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JOURNAL OF , VOL. , NO. , AUGUST 2016 1
Synthetic Modeling Method for Large Scale
Terrain Based on Hydrology
Huijie Zhang, Dezhan Qu, Yafang Hou, Fujian Gao, and Fang Huang*(Corresponding author)
Corresponding author: Fang Huang (huangf835@nenu.edu.cn)
Abstract—Generating large scale terrains that conform to the morphology of real scenes is a great challenge for terrain modelling, as
simulating complex geometric details is time-consuming and the realistic geographical features are hard to be controlled. In this paper, we
propose an efficient modeling method for large scale terrain visualization based on hydrology. To simulate real geographic features, we
introduce the hydrology based Tokunaga river network to guide the terrain generation, and propose a production rule set of river network
using procedural modeling. The distribution and structure of river network can be adjusted by user interactions. Ridges are extracted
based on river network to provide more skeleton features, and the enrichment method of skeleton features is presented to maintain
the morphology of valleys and ridges. Based on the enriched features, diffusion equation is exploited to compute the full elevation field,
which can achieve the nature transitions of the regions between skeleton features. Large scale terrain with real morphological features
can be generated on-line through the parallel implementation of diffusion equation. According to user requirements, the augmented
virtual terrain can be obtained by blending the selected real terrain with the synthesis terrain seamlessly. Experiments are conducted
on Digital Elevation Model (DEM), and the results show that the proposed methods can generate large scale terrains that conform to
morphology of real terrain and can well simulate various natural scenes.
Index Terms—Terrain visualization, hydrology, river network, procedural modeling, augmented virtual terrain.
F
1 INTRODUCTION
VIRTUAL scenes have been widely used on fields of
medicine and healthcare, geological analysis, video
games and movies. Among all parts of a virtual scene,
virtual terrain is one of the main components, and is a
dominant visual element. In recent years, a great progress
in terrain modeling has been made by many researchers,
and many effective terrain modeling methods have been
proposed. Existed techniques for terrain modeling can be
classified as follows: procedural modeling methods [1, 2],
modeling methods based on sketches [3, 4] and examples
[5], and modeling methods based on physics [6]. Procedural
modeling methods are usually efficient for generating large
scale terrain, but they usually don’t support good control of
terrain morphology because of the randomness. Modeling
methods based on sketches can generate terrain according
to the morphology denoted by user sketching. However,
manual editing is usually inefficient and requires the pro-
fessional knowledge of users. For methods based on terrain
examples, features of real terrain can be reserved in the syn-
thetic terrain by jointing the real terrain. But the efficiency
of terrain piecing is low, which is hard to generate large
scale terrain. Virtual terrains generated by methods based
on physics algorithm can well conform to the morphology
of real terrain. However, this kind of method is also hard
•Huijie Zhang, Dezhan Qu, Yafang Hou and Fujian Gao are with School
of Computer Science and Information Technology, Northeast Normal
University.
E-mail: zhanghj167@nenu.edu.cn, qudz862@nenu.edu.cn,
houyf397@nenu.edu.cn, 1055563905@qq.com.
•Fang Huang is with school of Geographical Science, Northeast Normal
University.
E-mail: huangf835@nenu.edu.cn
to control and can’t support the generation of large scale
terrains. So the lack of controllability is a general issue
for most of current methods. Moreover, how to efficiently
generate large scale terrains that fit in the requirements of
geology is a great challenge.
In this paper, we propose a procedural modeling method
based on hydrology theory to generate virtual terrains that
accord with the morphology of real terrain. Our system
allows users to control the terrain morphology interactively
through easy-to-use operations. According to hydrology
theory, river networks play important roles in terrain forma-
tion and are effective to ensure the real terrain morphology.
So, we introduce fractal theory to generate river network,
according to the self-similarity property of river network.
Based on the Tokunaga river network [7, 8], we propose a
set of production rules that make our river network accord
with hydrology theory. Based on the river network, we
extract the watersheds as ridge features, to constitute the
whole skeleton features of terrain. The elevations of these
skeletons are computed according to the distances from
rivers to the estuary. To enrich the features of mountains
and valleys, Midpoint Displacement method and its inverse
process are exploited to extend the skeleton features, which
can avoid that the features of mountains and valleys are
too steep. On the basis of extended skeleton features, we
exploit diffusion equation to compute the elevations of other
regions to obtain the elevation field of the whole terrain.
Moreover, we provide a terrain editing operation to embed
a block of real terrain into the virtual terrain. For example, if
users are interested in a block of real terrain, the augmented
virtual terrain with features of real terrain can be obtained,
through the terrain editing operation. Additionally, the per-
formance for generating large scale terrain is enhanced, by
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JOURNAL OF , VOL. , NO. , AUGUST 2016 2
the parallel implementation of elevation computation on
GPU. The main contributions of our work are threefold:
1. We propose a production rule set based on Tokunaga
self-similar network for generating river network that con-
forms to hydrology theory. The terrain generated based on
our river network meets the accuracy requirements of scene
modeling and supports the observations in the field of river
science.
2. We propose a terrain modeling framework to effi-
ciently generate large scale terrain with good controllability,
combining procedural modeling with skeleton features.
3. We propose a terrain blending method based on
Poisson equation to blend virtual terrain and real terrain
seamlessly. Users can embed a block of real terrain into an
arbitrary position of the generated terrain, which ensures
that the features in real terrain are blended into the virtual
terrain and the boundaries of real terrain are smoothly
connected with the original virtual terrain.
