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Academic Editor: Maxim
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Received: 8 December 2024
Revised: 21 January 2025
Accepted: 21 January 2025
Published: 23 January 2025
Citation: Zheng, Y.; Shcherbakova,
G.; Rusyn, B.; Sachenko, A.; Volkova,
N.; Kliushnikov, I.; Antoshchuk, S.
Wavelet Transform Cluster Analysis of
UAV Images for Sustainable
Development of Smart Regions Due to
Inspecting Transport Infrastructure.
Sustainability 2025,17, 927. https://
doi.org/10.3390/su17030927
Copyright: © 2025 by the authors.
Licensee MDPI, Basel, Switzerland.
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Article
Wavelet Transform Cluster Analysis of UAV Images for
Sustainable Development of Smart Regions Due to Inspecting
Transport Infrastructure
Yanyan Zheng 1, Galina Shcherbakova 2, Bohdan Rusyn 3,4 , Anatoliy Sachenko 3,5,* , Natalya Volkova 6,* ,
Ihor Kliushnikov 7and Svetlana Antoshchuk 2
1School of Electronics and Information Engineering, Taizhou University, Taizhou 318000, China;
zhengyanyan03@gmail.com
2Department of Information Systems, Odessa National Polytechnic University, 65044 Odessa, Ukraine;
galina_sherbakova@op.edu.ua (G.S.); asg@op.edu.ua (S.A.)
3Department of Informatics and Teleinformatics, Kazimierz Pulaski University of Radom,
26-600 Radom, Poland; b.rusyn.prof@gmail.com
4
Department of Information Technologies of Remote Sensing, Karpenko Physico-Mechanical Institute of NAS
of Ukraine, 79601 Lviv, Ukraine
5Research Institute for Intelligent Computer Systems, West Ukrainian National University,
46009 Ternopil, Ukraine
6
Department of Applied Mathematics and Information Technologies, Odessa National Polytechnic University,
65044 Odessa, Ukraine
7Department of Computer Systems, Networks and Cybersecurity, National Aerospace University “Kharkiv
Aviation Institute”, 61070 Kharkiv, Ukraine; i.kliushnikov@csn.khai.edu
*Correspondence: as@wunu.edu.ua (A.S.); volkova.n.p@op.edu.ua (N.V.);
Tel.: +380-95-609-2060 (A.S.); +380-50-731-0226 (N.V.)
Abstract: Sustainable development of the Smart Cities and Smart Regions concept is
impossible without the development of a modern transport infrastructure, which must be
maintained in proper condition. Inspections are required to assess the condition of objects
in the transport infrastructure (OTI). Moreover, the efficiency of these inspections can be
enhanced with unmanned aerial vehicles (UAVs), whose application areas are continuously
expanding. When inspecting OTI (bridges, highways, etc.) the problem of improving
the quality of image processing, and analysis of data collected by UAV, for example, is
particularly relevant. The application of advanced methods for assessing the quantity
of information and making decisions to reduce information uncertainty and redundancy
for such systems is often complicated by the presence of noise there. To harmonize the
characteristics of certain procedures in such conditions, authors propose conducting data
processing using wavelet transform clustering in three main phases: determining the
number of clusters, defining the coordinates of cluster centres, and assessing the quality
and efficiency of clustering. We compared the efficiency and quality of existing clustering
methods with one using wavelet transform. The research has shown that UAVs can
be used for OTI inspecting; moreover, the clustering method with wavelet transform is
characterised by an improved quality and efficiency of data processing. In addition, the
quality assessment enables us to assess the degree of approximation of the clustering result
to the ideal one. In addition, authors examined the specific challenges associated with
planning UAV flights during inspections to obtain data that will enhance the accuracy of
clustering and recognition. This is especially important for a comprehensive quantitative
assessment of adaptation degree for image processing procedures to the tasks of inspecting
OTI “Smart Cities/Regions” based on a pragmatic measure of informativeness.
Keywords: transportation infrastructure; inspection; UAV; clustering; wavelet transform
Sustainability 2025,17, 927 https://doi.org/10.3390/su17030927
Sustainability 2025,17, 927 2 of 27
1. Introduction and Related Work
1.1. Motivation
The development of technology and the desire to improve people’s quality of life
have led to the emergence of smart homes, smart cities, and smart regions. Each of these
concepts aims to use advanced technologies and innovations to ensure their sustainable
development [
1
–
6
], which requires the creation of an effective management system and the
provision of safe and comfortable living conditions for residents [1,2].
In turn, the sustainable development of Smart Cities/Smart Regions (SCs/SRs) is
impossible without a modern transport infrastructure (TI), which should ensure safe
and fast transport. The requirements for the condition of the transport infrastructure for
SCs/SRs should be more stringent than those for conventional cities and regions.
To maintain the proper condition of the transport infrastructure, it is necessary to
inspect/investigate the condition of all OTI structures, especially interchanges, bridges,
and viaducts (tunnels), the density of which is constantly increasing with the construction
of new transport infrastructure facilities. The latter are the most critical elements of TI, the
failure of which can lead to the inability to use large parts of the transport infrastructure,
with negative and/or catastrophic consequences.
The frequency of inspection/testing for such TI elements is determined, for example,
by the national regulatory documents of the states [
7
]. The procedure for carrying out
these activities is specified too. It involves a significant number of inspections using large
equipment [
8
], which may restrict movement around the inspected facilities. In addition,
existing inspection methods involve many tools and may limit the throughput of the TI.
The policy of sustainable development for transport in general, and transport in-
frastructure as its component, aims to combat the increase in congestion, noise, and
harmful emissions from transport in the context of the increasing traffic. The quality
of transport infrastructure affects the characteristics of road transport: average flow speed,
throughput, average fuel consumption, average cost per unit of freight, etc. Average traffic
speed should ensure minimum fuel consumption and emissions, and throughput should
avoid congestion.
The control and quality of assessment for the condition of OTI can be measured by
indicators, which determine the degree of TI sustainable development. Moreover, they are
affected directly or indirectly by these processes [
9
–
11
]. There are several indicators used,
as follows:
•Number of transports accidents;
•Average travel time/transport price;
•Throughput of transport facilities;
•Levels of harmful emissions.
The improvement of the inspection process and the quality of the assessment should,
on the one hand, have the least impact on road traffic and, on the other hand, ensure a high
degree of accuracy in determining the technical condition and, in the event of deviations
from the requirements, take measures promptly for its restoring.
The use of intelligent autonomous inspection systems based on UAVs allows certain
contradictions to be resolved and certain requirements to be met. One of the approaches
is to build intelligent systems using the multi-agent technologies [
12
]. In such systems,
UAV agents are equipped with various sensors collecting data sets with a certain frequency.
Based on it, a possibility to make decisions about OTI state appears. Moreover, the use of
such systems can offer several advantages, as follows:
•Firstly, there is no need to deploy large numbers of staff and equipment;
•Secondly, inspection does not create obstacles to the flow of vehicles;
Sustainability 2025,17, 927 3 of 27
•
Thirdly, the structure and composition of the systems can be determined for specific
tasks and conditions, considering functional and non-functional requirements for the
collection of different data sets and their subsequent analysis.
Inspection data obtained from UAVs are large and their quality depends on inspection
conditions, which are not always ideal. Therefore, there is a need to reduce the obtained
data to a form that is suitable for further analysis determining the state of the OTI. The
wavelet transform (WT) [
13
–
15
] is already becoming a traditional approach in processing
visual information. For example, the WT can reduce the noise influence during filtering [
16
].
