13
 Open Access. © 2018 Babič M, Huber m.A., Bielecka E, Soycan M, Przegon W, Gigović L, Drobnjak S, Sekulović D, Pogarčić I, 
Miliaresis G, Mikoš M, Komac M., published by Sciendo.  This work is licensed under the Creative Commons 
Attribution-NonCommercial-NoDerivatives 4.0 License.  
Received: Oct 18, 2018
Accepted: Dec 10, 2018
 DOI: 10.2478/rmzmag-2019-0006
Original scientific paper 
Abstract
Many problems in the analysis of natural terrain sur-
face shapes and the construction of terrain maps to 
model them remain unsolved. Almost the whole pro-
cess of thematic interpretation of aerospace informa-
tion consists of a step-by-step grouping and further 
data conversion for the purpose of creating a com-
pletely definite, problematically oriented picture of the 
earth’s surface. In this article, we present application 
of a new method of drawing 3D visibility networks 
for pattern recognition and its application on terrain 
surfaces. For the determination of complexity of 3D 
surface terrain, we use fractal geometry method. We 
use algorithm for constructing the visibility network 
to analyse the topological property of networks used 
in complex terrain surfaces. Terrain models give a fast 
overview of a landscape and are often fascinating and 
overwhelmingly beautiful works by artists who invest 
all their interest and an immense amount of work and 
know-how, combined with a developed sense of the 
portrayed landscape, in creating them. At the end, we 
present modelling of terrain surfaces with topological 
properties of the visibility network in 3D space.
Key words: Network theory, complex pattern recogni-
tion, terrain surface analysis, modelling
New Method of Visibility Network and Statistical Pattern 
Network Recognition Usage in Terrain Surfaces
Matej Babič
1,
*, Miłosz Andrzej Huber
2
, Elzbieta Bielecka
3
, Metin Soycan
4
, Wojciech Przegon
5
, Ljubomir Gigović
6
, 
Siniša Drobnjak
7
, Dragoljub Sekulović
7
, Ivan Pogarčić
8
, George Miliaresis
9
, Matjaž Mikoš
10
, Marko Komac
10
1, 
*Jožef Stefan Institute, Slovenia 
2
 Maria Curie –Sklodowska University, Lublin, Poland
3
 Military University of Technology, Faculty of Civil Engineering and Geodesy, Poland
4
 Yildiz Technical University, Faculty of Civil Engineering, Department of Geomatic Engineering Davutpasa Campus TR-34220 
Esenler-Istanbul-Turkey
5
 University of Agriculture in Krakow, Poland
6
 University of Defence, Military Academy, Serbia
7
 Military Geographical Institute, Belgrade, Serbia
8
 Juraj Dobrila University of Pula, Croatia
9
 University of Patras, Greece
10
 Faculty of Civil and Geodetic Engineering, University of Ljubljana Slovenia
POVZETEK
Obstaja mnogo nerešenih problemov s področja anali-
ze oblik terenov naravnih površin in izdelavo njihovega 
modela. Skoraj celoten proces tematske interpretacije 
letalskih in vesoljskih informacij je sestavljen iz skupi-
ne po korakih in nadaljnje pretvorbe podatkov z name-
nom oblikovanja popolnoma določene, problematič-
no usmerjene slike zemeljske površine. V tem članku 
predstavljamo uporabo nove metode konstruiranja 
3D omrežij vidljivosti za prepoznavanje vzorcev in 
njeno uporabo na površinah terena. Za določanje kom-
pleksnosti 3D površine terena uporabljamo metodo 
fraktalne geometrije. Uporabljamo algoritem omrežja 
vidljivosti za analizo topoloških lastnosti omrežij, ki 
se uporabljajo na kompleksnih terenih. Terenski mo-
deli omogočajo hiter pregled pokrajine in so pogosto 
očarljiva in lepa dela umetnikov, ki vlagajo vse svoje 
zanimanje in ogromno dela in znanja, skupaj z razvi-
tim občutkom za predstavljene pokrajine pri njihovi 
ustvarjanju. Na koncu predstavljamo modeli površine 
terena s topološkimi lastnostmi 3D omrežij vidljivosti.