The rest of this paper is organized as follows: we first re-
view the related researches in Section 2. Then, the overview
of our methods is described in Section 3. The details of
our methods are demonstrated in Section 4, Section 5 and
Section 6. The results of our method are presented in Section
7. The performance is discussed in Section 8, followed by the
conclusion and future work in Section 9.
2 RE LATED WORK
2.1 Procedural Modeling Methods
Procedural modeling is a classical parametric method that
is easy to operate and is suitable for generating large s-
cale terrain. A detailed overview of procedural modeling
method was given by Ebert et al [9]. Among the approaches
of procedural modeling, the most representative one is the
adaptive subdivision method, proposed by Fournier et al
[10], which provided a set of stochastic models to present
various natural phenomena. Perlin et al [11] proposed a
noise-based procedural method, which combined the noise
function to present various details under different scales.
Though fractal based approaches have been commonly used
to generate details of large scale terrain, the procedural
modeling methods lack the spatial information based on the
topography features, which is hard to precisely present the
morphological characteristics of landforms.
2.2 Terrain Modeling Based on River Network and S-
ketch
Utilizing river network and sketch are two effective ways
to generate terrains with specific morphology features. Ac-
cording to geology theory, real terrain is generally produced
through erosion and weathering. So introducing the infor-
mation of the river network to the terrain modeling process
is a reasonable solution for generating realistic terrain [12].
Based on the Horton-Strahler river network classification
method, Zhang et al [13] concluded a description of Tokuna-
ga fractal river networks on statistical average sense. Claps
et al [14] generated self-similar river networks through
recursive and iterative operations. Researches on geologi-
cal river network can provide morphological guidance for
modeling realistic terrains. Terrains were generated through
constructing river network in the work of Teoh et al [15],
but the model of river network is not based on hydrology.
G´
enevaux et al [12] proposed a terrain generation method
based on hydrology, using procedural models. They first
modelled river network using a probabilistic way, and then
constructed watersheds based on the river network. The
final terrain was generated by blending river patches with
the procedural terrain. The idea of their method is similar to
ours. However, from the point of hydrology, our method
introduces Tokunaga river network to simulate the real
morphology of drainage basins, which better accords with
hydrology theory. For terrain generation, we use diffuse
equation to generate realistic landform under the guidance
of river network and ridges, which can present the details
of terrain more realistically. The context-sensitive L-system
and Midpoint Displacement method were introduced by
Prusinkiewicz et al [16], to embed the morphology of rivers
into terrain by using fractal modeling system. Kelley et al
[17] proposed a procedural method to generate river basins.
Using this method, the local sketch can be produced as the
input data of global terrain. However, only several abstract
parameters can be used to control the river network, without
providing user interactions. A fractal based algorithm was
proposed by Belhadj and Audibert [18] to generate realistic
landscapes. In their work, the terrain is constrained by the
river network and ridge lines. However, the river network
in this work was uncorrelated with information of the river
basins. In 2014, according to perceptual cues, Tasse et al [4]
have found the best match between profile sketches and the
input terrain profile of ridges. In their work, to ensure that
the user-defined profiles were not occluded by other parts of
terrain, deformation was exploited for terrain to completely
match the sketches from a perspective view.
2.3 Terrain Modeling Methods Based on Physics
This type of methods usually exploits physical models to
simulate the process of terrain generation. In view of the
fact that the morphology of land surface is affected by
multiple environmental factors, such as water erosion and
change of temperature, modeling method based on physics
was proposed by Musgrave et al [19], who introduced a
simple physical erosion model to simulate the details on
land surface. By using different erosion algorithm, mountain
sceneries were well simulated in the work of Chiba et al [20].
An erosion model that provided high level of control has
been presented in the work of [21], and a hydraulic erosion
model inspired by fluid mechanics was deduced in the work
of [22]. Wojtan et al [23] introduced a morphology algorithm
of corrosion simulation, which supports user interactions to
control the shapes of objects. Kristof et al [24] introduced
smooth particle hydrodynamics to the erosion method. The
algorithm efficiency is a nonnegligible issue for this kind
of methods. Although GPU implementation can settle this
matter to a large extent [25, 26], it is still hard to generate
large scale terrain, because of the algorithm complexity.
2.4 Controllability and Interactivity of Terrain Modeling
The controllability and interactivity are important for gen-
erating terrains that satisfy the user requirements. In order
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JOURNAL OF , VOL. , NO. , AUGUST 2016 3
to better control the terrain construction, modeling meth-
ods with interactive editing have been studied by many
researchers. Hnaidi et al [27] described an algorithm to edit
the river trajectories directly. In this method, terrain was
generated based on a given set of topographic features,
using multigrid diffusion equation and gradient constraint
for matching the corresponding control curve. Although
this method is based on the differential properties of grid,
and can generate large high-resolution terrain, the problem
of physiognomy representation in rivers modeling is still
existed. In order to solve this problem, Skytt et al [28]
exploited the novel LR B-splines [29] in the processing of the
huge geographical data. Surfaces modeled by LR B-splines
can provide the adjustment mechanism of local details and
the compact representation of global data. Rusnell et al [30]
proposed a feature based synthetic modeling method to
add controllability of terrain morphology. According to the
sketch features, Zhou et al [31] generated the final terrain
based on the synthesis of example terrains. Although very
well results have been achieved in this work, an input
terrain must be given in advance, which is also the common
problem of terrain synthesis methods based on example
terrains. Under the constraints of user sketches, Gain et al
[32] utilized multiresolution surface deformation to generate
terrains that satisfied user requirements. However, incorrect
results may be produced from a geologic perspective if
the users don’t have professional knowledge. To address
this issue, some researchers have combined erosion and
deposition algorithms with interactive editing [6]. But this
kind of method cannot support the generation of large
scale terrains. In this paper, we provide flexible user in-
teractions to control the start position and grow direction
of the river network in a large scale scene. Furthermore,
the morphology of rivers and mountains is enriched by
Midpoint Displacement method and its inverse process.