Moreover, an important property of the WT is the change in sign when crossing an ex-
tremum, which is also typical for optimisation methods based on the estimation of the
first derivative.
1.2. State–of–the–Arts
An analysis of the utilisation UAVs in SCs [
1
,
17
] shows that great attention should be
paid to planning and controlling UAV use to improve both efficiency and safety. When
UAVs are used for sustainable city and society tasks [
18
], one of the important tasks is
to visualise the data obtained and perform an effective analysis [
19
]. In such analysis,
it is necessary to perform the procedure of segmentation and/or clustering of the data
obtained at the output of UAV cameras, which is provided commonly by convolutional
neural networks [
20
–
25
]. As a result, after clustering, a formalised data vector is obtained,
which can be further used in training a classifier for making the diagnostic solutions.
The paper [
26
] describes the procedure for using a monitoring system with UAVs, which
consists of several steps, as follows:
•Data collection;
•Preprocessing the collected data;
•Performing classification tasks;
•Presentation of results and decision making.
However, authors [
26
] describe the procedures for obtaining and presenting results
only in general terms, without detailing the individual steps. Usually, in the process of
obtaining the result using the proposed approach, it is necessary to perform the procedure
of segmentation and/or clustering of data from UAV cameras. As a result, a formalised
feature vector is obtained, which can be further used in training a classifier to make
diagnostic decisions about the state of objects.
Furthermore, the frequency of OTI inspections, the number of inspections required,
and the size of the area to be inspected [
7
] necessitate the development of procedures for
the synthesis of UAV-based inspection systems, which is beyond the scope of the authors’
research [26].
The synthesis of monitoring systems using multi-agent technologies is a topic that
has been extensively explored in academic literature. In [
27
], the authors put forth a gen-
eral algorithm for the formation of a monitoring system capable of performing a range
of tasks, with due consideration of the requisite requirements. A review of the utilisa-
tion of multi-agent technologies for the completion of tasks across a range of industrial
sectors is presented in [
12
,
28
,
29
]. However, these works do not devote sufficient atten-
tion to the process of synthesising the structure and composition of systems to perform
inspection tasks.
Decision making in such inspection systems is aimed at reducing information uncer-
tainty, redundancy, and diversity. For example, these could be approaches [
30
,
31
], as well
as formal models and methods for analysing the information component of signals and/or
images [32–36].
Sustainability 2025,17, 927 4 of 27
However, it should be noted that the task of reducing uncertainty that the data
processed in OTI status assessment, for example, in image processing from UAVs during
the inspection of critical infrastructure objects, is complicated by the presence of noise
in the data. The image quality also decreases when shooting in low-light conditions (for
example, under a bridge). Blurring of the image when the UAV moves caused by vibration
and wind also reduces image quality. This can lead to inaccurate detection of the shapes,
sizes, and locations of defects in critical infrastructure objects, such as cracks, defects in
riveted plates with cracks in rivet holes, and welding cracks. In the case when defects are
in areas of reduced visibility, such as under the bridge, difficulties may arise in receiving
GPS signals for UAVs [
37
]. This may cause problems in obtaining the required number of
images for the object [38,39].
The selection of procedures for visual information processing in the OTI status as-
sessment is conducted through mathematical modelling and assessing the procedure’s
adaptation to achieve its goal (efficiency). The task of assessing the effectiveness of the
procedure is to extract meaningful information about the object, considering the probabilis-
tic nature of the visual information and the importance of this information. The quality
score (QS) assesses the degree of closeness between the simulation and the ideal result.
The quality score of the system is formed based on the procedure indicators. Algorithms,
technical devices, and the influence of external factors on the process of obtaining and/or
converting information determine the quality score of the system. Improving the quality of
the procedure does not always lead to increased efficiency. Therefore, in addition to quality
indicators, efficiency indicators are used as well [40].
However, tasks in OTI with visual information are characterised by the presence of
noise in the data [
41
]. Moreover, the number of clusters may not be known, and clusters
may have a complex shape, intersect, and vary in size and density [
42
]. The number of
patterns in groups may also be small due to the limited data volumes [
43
], for example,
when UAVs inspect critical infrastructure facilities such as bridges, energy facilities, and
petroleum product warehouses in hard weather conditions.
When clustering using a small sample of parameters, it may be impossible to estimate
adequately the density probability characterising the belonging of the object to a cluster. In
such conditions, existing clustering methods, whether hierarchical or iterative ones with
clear and fuzzy partition, do not always provide high-quality results [44–49].
The main disadvantage of hierarchical methods is the low noise immunity. The main
drawbacks of iterative optimisation methods, on which clustering is based, are sensitivity
to the starting point of the search and data noise while finding a local minimum. This
problem is solved by optimisation based on genetic, evolutionary, and swarm algorithms.
To reduce computation time, parallelisation of computations and the increased number of
processors are used [50].
Two main tasks that are solved during clustering are following: determining the num-
ber of clusters and selecting an appropriate clustering method for the specific application
problem. A huge number of methods [
51
–
53
] is determined by the variety of applied tasks,
the presence of data noise, and the nonstationary cluster parameters over time.
To estimate the number of groups in the data, metrics were introduced based on assess-
ing the ratio of data variance within the cluster to the distance between clusters. These met-
rics have several disadvantages. The procedure for calculating Hubert’s
statistics [47–49]
is characterised by a relatively low degree of formalisation (the number of clusters is de-
termined by the coordinate of the most acute angle between the segments of a piecewise
linear curve on the graph). Dunn’s indices [
54
] and several Bezdek–Pal indices [
55
] have
low noise immunity; they are mainly focused on separating hyper spherical clusters. To
determine the number of clusters with more complex shapes, estimation methods based
Sustainability 2025,17, 927 5 of 27
on the analysis of the shortest open path connecting points in the feature space have been
developed [
56
]. However, this approach reduces the noise immunity. In [
55
], the number of
clusters for the test data set was assessed using more than 20 metrics. The explored results
confirmed that approximately 50% of the metrics only showed the correct division of data
into clusters.
Some approaches determine the number of clusters by searching for the extremum
of functionals that consider the compactness of the data distribution in the cluster and
the distance of the different clusters [
46
]. When we have a small data sample with noisy
influence, such a functional can be multi-extremal. Therefore, when searching for an
extremum, some problems arise associated with insufficient noise immunity, high error,
sensitivity to local extrema, and the starting point of searching for optimisation methods.
To reduce the impact of these problems, the number of clusters is selected with expert
involvement [57].
Several methods have been developed to estimate the number of clusters based on
the information approach. For example, authors [
58
] emphasise, that information-theoretic
measures form a fundamental class of measures for comparing clusters. At the same time, a
few issues remain unresolved, including adjusting information-theoretic measures in cases
of clustering a small data sample, when the sample size is small compared to the number
of selected clusters.
Besides, some authors propose to use the well-known Shannon entropy formula to
measure information content [
59
] for the selection parameters of certain procedures in
OTI by comparing their characteristics. For example, this approach has been proposed for
selecting the segmentation [
60
] and classification procedures [
30
,
50
,
61
–
63
]. Authors [
32
]
used this approach when choosing WF parameters for the classifier.
Despite the extensive research on cluster methods, some unsolved problems remain,
as follows:
•Sensitivity to the starting point of searching for the cluster centre coordinates;
•Sensitivity to noise (interference) in the data proposed for clustering;
•
To address the above problems, we propose the processing of data during clustering
using wavelet transform in three main stages and develop the method for assessing
the efficiency and quality of clustering.
