A Voronoi interior adjacency-based approach for generating a contour tree

Computers & Geosciences 30 (2004) 355–367

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*Correspond

10-6842-4101.

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doi:10.1016/j.ca

A Voronoi interior adjacency-based approachfor generating a contour tree

Jun Chena,*, Chaofei Qiaoa,b, Renliang Zhaoa

aNational Geomatics Centre of China, No. 1, Baishengcun, Zizhuyuan, Beijing 100044, ChinabChina University of Mining and Technology (Beijing Campus), Beijing 100083, China

Received 19 February 2003; accepted 23 June 2003

Abstract

A contour tree is a good graphical tool for representing the spatial relations of contour lines and has

found many applications in map generalization, map annotation, terrain analysis, etc. A new approach for

generating contour trees by introducing a Voronoi-based interior adjacency set concept is proposed in this

paper. The immediate interior adjacency set is employed to identify all of the children contours of each contour

without contour elevations. It has advantages over existing methods such as the point-in-polygon method and

the region growing-based method. This new approach can be used for spatial data mining and knowledge dis-

covering, such as the automatic extraction of terrain features and construction of multi-resolution digital elevation

model.

r 2004 Elsevier Ltd. All rights reserved.

Keywords: Contour tree; Voronoi diagram; Spatial relations; Immediate adjacency; Interior

1. Introduction

As a fundamental dimension of our living environ-

ment, topography reflects the shape of the earth’s

surface and plays a very important role in shaping or

mediating many other environmental flows or functions

(Mark and Smith, 2001). After many years of intensive

investigation and mapping of the topography, a large

number of topographic maps were produced by national

mapping agencies. These have been converted into

digital formats in the last 20 years. For example, over

24,000 topographic map sheets at 1:50,000 scale cover-

ing the territory of mainland China were mapped

by the State Bureau of Surveying and Mapping of

China. These topographic maps have been digitized and

a nation-wide 25m� 25m digital elevation model

ing author. Tel.: +86-10-6842-4072; fax: +86-

ess: [email protected] (J. Chen).

e front matter r 2004 Elsevier Ltd. All rights reserve

geo.2003.06.001

(DEM) data set has also recently been completed (Chen

et al., 2002). The explosive growth and wide availability

of topographic data and other geo-referenced data has

created a data-rich environment for a variety of GIS

users. How to make the best use of such topographic

data has become a basic but daunting task (Miller and

Han, 2001).

Among the digital topographic data, contour line is

the most fundamental element representing an imagin-

ary line on the ground, all points of which are of equal

elevation in reference to a specified common datum

plane (Wu, 1993, pp. 22; Li and Sui, 2000). A group of

contour lines provides a 2D representation of 3D

terrain, and can be used for generating a realistic

portrayal of actual 3D terrain. A significant amount of

DEM data were generated from these digitized contours

using an appropriate approach employing a triangulated

irregular network (TIN) and interpolation (Moore et al.,

1992; Li and Zhu, 2000). Terrain landform features

(including peak, pit, saddle, ridge and valley) can also be

d.

ARTICLE IN PRESSJ. Chen et al. / Computers & Geosciences 30 (2004) 355–367356

extracted from contours (Tang, 1992; Thibault and

Gold, 2000). A distinct advantage of contour line data

over a DEM is that the vector representation lends itself

to the object-oriented modeling of terrain and provides a

natural mechanism to sort out terrains, a feature that

facilitates the search through a contour map. However,

the interpretation of these contour maps requires

abstraction processes and deep domain knowledge that

only human experts have, such as in the detection of

morphologies characterizing the landscape, the selection

of important environmental elements, both natural and

artificial, and the recognition of forms of territorial

organization (Cronin, 1995). The research challenge

now is to develop and use suitable computational

tools to discover new and interesting patterns, trends

and relationships that might be hidden deep within

the very large national topographic datasets (Malerba

et al., 2001).