3 OVERVIEW
In this paper, we propose a terrain synthetic modeling
method to generate realistic terrain that conforms to the
geographical morphology through a simple and efficient
process with high user controllability. To make the gener-
ated terrain accord with the morphology of real terrain,
we create self-similar river network based on hydrology
theory, as the base of terrain modeling. The height values
of the river network are determined by the distances from
rivers to estuary. To contain more abundant features in our
generated terrain, we build Voronoi diagram based on the
river network and extract ridges to form the whole skeleton
features. Discretization of the skeleton features is performed
by Bresenham algorithm. Then the elevations of ridges are
calculated according to the heights of river network. To
generate the whole terrain based on skeleton features, the
morphology of mountains and valleys should be further
described. So Midpoint Displacement (MD) and Midpoint
Displacement Inverse (MDI) are utilized to generate more
data points for enriching the morphology features of river
network and ridges. After obtaining elevations of enriched
skeleton features, the whole elevation field of the generat-
ed terrain is computed through diffusion equation based
on Laplace. At last, we provide a flexible terrain editing
Fig. 1. Workflow of our method. The framework is composed of the
generation of river work, construction of ridges, construction of mountain
and terrain blending. Final terrain can simulate morphology of real ter-
rain scenes and can be edited by users according to the requirements.
operation based on Poisson equation to integrate existed
real terrain and the generated virtual terrain. The user con-
trollability contains two aspects, including drawing vector
lines for controlling the generation of river network and
selecting terrain features for terrain blending. The schematic
representation of our workflow is shown in Fig. 1.
4 RI VER NETW ORK CONSTRUCTION
In this section, we describe the details of the river network
construction based on hydrology theory. Horton-Strahler
river network classification is a classical method that ex-
ploits multiple measures to quantitatively describe the river
network [33]. Through introducing the fractal theory, re-
searchers have described the topology of self-similar net-
work (SSN) through links. Tokunaga river network [7, 8] is
a kind of self-similar river networks based on the Horton-
Strahler river network classification. It can satisfy the plane-
filling property of river network by using its fractal rules
and can well describe the morphology of real river network.
Zhang et al [13] have given a basic generator sequence of
SSN and deduced the parameter of Tokunaga river network
based on this generator sequence. They found that the pa-
rameter can well describe the morphology of river network.
In this paper, we propose a set of fractal rules to generate
river networks that are consistent with real natural land-
scape, through combining the basic generator sequence with
the parameter of Tokunaga river network. The generated
river network is the base of our terrain modeling method.
4.1 Quantization of River Network Morphology
Self-similar river network is composed of a set of basic gen-
erators. Based on specific fractal rules, river network is gen-
erated from the initial generator through replacing the links
with generators iteratively. Since most of river networks
are formed as tree structure, we define generators of our
river network based on binary tree. The generators of river
network are made up by interior links and exterior links
that respectively describe the main streams and tributaries.
Similar to the work by Zhang et al [13], we also adopt an
asymmetric structure for our basic generators. Furthermore,
we assign different angles to exterior links for distinguishing
the tributaries in different phases. Four fractal generators
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JOURNAL OF , VOL. , NO. , AUGUST 2016 4
TABLE 1
Relationship of ϕand R
ϕ1 2 3 4
Generators
P−− 1.00 0.768 0.640
R−− 1.00 1.535 1.921
with different numbers of links are shown in Table 1, in
which the arrow orientation points out the direction of de-
bouchure, the vertical links denote the interior links, and the
non-normal links denote the exterior links. We let the two
exterior links farthest from debouchure rotate θangle about
interior links, and let other exterior links rotate 2θangles
about interior links, which better conforms the morphology
of real river network through practical observations. Based
on the basic characteristic of binary tree, we can obtain
that the numbers of exterior links are one more than the
number of interior links. To quantize the description of river
network, we arrange the basic generators as a sequence in
the ascending order of link numbers. Thereby, the growth
pattern of river network can be described numerically by
the generator index ϕ. After determining the generator
sequence, we use Tokunaga fractal rules to generate the
self-similar river network. Tokunaga fractal rules ensure
that the growth of exterior links is prior to interior links,
which conforms to the growth rules of real river network. So
the exterior links are replaced by the generators with high
serial numbers, whereas the interior links are replaced by
the generators with relatively low serial numbers. Besides,
the plane-filling property can also be satisfied by combining
the generators with Tokunaga fractal rules according to the
verification by Zhang et al [13], which is essential for river
network modeling. Given the parameter ϕthat denotes the
serial numbers of the initial generator, we use generator
ϕto replace exterior links and replace interior links with
generator Rthat is computed as follow:
R=P∗(ϕ−1),(1)
in which P=1+√1+4ϕ
2ϕ. If Ris not an integer, the decimal
part of Ris used as the probability to select a generator
between its floor and ceil. For example, the value of Ris
1.535 with ϕ= 3 as shown in Table 1. In this condition, we
have the probability of 53.5%to select generator 2 and the
probability of 46.5%to select generator 1. The values of R
for basic generators are presented in Table 1.
