merging the lists

of edges to other nodes of neighboring data intervals.

Moreover, references to merged nodes have to be ad-

justed. The main advantage of this approach is that

the resulting merged tree can be visualized in a simi-

lar way as described in (Kraus, 2010).

5 VISUALIZING MERGED

CONTOUR TREES

Our visualization of the merged contour tree is based

on a visualization technique for contour trees of error-

free data, which was presented in (Kraus, 2010). This

method computes x coordinates of the nodes based

on the data intervals of the corresponding connected

components. The y coordinates are computed by ﬁrst

sorting the nodes of each data intervalin order to min-

imize the number of crossing edges, in particular of

edges that include at least one relatively large con-

nected component. Then, partial sums of the surface

area of the connected components are computed to

determine the actual y coordinates of the nodes.

The resulting grid of vertices is used to draw sep-

arating lines between nodes that are not connected

by edges, which correspond to recorded overlaps be-

tween connected components as discussed in Sec-

tion 4. More details and several examples of this vi-

sualization technique are presented in (Kraus, 2010).

In order to adapt this visualization to the merged

contour tree, we distinguish between separating lines

that mark uncertain structures and all others. To clas-

sify a separating line, the two connected components

that are separated by the line are considered. If they

are both part of the “opened tree” (see Section 4) and

the primary overlaps of both components refer to the

same connected component of the “closed tree” then

the separating line is considered uncertain since it is

an internal separation within the same certain compo-

nent of the closed tree.

It should be noted that arbitrarily many compo-

nents of the opened tree can be associated with a sin-

gle component of the closed tree. Moreover, these un-

certain separations can also extend over several data

intervals and thus form arbitrarily deep tree structures.

IVAPP 2010 - International Conference on Information Visualization Theory and Applications

136

Table 1: Timings for computing and rendering the visualizations in Figure 6. All visualizations decompose the data range into

200 intervals. The fuel data set and the silicium data set are trilinearly upsampled versions of the publicly available data sets

(Bartz, 2005; Levoy, 2001).

time in seconds

data set size original original + closed original + opened closed + opened

fuel 127× 127× 127 5 110 112 79

silicium 195 × 67× 67 16 188 219 197

CT head 256 × 256× 113 86 384 470 452

The lines between uncertain structures are ren-

dered in a different color than the other separating

lines. The former color should be closer to the back-

groundcolor in order to convey the idea of a weaker or

less certain separation. In this work, however, bright

red is used in order to emphasize these lines. Exam-

ples of the visualization are presented in the next sec-

tion.

6 RESULTS

We present results for three publicly available data

sets (Levoy, 2001; Bartz, 2005). Figure 6 shows a

comparison between the original contour tree visual-

ization (Kraus, 2010) for these data sets in the second

row and the proposed generalization of this visualiza-

tion in the third, fourth, and ﬁfth row. The differences

between pairs of contour trees are marked by red lines

in order to emphasize them.

The third row depicts the differences between the

contour trees of the original data and the contour trees

of the morphologically closed data. Since the clos-

ing operation can join separated isosurface compo-

nents, some parts of the original separating lines are

no longer valid. These parts are marked in red in the

visualization. In most cases only the ends of the sep-

arating lines are affected. There are, however, also

some cases (in particular in complex data sets) where

a separation vanishes for all relevant isovalues. Since

a single closing operation will only join hardly sep-

arated components, the red lines indicate uncertain

separations.

In the fourth row the differences between the con-

tour trees of the original data and the contour trees of

the morphologically opened data is shown. The open-

ing operation can separate connected isosurface com-

ponents; thus, additional separating lines are intro-

duced, which are marked in red. These lines usually

extend lines of the original visualization to a larger

range of isovalues. However, there are also cases of

additional structures, which are only revealed by the

opening operation. The opening operation can sepa-

rate components, which are connected in the original

data. Thus, the additional separating lines are only

conjectured, i.e., they are less certain than the separa-

tions in the original visualization.

For the sake of completeness, the ﬁfth row com-

bines the contour trees of the morphologically closed

and opened data set. Again, the differences are

marked in red.

Our prototypical implementation was tested on a

PC equipped with 2 GB RAM and two 3.6 GHz Pen-

tium 4 CPUs (only one was used to run the program).

Timings for the three data sets depicted in Figure 6

are summarized in Table 1. The computational costs

of merging two contour trees appear to be rather high;

however, no attempts have been made to optimize the

code.

7 CONCLUSIONS AND FUTURE

WORK

By employing grayscale morphology we can compute

multiple versions of a data set, which include either

more or less certain separations of isosurface com-

ponents than the original data set. Using these ver-

sions of the data, the proposed visualization of multi-

ple contour trees in a single image allows us to vi-

sually distinguish the more certain parts of a con-

tour tree from the less certain parts. Thus, it enables

us to visualize uncertain structures in contour trees.

Apart from this interpretation, our proposed visual-

ization can also be considered a preview of the effects

of grayscale morphological ﬁlters.

This work demonstrates that uncertainty visual-

ization is feasible even for very large graphs. We

achieved this goal by matching the objects repre-

sented by the nodes of two graphs (i.e., measuring the

overlap of isosurface components) instead of match-

ing the abstract structure of the two graphs, which

would introduce additional ambiguities and uncer-

tainties.

Future work includes performance optimizations,

the adjustment of the visualization to the actual de-

VISUALIZATION OF UNCERTAIN CONTOUR TREES

137

logarithmic surface area

isovalue

logarithmic surface area

isovalue

logarithmic surface area

isovalue

logarithmic surface area

isovalue

logarithmic surface area

isovalue

logarithmic surface area

isovalue

logarithmic surface area

isovalue

logarithmic surface area

isovalue

logarithmic surface area

isovalue

logarithmic surface area

isovalue

(a)

logarithmic surface area

isovalue

(b)

logarithmic surface area

isovalue

(c)

Figure 6: Visualizations of contour trees in plots of the logarithmic surface area for (a) the fuel data set, (b) the silicium

data set, and (c) the CT head data set. From top to bottom: an isosurface from the data set (ﬁrst row), the standard contour

tree visualization with a vertical bar indicating the isovalue corresponding to the isosurface (second row), differences to the

morphologically closed data set marked in red (third row), differences to the morphologically opened data set marked in red

(fourth row), and differences between the closed and the opened data set marked in red (ﬁfth row).

gree of uncertainty of the data, and the computation

and visualization of the degree of uncertainty. Further

plans include generalizations to different image pro-

cessing operations, and the integration of alternative

computations of contour trees.

IVAPP 2010 - International Conference on Information Visualization Theory and Applications

138

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