The shape and relationship of contour lines play a key

role when human beings recognize topographic features

from a contour map without knowing the detailed

elevation of the terrain (Kweon and Kanade, 1994). For

instance, contour lines have some particular properties

as a group of nested closed curves. Any given contour

line encloses an arbitrary number of other contour lines,

but is in turn enclosed by only one contour line, and

these lines never intersect (Freeman and Morse, 1967).

Peaks or pits can be defined as a series of closed

contours, having either ascending or descending eleva-

tions, and corresponding to local maxima and minima,

respectively. In order to model and represent the

relationships between contour lines, in 1963 Boyell and

Reston proposed a contour tree generated by mapping

contour lines as edges and interstitial spaces as nodes

(Boyell and Reston, 1963). This tree structure has found

applications in many geo-science related areas, such as

the finding of terrain profiles (Freeman and Morse,

1967), the automated labeling the elevation of contours

(Wu, 1993; Liu and Ramirez, 1997), the modeling of

topographic changes (Kweon and Kanade, 1994),

automated reasoning with contour maps (Cronin,

1995), and fast color filling between contours (Zhang

et al., 2001). The authors believed that the contour tree

can serve to represent basic geographic facts and rules

for deducing information from the very large national

contour databases.

There has been a great deal of previous work

representing the contour map as a tree structure, and

various strategies have been proposed to transfer a

contour map into a graph. The contour tree in Fig. 1D

was generated by mapping the contour lines in Fig. 1A

into nodes and the inter-contour regions into edges. L9

has the lowest elevation and encloses all of the other

contours. It has five immediate descendants of the same

elevation, L3;L4;L5;L6 and L8; representing five

branches in the tree. L3 has two descendants, L1 and

L2; and L4;L5 and L6 have no children. Other

alternative approaches are to map the contour lines

into edges and the inter-contour regions into nodes, such

as shown in Fig. 1B; and mapping both inter-contour

regions and contour lines as nodes, such as shown in

Fig. 1C. Although these strategies differ in that contour

map objects and their relationships are represented as

nodes or arcs of the graph, the manner in which the

contour tree is generated remains the same: by mapping

and representing the enclosure relation between contour

lines (Roubal and Poiker, 1985; Liu and Ramirez, 1997;

Cronin, 2000). The identification of the enclosure

relation is therefore a crucial issue, and two different

kinds of methods to do so have been developed: the

point-in-polygon-based method and the region growing-

based method. The basic idea of the former is to sort

contour lines into n levels according to their elevation

and to check the enclosure relation for each pair of

contour lines between two successive levels via point-in-

polygon processing. The lack of elevations of contour

lines and low computational efficiency of point-in-

polygon processing are two serious deficiencies of this

method. The region growing-based method checks all

possible adjacent neighboring contour lines whose

growing regions share a common boundary, and the

length of the boundary is used as a weight for identifying

the most significant neighbors. Two major problems are

the subjective judgment of the common boundaries and

the fact that the spatial adjacency is not equal to the

enclosure relation.

This paper aims to describe a new method for the

computation of a contour tree using the immediate

interior adjacency set (IAS1), which is based on the

Voronoi 1-order neighbors. The next session reviews the

basic ideas and deficiencies of the point-in-polygon-

based method and the region growing-based method.

An immediate interior adjacency set (IAS1)-based

method is proposed, and its implementation is described

in Section 3. In Section 4, the preliminary tests are

presented. Discussions and further investigations are

given in Section 5.

2. Previous work on generating contour trees

2.1. Point-in-polygon-based method

Since a given contour line encloses an arbitrary

number of other contour lines, but is in turn enclosed

by only one contour line, and these lines never intersect,

one natural way is to use a point-in-polygon algorithm

to identify the enclosure relation between contour lines.