KLJUČNE BESEDE: Teorija omrežij, razpoznavanje 
kompleksnih vzorcev, analiza površine terena, mode-
liranje

Babič M, Huber m.A., Bielecka E, Soycan M, Przegon W, Gigović L, 
Drobnjak S, Sekulović D, Pogarčić I, Miliaresis G, Mikoš M, Komac M.
14
RMZ – M&G | 2019 | Vol. 66 | pp. 013–026
Introduction
Network theory and graph theory are very use-
ful in Geography. Landscape network models 
can be a useful tool (procedure) in landscape 
aesthetic value management and spatial plan-
ning processes [1]. Visibility calculations [2] 
are central to any computer graphics and geoin-
formation system (GIS) application. Visibility 
graph analysis [3] is a spatial analysis tech-
nique that can also be applied to terrain sur-
face analysis. Existing or newly designed hous-
es, structures, quarters, etc. are considered as 
peaks, and the connecting roads, engineering 
networks, transmission lines, etc., are like ribs. 
The process of creating terrain and landscape 
models [4] is important in various computer 
graphics and many visualisation applications. 
Figure 1 presents a terrain surface. Terrain 
surfaces [5] represent phenomena that have 
definite values at each point along the entire 
range of their extent. The values of an infinite 
set of points on the entire surface are extract-
ed from a limited set of initial values. They can 
be based on direct measurements, for example, 
elevation height values or temperature values 
for temperature surfaces. The values for the 
surface between the measurement points are 
assigned by interpolation. Surfaces can also be 
mathematically calculated based on other data, 
for example, surface slopes or exposures, de-
rived from terrain surface data, the surface of 
distances from bus stops in a city or a surface, 
showing the concentration of criminal activity 
or the possibility of strikes.
In problems of transforming the terrain into 
a project surface, mathematical methods are 
often used. In modern conditions of design-
ing in real time or object-oriented design, the 
solution of some problems is eliminated with 
a geometric approach. This approach is called 
geometric modelling. More and more popu-
lar are the methods of geometric modelling in 
engineering design through computer model-
ling, since the solution to the problem acquires 
a spatial, visual appearance. In the design 
of the project surface relative to the terrain, 
a mathematical method is applied to the input 
data x, y, and z coordinates of discrete points 
of the relief. Transformation of the relief into 
the project surface requires a high-level engi-
Figure 1: Terrain surface.
neer of the designer of mathematical skills that 
complicate the design process. Determination 
and selection of the optimal design surface 
along the structural relief lines [6] are becom-
ing more and more in demand in the practice 
of engineering design, as the design takes place 
in real time and acquires a creative character. 
In modern systems of engineering design such 
as Compass, AutoCAD having input data on 
the topographic surface, it is possible to con-
struct structural relief lines-profile and cross 
sections, isolines, slope lines, watersheds, and 
thalwegs [7]. All this is visualised in real time 
and gives the widest possibilities for analysis to 
a design engineer even of the middle class.
Consider an inclined line of an arbitrary flat 
curve, which is the profile or cross section of 
the relief. Having a number of such profile or 
cross sections interspersed at a certain dis-
tance (Dx, Dy), we can represent the framework 
model of the relief. The main task of the analy-
sis of the relief is to determine the general slope 
of the terrain, and on the basis of it the design 
surface is selected. At the word “geometry”, we 
have cylinders, triangles, hypotenuses, bisec-
tors of corners, “find the area of a figure”, slate 
boards, and breaking chalk from the depths of 
memory. The problem is that everything that 
comes to the mind is a language for describing 
an extremely narrow set of phenomena of the 
surrounding world. At home, sometimes, they 
are close to a parallelepiped, but trees – not 
the cylinders, the mountains are not cones, 
but the shape of the cloud is incomprehensible 
with what to compare. If we look closely, in the 
world around us this school geometry (we will 

New Method of Visibility Network and Statistical Pattern Network Recognition Usage in Terrain Surfaces
15
call it Euclidean) describes not so much. The 
scientists asked this question for a long time, 
but since they did not find a convincing answer, 
they wrote down these forms as “disordered”, 
“monstrous”, and “unexplored”. A global break-
through occurred only in the 1960–1970s, when 
the French mathematician Benois Mandelbrot 
invented and developed his theory of frac-
tals [8]. It was a new, fractal geometry, which 
took for the object of research all that uneven, 
broken, and rough that surrounds us (that is, 