Extending the results of existing studies, the paper considers the solution of two tasks,
as follows:
•
The determination of the structure and composition of UAV-based system for inspect-
ing the OTI of SCs (UAV-IS-OTI-SCs) and the peculiarities of UAV application;
•
The cluster analysis of a pre-processed set for collected data and quality assessment
using wavelet transform.
2. Materials and Methods
2.1. Generalised Structure of OTI Inspection Stages
An important issue for the accident-free operation of bridges and other objects of
transport infrastructure is the timely detection of cracks and other surface defects based
on image processing [
64
]. These defects may be related to excessive loads, deflection of
the load-bearing structure, weathering, uneven heating, vibration, or moisture exposure.
The main stages of the OTI inspection for solving this issue within the context of the
above-mentioned tasks are presented in Figure 1, with blocks 1, 2, and 7, on which this
work is focused.
Sustainability 2025,17, 927 6 of 27
Sustainability 2025, 17, x FOR PEER REVIEW 6 of 27
Figure 1. Generalised structure of the image inspection stages.
At the initial stages, a UAV mission for OTI inspection and obtaining a set of RGB
images (see Figure 1, blocks 1, 2, 3) involves the removal of noise that complicates the
effective detection of surface defects. Typically, the source of noise distorting the images
(defocusing, low contrast) is uneven illumination of the objects surface, as well as UAV
vibrations. To eliminate the effect of such noise, it is advisable to use low-frequency filter-
ing [65] and highlight the edges of the surface defects by high-frequency filtering (see Fig-
ure 1, block 4) [14]. Then, after the binarisation of the image and morphological operations,
specifically dilation and erosion (see Figure 1, block 5), a defect feature vector is deter-
mined, which includes parameters such as area, length, and others (see Figure 1, block 6).
The final stages are clustering (block 7) and classification (block 8). The purpose of
the first one is to partition the paerns of objects into clusters that optimise a convex qual-
ity functional, characterising compactness. In cases where the object is inspected for the
first time or there are a few images of certain fragments of its surface, the authors propose
using wavelet optimisation to determine the number of clusters. In cases where the object
is inspected periodically and a certain number of new defects may appear, it is advisable
to use the Elbow method [66] to determine the number of clusters.
The choice of clustering method is significantly influenced not only by the volume of
the input data but also by its noise immunity and processing speed. For example, the well-
known k-means method has a high speed [67]. Although clustering using wavelet trans-
formation is slower, it offers beer noise immunity.
Therefore, the selection of clustering methods generally requires an assessment of
clustering quality. Even in the case of clusters with a bulbs form and big dataset, accord-
ing to known studies [55], such an assessment requires, firstly, iterative evaluations. Sec-
ondly, it needs to compare the results obtained by using different criteria (Dunns, Ca-
linsky–Harabasz, Bezdek–Pal, Davies–Bouldin, Silhouee index). Therefore, with a small
dataset and unknown cluster shape, we employ both the above-mentioned clustering-ef-
ficiency metrics and informational characteristics.
Furthermore, the dataset may be relatively small because of the complexity of data
collection procedures. In that case, the authors suggest using a noise-robust wavelet-based
approach for clustering. In turn, this can help to reduce errors at the next processing
stage—classification with training—even when the dataset increases, and CNN classifica-
tion is running.
2.2. The Use of UAV for Sustainability of Smart Regions TI
The sustainability of TI is one of the pivotal criteria for the advancement of SRs. To
guarantee sustainable development, smart regions employ the use of intelligent services
that are specifically designed to address a range of monitoring and management tasks
Figure 1. Generalised structure of the image inspection stages.
At the initial stages, a UAV mission for OTI inspection and obtaining a set of RGB
images (see Figure 1, blocks 1, 2, 3) involves the removal of noise that complicates the
effective detection of surface defects. Typically, the source of noise distorting the images
(defocusing, low contrast) is uneven illumination of the object’s surface, as well as UAV
vibrations. To eliminate the effect of such noise, it is advisable to use low-frequency
filtering [
65
] and highlight the edges of the surface defects by high-frequency filtering
(see Figure 1, block 4) [
14
]. Then, after the binarisation of the image and morphological
operations, specifically dilation and erosion (see Figure 1, block 5), a defect feature vector
is determined, which includes parameters such as area, length, and others (see Figure 1,
block 6).
The final stages are clustering (block 7) and classification (block 8). The purpose of the
first one is to partition the patterns of objects into clusters that optimise a convex quality
functional, characterising compactness. In cases where the object is inspected for the first
time or there are a few images of certain fragments of its surface, the authors propose using
wavelet optimisation to determine the number of clusters. In cases where the object is
inspected periodically and a certain number of new defects may appear, it is advisable to
use the Elbow method [66] to determine the number of clusters.
The choice of clustering method is significantly influenced not only by the volume
of the input data but also by its noise immunity and processing speed. For example, the
well-known k-means method has a high speed [
67
]. Although clustering using wavelet
transformation is slower, it offers better noise immunity.
Therefore, the selection of clustering methods generally requires an assessment of
clustering quality. Even in the case of clusters with a bulb’s form and big dataset, according
to known studies [
55
], such an assessment requires, firstly, iterative evaluations. Secondly,
it needs to compare the results obtained by using different criteria (Dunn’s, Calinsky–
Harabasz, Bezdek–Pal, Davies–Bouldin, Silhouette index). Therefore, with a small dataset
and unknown cluster shape, we employ both the above-mentioned clustering-efficiency
metrics and informational characteristics.
Furthermore, the dataset may be relatively small because of the complexity of data
collection procedures. In that case, the authors suggest using a noise-robust wavelet-based
approach for clustering. In turn, this can help to reduce errors at the next processing stage—
classification with training—even when the dataset increases, and CNN classification
is running.
2.2. The Use of UAV for Sustainability of Smart Regions TI
The sustainability of TI is one of the pivotal criteria for the advancement of SRs. To
guarantee sustainable development, smart regions employ the use of intelligent services
Sustainability 2025,17, 927 7 of 27
that are specifically designed to address a range of monitoring and management tasks across
various sectors, including energy efficiency, environmental conservation, transportation,
security, and others [
1
,
3
–
5
]. In recent years, the deployment of various services has extended
beyond the utilisation of information technology as the primary infrastructure. The role
of mobile technologies is now of equal importance, and the combination of information
and mobile technologies creates a powerful synergy that will determine the future of smart
cities and regions.
Unmanned aerial vehicles (UAVs) are at the vanguard of the provision of smart city
services and are effecting beneficial changes to urban life [
17
,
68
,
69
]. Unmanned aerial
vehicles (UAVs) are particularly suited to situations where it is necessary to measure
and survey objects in inaccessible and dangerous locations. The utilisation of UAVs as a
component of the sustainable development of smart systems serves to mitigate risks and
reduce the cost of services, thereby underscoring their economic efficiency [5,6,70–72].
Structurally, all the UAVs that can be used to perform different tasks form a UAV fleet,
which is a collection [1], such as the following:
•
Swarms of UAVs: a set of swarms of UAVs, where each swarm consists of UAVs
working together to achieve a common goal or service;
•
UAVs: a collection of individual UAVs of different types used to perform individual
services or to supplement swarms and flocks as required;
•
UAV control systems: a control system includes a network of control stations that
manage a fleet of UAVs and their individual components.