This the method typically used by researchers to

determine the enclosure relation between contour lines

(Boyell and Reston, 1963; Freeman and Morse, 1967;

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100

L1

L2

L3

L4

L9

L7

L8

L5

L6

200 300

R0

R1

R2

R3

R4

R5

R6

R8

R7

R9

L9

L1 L2

L3 L4 L5 L6

L7

L8

R9

R1 R2

R3 R4 R5 R6

R7

R8

L3

R0

L9

L4 L5 L6 L8

L1 L2L7

R9

L1 L2

R3 R4 R5 R6

L7

R8

L3

L9

L4 L6 L8

R0

R1 R2 R7

L5

(A)

(B) (C) (D)

Fig. 1. Different strategies for transferring a contour map into a contour tree: (A) example of nested closed contour lines (100, 200, 300

are elevation values); (B) contour tree using inter-contour regions as nodes and contour lines as edges (Boyell and Ruston, 1963;

Freeman and Morse, 1967); (C) contour tree using both inter-contour regions and contour lines as nodes (Morse, 1969); and (D)

contour tree using contour lines (and their enclosed regions) as nodes and contour containment as edges (Cronin, 1995, 2000; Zhang

et al., 2001).

J. Chen et al. / Computers & Geosciences 30 (2004) 355–367 357

Kweon and Kanade, 1994; Cronin, 1995; Zhang et al.,

2001).

Suppose that Hmax and Hmin are the maximum and

minimum contour elevations of the studied area, and Dh

is the contour interval. The elevation of each level in the

contour tree can be determined by the sequence of

elevation Hmin;HminþDh;Hminþ2Dh;yand Hmax: All ofthe contours will be sorted according to this sequence

and stored as nodes of the contour tree. In the example

in Fig. 1A, 100, 200 and 300 are the elevations of the

three levels of contour. All of the contour lines L1;y;L9

are sorted into one of the three levels and considered as

nodes of that level, as shown in Fig. 2A. A point-in-

polygon algorithm is then used to determine whether

L1;L2;L7 fall into L3 (Fig. 2B). The same processing will

be performed for any contour line pair between two

successive levels.

One of the limitations of the point-in-polygon

based method is that it is necessary to elevate the

contour lines to sort them into different levels in the

tree. But the elevations might be unknown or false

in some cases, such as in the process of labeling the

digital contour (Wu, 1993). Another problem comes

from the low computational efficiency as point-in-

polygon processing should be executed for each pair of

contour lines between two successive levels. When the

shape of the contours is very complex and the polyline

numbers of the contours is large, the computational

efficiency will also decline. A minimum bounding

rectangle (MBR) of each contour was proposed to

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Level Elevation Nodes

1 100

2 200

3 300

L9

L1 L2

L3 L4 L5 L6

L7

L8

L1 L2 L7

L3

Holding enclosurerelation or not ?

(A) (B)

Fig. 2. Sorting and point-in-polygon processing steps: (A) sorting contours based on elevation; and (B) identifying enclosure relation.

L1 L2

L3

L4

L9

L7 L8

L5

L6

L1 L2 L3

L4 L5 L6

L7 L8 L9

L1

L3

L9

L2

L3

L9

L5

L9

L4

L9

L6

L9

L7

L8

L9

(A) (B) (C)

Fig. 3. Examination of contour enclosure relation with adjacency graph and associated weights defined by growing regions: (A)

regions growing from contours (dashed lines); (B) adjacency graph of contour lines (an arc between two contour lines means they are

adjacent); and (C) lists of successive contours derived from adjacency graph and associated weights.

J. Chen et al. / Computers & Geosciences 30 (2004) 355–367358

improve the efficiency of identifying the enclosure

relation (Cronin, 2000). The two contours will hold an

enclosure relation if the MBR of one contour falls in the

MBR of the other contour. However, this MBR

technique does not work properly in some cases, such

as with a contour enveloped by a ‘horseshoe-shaped’

contour (Cronin, 2000).