Besides, the lengths of links are same in one iteration
step and are changed as the river network grows. They are
computed as:
lt= (1/P ·ϕ)t−1,(2)
in which tdenotes the current iteration step, ltis the link
length in iteration step t. Through performing the fractal
TABLE 2
Basic Operations of Generator Links Evolution
symbol operation
→a replacement operation during iteration
[ a push operation of location stack
] a pop operation of location stack
+rotate an angle clockwise
−rotate an angle counter clockwise
process iteratively, the self-similar river network that con-
forms to hydrology theory is generated.
4.2 Procedural Modeling for River Network Based on
L-System
Based on the quantization of river network, we propose a
rule set to describe the evolution of river network through
fractal L-system. Fractal L-system is a modeling technique
to generate scenes by exploiting production rules, which can
be embedded into algorithms. According to self-similar and
fractal theory, we propose a production rule set based on
the generators to describe the evolution of river network
(interior links and exterior links). The basic operations in
the rules are described in Table 2 and the rule set of river
network generation are defined as follows:
Definition 1. Let Fbe a line segment that represent an
interior link, Gbe a line segment that represents an exterior
link, and Dep be the maximum iteration depth. lis the
length of line segments that are corresponding to Fand
G. Let F0be the interior links in the previous iteration step,
and G0be the exterior links in the previous iteration step,
then Ωis the following rule set for river network generation
in one iteration.
I For the case of ϕ= 2:
Exterior links G0→F[− − G]F[−G][+G];
Interior links F0→F[+ + G]For F0→F F .
II For the case of ϕ= 3:
Exterior links G0→
F[+ + G]F[− − G]F[−G][+G];
Interior links F0→F[−G]F[−G][+G]or
F0→F[+ + G]For F0→F F .
III For the case of ϕ > 3:
Exterior links G0are replaced by generator ϕ;
Interior links F0can be replaced by a generator
according to the value of Ror use the rule F0→
F F .
To ensure the generality and naturality of the produced
river network, exterior links are replaced by generator ϕ,
and interior links are substituted for one of the available
generators according to a probability computed by the value
of Ror are lengthened directly.
The river network generated by rule set Ωis shown in
Fig. 2, in which the red vector lines are input by users to
denote the start position and the growth direction of river
network. Through setting the initial generator index ϕand
the maximum iteration depth Dep, the morphology of our
generated river network can be controlled. Fig. 2(a) is the
river network generated with ϕ= 2 and M axDepth = 7,
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which can describe the tree structure of real river system. As
larger as the value of ϕ, the morphology of river network
tends to be more featherlike. Conversely, the dendritic river
network is more possible to be presented. As shown in Fig.
2(b), multiple river networks can be generated in one scene.
The reliability of our generated river network is verified
in Section 7, by the comparison with river network of real
terrain extracted by ArcGIS.
(a) (b)
Fig. 2. River network produced by our generation rules. (a) River network
generated by one user-input vector line with ϕ= 2 and MaxDepth = 7.
(b) River network generated by four user-input vector lines.
4.3 Height Computation of River Network
To present the tendency of water flow, we compute the
slopes of river segments according to the Geodesic distances
from river network nodes to the estuary point. The elevation
of river network in estuary point is set as 0. The heights
of river network nodes are directly proportional to their
Geodesic distances to the estuary point. As the fractal river
network grows, heights of river network nodes are calculat-
ed step by step according to their distances to the estuary
point. We provide an easy-to-use interactive tool to set the
estuary point and general flow direction of river network,
through users drawing straight line segments simply on a
null height field. In this way, realistic river network with
heights can be generated, by setting the start position and
the growth direction of river network in L-system as the
start point and orientation of the line segment. The iteration
depth depthLis computed as follow:
depthL=lengthab /widthT×M axDepth, (3)
in which lengthab ,widthTand M axDepth respectively
denote the length of the line segment drawn by users, width
of the terrain DEM and the given maximum iteration depth.
5 TERRAIN GENERATION
In this section, to obtain more morphology features for
modeling realistic terrain, we further produce ridges of
terrain based on the generated river network. In geography,
ridges are also important features for terrain generating.
At the same time, for many real scenes, watersheds are
corresponding to ridges. So the ridges can be obtained
by extracting watersheds of the river network. Similar to
the work in [12], we construct Voronoi diagram based on
the nodes of our Tokunaga river network, and watersheds
are the edges of Voronoi diagram among different river
basins. Due to the continuous edges of skeleton features are
difficult to be used in the grid data of DEM, we discretize
the skeleton features onto the DEM using Bresenham al-
gorithm. Furthermore, neighbor points of skeleton features
are interpolated using Midpoint Displacement (MD) and its
inverse process (Midpoint Displacement Inverse, MDI) [18]
to ensure authentic shapes of rivers and mountains. Finally,
diffusion equation based on Laplace is exploited for all the
generated elevation points to compute the whole elevation
field.
5.1 Construction of Ridges
In the field of geographical information system, Voronoi
diagram is commonly used for efficient neighbor interpo-
lation and analysis of the influenced regions of geographic
entities. Voronoi diagram satisfies the properties as follows:
each Voronoi cell has only one point; and two adjacent
points have the same distance to the edge between the two
points. According to these properties, we construct Voronoi
diagram based on the nodes of the river network using the
classical Sweep Line algorithm [34], as shown in Fig. 3(a).