This paper considers aspects of the application of UAV-based intelligent mobile sys-
tems as a data collection object during the inspection of transport infrastructure (OTI) in
smart cities and regions.
Thus, the UAV fleet is a source of UAV resources from which UAV swarms are formed
to perform OTI inspection tasks (Figure 2).
Sustainability 2025, 17, x FOR PEER REVIEW 7 of 27
across various sectors, including energy efficiency, environmental conservation, transpor-
tation, security, and others [1,3–5]. In recent years, the deployment of various services has
extended beyond the utilisation of information technology as the primary infrastructure.
The role of mobile technologies is now of equal importance, and the combination of infor-
mation and mobile technologies creates a powerful synergy that will determine the future
of smart cities and regions.
Unmanned aerial vehicles (UAVs) are at the vanguard of the provision of smart city
services and are effecting beneficial changes to urban life [17,68,69]. Unmanned aerial ve-
hicles (UAVs) are particularly suited to situations where it is necessary to measure and
survey objects in inaccessible and dangerous locations. The utilisation of UAVs as a com-
ponent of the sustainable development of smart systems serves to mitigate risks and re-
duce the cost of services, thereby underscoring their economic efficiency [5,6,70–72].
Structurally, all the UAVs that can be used to perform different tasks form a UAV
fleet, which is a collection [1], such as the following:
• Swarms of UAVs: a set of swarms of UAVs, where each swarm consists of UAVs
working together to achieve a common goal or service;
• UAVs: a collection of individual UAVs of different types used to perform individual
services or to supplement swarms and flocks as required;
• UAV control systems: a control system includes a network of control stations that
manage a fleet of UAVs and their individual components.
This paper considers aspects of the application of UAV-based intelligent mobile sys-
tems as a data collection object during the inspection of transport infrastructure (OTI) in
smart cities and regions.
Thus, the UAV fleet is a source of UAV resources from which UAV swarms are
formed to perform OTI inspection tasks (Figure 2).
Figure 2. Sequence of steps in performing the OTI inspection tasks.
Figure 2. Sequence of steps in performing the OTI inspection tasks.
Sustainability 2025,17, 927 8 of 27
It should be noted that in the process of monitoring by both single and swarm
UAVs [
73
,
74
], various tasks related to the collection and processing of various data, includ-
ing multispectral images of OTI, are solved (using sustainable smart technologies) in real
time. Based on the received information flows, classification, and clustering are usually
performed, which allows to divide of objects into classes. It is known from decision-making
theory that when an object is assigned to a particular class, errors of the first (errors
α
, false
positives (FP)) and second (errors
β
, false negatives (FN)) kind usually occur. On the one
hand, this leads to the loss of UAV resources, which cannot be wasted due to their limited
availability. On the other hand, the presence of these errors can lead to a situation in which
reliable information about the object will be lost since no work has been done to restore its
condition. To minimise losses, the following conditions should be met:
L(α,β)≤minαminβ, (1)
where Lis the loss function.
If condition (1) is fulfilled, then the clustering will be performed with high accuracy
and the system using UAVs and smart technology will be robust. It should be noted that
FNs and FPs are used to construct the confusion matrix and to evaluate the accuracy.
2.3. Forming the Structure and Composition of UAV-IS-OTI-SCs and the Peculiarities
of UAV Application
Due to the process of creating UAV-IS-OTI-SCs, it is necessary to consider the fact that
a single UAV has relatively small capabilities to perform tasks (short flight time, limited
by onboard power resource; small number of functions performed; low probability of
performing a task in extreme situations, etc.). Therefore, the efficiency of the system should
be increased by the group application of their components.
The construction of an adaptive monitoring system should be based on the use of
technologies that ensure a few things, as follows:
•Joint (group) performance of tasks;
•Adaptation to new requirements and conditions;
•Ability to expand (scale).
These requirements can be met by deploying UAV-IS-OTI-SCs as a multi-agent system,
where UAVs or groups of UAVs are considered as intelligent agents.
In the multi-agent approach, UAVs act as ‘agents’ that collect data, assess the situation,
make decisions about actions, and interact with other ‘agents’ using specialised software
and sensors.
The situational creation of the structure of multiagent systems for performing specific
tasks can be carried out by considering various parameters in a particular object area. A
promising area for formalising such knowledge is the development of ontologies [75].
Two ontologies and a method of defining system base composition are used in the
process of forming the structure and composition of UAV-IS-OTI-SCs, as follows:
•
The basic ontology of UAV-IS-OTI-SCs describes the structure and interaction of the
system’s components when performing tasks in different conditions. It allows the
definition of UAV types, and their payloads, which are suitable for the task in the
current conditions, as well as, if necessary, the types of UAV maintenance systems.
•
Low-level model ontology, which allows you to select a model that considers the
largest number of attributes necessary to assess the compliance of the system with
non-functional requirements, for example, reliability and safety requirements. This
ontology is intended to determine the number of UAVs and their maintenance systems,
considering the fulfilment of certain non-functional requirements.
Sustainability 2025,17, 927 9 of 27
•
A method of defining the base composition of the system and mission parameters
used to determine the number of UAVs and flight parameters of UAVs which ensure
the fulfilment of the functional requirements for the inspection mission.
A diagram of the process for forming the structure and composition of UAV-IS-OTI-
SCs is shown in Figure 3.
Sustainability 2025, 17, x FOR PEER REVIEW 9 of 27
• A method of defining the base composition of the system and mission parameters
used to determine the number of UAVs and flight parameters of UAVs which ensure
the fulfilment of the functional requirements for the inspection mission.
A diagram of the process for forming the structure and composition of UAV-IS-OTI-
SCs is shown in Figure 3.
Figure 3. Diagram of the process for forming the structure and composition of UAV-IS-OTI-SCs.
To achieve the accurate clustering of objects with low-dimensional features, it is es-
sential to ensure that the requisite pixel density is provided. The use of UAVs allows the
collecting of more data and ensuring beer data quality, as well as adapting these data to
modern data processing methods. For example, when creating a digital twin of a small
bridge with a resolution of 0.5 mm/pixel, the three UAVs can be used in 4–5 h.
Inspection requires periodic surveys of the OTI, which requires UAV control, navi-
gation, and flight control to ensure accurate and repeatable flight paths for infrastructure
inspections.
Optimising UAV flight paths involves balancing UAV capabilities, environmental
constraints, data quantity (coverage of the structure and/or amount of data at specific lo-
cations on the structure), data quality (resolution appropriate for the use case), and Smart
Citys needs.
Consequently, data quality depends on the payload characteristics and flight param-
eters of the UAV, such as the frames overlap (horizontal and vertical), UAV flight speed,
and UAV flight altitude.
Figure 4 shows the UAV flight paern during inspection mission ensuring frame
overlap.
Figure 3. Diagram of the process for forming the structure and composition of UAV-IS-OTI-SCs.
To achieve the accurate clustering of objects with low-dimensional features, it is
essential to ensure that the requisite pixel density is provided. The use of UAVs allows the
collecting of more data and ensuring better data quality, as well as adapting these data to
modern data processing methods. For example, when creating a digital twin of a small
bridge with a resolution of 0.5 mm/pixel, the three UAVs can be used in 4–5 h.
Inspection requires periodic surveys of the OTI, which requires UAV control, navi-
gation, and flight control to ensure accurate and repeatable flight paths for infrastructure
inspections.