2.2. Region growing-based method

It can be seen from Fig. 3A that regions growing from

two enclosed/enclosing contour lines share a common

boundary. The region growing inward from the lowest

contour line, L9; has common boundaries with the

outward-growing regions of its immediate descendants,

L3;L4;L5;L6 and L8: The inward-growing region of

L3 has common boundaries with the outward-growing

regions of its children, L1 and L2; and its outward-

growing region has common boundaries with the

inward-growing region of its parent, L9; and the

outwards growing region of L4: The basic idea of

the region growing-based method is therefore to

check whether or not the regions growing from

two adjacent neighboring contour lines share a

common boundary with a relatively significant length,

as many enclosed/enclosing contour lines are adjacent

neighbors.

An adjacency graph among contour lines has been

derived from the growing regions and is shown in

Fig. 3B. L1 is adjacent to L2 and L3: L3 is adjacent to

L1;L2 and L9: L4 is adjacent to L3;L8 and L9: Thelength of the common boundary of the two growing

regions is taken as the weight of the corresponding

adjacent contour line pair (such as L1 and L2;L1 and

L3). The weight of contour line pair (L1=L3) is several

times than that of (L1=L2). On the basis of the adjacency

graph and its associated weights, six successive contour

lists with descending elevations are derived from the

local peaks L1;L2;L4;L5;L6 and L7; which do not

contain any contour lines. Once a list has been started,

contour lines are added to the list one by one by

examining the weights of the adjacent relation and using

some rules (see Roubal and Poiker, 1985). The pair of

contours with the largest weight is regarded as holding

ARTICLE IN PRESS

L1

L2

L3

L4

L9

L7L8L5

L6

Fig. 5. Examples of Voronoi regions of contours (shadow

regions).

L1

L4 L2

L5

L3

L6

L1

L4 L2

L5

L3

(A) (B)

Fig. 4. Contours holding enclosure relations might not be adjacent: (A) L1 enclosing L3 without common boundary; and (B) L1

enclosing L3 with relatively shorter common boundary.


the enclosure relation. As shown in Fig. 3C, every list

consists of successively lower contours, each one

enclosing the contour above it. The final branches of

the contour tree will be determined after a further

careful examination of these successive contour lists and

the identification of the lower neighbors of each contour

(Roubal and Poiker, 1985).

The major disadvantage of the region growing-based

method comes from the fact that the spatial adjacency

does not equal the enclosure relation. For example, L1

and L2 are adjacent to each other, but one is not

enclosed by the other. It was only with the help of the

relatively shorter length of the common boundary of

their growing regions that the adjacent link L1=L2 was

not interpreted as an enclosure relation and L2 was not

included in the list of the successive contours of L1:Making such a subjective judgment might be difficult,

and lead to incorrect results. Moreover, some of the

contours holding an enclosure relation might not be

adjacent, such as L1 and L3 in Fig. 4A, whose growing

regions do not have a common boundary. Fig. 4B

illustrates another situation, where L3 is enclosed by L1;but the length of common boundary of their growing

regions is shorter than the common boundaries of L3

with L2;L4;L5: So, in this case the relations between L1

and L3 will not be the enclosure relations, i.e., the

relations between L1 and L3 will not been detected

correctly.

3. An immediate interior adjacency set-based method

It is apparent from the above discussion that both the

point-in-polygon and region growing-based methods are

based on the assumption that a contour line defines a

closed region. According to the spatial relation theory,

such a contour line is a line feature in IR2 with an empty

interior and only a boundary, which is the line itself but

not the two end-points any longer (Li et al., 2000). A

group of contour lines can then be considered as a set of

disjointed spatial adjacent neighbors and be further

defined as Voronoi-based neighbors (Aurenhammer,

1991; Gold, 1992; Chen et al., 2001).