almost all). Mandelbrot found his wonderful or-
der in complex forms of nature. Machine learn-
ing [9] is a highly specialised field of knowledge 
that is part of the main sources of technologies 
and methods used in the fields of large data and 
the Internet of things that studies and develops 
algorithms for automated extraction of knowl-
edge from a raw data set, learning software 
systems based on the data received, generation 
of predictive and/or prescriptive recommen-
dations, pattern recognition, etc. Statistics [10] 
is a science that uses many effective methods 
(including the method of mass observations, 
the method of groupings, and the method of 
generalising indicators) for the achievement of 
accurate results for the study of an object (sub-
ject, phenomenon, and process) and structur-
ing them in a form convenient for the subject 
text, table, graph, and diagram with the subse-
quent analysis of the received data; the extract-
ed information forms a statistical vision of the 
situation, the element of which is the object 
under study. Statistics is a general theoretical 
science (a complex of scientific disciplines) that 
studies the quantitative side of qualitatively de-
fined mass socioeconomic phenomena and pro-
cesses, their composition, distribution, spatial 
placement, and movement in time, revealing 
the current interdependencies and patterns in 
specific conditions of place and time. So, sta-
tistics is a branch of practical activity (“statis-
tical accounting”) for the collection, processing, 
analysis, and publication of mass digital data 
on various phenomena and processes of pub-
lic life. Modern information systems and tech-
nologies include a large number of procedures 
that model or support data mining process. To 
the simplest procedures any type of classifica-
tion quantitative data on given to users criteria, 
more complex provide analysis scenes, pro-
cesses, phenomena for the purpose of selecting 
objects with given characteristics or properties. 
Procedures of this type are present not only 
in analysis problems in aerospace images but 
also when processing signals in technical sys-
tems, in medical diagnostics, biology, sociology, 
banking business, and other areas of human 
activity. As you expand sphere of application of 
geoinformation technologies and complicating 
procedures, geoinformation modelling proce-
dures for analysis and classification aggregates 
of data, objects, and structures are very import-
ant in the new generation of GISs. For designing 
any system of thematic analysis classification of 
information objects and structures, its applica-
tion requires a specialist knowledge. Research 
and development methods, algorithms, and 
systems for solving such tasks on a computer 
are engaged in a discipline called pattern rec-
ognition [11]. Our purpose in the present paper 
is to investigate different topological properties 
of visibility graphs in 3D space, which are be-
ing viewed as attractive alternatives in terrain 
surface analysis. The aim of the study is to use 
a visibility network algorithm for statistical 
pattern recognition of 3D classification for the 
prediction of complex terrain surfaces.
Materials and methods
A new algorithm for the construction of vis-
ibility networks in 3D space was presented 
in the study by Stempien [12]. This algorithm 
was used to analyse the topological properties 
of a complex surface (Figure1). In the visibili-
ty graph in 3D space, we calculated topological 
properties of the graph with the programme 
Pajek [13]. To analyse the topographical prop-
erty of the terrain surface, we use the construc-
tion of the visibility graph in 3D space. Also, the 
problem in visibility points that we can connect 
together. In Figure 2, the problem of visibili-
ty network in 3D space is presented. Figure 3 
presents the 3D surface and Figure 4 solution 
for constructing visibility graphs in 3D space 
for Figure 3.
We analysed a set of 22 randomly created ter-
rain surfaces (Appendix 1). For each terrain 
surface, we constructed visibility graphs in 3D. 
After that, we calculated topological properties 

Babič M, Huber m.A., Bielecka E, Soycan M, Przegon W, Gigović L, 
Drobnjak S, Sekulović D, Pogarčić I, Miliaresis G, Mikoš M, Komac M.
16
RMZ – M&G | 2019 | Vol. 66 | pp. 013–026
of visibility graph in 3D for each terrain surface. 
Topological properties can apply to the net-
work as a whole or to individual nodes and edg-
es. Some of the most used topological proper-
ties and concepts are # extremes, # edges, and 
# k-core, All Degree Centralisation, Network 
Clustering Coefficient, Average Degree, and tri-
adic census type 16–300 of visibility graphs in 
3D for each terrain surface. We calculated sta-
tistical properties of topological properties of 
the 3D visibility network of all 22 terrain sur-
faces. For the determination of complexity of 
terrain surfaces, we use fractal geometry [14]. 