Optimising UAV flight paths involves balancing UAV capabilities, environmental
constraints, data quantity (coverage of the structure and/or amount of data at specific
locations on the structure), data quality (resolution appropriate for the use case), and Smart
City’s needs.
Consequently, data quality depends on the payload characteristics and flight parame-
ters of the UAV, such as the frame’s overlap (horizontal and vertical), UAV flight speed,
and UAV flight altitude.
Figure 4shows the UAV flight pattern during inspection mission ensuring
frame overlap
.
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Figure 4. UAV flight paern ensuring frame overlap.
The algorithm of defining the base composition and mission parameters (flight pa-
rameters of the UAV) to obtain images with a given resolution (Res) in a set time of
inspection (T) for an OTI with the given dimensions (Size_;Size_) is running by
following nine steps (Figure 5):
Figure 5. Algorithm for determining mission parameters and UAV flight characteristics to obtain
images of OTI with specified requirements.
Step 1. Obtaining input data and requirements.
Step 2. Calculate the required viewing width (h) for the camera with the number
of pixels of the matrix horizontally Res
:
Figure 4. UAV flight pattern ensuring frame overlap.
The algorithm of defining the base composition and mission parameters (flight pa-
rameters of the UAV) to obtain images with a given resolution (
Resreq
) in a set time of
inspection (
Treq
) for an OTI with the given dimensions (
SizeOTI_v; SizeOTI_h
) is running by
following nine steps (Figure 5):
Sustainability 2025, 17, x FOR PEER REVIEW 10 of 27
Figure 4. UAV flight paern ensuring frame overlap.
The algorithm of defining the base composition and mission parameters (flight pa-
rameters of the UAV) to obtain images with a given resolution (Res) in a set time of
inspection (T) for an OTI with the given dimensions (Size_;Size_) is running by
following nine steps (Figure 5):
Figure 5. Algorithm for determining mission parameters and UAV flight characteristics to obtain
images of OTI with specified requirements.
Step 1. Obtaining input data and requirements.
Step 2. Calculate the required viewing width (h) for the camera with the number
of pixels of the matrix horizontally Res
:
Figure 5. Algorithm for determining mission parameters and UAV flight characteristics to obtain
images of OTI with specified requirements.
Sustainability 2025,17, 927 11 of 27
Step 1. Obtaining input data and requirements.
Step 2. Calculate the required viewing width (
hframe
) for the camera with the number
of pixels of the matrix horizontally (Rescamh):
hframe =Resreq·Rescam_h , (2)
Step 3. Calculate the required UAV flight altitude (
hUAV
) with a camera that has a
viewing angle of α:
hUAV =hframe/(2+tg(α/2)), (3)
Step 4. Calculate the camera frame height (
vframe
) for the camera with the number of
pixels of the matrix vertically (Rescam_h :
vframe =Resreq·Rescamv, (4)
Step 5. The flight speed of the UAV (
VUAV
) is determined by the following condition:
VUAV ≤freqcam·(Size OTI_v·(vframe −∆vframe_overlap )), (5)
Step 6. The flight time of the UAV to fly over a given space of OTI is determined:
tUAV_fl_total =VUAV·SizeOTI_v /(Size OTI_h/(hframe −∆hframe_overlap )), (6)
Step 7. The condition
tUAV_fl_total ≤Treq
is checked. If the condition is not met, the
number of UAVs (required to fulfil it) is determined (Step 8):
NUAV =tUAV_fl_total/Treq. (7)
Step 9. Output of the results.
Data collected in the same location and from different orientations at different intervals
allows for the analysis of changes using artificial intelligence. Increasing the amount of data
collected can facilitate regular repeat surveys to provide the data needed to train artificial
intelligence and machine learning change detection algorithms.
Moreover, it should also be understood that UAVs are aircraft, and their use should
be regulated by the relevant regulations. In recent years, international rules for the use
of UAVs have become more clearly defined, and inspection missions (including missions
in overcrowded areas or critical infrastructure, operations using multiple UAVs, etc.) are
becoming more commonplace for users and more acceptable to authorities. Both authorities
and end users are reaching a mutual understanding of the security risks associated with
these operations. Bodies such as the European Aviation Safety Agency are disseminating
resources and tools for users to quantify the safety risks associated with the use of UAVs
for the future sustainability of Smart Cities.
2.4. Clustering Based on WT
Considering the described in Section 2.1 above, we propose conducting data processing
during WT clustering in three main phases (Figure 6). Phase 1: Determining the number of
clusters using WT, Phase 2: Determining the coordinates of cluster centres, and Phase 3:
Assessing the quality and efficiency of clustering.
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Figure 6. Scheme of clustering algorithm based on WT.
2.4.1. Determining the Number of Clusters Using WT
Phase 1 is based on the search for the extrema of the multi-extremal objective function
based on processing WT. The iterative search for the optimum of the objective function is
implemented according to the scheme [32,46]:
𝒄[𝑛]=𝒄[𝑛−1] − 𝛾[𝑛]𝑊𝑇(𝑸(𝒙[𝑛],𝒄[𝑛−1])), (8)
where 𝑸(𝒙,𝒄) is a functional that depends on the vector of parameters 𝒄=(𝑐,…,𝑐)
and 𝒙=(𝑥,…,𝑥); 𝛾[𝑛] is a step; 𝑛 is the iteration number; 𝑘 is the start number;
𝑊𝑇𝑸(𝒙[𝑛],𝒄[𝑛−1])=𝐺,𝐺,…,𝐺, (9)
where 𝑊𝑇𝑸(𝒙[𝑛],𝒄[𝑛−1]) is a WT, which determines the movement direction to-
ward the extremum;
𝐺 =(𝑸𝒙[𝑛],𝒄+𝑖𝑎)∙𝜓(𝑖)
𝑠
, , (10)
where 𝑠 is the length of the WF carrier at the 𝑘-th start (𝑠 is an even number); 𝑎 is the
discretisation step of the WF; 𝜓(𝑖) is the WF at the 𝑘-th start; 𝑗 = 1, …,𝑁 is the dimen-
sion of the parameter vector.
The step 𝛾[𝑛] is selected to find the optimum with the gradient estimate: 𝛾[1] = 0.4,
…, 0.6. If the sign of 𝑊𝑇𝑸(𝒙[𝑛],𝒄[𝑛−1]) is changing when passing through the opti-
mum at the 𝑛−1 step, then 𝛾[𝑛]=0.5∙ 𝛾[𝑛−1].
To evaluate the direction of the search for the optimum (9) symmetric and non-sta-
tionary WF are selected. In [32] features of the search for WT-based extremum, WF Haar
impulse response and fragment of the objective function, are shown. In the first step, the
Haar WF was chosen, and the impulse response for 𝑠=12:
𝜓(𝑖)=1, 𝑖=1,…,𝑠
2
−1, 𝑖=−1,…,𝑠
2. (11)
Then the hyperbolic wavelet function
Figure 6. Scheme of clustering algorithm based on WT.
2.4.1. Determining the Number of Clusters Using WT
Phase 1 is based on the search for the extrema of the multi-extremal objective function
based on processing WT. The iterative search for the optimum of the objective function is
implemented according to the scheme [32,46]:
c[n]=c[n−1]−γ[n]WTk(Q(x[n],c[n−1])), (8)
where
Q(x,c)
is a functional that depends on the vector of parameters
c=(c1, . . . , cN)
and
x=(x1, . . . , xM);γ[n]is a step; nis the iteration number; kis the start number;
WTk(Q(x[n],c[n−1])) ={G1k,G2k, . . . , GNk}, (9)
where
WTk(Q(x[n],c[n−1])) is a
WT, which determines the movement direction toward
the extremum;
Gjk =
sk
2
∑
i=sk
2,i=0Qx[n],cj+ia·ψk(i)
sk
, (10)
where
sk
is the length of the WF carrier at the
k
-th start (
sk
is an even number);
a
is the
discretisation step of the WF;
ψk(i)
is the WF at the
k
-th start;
j=
1,
. . .