The edges of Voronoi diagram can be processed differently
according to whether the edges intersect with river network
or not. For the edges that don’t intersect with the river net-
work, they can be considered as watersheds because these
edges usually divide two different river basins. Therefore,
we cut the edges that intersect with the river network, and
preserve the uncrossed edges as ridge lines, as shown in
Fig. 3(b). In Fig. 3(c), the ridge lines denoted by yellow
lines and the river network denoted by cyan line segments
constitute together the whole skeleton features of terrain.
The generated ridges can well simulate the morphology of
real terrain, which is demonstrated in Section 7.
(a) (b) (c)
Fig. 3. Constructing ridges by building Voronoi diagram based on nodes
of river network and cutting the edges crossed with the river network.
Red points are the nodes of river network. Yellow lines are the edges
of Voronoi diagram. Cyan line segments denote the river network. (a)
Building Voronoi diagram based on nodes of river network. (b) Cutting
the edges crossed with the river network. Watersheds separating differ-
ent river basins are obtained. (c) Skeleton features composed of river
network and ridges.
For the heights of ridge nodes, we calculate them accord-
ing to the heights of river network nodes as follows [12]:
Heightq=max(Heightp1, H eightp2, H eightp3)+α·d, (4)
where qis a ridge node, p1,p2and p3are the first three
nearest nodes on river network to node q, and Heightq,
Heightp1,Heightp2and H eightp3are respectively the ele-
vation values of nodes q,p1,p2and p3. Besides, d represents
the distance from qto its nearest river network node p1,
and αis a parameter for adjusting heights. The 3D skeleton
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Fig. 4. Produced 3D terrain skeleton composed of river network and
ridges after the computation of heights. Cyan segments denote the river
network with elevations. Red segments are the ridges.
features of our virtual terrain are shown in Fig. 4. We can see
that the river network is located around the ridges, which
conforms to the cases of real scenes.
Terrain is usually represented by discrete data points.
Skeleton features play an important role in presenting ter-
rain morphology. So, for obtaining the whole terrain based
on skeleton features, we discretize the segments of skeleton
features onto grids, using classical Bresenham method. In
order to make ridges better conform to the natural case, the
heights of the discrete points are added by random values
that accord with Perlin noises [11], and are computed as
follows:
HeightP=HeightPs+kP−Psk
kPe−Psk(HeightPe−HeightPs)
+F(P),
(5)
in which Pis an interpolation point, Psand Peare respec-
tively the start point and end point of a ridge line, and
HeightPand HeightPsand H eightPeare respectively the
elevation values of P,Psand Pe.Fdenotes the fractal Perlin
noise function.
5.2 Data Extension of Skeleton Features
Based on the discrete skeleton features, the whole terrain
can be computed through many methods. However, the
terrain interpolated directly has too steep ridges and valleys.
So, we extend the skeleton features to generate more feature
points, exploiting Midpoint Displacement and its inverse
process. Through this process, the morphology of moun-
tains and rivers can be better presented and more realistic
terrain is produced. Moreover, the whole elevation field can
be generated more efficiently, because the elevations of more
discrete points are known previously.
Midpoint Displacement (MD) is commonly used to gen-
erate fractal terrains. Through iterative dividing, terrain can
be generated with the resolution required by users. The
MD process can interpolate the square center and four mid-
points of the edges based on four corner points. Midpoint
Displacement Inverse (MDI) interpolates the corner points
of a square based on the known data points inside the
square. The process of MDI and MD are illustrated in Fig. 5.
Through combining these two methods, skeleton features
can be extended with dense neighbor points. Compared
with linear interpolation, the extended mountains and rivers
better conform to the morphology of real terrain.
(a)
(b)
Fig. 5. Schematic diagram of MD and MDI methods. Black block dots
denote the discrete points with computed elevation. White circular dots
represent the null points need to be interpolated. (a) Midpoint displace-
ment method. Directions of solid arrowed lines describe the interpolation
process of MD. (b) Midpoint displacement inverse method. Directions of
dashed arrowed lines illustrate the interpolating process of MDI.
Since the discrete data points of skeleton features are
adjacently located on the grid with the required resolution,
we first perform MDI process for these points to extend
the regions of skeleton features. However, in the skeleton
features region extended by MDI, some points inside square
might be absent. So we next perform MD process to fill
in the null points inside the region of skeleton features.
Through continually performing MDI process and MD pro-
cess in turn, skeleton features can be significantly enriched.
The more times the process performs, the more data points
with elevations can be obtained. However, as the distance
dbetween the two points becomes long, error of the inter-
polation using these two points becomes large. To address
this issue, a parameter Lis introduced to reduce the number
of anomalous points in the interpolation. When d > L, we
stop the interpolation. The results of feature enrichment are
shown in Fig. 6 with resolution 1024×1024. Fig. 6(a) and Fig.
6(c) show the skeleton features before feature enrichment
respectively in the global and detail view. From Fig. 6(b)
and (d), we can see that dense points have been generated
around the skeletons.
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JOURNAL OF , VOL. , NO. , AUGUST 2016 7
(a) (b) (c) (d)
Fig. 6. Enriching the skeleton features by processes of MD and M-
DI. Many discrete points are interpolated according to the heights of
skeleton features. (a) Skeleton features without feature enrichment. Red
lines are ridge lines. Cyan lines are river network. (b) Result of the
enriched skeleton features from 2D perspective. Dense cyan points
are interpolated based on skeleton features. (c) Details of the skeleton
features before feature enrichment. (d) Details of the skeleton features
after feature enrichment. More discrete points can be obtained.