Let L be a set of contour line objects L1;L2;yLn in a

finite convex in R2; Li;LjALðiaj; i; j ¼ 1; 2;ynÞ: As

there is no common boundary between these disjointed

contour line objects, it is natural to use the Voronoi

regions of spatial objects to enhance the spatial

interaction between these objects themselves. It can be

seen from Fig. 5 that the Voronoi region of contour Li is

a ring and is represented by VLi: This Voronoi region is

composed of two parts: one belonging to the exterior of

the regions enclosed by the contour, called the exterior

Voronoi region and denoted by V�Li; the other belonging

to the interior, called the interior region and denoted by

V0Li: The Voronoi region of contour Li can therefore be

described as follows:

VLi¼ V0

Liþ V�

Li; ð1Þ


where

V0LiCL0

i ;V�LiCL�

i :

The contours representing peaks or pits have no

interior Voronoi regions; i.e., V0Li¼ +: For example,

contours L4;L5 and L6; which represent local peaks in

Fig. 5, have no interior Voronoi regions. Correspond-

ingly, the Voronoi edges of V0Liare here called interior

Voronoi edges, denoted by EV0Li; and the Voronoi edges

of V�Li

are called exterior Voronoi edges, denoted by

EV�Li:

Following the basic definition of Voronoi-based k-

order neighbors (Zhao et al., 2002), the Voronoi-based

k-order neighbors of a contour line Li can be described

as:

NkðLiÞ ¼ Lj jvdðLi;LjÞ ¼ k; k > 0� �

: ð2Þ

With this measure being semi-quantitative in nature,

the interior adjacency set (IAS) of a given contour line

Li can be further defined as:

IASk ¼ NkðLiÞ ¼ fLj jvdðLi;LjÞ ¼ k;LjCLig: ð3Þ

In particular, we can express the immediate

interior adjacency set (IAS1) of a given contour line

Li as:

IAS1 ¼ N1ðLiÞ ¼ fLj jvdðLi;LjÞ ¼ 1; LjCLig: ð4Þ

Contour lines are essentially a group of nested closed

curves. The outside contour lines enclose some inside

contour lines. The enclosure relations among contours

can be regarded as a kind of hierarchical relation that is

well represented in the contour tree. For a given contour

Li; its enclosing first-layer contours are its children

contours. In a contour tree, the nodes representing the

children contours of Li are located in the level

immediately below that of the node representing Li:The children nodes are linked with the parent node

by edges. Therefore, the contour tree can be constructed

by gradually identifying all of the children contours of

each contour in the contour map.

In Fig. 4A, contour L1 has five children contours; i.e.,

L2;L4;L5;L6 and L3: The former four children contours

are 1-order adjacent with L1; so that they belong to the

IAS1 of L1:L3 is 2-order adjacent with L1: From this

example it can be seen that the IAS1 of a given contour

Li is a subset of the set of all children contours of Li: Allof the children contours of a given contour L0 can be

described as follows:

childrenðL0Þ ¼ IAS1ðL0Þ þ[n

k¼2

Li jV�Li-V�

Lja+;

n

� LiAIASkðL0Þ;LjAIASk�1ðL0Þo: ð5Þ

From Eq. (5) it can be determined that to identify

all of the children contours of a given contour Li ;one should first identify all of the contours belonging to

the IAS1 of Li; then identify the children contours that

do not belong to the IAS1: The contours in the IAS1 canbe easily identified by searching those contours that are

1-order adjacent with Li and enclosed by Li: However,the identification of other children contours is more

complex.

To identify those children contours, one can con-

tinually search the contours that are 1-order adjacent

with at least two contours identified as children

contours, until no more contours meeting the

condition can be found. For example, suppose that

all of the children contours of contour L1 shown in

Fig. 4A need to be identified. First, the contours

belonging to the IAS1 of L1 are detected by searching

the contours that are 1-order adjacent with L1:The resulting contours are L2; L4;L5 and L6: Thesecontours are stored as children contours of L1:Then, those contours that are 1-order adjacent

with at least two contours identified as children

contours are further searched. The resulting contour

is L3: It is also stored as a child contour of L1:This searching process is repeated. The result is null,

which indicates that all of the children contours of L1

have been identified.