,
N
is the dimension
of the parameter vector.
The step
γ[n]
is selected to find the optimum with the gradient estimate:
γ[1]
= 0.4,
. . .
,
0.6. If the sign of
WTk(Q(x[n],c[n−1]))
is changing when passing through the optimum
at the n−1 step, then γ[n]=0.5·γ[n−1].
To evaluate the direction of the search for the optimum (9) symmetric and non-
stationary WF are selected. In [
32
] features of the search for WT-based extremum, WF Haar
impulse response and fragment of the objective function, are shown. In the first step, the
Haar WF was chosen, and the impulse response for s1=12:
ψ1(i)=(1, i=1, . . . , s1
2
−1, i=−1, . . . , s1
2
. (11)
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Then the hyperbolic wavelet function
ψk(i)=(1
ak(|i|+1),i>0,
−1
ak,(|i|+1),i<0,iϵh−sk
2,+sk
2i,i=0 (12)
is selected.
With such a multi-step processing, the search on the first step, using the Haar WF,
moves with a high probability to the region of the global extremum.
At the next steps of searching for an extremum, the coordinates of the extremum are
“refined” according to (8), using a hyperbolic WF with a decreasing carrier length. In this
case, noise immunity gradually decreases as the value of
ak
increases according to (12) from
1 to 5. The carrier length of the Haar WF was 12, 10, 6, 4, 4, and 2 for the corresponding
start numbers
k
from 2 to 7. At the final step, the search direction is estimated using a
finite-difference derivative estimate [32].
The results of an experimental study of the optimisation method with WT on the test
functions of the Schwefel, De Jong 1, and Rosenbrock “ravine” [
38
] are shown in Figure 4. In
particular, the convergence speed of optimisation using WT was investigated in comparison
with the gradient descent method using the “ravine” Rosenbrock function (Figure 7a):
f1(x) = 100(x2−x2
1) + (1−x1)2. (13)
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(a)
(b)
(c)
Figure 7. Assessment of the capabilities of optimisation method with WT: (a) Rosenbrock “ravine”
test function; (b) Schwefel test function; (c) De Jong 1 test function with the addition of noise.
Based on the above, the following five steps are run to determine the number of clus-
ters in the data (see Figure 6, blocks 1–5):
Step 1. Display data parameters from Euclidean to 𝜆—space including the following
procedures:
• Constructing a complete graph in Euclidean space;
• Calculating the normalised distance between all pairs of its vertices 𝑑=
;
• Calculating the characteristic of the local density of the set in the neighbourhood of
the 𝑖-th edge: 𝜏=
,
Figure 7. Assessment of the capabilities of optimisation method with WT: (a) Rosenbrock “ravine”
test function; (b) Schwefel test function; (c) De Jong 1 test function with the addition of noise.
Sustainability 2025,17, 927 14 of 27
In the function (13) at
x∈(−2.048; 2.048)
, the global minimum
f1(x)=
0 at
x1=
1,
x2=
1. During the study, the sampling step of the WF
a=
0.0005 and the
carrier length sk=10.
As a result of the study, it was determined that the optimisation method using WT
allowed it to reach the extremum in 1.7 times faster (in terms of the number of iterations)
compared to the gradient descent method. The sensitivity of the developed optimisa-
tion method to local extrema and the starting point of the search using the Schwefel
function (13) (with a false global minimum) was investigated (Figure 7b):
f2(x)=418.9829 + (−x·sinq|x|). (14)
Here
x∈(−500; 500)
, the global minimum
f2(x)=
0 at
x=
420.9829. The starting
point was chosen randomly. The gradient descent method made it possible to find the
minimum closest to the starting point. When optimising with WT, the global minimum
was reached with an error of
δ≤
10
−2
in 123 out of 150 cases. The global minimum was
not found when starting point values were selected outside the interval x∈(−420; 470).
The noise immunity of the method using the De Jong function with the addition of
noise has been studied (at x∈(−205; 205)) (Figure 7c):
f3(x)=x2. (15)
The noise was distributed according to a normal distribution with a zero mean
and standard deviation from 0 to 40,000. The maximum value of the function was
f3(x)=
42, 000. With a signal-to-noise ratio in amplitude up to 1.05, the method made it
possible to reach the neighborhood of the global minimum with an error
δ≤
10
−2
. The
obtained results confirmed the high noise immunity of the method as well as the reduced
sensitivity to local extrema and the starting point of the search.
Based on the above, the following five steps are run to determine the number of
clusters in the data (see Figure 6, blocks 1–5):
Step 1. Display data parameters from Euclidean to
λ
—space including the following
procedures:
•Constructing a complete graph in Euclidean space;
•Calculating the normalised distance between all pairs of its vertices di=αi
βimax ;
•
Calculating the characteristic of the local density of the set in the neighbourhood of
the i-th edge:
τi=αi
βimin τmax ,
where
αi
is the distance between the
i
—th pair of vertices;
βimax
and
βimin
are the
lengths of the longest and shortest edges, respectively;
τmax
is the maximum value of
τ∗
i=αi
βimin ;
•Calculating the length of edges in the graph in λ-space [60]:
λi=τ2
i×di.
Step 2. Construction of the
λ
-graph for the shortest open (non-circular) path according
to the
λp
algorithm (see Figure 6) and considering the probability of breaking its edge [
62
]:
piz =λiz
∑k
j=1λiz
,
Sustainability 2025,17, 927 15 of 27
where
piz
is the probability of breaking the
i
-th edge connected to the vertex
z
;
λiz
is the
λ
distance corresponding to the
i
-th edge connected to the vertex
z
in the
λ
-graph of the
shortest open (non-circular) path.
Step 3. Calculation of the parameter (see Figure 6)
qi=f(i+1)
f(i),i∈[1, n−1], (16)
where fiis the average λ—distance when adding the i-th value to the data cluster.
Step 4. Search for the global maximum of the multi-extremal
Q=max (qi)
by
optimisation with WT according to (8)–(12) (see Figure 6). At this stage, the initial set of
objects is first divided into two clusters.
Step 5. The graph edge that connects the first cluster to the rest of the data is broken
(see Figure 6). If the condition in block 6 is satisfied, then we proceed to block 7; otherwise,
we take it back to block 4.
To assess the capabilities of the proposed method, the number of clusters for the
test data set
X30
was estimated (Figure 8a). These data are unnamed and consist of three
compact groups of ten points in a two-dimensional feature space. Figure 8b shows the
Log(qi)
data (curve 1) calculated using the method [
55
] and calculated using the proposed
method (curve 2). As can be seen from Figure 8b, the amplitude of the first mode on
curve 2 is significantly higher than that of the other modes. This allows determining the
composition of the first cluster (10 patterns in the feature space). Curve 1 in Figure 8b
has significantly more modes with high amplitude, which may require more complex
calculations to determine the number of clusters. In the proposed method, the amplitudes
of the second and subsequent modes may be several tens of times smaller than the first
one. Therefore, it is proposed, after identifying the first cluster, to find the maxima of the
criterion
Q=max(qi)
successively, excluding from consideration the data assigned to the
previous cluster when evaluating qi.