5.3 Terrain Generation Based on Diffusion Equation
After obtaining enough discrete data points of skeleton
features, we generate the elevation field of the whole terrain,
using diffusion equation based on Laplace. In the field of
physics, diffusion equation is exploited to study the temper-
ature distribution for homogeneous objects with isotropic
material, based on the heat transfer with environmental
media. In this paper, we introduce the idea of diffusion
equation to obtain the whole terrain data based on the differ-
ence in elevations. On the base of our feature model, the 3D
skeleton features and non-null data points generated by MD
and MDI are considered as the diffusion sources. Then, the
elevations of null points can be calculated iteratively from
the diffusion sources. The mathematical model of diffusion
equation that describes the distribution of elevations at
iteration step tcan be demonstrated as follows:
∂u(x, y , z, t)
∂t =D4u(x, y , z, t) + f(x, y, z, t),(6)
in which u(x, y, z, t)represents the elevation on position
(x, y, z)at iteration step t,f(x, y, z, t)denotes the growth
number of non-null points in current iteration step, 4is the
Laplacian operator, and Dis the diffusion coefficient. The
Laplacian operator used in Equation (6) is
4= 4Ui,j −Ui+1,j −Ui−1,j −Ui,j−1−Ui,j+1,(7)
in which Ui,j is the elevation of a point with coordinate
(i,j) on discrete grid, and Ui+1,j ,Ui−1,j ,Ui,j−1and Ui,j+1
are elevations of the 4 adjacent points of point (i, j). By
performing the diffusion iteratively, the whole elevation
field is obtained. Moreover, morphology of real terrain can
also be ensured, because the diffusion process is constrained
by the elevations of ridges and river network. Additionally,
Perlin noises or Gaussian smoothing can be performed onto
the generated terrain according to the requirements of users.
6 GENERATION OF AUGMENTED VIRT UAL TER -
RA IN
In order to satisfy the requirements of containing features
of real terrain in the virtual terrain, we propose a terrain
blending method based on Poisson equation. It allows users
to embed a block of real terrain into our virtual terrain, and
morphology features of real terrain can be reserved, with
the seamless connection between real terrain and virtual
terrain. To achieve this goal, we compute the gradient field
of real terrain block to describe the terrain features and
exploit Poisson equation to blend the terrains according to
the elevations of the boundaries on the virtual terrain.
6.1 Poisson Equation
Poisson equation has been used in mesh processing to
calculate the smoothly deformed mesh under the guidance
of a gradient field [35]. In this paper, we utilize Poisson
equation to blend our virtual terrain with a block of real
terrain seamlessly, and take the block of real terrain, the
block of virtual terrain and the block of blended terrain
respectively as reference terrain R, source terrain Sand
target terrain T. These three terrain blocks have the same
length len, the same width wid and the same points number
m=lenwid. Since the height field of DEM is a piecewise
linear model, it can be considered as a discrete potential
field. Meanwhile, Sand Rhave the same topology, because
the grid of DEM is uniform. So, Poisson equation is suitable
for this paper to construct the augmented virtual terrain. For
target terrain T, its boundaries are set as the boundaries of
S, and the elevations of its inner regions can be obtained by
solving Poisson equation based on the divergence field of R
and boundaries of S. To obtain the unknown elevations in
T, the divergence field Div of Ris introduced in Poisson
equation to describe the tendency of elevations in R. The
divergence value Divi,j of point with coordinate (i, j)can
be computed based on the gradients of its neighbor points,
as follows:
Divi,j =(GarXi,j−1−GarX i,j +1)/2
+ (GarY i+1,j −GarY i−1,j )/2,(8)
in which GarX and GarY are respectively the gradient
field of Rin Xdirection and Ydirection of the terrain
plane. GarXi,j−1,GarXi,j+1 ,GarY i+1,j and GarY i−1,j
are the gradient values of the neighbor points, which are
computed by the values of elevation differences in the corre-
sponding directions. These divergence values are formed as
a vector b= (b1, ..., bk, ..., bm)0, in which bk=Divi,j ,(k=
i×wid +j). Additionally, the boundaries of Sare taken
as the constraints to ensure the smooth connection between
Tand the original virtual terrain. For a point with coor-
dinate (i, j)in T, if it has neighbor points that are locat-
ed on the boundaries of S, the corresponding component
bk(k=i×wid +j)in vector bis further subtracted by the
elevations of these boundary points in S. Thereby, vector b
combines the divergence field of Rwith boundaries of S.
Then, the discrete Poisson equation can be represented as
the linear system
Ax =b, (9)
in which Ais a sparse coefficient matrix with size m×m,
and vector x= (x1, ...xk, ..., xm)0,(k=i×wid +j)denotes
the unknown elevation values of T. In this paper, we adopt
the Laplace operator described in Equation (7) in Poisson
equation, so matrix Ais the corresponding Laplace matrix.
This means that the result of processing Laplace operator
onto the unknown elevation values in Tshould conform
2169-3536 (c) 2016 IEEE. Translations and content mining are permitted for academic research only. Personal use is also permitted, but republication/redistribution requires IEEE permission. See
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JOURNAL OF , VOL. , NO. , AUGUST 2016 8
to the divergence field of Rand boundaries of S. Through
solving the linear system, elevations of the data points in T
can be obtained.
For the application of multi-resolution terrain, a mesh
parameterization can be first performed. Thereby, the gradi-
ent fields can be computed for the parameterization discrete
fields of each point. The work for multi-resolution case is
left to future work.
(a) (b) (c)
Fig. 7. Results of terrain blending. Through solving Poisson equation, the
wanted feature in another terrain can be blended into our virtual terrain.