In fractal geometry, fractal dimension is the key, 
which determines the complexity of an object.
Random forest (RF) [15] is one of the most 
stunning machine learning algorithms invent-
ed by Leo Bryman and Adele Cutler in the last 
century. He came to us in an “original form” 
(no heuristics could not substantially improve 
it) and is one of the few universal algorithms. 
The universality consists, first, that it is good in 
many tasks, second, that there are RFs for solv-
ing problem classification, regression, cluster-
ing, search of anomalies, selection of signs, etc. 
RF is a lot of decisive trees. In the task of regres-
sion, their answers are averaged, in the task of 
classification the decision is made by voting for 
the majority. Figure 5 presents five illustrative 
decision trees forming a (very small) RF for 
classification. We use parameters 30 fixed seed 
for random generator and growth control, not 
split subsets smaller than 5.
Figure 2: Visibility points (black line) and unrelated points 
(orange line).
Figure 3: 3D surface.
Figure 4: Visibility network in 3D space for Figure 3.
Figure 5: Five illustrative decision trees forming a (very small) 
random forest for classification.

New Method of Visibility Network and Statistical Pattern Network Recognition Usage in Terrain Surfaces
17
The K-nearest neighbours model [16] is a meth-
od of machine learning, which saves itself as the 
knowledge of learning. For a given new exam-
ple, the algorithm finds a given set of learning 
examples to the nearest, most similar cases 
and estimates the probability distribution from 
the relative distribution of these to examples 
by grade. In the simplest variant of this meth-
od, the algorithm is compiled by the new case 
as the class to which most of the closest neigh-
bours belong. The advanced method takes into 
account the weighting of the impact of learning 
examples on the classification of a new case by 
distance. Figure 6 presents the k-nearest neigh-
bours method. We use parameters 2 neigh-
bours, Chebyshev metric and uniform weight.
The solution of the problem of binary classifica-
tion using the support vector method [17] con-
sists of finding a linear function that correctly 
divides the data set into two classes. The prob-
lem can be formulated as the search for a func-
tion f(x) that takes values less than zero for 
vectors of one class and greater than zero for 
vectors of another class. As initial data for the 
solution of the task, that is, search for the classi-
fying function f(x), given a training set of space 
vectors for which they are known to belong 
to one of the classes. The family of classifying 
functions can be described in terms of the func-
tion f(x). The hyperplane is defined by the vec-
tor a and the value b, i.e. f(x) = a.x + b. As a result 
of solving the problem, i.e. constructing an SVM 
model, a function is found that takes values less 
than zero for vectors of one class and greater 
than zero for vectors of another class. For each 
new object, a negative or positive value deter-
mines whether the object belongs to one of the 
classes. Figure 7 presents the support vector 
method. We use v-SVM type with regression 
cost (C) 1.05. Optimisation parameters, we use 
100 iteration limit and numerical tolerance 
0.002. We use Kernel (g × x × y + 0.15)
3
 and g 
was auto.
Results and discussion
In Table 1, the topological properties of 3D visi-
bility network are presented. Topological prop-
erties present input of predictive models of 
complexity of terrain surfaces. We mark terrain 
surfaces from T1 to T22. Column one presents 
topological property # extremes, column two 
presents topological property # edges, column 
three presents topological property # k-core, 
column four presents topological property All 
Degree Centralisation, column five presents 
topological property Network Clustering Coef-
ficient, column six presents topological prop-
erty Average Degree, and the last column pres-
ents topological property triadic census type 
16–300. The last column presents complexity 
of terrain surfaces. Terrain surface T22 has the 
best topological properties of the 3D visibility 
network, because T22 has maximal number 
of edges, Network Clustering Coefficient, and 
triadic census type 16–300. Terrain surface 
Figure 6: K-nearest neighbours.
Figure 7: Support vector method.

Babič M, Huber m.A., Bielecka E, Soycan M, Przegon W, Gigović L, 
Drobnjak S, Sekulović D, Pogarčić I, Miliaresis G, Mikoš M, Komac M.