Sustainability 2025, 17, x FOR PEER REVIEW 15 of 27
where 𝛼 is the distance between the 𝑖 —th pair of vertices; 𝛽 and 𝛽 are the
lengths of the longest and shortest edges, respectively; 𝜏 is the maximum value of
𝜏∗=
;
• Calculating the length of edges in the graph in 𝜆-space [60]:
𝜆=𝜏𝑑.
Step 2. Construction of the 𝜆-graph for the shortest open (non-circular) path accord-
ing to the 𝜆𝑝 algorithm (see Figure 6) and considering the probability of breaking its edge
[62]: 𝑝=
∑
,
where 𝑝 is the probability of breaking the 𝑖-th edge connected to the vertex 𝑧; 𝜆 is
the λ distance corresponding to the 𝑖-th edge connected to the vertex 𝑧 in the 𝜆-graph
of the shortest open (non-circular) path.
Step 3. Calculation of the parameter (see Figure 6)
𝑞=
𝑓
(𝑖+1)
𝑓
(𝑖) ,𝑖∈ [1, 𝑛−1], (16)
where 𝑓 is the average 𝜆—distance when adding the 𝑖-th value to the data cluster.
Step 4. Search for the global maximum of the multi-extremal 𝑄=max (𝑞) by opti-
misation with WT according to (8)–(12) (see Figure 6). At this stage, the initial set of objects
is first divided into two clusters.
Step 5. The graph edge that connects the first cluster to the rest of the data is broken
(see Figure 6). If the condition in block 6 is satisfied, then we proceed to block 7; otherwise,
we take it back to block 4.
To assess the capabilities of the proposed method, the number of clusters for the test
data set 𝑋 was estimated (Figure 8a). These data are unnamed and consist of three com-
pact groups of ten points in a two-dimensional feature space. Figure 8b shows the 𝐿𝑜𝑔(𝑞)
data (curve 1) calculated using the method [55] and calculated using the proposed method
(curve 2). As can be seen from Figure 8b, the amplitude of the first mode on curve 2 is
significantly higher than that of the other modes. This allows determining the composition
of the first cluster (10 paerns in the feature space). Curve 1 in Figure 8b has significantly
more modes with high amplitude, which may require more complex calculations to de-
termine the number of clusters. In the proposed method, the amplitudes of the second and
subsequent modes may be several tens of times smaller than the first one. Therefore, it is
proposed, after identifying the first cluster, to find the maxima of the criterion 𝑄=
𝑚𝑎𝑥(𝑞) successively, excluding from consideration the data assigned to the previous
cluster when evaluating 𝑞.
(a)
Figure 8. Cont.
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(b)
Figure 8. Determining the number of clusters: (a) estimating the number of clusters for the test da-
taset 𝑋
; (b) 𝐿𝑜𝑔(𝑞
) data calculated using the method [55] (curve 1) and calculated using the pro-
posed method (curve 2).
Thus, based on finding optima with WT, the number of clusters is determined after
transformation to the 𝜆–space.
2.4.2. Determining the Coordinates of Cluster Centres
At Phase 2 it is necessary to determine the vector of coordinates for cluster centres
𝒄=𝑐. That provides the extreme value of the quality functional 𝑄(𝒙, 𝒄) (see Figure 6,
block 7) relative to the vector of variables 𝒄=(𝑐,…,𝑐), depending on the vector of ran-
dom sequences 𝒙=(𝑥,…,𝑥). To increase noise immunity and reduce sensitivity to lo-
cal extrema, as well as assess the movement direction towards the optimum, a set of WF
according to (11), (12) with sequentially decreasing carrier length was used (see Figure 6,
block 8).
Then coordinates of cluster centres for the two clusters are defined (see Figure 6,
block 10): 𝑐[𝑛]=𝑐[𝑛−1]−𝛾[𝑛]∇
𝑄(𝒙[𝑛], 𝑐[𝑛−1], 𝑐[𝑛−1])
𝑐[𝑛]=𝑐[𝑛−1]−𝛾[𝑛]∇
𝑄(𝒙[𝑛], 𝑐[𝑛−1], 𝑐[𝑛−1]),
where 𝛾[𝑛],𝛾[𝑛] are the step sizes;
𝑛 is the iteration number;
∇
𝑄(𝑥[𝑛], 𝑐[𝑛−1], 𝑐[𝑛−1]) is the assessment of the movement direction to-
wards the extremum with WT for the first cluster (see Figure 6, block 9);
∇
𝑄(𝑥[𝑛], 𝑐[𝑛−1], 𝑐[𝑛−1]) is the assessment of the movement direction to-
wards the extremum with WP for the second cluster (see Figure 6, block 9);
𝑄(𝑥,𝑐,𝑐)=∑𝜀(𝑥,с,с)𝐹(
𝑥,𝑐,𝑐) is the implementation of the quality func-
tional;
𝐹(𝑥,𝑐,𝑐) is the distance function of elements x from the set X to the centres of
the clusters;
𝜀(∙) are characteristic functions, where:
Figure 8. Determining the number of clusters: (a) estimating the number of clusters for the test
dataset
X30
; (b)
Log(qi)
data calculated using the method [
55
] (curve 1) and calculated using the
proposed method (curve 2).
Thus, based on finding optima with WT, the number of clusters is determined after
transformation to the λ–space.
2.4.2. Determining the Coordinates of Cluster Centres
At Phase 2 it is necessary to determine the vector of coordinates for cluster centres
c=copt
. That provides the extreme value of the quality functional
Q(x,c)
(see Figure 6,
block 7) relative to the vector of variables
c=(c1, . . . , cN)
, depending on the vector of
random sequences
x=(x1, . . . , xM)
. To increase noise immunity and reduce sensitivity to
local extrema, as well as assess the movement direction towards the optimum, a set of WF
according to (11), (12) with sequentially decreasing carrier length was used (see Figure 6,
block 8).
Then coordinates of cluster centres for the two clusters are defined (see Figure 6, block 10):
c1[n]=c1[n−1]−γ1[n]∼
∇c1+Q(x[n],c1[n−1],c2[n−1])
c2[n]=c2[n−1]−γ2[n]∼
∇c2+Q(x[n],c1[n−1],c2[n−1])
,
where γ1[n],γ2[n]are the step sizes;
nis the iteration number;
∼
∇c1+Q(x[n],c1[n−1],c2[n−1])
is the assessment of the movement direction
towards the extremum with WT for the first cluster (see Figure 6, block 9);
∼
∇c2+Q(x[n],c1[n−1],c2[n−1])
is the assessment of the movement direction
towards the extremum with WP for the second cluster (see Figure 6, block 9);
Q(x,c1,c2) = ∑2
b=1εk(x,c1, c2)Fb(x,c1,c2)is the implementation of the quality functional;
Fb(x,c1,c2)
is the distance function of elements
x
from the set
X
to the ‘centres’ of
the clusters;
Sustainability 2025,17, 927 17 of 27
εb(·)are characteristic functions, where:
εb(x,c1,c2)=(1, x∈Xb,
0, x/∈Xb..
After initialising the parameters per each of the
i
elements for the weighted sum with
WP, the values of the characteristic functions εb(x, c1,c2)at b=1,2 are determined.