(a) An artificial terrain with features that are easy to be recognized.
Red square denotes the source terrain. (b) Our virtual terrain. Yellow
square denotes the target terrain. (c) Result of target terrain. The wanted
features are embedded into our virtual terrain smoothly.
6.2 Terrain Blending
In this section, we describe the process of blending the
virtual terrain and real terrain block. If users are interested
in a specific feature in real terrain, they can select a terrain
block containing this feature as reference terrain R. Then the
divergence field of Ris calculated to simulate the variation
trend of the specific features. Through manipulating the
divergence field of reference terrain, users can control the
size of the features in R. Transformations such as scaling
can be performed onto the gradient field. As the gradient
field of reference terrain block is obtained, we choose a block
region on the source terrain and blend the reference terrain
and source terrain on this region. Algorithm 1 explained the
process of terrain blending.
Algorithm 1 Terrain blending
Input: Reference terrain block Rand source terrain block
S
Output: Results of target terrain block T
1: Compute the gradient field Dof R;
2: Users transform the gradient field Daccording to their
requirements;
3: Set the boundaries Bsof S;
4: Perform Poisson equation:
4.1 Calculate the divergence field of Rbased on D;
4.2 Solve the linear system in Equation (9);
5: If the result of Tcan not satisfy requirements of users,
go to Step 2;
6: Algorithm end.
We take an artificial terrain with obvious features as the
reference terrain to illustrate the effectiveness of terrain edit-
ing. As shown in Fig. 7(a), the reference terrain Rselected by
users is marked by the red square. The boundaries of source
terrain Sare marked by the yellow square in Fig. 7(b), and
the target terrain Tis shown in Fig. 7(c). We can see that the
features of Rin the red square of Fig. 7(a) are contained in
the target terrain Tmarked by the yellow square in Fig. 7(c).
Besides, the connection regions of two terrains are seamless,
because the boundaries of source terrain block are used to
ensure the smooth transition between two terrains.
7 RE SULTS
The system in this paper is developed using C++ to imple-
ment the algorithms. OpenGL Shading Language (GLSL) is
utilized to render the terrain. Additionally, OpenCL is used
to enhance the performance of terrain diffusion, by using
parallel resources on GPU.
ArcGIS is a popular GIS software that can extract ridge
lines and valley lines accurately. To verify the reliability of
the river network and ridges generated by our methods,
we compare our results with the valley lines and ridge
lines extracted by ArcGIS. River networks in real scenes are
usually presented as tree structure that has a main stream
and some alternately arranged tributaries. River network
generated by our method can also conform to this pattern.
A typical example of valley lines in real terrain scenes are
shown in Fig. 8(b), whose structure is very similar to binary
tree. By comparing Fig. 8(a) and Fig. 8(b), we can find that
the two river networks have similar patterns, because our
production rules of river network conform to hydrology
theory. Fig. 8(d) is the corresponding ridge lines of the
terrain in Fig. 8(b). From Fig. 8(c) and Fig. 8(d), we can
both find some continuous ridges with similar morphology,
which actually separate different river basins. So, the skele-
ton features generated by our methods can well describe the
skeleton features in real terrain scenes and provide a good
base for generation of the whole terrain.
(a) (b)
(c) (d)
Fig. 8. Comparison between results of our method and real feature lines
extracted by ArcGIS. (a) Valley lines generated by our methods. (b) A
typical example of valley lines in real terrain scenes extracted by ArcGIS.
(c) Ridge lines generated by our methods. (d) Corresponding ridge lines
extracted by ArcGIS.
To demonstrate the controllability of our method for
terrain modeling, we compare our methods with the clas-
sical Midpoint Displacement method. As shown in Fig. 9(a),
the terrain generated by Midpoint Displacement method
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JOURNAL OF , VOL. , NO. , AUGUST 2016 9
present no specific geomorphic features, and users can only
control the terrain generation by setting the elevations of
four corner points. As shown in Fig. 9(b), the terrain generat-
ed by our method clearly present realistic geomorphological
characteristics, such as riverways and watersheds, because
our terrain is generated based on the skeleton features of
rivers and mountains. Moreover, the skeleton features can
be controlled by users through easy-to-use interactions and
parameters setting.
(a) (b)
Fig. 9. Comparing the terrain generated by our methods with the terrain
generated by classical Midpoint Displacement. (a) Terrain generated by
classical Midpoint Displacement method. Nearly no specific geomorpho-
logical characteristics can be presented. (b) Terrain generated by our
methods. Riverways and mountains can be recognized.
The results of one final terrain rendered by point cloud
with resolution 1024 ×1024 are shown in Fig. 10, in which
subfigures (a) and (b) are observations from different per-
spectives. From the perspective of Fig. 10(a), we can clearly
see the complex mountain features containing ridges and
the outspread mountain ranges. As in the perspective of
Fig. 10(b), the hierarchy structure of the main streams and
tributaries can be clearly recognized. So, the requirements
of both mountains and valleys in real scenes can be met
by our terrain modeling method. Moreover, the shapes of
mountains and valleys in the generated terrain are realistic,
because our modeling method conforms to hydrology theo-
ry and the skeleton features are enriched by MD and MDI.
Compared with the work of G´
enevaux et al [12] that can
also produce large scale terrains based on hydrology, our
method generate Tokunaga river network that can make
our terrain better conform to the real terrain morphology.
Moreover, we use parallel diffusion equation to compute
elevations of data points with high resolution in real time.
So our method is more suited for generating very large scale
terrain with complex features than the work of G´
enevaux et
al. The performance of our method is discussed in Section 8.