18
RMZ – M&G | 2019 | Vol. 66 | pp. 013–026
T17 has a maximal number of extremes of the 
3D visibility network. Terrain surface T14 has 
maximal number of k-core and triadic census 
type 16–300 of the 3D visibility network. Ter-
rain surface T11 has a maximal number of All 
Degree Centralisation of 3D visibility network. 
Table 2 presents statistical properties of topo-
logical properties of the 3D visibility network of 
terrain surfaces. We calculated statistical prop-
erties such as mean, standard deviation, stan-
dard error, median, geometric mean, harmonic 
mean, variance, skewness, Kurtosis, Fisher’s g1, 
Fisher’s g2, coefficient of variation, coefficient 
of dispersion, communality, area under curve, 
mean direction (Theta), mean resultant length, 
circular variance (V), circular standard devia-
tion (v), circular dispersion (Delta), von Mises 
concentration (Kappa), Phi Pearson’s Contin-
gency, Coefficient Tschuprow’s T, lambda B, 
symmetric lambda, Kendall’s tau-B, Kendall’s 
tau-C, and gamma of topological properties of 
the 3D visibility network of terrain surfaces. 
Terrain surface T21 has maximal # k-core.
Mostly, we have positive amount in Table 2 un-
less Skewness and Fisher’s g1 for # extremes, 
average degree and triadic census type 16–300, 
kurtosis for average degree and triadic census 
type 16–300, Kendall’s tau-B, Kendall’s tau-C 
and Gamma for # extremes, All Degree Central-
isation, average degree, and triadic census type 
16–300. Terrain surface T5 has most complex-
ity. Terrain surface T21 has minimal complexi-
ty. This statistic measures the heaviness of the 
tails of a distribution. The usual reference point 
Table 1: Topological properties of the 3D visibility network and fractal dimension.
SP # extremes # edges # k-core
All Degree 
Centralisation
Network 
clustering 
coefficient
Average 
degree
Triadic 
census type 
16–300
Fractal 
dimension
T1 120823 3500351 16 0.00020627 0.364 6.675 4865624 2.7365
T2 125787 3308776 12 0.00019998 0.364 6.311 4191425 2.668
T3 123943 3335861 13 0.00017608 0.371 6.363 4267175 2.6871
T4 124833 3355735 13 0.00017319 0.379 6.401 4353872 2.7458
T5 124626 3314397 12 0.00016182 0.378 6.322 4212248 2.8065
T6 131540 3190001 12 0.00013725 0.364 6.084 3796016 2.4956
T7 126962 3311163 12 0.00017518 0.373 6.316 4196282 2.1664
T8 130799 3173601 12 0.00018496 0.36 6.053 3741603 2.4784
T9 123393 3355056 11 0.00016842 0.385 6.399 4256560 2.622
T10 126395 3386391 13 0.00015883 0.378 6.459 4483986 2.7426
T11 124296 3315948 11 0.00026577 0.367 6.325 4207031 2.6142
T12 123829 3355735 13 0.00017319 0.379 6.401 4353872 2.6982
T13 128143 3451450 16 0.00018446 0.352 6.583 4862060 2.6743
T14 122500 3685175 20 0.00027177 0.372 7.029 5877473 2.3634
T15 120818 3338595 11 0.00012649 0.386 6.368 4199754 2.5155
T16 116812 3733624 18 0.00020016 0.364 7.121 5848517 2.5342
T17 133031 3178192 12 0.00013822 0.36 6.062 3774789 2.6528
T18 130974 3182544 15 0.00014298 0.357 6.070 3819193 2.2996
T19 131043 3170121 13 0.00014205 0.359 6.047 3746658 2.6427
T20 95090 4151533 16 0.00010784 0.387 7.918 7284078 2.1693
T21 106916 5653616 33 0.00021764 0.363 10.783 1764141 2.1042
T22 86871 5735036 19 0.00015646 0.394 10.939 1466536 2.1332

New Method of Visibility Network and Statistical Pattern Network Recognition Usage in Terrain Surfaces
19
Table 2: Statistical properties of topological properties of the 3D visibility network.