Although the children contours that are Voronoi-

based k-order (k > 1) adjacent with a given contour

must be identified during the process of identifying

the children contours, it is fortunate that those

contours only occur in places where the density

of contours is relatively high. In most places in the

contour map, all of the children contours of a

given contour belong to the IAS1 of this contour.

Take the contours in Fig. 5 as an example. All

of the children contours of contour L9; which are

L3;L4;L5;L6 and L8; are Voronoi-based 1-order

adjacent with L9; i.e., they are the elements of

the IAS1 of L9: Also, all of the children contours

of L3; which are L1 and L2; are the elements of the

IAS1 of L3: Therefore, in most cases, all of the children

contours of a given contour are simply the IAS1 of

this contour.

Thus far, a new strategy for generating contour trees

has been drawn out. That is, taking the furthest contour

line on the map as the root and starting from this

contour, all of the children contours of each contour are

gradually detected by means of Voronoi-based 1-order

adjacency and stored as nodes located on different levels

of the tree, until all the contours on the map are stored

as nodes. The final structure created is just the contour

tree.

The procedure of this immediate interior adjacency

set-based method can be described with using following

pseudo-codes:

ARTICLE IN PRESS

Initial: Let the boundary of the map be stored as

the root node of the whole tree, and let

Current node point to this root node.

Let the Current level=1.

Step 1: Detect first-level children nodes

Identify the contours that are 1-order

adjacent with the boundary of the

map.

Record all of the resulting contours as

children nodes of Current node and mark

these contours.

Do while TRUE

Identify all unmarked contours that

are 1-order adjacent with at least two

children contours.

If the result is null

Then Exit Do

Else

Record all the resulting contours as

children nodes of Current node and

mark these contours.

Endif

Loop

Let Current level=2

Step 2: Determine the children nodes iteratively

Do While TRUE

If all contours are marked

Then Exit Do

Else

For each node at Current level

Assign it to Current node.

Identify all unmarked contours

that are 1-order adjacent with Curren-

t node.


Then Current node is recorded

as a leaf node

Else

Record all of the resulting

contours as children nodes of Curren-

t node and mark these contours.

Do while TRUE

Identify all unmarked contours

that are 1-order adjacent with at least

two children contours.


Then Exit Do

Else

Record all of the resulting

contours as children nodes of Curren-

t node and mark these contours.

Endif

Loop

Endif

Next

Let Current level=Current level +1.

Endif

Loop

Step 3: END; i.e., the contour tree is formed and

outputted.


4. Experiments

This section contains the experimental tests of the

proposed approach. In the first test, the new method is

applied to the contour maps portraying three typical

kinds of landforms. In the second test, a comparison was

made between the new method and the existing methods

using the same contour data set.

Before generating a contour tree using the procedure

introduced in Section 3, the following three pretreatment

steps should be implemented:

(a) Preprocessing of contour data: The contour lines

dealt with should fulfill two requirements. First,

contours should be closing lines, which ensures the

correctness of the identification of adjacent relations.

Second, every contour has only one unique identifier,

which ensures that every contour appears as one unique

node in the tree. Those contours that do not fulfill the

two requirements should be preprocessed. A more

detailed discussion of preprocessing lies outside the

scope of this paper.

(b) Generating a Voronoi diagram of contours: Thus

far, there have been many methods of generating a

Voronoi diagram. However, most are vector-based. On

the one hand, it has been realized that vector-based

methods are good only for point sets and are compli-

cated for line and area sets, although they can be

approximated. On the other hand, in a raster mode, the

spatial objects can be treated as entities and a Voronoi

diagram for entities can be formed easily. In our work,

the dynamic distance transformation method is used to

compute the Voronoi diagram of contour lines (Li et al.,

1999).