(a) (b)
Fig. 10. Results of final terrain generated by our methods. Red segments
are ridge lines. Cyan segments denote the river network. (a) Observing
the terrain in the perspective that focuses on ridges. (b) Observing the
terrain in the perspective that focuses on valleys.
Through using open software Terragen, we generate
various scenes based on our produced terrains. As shown
in Fig. 11, the results are realistic and present various
geomorphic features. Additionally, the generated terrain can
describe a large scale terrain scene, through our proce-
dural modeling method. Obviously, our generated terrain
can meet demands in various fields, such as researches in
geography, simulation systems and movies.
(a) (b) (c)
Fig. 11. Terrain scenes generated by software Terragen based on our
produced terrains. (a) Terrain scene focused on the ridges. (b) Terrain
scene focused on the rivers. (c) Terrain scene from perspective of
overlook.
8 PARALLEL IMPL EME NTATIO N AND PERFOR-
MANCE
Due to the computations of diffusion equation for each
grid point are independent, we parallelize the computation
of each point in one iteration step on GPU to enhance
the performance of terrain diffusion, using OpenCL. We
consider the discrete elevation points of terrain as pixels of
an image, and store these elevations in the cache of GPU.
As the row index and column index of a point are obtained,
the computation of Laplace equation is performed in local
memory, and the result is written back to the global memory.
We test the performance of diffusion process respectively
on Intel core i5-2410M CPU with 2.3Ghz, GPUs of NVIDIA
GeForce GT 550M, and NVIDIA GeForce GTX 780Ti, us-
ing two terrains with sizes of GD1024(1024 ×1024) and
GD2048(2048 ×2048). We compare the time of executing
diffusion equation 5500 times for three hardware environ-
ments. The running time on CPU for GD1024 and GD2048
are respectively 323.5s and 21 minutes. The running times
on GPUs are shown in Fig. 12. We can see that on NVIDIA
GeForce GT 550M using 2 computing units, the average
running times are reduced respectively to 6.582s for GD1024
and 24.033s for GD2048. The performance increased nearly
50 times over the performance on CPU. And the running
times on NVIDIA GeForce GTX 780Ti using 15 computing
units are decreased respectively to 0.71s for GD1024 and
1.726s for GD2048. So our implementation can meet the
requirements of real-time terrain generation.
About other parts in our method, we implement them in
CPU without parallel acceleration, because the algorithms
are intrinsically efficient. We tested the performance on
an Intel core i7-4710MQ CPU with 2.5Ghz. Table 3 lists
the performance of each part for generating terrains with
different sizes. Nrand Nvrespectively denote the number
of river network nodes and Voronoi diagram edges. Tr,
Tv,Td,Tfand Tgrespectively denote the computation
times of river network construction, ridges construction,
discretization, feature enrichment and performing Gaussian
smoothing 20 times. The timings are the means of perform-
ing our method 8 times. The resolution mainly effects the
computation time of feature enrichment, and the numbers
of river network nodes and Voronoi diagram edges mainly
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JOURNAL OF , VOL. , NO. , AUGUST 2016 10
Fig. 12. Comparison of optimization effects on two different GPUs.
The computation time of GTX780Ti is much less than GTX 550M. Our
method has good scalability.
TABLE 3
Performance of Each Part in Our Method
Resolution NrNvTr(ms) Tv(ms) Td(ms) Tf(ms) Tg(ms)
GD1024 2033 5968 3.875 820 0 58.6 927.6
GD2048 12267 33291 114.1 32944 13.5 280.1 3642
affect the computation for the constructions of river network
and ridges. From Fig. 12 and Table 3, we can see that our
method can generate large-scale terrain with complex river
network efficiently and can meet the requirements of on-line
terrain generation.
9 CONCLUSIONS AND FUTURE WORK
In this paper, we present a modeling method to generate
large scale realistic terrains, by combining hydrology theory
with procedural modeling. Our method has the following
advantages:
•Large scale terrain can be generated efficiently by
using procedural modeling.
•Providing good controllability for the morphology of
generated terrains.
•Supporting the augmented visual terrains that con-
tain real terrain features.
•Providing easy-to-use interactive means to set the
properties of terrain generation.
In our method, we propose a rule set of L-system to
generate Tokunaga river network. Based on the river net-
work, we extract ridges to form more skeleton features that
are pivotal to constrain the morphology of the generated
terrain. To obtain natural mountains and valleys, we exploit
Midpoint Displacement method and its inverse process to
enrich the skeleton features. Based on the discrete data
points of enriched skeleton features, diffusion equation is
utilized to obtain the whole terrain. We develop a technique
of generating augmented visual terrains, which allows users
to embed real terrain features into the virtual terrain seam-
lessly through Poisson equation. The performance of our
method is enhanced by parallel implementation of terrain
diffusion. The terrains generated by our method can present
good authenticity, and can describe various terrain scenes,
such as mountains, valleys, and waterways.
Since our modeling method only considers the structure
information of river network, we want to further quantize
the flow rate of the river network in future, for generating
more realistic riverways. Based on the methods proposed in
this paper, we want to further study the reconstruction of
real river scenes.
ACKNOWLEDGMENTS
This work is supported by Natural Science Foundation of
Jilin Province under grant number 20140101179JC, National
Natural Science Foundation of China (41671379), National
Natural Science Foundation of China for Young Scholars
(41101434), Research Fund for the Doctoral Program of
Higher Education of China (20130043110016), and National
Natural Science Foundation of China (41571405).
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