SP # extremes # edge
# 
k-core
All Degree 
Centralisation
Network 
clustering 
coefficient
Average 
degree
Triadic 
census type 
16–300
Mean 121792 3599223 14.68 17586.41 3.71E+07 5.98E+08 4253132
Standard deviation 11531.35 714391.3 4.86 4048.007 1119925 1.64E+08 1201353
Standard error 2458.49 152308.8 1.04 863.038 238768.9 3.50E+07 256129.4
Median 124461 3346826 13.00 17319 3.69E+07 6.34E+08 4209640
Geometric mean 121194 3546497 14.14 17168.05 3.71E+07 5.49E+08 4055020
Harmonic mean 120509 3505535 13.75 1.68E+04 3.70E+07 4.45E+08 3790200
Variance 1.3E+08 5.10E+11 23.66 1.64E+07 1.25E+12 2.69E+16 1.44E+12
Skewness -1.89 2.409759 2.60 0.8111144 0.3436712 -2.462485 -0.026658
Kurtosis 5.90 7.454659 10.23 3.508959 2.195798 8.01742 4.719475
Fisher's g1 -2.04 2.58979 2.79 0.8717122 0.3693466 -2.646456 -0.028649
Fisher's g2 4.03 5.993685 9.52 0.9784921 -0.6906038 6.708983 2.517122
Coefficient of 
variation
0.094 0.1984849 0.33 0.2301782 0.03020786 0.2743574 0.2824632
Coefficient of 
dispersion
0.05 0.1024648 0.21 0.1684723 0.02531182 0.114327 0.1701942
Communality 1.007201 1.000102 0.55 0.062183 0.474759 0.964151 0.570508
Area under curve 2575577 7.46E+07 305.50 368764.5 7.78E+08 1.28E+10 9.04E+07
Mean direction 
(Theta)
103.14 178.3926 14.67 120.4838 157.2197 148.077 188.3852
Mean resultant 
length 
0.11 0.1746 1.00 0.0943 0.2903 0.3947 0.1533
Circular variance 
(V)
0.88 0.8254 0.00 0.9057 0.7097 0.6053 0.8467
Circular standard 
deviation (v)
118.11 107.0477 4.74 124.5001 90.1178 78.124 110.9654
Circular dispersion 
(Delta)
32.66 12.9282 0.01 48.9952 5.2048 2.7628 18.7855
von Mises 
concentration 
(Kappa)
0.24 0.3546 146.45 0.1895 0.6069 0.8603 0.3103
Phi 4.58 4.4721 2.83 4.4721 4.4721 4.4721 4.4721
Pearson's 
contingency 
coefficient
0.97 0.9759 0.94 0.9759 0.9759 0.9759 0.9759
Tschuprow's T 1 0.9879 0.79 0.9879 0.9879 0.9879 0.9879
Lambda B 1 0.9524 0.38 0.9524 0.9524 0.9524 0.9524
Symmetric lambda 1 0.9756 0.65 0.9756 0.9756 0.9756 0.9756
Kendall's tau-B -0.19 0.2338 0.28 -0.2165 0.026 -0.1126 -0.1558
Kendall's tau-C -0.19 0.2338 0.28 -0.2165 0.026 -0.1126 -0.1558
Gamma -0.19 0.2348 0.32 -0.2174 0.0261 -0.113 -0.1565

Babič M, Huber m.A., Bielecka E, Soycan M, Przegon W, Gigović L, 
Drobnjak S, Sekulović D, Pogarčić I, Miliaresis G, Mikoš M, Komac M.
20
RMZ – M&G | 2019 | Vol. 66 | pp. 013–026
in kurtosis is the normal distribution. If this 
kurtosis statistic equals three and the skew-
ness is zero, the distribution is normal. Also, we 
found positive significance between Kurtosis 
and topological properties of visibility graphs 
in 3D space of terrain surface. Pearson’s con-
tingency coefficient is a measure of association 
independent of sample size. It ranges between 
0 (no relationship) and 1 (perfect relationship). 
For any particular table, the maximum possible 
depends on the size of the table (a 2 × 2 table 
has a maximum of 0.707), and so it should 
only be used to compare tables with the same 
dimensions. Also, in our result mostly topo-
logical properties have Pearson’s contingency 
coefficient 0.97. Tschuprow’s T is a measure of 
association independent of sample size. This 
statistic is a modification of the Phi statistic so 
that it is appropriate for larger than 2 × 2 ta-
bles. T ranges between 0 (no relationship) and 
1 (perfect relationship), but 1 is only attainable 
for square tables. In our result, mostly topolog-
ical properties have Tschuprow’s T = 0.98.