(c) Identifying adjacent relations based on a Voronoi

diagram: In a Voronoi diagram of contours, each

Voronoi edge is the common boundary of the Voronoi

regions of two adjacent contours. Each Voronoi edge is

associated with the identifiers of the two contours

sharing this Voronoi edge. For a given contour, all of

the contours sharing common Voronoi edges with it are

recorded as its neighbors. All of the adjacent relations of

contours are stored in a table.

4.1. Generating a contour tree using the new approach

In this test, contour maps portraying three kinds of

landforms, a mesa, loess ridge and a Karst hill are used.

The contours, which are at the scale of 1: 50,000 and are

all 5 km� 5 km in size, were obtained from the National


Geomatics Center of China (NGCC). The contour

intervals are 20, 20 and 40m, respectively. The

immediate interior adjacency set-based method is

applied to the three contour maps. The trees generated

are shown in Figs. 6–8. From these figures, it is evident

that the new method works well, even for a large number

of contour lines.

From the contours representing a mesa and the

corresponding tree (Fig. 6), it can be seen that:

(a)
The hills in the left are much more intact than other
hills on the map.

(b)
There are altogether four intact hills; i.e., hills A, B,
C and D.

(c)
Contour 253 is the common foot of hills A and C.
Hill C is composed of the contour nodes in the left

branch below contour 253. Hill A is composed of

the contour nodes in the right branch below contour

253. The level of hill C is larger than that of hill A,

which indicates that hill C is higher than hill A.

(d)
The hilltops of hills B and D are shown in the tree.
From the contours representing a loess ridge and the

corresponding tree (Fig. 7), it can be seen that there are

altogether four large loess ridges, which are on the left,

top right, bottom right and bottom middle. The

boundaries of the loess ridges are contours 62, 107, 70

and 102, respectively. Contour 105 is the common foot

of these loess ridges. Contour 106 has the minimum

elevation, which is the root of the tree.

From the contours representing a Karst hill and

the corresponding tree (Fig. 8), it can be seen that the

Karst hill is very cracked. The number of nodes

counting backwards from the sixth level is the biggest

A

C

Fig. 6. Contour map of a mesa

of all of the levels, which indicates that the number of

contours with this elevation is the biggest among all of

the contours.

From the contour tree, some useful items can be

obtained. For instance, the number of nodes of the tree

represents the number of contour lines. The number of

leaf nodes represents the number of local peaks.

The level of the tree represents the range of elevation,

and so on.

The statistical results of the contour trees corres-

ponding to the three kinds of landforms shown in

Figs. 6–8 are shown in Table 1 and Fig. 9. It can be

observed that the tree of the Karst hill has the largest

values of all of the items, except for the item ‘the level of

the tree.’ This indicates that the Karst hill is the most

cracked of the three kinds of landforms. By contrast,

the loess ridge is the most intact landform. These

interesting results can be further used in the field of

landform classification, which is now being investigated

by the authors.

4.2. Comparison of the new method with existing methods

In this test, the new method and the existing two

methods are applied to the same contour map to

compare the efficiency of these methods.

The contour map used is shown in Fig. 4A. The

elevations of the contours are unknown. Because the

elevations of these contours are unknown, they cannot

be sorted in terms of elevation, which leads to the

invalidation of the point-in-polygon method. The

contour trees generated by the region growing-based

method and the new method are shown in Figs. 10A

and B, respectively. It can be noted that in Fig. 10A,

D

B

and corresponding tree.

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The common foot ofhill A and C

The left branch belowconstitutes hill C

Hilltop of hill D

Hilltop of hill B

The right branch belowconstitutes hill A

Fig. 6 (continued).


contour L3 is in the wrong location in the tree generated

by the region growing-based method. The main reason

for this is that contour L3 is regarded as being enclosed

by contour L5 on the basis of the length of the common

boundaries of growing regions of contours.