For this, pairs of values are:
c1[n−1],c2[n−1],
c1[n−1]±ie1a[n],c2[n−1],
c1[n−1],c2[n−1]±ie2a[n]i=1, N,
where
a[n]
is a scalar,
N
is the length of the wavelet function carrier. Then the scalar for a
given x[n]is substituted in the expression:
f(x, c1,c2)=∥x[n]−c1∥2−∥x[n]−c2∥2.
The function
f(x,c1,c2)
is zero on the boundary between two clusters and it has
different signs on either side of the boundary. Therefore, if the value of
f(x,c1,c2)
is
negative, then ε1=1, ε2=0. In the opposite case ε1=0, ε2=1.
If the condition in block 11 is satisfied, the coordinates of cluster centres are determined,
and we proceed to block 12. In the opposite case, we take back to block 7.
2.4.3. Assessing the Quality and Efficiency of Clustering
At Phase 3, the quality attributes are calculated based on the closeness of the processing
result to the ideal option, which is determined from a specially generated test sample. In
such a sample, patterns of objects
Ctm
zj
and supervised labels are given, indicating that
the patterns belong to the
m
-th cluster out of
M
clusters. Here
z=1, Z
, where
Z
is the
dimension of the feature space;
j=1, L
is the pattern number belonging to the cluster;
L
is
the total number of cluster patterns;
t=1, T
is the number of the clustering method, and
T
is the number of clustering methods under study (t=0 for the test sample).
To assess the quality of clustering, we propose to use the Hamming distance [
76
]
between the test sample and the clustering result (see Figure 6, block 12) obtained by the
method being studied:
drs (Ctm
zr ,C0m
zs ) = (0, i f C tm
zr =C0m
zs
1, i f Ctm
zr =C0m
zs
, (17)
where
C0m
zs
is the supervised labels indicating that the pattern belongs to the cluster;
Ctm
zr
is
the result of clustering by the method being studied.
Let us define the normalised distance between vectors, which characterise the dif-
ference between the generated partition of patterns into clusters in the feature space and
the alternative partitioning of them into clusters (see Figure 6, block 13). We calculate the
normalised distance Dfor the applied clustering method tusing expression:
Dt,0 =1
L
L
∑
j=1
M
∑
m=1
Z
∑
z=1
djj Ctm
zj ,C0m
zj . (18)
The assessment of the degree to which a clustering procedure achieves its processing
goal can be conducted using efficiency indicators. Those indicators can be of a statistical or
informational nature, and it is evaluated often using expert methods. We propose using
Sustainability 2025,17, 927 18 of 27
the average gain criterion within the framework of a statistical approach to assess the
effectiveness of clustering (see Figure 6, block 14). In this case, the probability of obtaining
the correct clustering result is equal to:
P=P1+P2, (19)
where
P1
is the probability of correctly assigning a pattern to a cluster, characterising type 1
error;
P2
is the probability of correctly not assigning a pattern to a cluster, characterising
type 2 error.
When assessing the average gain, we propose using the concept of the pragmatic
measure of information. The last one determines the usefulness (value) of the information
for the user to achieve a goal [
77
]. Besides, this measure is a relative one, and it depends on
the specifics of how it is used in a particular automated system. Within the information
approach to clustering, we propose to employ a semantic measure (see Figure 6, block 15):
R=log2Ri−log2R0=log2
Ri
R0, (20)
where
Ri
and
R0
are the average gain for the i-th (being studied) and 0-th (baseline)
clustering variants, respectively.
As it follows from (20), when comparing two clustering procedures, the following
scenarios are possible:
R=
0—procedures are identical in effectiveness; preference is given to the procedure
of higher quality when making a choice;
R>
0—the procedure being studied is better than the baseline in terms of effectiveness,
and therefore, it should be chosen;
R<
0—the baseline procedure is better than the one being studied, and therefore, it
should be chosen.
Thus, the final decision on selecting the clustering method and its parameters should
be made after exploring their impact on the efficiency of the system.
3. Case Study
3.1. Determination of the UAV Light Time Required for Inspection and the Volume of Data to Be
Processed and Analysed
Time expenditure calculations for inspecting a small bridge with an inspection area of
10,000 m
2
and the expected volume of raw video data for inspection at 4k resolution are
performed. The calculations are based on the specifications of the DJI Mavic 3 UAV.
The determination of the time required for inspecting a small bridge using a single
UAV is conducted using the following formula:
t=Si
0.8WsV, (21)
where
Si
is the defined area of the inspection region,
Ws
is the width of the viewing strip
considering a 10% frame overlap, Vis the UAV flight speed.
The width of the seeing area depends on the technical characteristics of the equipment
being used and the distance from the sensor (camera) to the surface of the object being
inspected by the UAV:
Ws=2h∗tg α
2, (22)
where
α
is the field of view of the UAV’s sensor (camera),
h
is the distance from the UAV’s
sensor (camera) to the surface of the inspected object.
Sustainability 2025,17, 927 19 of 27
The values used for the calculations: camera field of view—15
◦
; distance from the
camera to the object’s surface—3 m; flight speed—1 m/s.
For these values, the inspection time is approximately 4 h and 24 min, which signif-
icantly exceeds the flight time of commercial UAVs, such as the DJI Mavic 3, which has
a flight time of up to 45 min. Therefore, during the inspection, the UAV will require six
battery replacements or recharges, leading to an increase in the overall inspection time. At
4k resolution, the camera captures 120 frames per second, and in this case, the total number
of frames to be processed will be 1,898,939, requiring further data processing.
If there are limitations on the duration of missions, a UAV team is required. The
number of UAVs in the team is determined by the expression:
NUAV =t
Treq , (23)
where Treq is the required time for the OTI inspection.
In the considered case, a swarm of three UAVs is required to carry out the bridge
status inspection in a time not exceeding 2 h.
For preliminary video stream processing, convolutional neural networks are recom-
mended, as they can output a feature vector that can be used for further clustering [26].
3.2. Comparison of Quality and Efficiency of Clustering Methods
Clustering methods have been studied using the developed methods for assessing
quality and efficiency. We examined changes in the relative value for the standard deviation
of data in clusters ranging from 0.05 to 0.25 (ensuring non-overlapping clusters in the
feature space) to experimentally test the proposed WT clustering method. In this case, the
standard deviation:
q=qp
q0·∆,
where
qp
and
q0
are the standard deviations of parameters in clusters in the testing and train-
ing phase, respectively,
∆
is the distance between the centres for clusters of the
training set.
Figure 9a shows the results of studies for the quality of clustering methods:
c
-means
(curve 1), fuzzy clustering using WT (curve 2),
k
-means (curve 3), and crisp clustering
method using WT (curve 4). A comparison with the
k
-means and
c
-means methods was
carried out using common representations of groups of clustering methods actively used
by developers of transport systems and UAVs. Figure 9b shows the results of efficiency
studies: curve 1—the efficiency of the crisp clustering method with WT compared to the
k
-means method, curve 2—the efficiency of the fuzzy clustering method with WT compared
to the c-means method.
Let us analyse the case when the standard deviation in clusters is changing. As it can
be seen from Figure 6the quality of the WT clustering method compared to
k
-means is
higher on average by 10%; the quality of fuzzy clustering with WT is higher than that of
the c-means method by about 8%.
It is known that the silhouette score and the Davies–Bouldin index are used to evaluate
clustering quality. Based on the analysis of the initial data, it was found that the silhouette
score exceeds 0.7, and the Davies–Bouldin index does not exceed 0.12. This confirms the
high quality of the proposed WT clustering method.