Calculated and predicted data are presented in 
Table 3. The RF model presents a 6.28% devi-
ation from the measured data. The k-nearest 
neighbours model presents a 6.88% deviation 
from the measured data. The support vector 
machine model presents a 6.82% deviation 
from the measured data. The calculated and 
Figure 8: The calculated and predicted surface terrain complexity.
Table 3: Calculated and predicted data.
S ED P RF kNN P SVM
T1 2.7365 2.522 2.710 2.589
T2 2.668 2.470 2.545 2.617
T3 2.6871 2.683 2.545 2.625
T4 2.7458 2.592 2.673 2.621
T5 2.8065 2.538 2.530 2.619
T6 2.4956 2.460 2.537 2.596
T7 2.1664 2.678 2.645 2.705
T8 2.4784 2.601 2.552 2.546
T9 2.6220 2.612 2.558 2.646
T10 2.7426 2.730 2.697 2.650
T11 2.6142 2.563 2.590 2.462
T12 2.6982 2.668 2.721 2.649
T13 2.6743 2.644 2.722 2.573
T14 2.3634 2.463 2.564 2.462
T15 2.5155 2.535 2.590 2.694
T16 2.5342 2.460 2.537 2.596
T17 2.6528 2.425 2.422 2.384
T18 2.2996 2.572 2.587 2.563
T19 2.6427 2.425 2.422 2.396
T20 2.1693 2.554 2.610 2.396
T21 2.1042 2.366 2.481 2.407
T22 2.1332 2.258 2.439 2.400

New Method of Visibility Network and Statistical Pattern Network Recognition Usage in Terrain Surfaces
21
predicted surface terrain complexity is shown 
in the graph in Figure 8.
Conclusions
The visibility graph problem itself has long 
been studied in computational geometry and 
has been applied to various areas. Finally , in this 
work the visibility network in 3D space, which 
contains more information than the visibility 
graph, has been used to analyse the terrain sur-
faces. Furthermore, we used the new method 
of construction of the visibility graphs in 3D 
space to describe the different terrain surfaces 
for possible further applications in studies of 
temporal and spatial landscape and/or terrain 
changes and evolution [18]. The main findings 
can be summarised as follows:
 ― We describe the different terrain surfaces by 
using the topological properties of the visi-
bility graphs in 3D space.
 ― We present statistical properties of topolog-
ical properties of the 3D visibility network.
 ― We use method of machine learning to pre-
dict complexity of terrain surfaces.
 ― With topological properties of 3D visibility 
network of terrain surface, we model com-
plexity of terrain surface.  
 ― With statistical statistical properties of to-
pological properties of 3D visibility network 
of terrain surface, we can better analyse and 
understand complexity of terrain surfaces.
 ― With statistical properties of 3D visibility 
network of terrain surface, we can better un-
derstand structure of terrain surfaces.
 ― Terrain surface with minimum complexity 
has maximal #k-core.
Possible further testing of the proposed meth-
od in the field of landscape/terrain morpholo-
gy would be:
 ― Description of temporal changes resp. devel-
opment of a landscape/terrain using tempo-
ral changes in visibility graphs produced on 
multitemporal LiDAR data [19].
 ― Application in LiDAR derived high-resolu-
tion topography for landform recognition 
and analysis [20].
 ― Comparison of visibility graphs with other 
techniques for analysing terrain texture [21] 
or landform recognition, such as for land-
slides [22].
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Appendix 1
Terrain surface T1
Terrain surface T2
Terrain surface T3
Terrain surface T4
Terrain surface T5
Terrain surface T6
Terrain surface T7

Babič M, Huber m.A., Bielecka E, Soycan M, Przegon W, Gigović L, 
Drobnjak S, Sekulović D, Pogarčić I, Miliaresis G, Mikoš M, Komac M.
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Terrain surface T8
Terrain surface T9
Terrain surface T10
Terrain surface T11
Terrain surface T12
Terrain surface T13
Terrain surface T14
Terrain surface T15

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Terrain surface T17
Terrain surface T18
Terrain surface T19
Terrain surface T20
Terrain surface T21
Terrain surface T22
Terrain surface T16