In fact, the Voronoi-based 1-order adjacency is used

in both the new method and the region growing-base

method. In the latter method, the regions growing from

contour lines are indeed the Voronoi regions of contour

lines. However, the 1-order adjacency plays different

roles in the two methods. In the region growing-based

method, the adjacent relations among contours are used

to identify the enclosure relations among contours and

leads to a theoretical default. In contrast, in the new

method, the 1-order adjacency is used to find the

immediate interior adjacency set (IAS1) of one contour

and further identify all of the children contours and

guarantee the completeness of the solution.

5. Conclusions and further investigations

A contour tree is a good graphical tool for represent-

ing the spatial relations of contour lines. It has found

many applications in map generalization, map annota-

tion, terrain analysis, etc. In this paper, we developed a

new method for generating contour trees based on a

Voronoi interior adjacency set.

In this new method, a Voronoi-based immediate

interior adjacency set is employed to identify all of the

children contours of each contour without contour

elevations. The new method has the following several

advantages over existing methods such as the point-in-

polygon method and region growing-based method:

(a)
It can be used in situations where the elevation of
contours is unknown or false. Moreover, since the

enclosure relations among contours do not need to

be identified, the new method is faster in terms of

computation than the point-in-polygon method.

(b)
The use of the immediate interior adjacency set
ensures that all of the children contours of each

contour can be completely identified. However, in the

region growing-based method, some children con-

tours of a given contour cannot be identified in some

conditions, such as the cases illustrated in Fig. 4.

(c)
In the new method, the nodes on each level of the
contour tree are specified by means of gradually

identifying the children contours of each contour

from the outside to the inside of the contour map,

which ensures that the enclosure relation between

the child contour and the father contour be

correctly identified. By contrast, in the region

growing-based method, whether two contours have

enclosure relation or not is judged on the basis of

the weight of the adjacent relation between them.

Making such a subjective judgment might be

difficult, and lead to incorrect results.

Two tests further indicate that the new method works

well for real contour data and can overcome the

deficiencies of existing methods.

Having a correct and efficient generating contour tree

is a crucial issue for mining and discovering knowledge

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106

105

102

70

107 62 The boundary of hill

The root node

The node in thesecond level

The boundary of hill



Fig. 7. Contour map of a loess ridge and corresponding tree.


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Fig. 8. Contour map of a Karst hill and corresponding tree.


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Table 1

Statistic result of contour trees of different kinds of landforms

Landform Item

Level of the

tree

Number of

nodes

Maximum number

of children nodes

Maximum number of

nodes in one level

Number of leaf

node

Mesa 42 314 17 36 152

Loess 13 107 7 25 48

Karst hill 18 337 19 43 193

0

100

200

300

400

Mesa Loess Karst hill

Level of the tree

Number of nodes

Maximum number ofchildren nodes

Maximum number ofnodes in one level

Number of leaf node

Fig. 9. Chart of statistical results of contour trees.

L1

L5

L3

L4 L2 L6

L1

L6 L5 L3 L4L2

(A) (B)

Fig. 10. Trees of contours shown in Fig. 4A, which have been generated with region growing-based method and new method: (A)

contour tree generated with the region growing-based method; and (B) contour tree generated with new method.


hidden deep within the vast number of contour maps.

The immediate interior adjacency set-based method

proposed in this paper ensures that the contour tree is

constructed correctly and efficiently. Further research

into how a contour tree can be used in geographical

knowledge-based applications such as the automatic

extraction of terrain features and construction of multi-

resolution DEM and so on is being carried out by the

authors.

Acknowledgements

The work described in this paper was substantially

supported by the National Nature Science Foundation

of China (under Grant No. 40025101). The authors

would like to thank Prof. Zhilin Li for his valuable

suggestions and comments.

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A Voronoi interior adjacency-based approach for generating a contour tree