Wavelet Transform Cluster Analysis of UAV Images for Sustainable Development of Smart Regions Due to Inspecting Transport Infrastructure

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.

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 [1,2,3,4,5,6], 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 infrastructure 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 [9,10,11]. 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;
• 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) [13,14,15] 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 extremum, 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 [20,21,22,23,24,25]. 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 general 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 utilisation 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 attention 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 uncertainty, 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 [32,33,34,35,36].

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 assessment 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 probabilistic 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 [44,45,46,47,48,49].

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 number of clusters and selecting an appropriate clustering method for the specific application problem. A huge number of methods [51,52,53] 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 assessing the ratio of data variance within the cluster to the distance between clusters. These metrics have several disadvantages. The procedure for calculating Hubert’s statistics [47,48,49] is characterised by a relatively low degree of formalisation (the number of clusters is determined 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 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 [30,50,61,62,63]. 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 inspecting 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.

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 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 [1,3,4,5]. 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 [5,6,70,71,72].

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 systems 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).

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, including 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 $\alpha$, false positives (FP)) and second (errors $\beta$, 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:

$$L(\alpha ,\beta )\le min\alpha min\beta ,$$

(1)

where $L$ is 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.
• 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.

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, navigation, 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 parameters of the UAV, such as the frame’s overlap (horizontal and vertical), UAV flight speed, and UAV flight altitude.

Figure 4 shows the UAV flight pattern during inspection mission ensuring frame overlap.

The algorithm of defining the base composition and mission parameters (flight parameters of the UAV) to obtain images with a given resolution (${\mathrm{R}\mathrm{e}\mathrm{s}}_{\mathrm{r}\mathrm{e}\mathrm{q}}$) in a set time of inspection (${\mathrm{T}}_{\mathrm{r}\mathrm{e}\mathrm{q}}$) for an OTI with the given dimensions (${\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}}_{\mathrm{O}\mathrm{T}\mathrm{I}\_\mathrm{v}}{;\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}}_{\mathrm{O}\mathrm{T}\mathrm{I}\_\mathrm{h}}$) is running by following nine steps (Figure 5):

Step 1. Obtaining input data and requirements.

Step 2. Calculate the required viewing width (${\mathrm{h}}_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}}$) for the camera with the number of pixels of the matrix horizontally $\left({\mathrm{R}\mathrm{e}\mathrm{s}}_{{\mathrm{c}\mathrm{a}\mathrm{m}}_{\mathrm{h}}}\right)$:

$${\mathrm{h}}_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}}={\mathrm{R}\mathrm{e}\mathrm{s}}_{\mathrm{r}\mathrm{e}\mathrm{q}}·{\mathrm{R}\mathrm{e}\mathrm{s}}_{\mathrm{c}\mathrm{a}\mathrm{m}\_\mathrm{h}},$$

(2)

Step 3. Calculate the required UAV flight altitude (${\mathrm{h}}_{\mathrm{U}\mathrm{A}\mathrm{V}}$) with a camera that has a viewing angle of $\alpha$:

$${\mathrm{h}}_{\mathrm{U}\mathrm{A}\mathrm{V}}={\mathrm{h}}_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}}/(2+tg\left(\alpha /2\right)),$$

(3)

Step 4. Calculate the camera frame height (${\mathrm{v}}_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}}$) for the camera with the number of pixels of the matrix vertically ${(\mathrm{R}\mathrm{e}\mathrm{s}}_{\mathrm{c}\mathrm{a}\mathrm{m}\_\mathrm{h}})$:

$${\mathrm{v}}_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}}={\mathrm{R}\mathrm{e}\mathrm{s}}_{\mathrm{r}\mathrm{e}\mathrm{q}}·{\mathrm{R}\mathrm{e}\mathrm{s}}_{{\mathrm{c}\mathrm{a}\mathrm{m}}_{\mathrm{v}}},$$

(4)

Step 5. The flight speed of the UAV (${\mathrm{V}}_{\mathrm{U}\mathrm{A}\mathrm{V}}$) is determined by the following condition:

$${\mathrm{V}}_{\mathrm{U}\mathrm{A}\mathrm{V}}\le {\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{q}}_{\mathrm{c}\mathrm{a}\mathrm{m}}·{(\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}}_{\mathrm{O}\mathrm{T}\mathrm{I}\_\mathrm{v}}·({\mathrm{v}}_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}}-{\Delta \mathrm{v}}_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\_\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{l}\mathrm{a}\mathrm{p}})),$$

(5)

Step 6. The flight time of the UAV to fly over a given space of OTI is determined:

$${\mathrm{t}}_{\mathrm{U}\mathrm{A}\mathrm{V}\_\mathrm{f}\mathrm{l}\_\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}={\mathrm{V}}_{\mathrm{U}\mathrm{A}\mathrm{V}}·{\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}}_{\mathrm{O}\mathrm{T}\mathrm{I}\_\mathrm{v}}/{(\mathrm{S}\mathrm{i}\mathrm{z}\mathrm{e}}_{\mathrm{O}\mathrm{T}\mathrm{I}\_\mathrm{h}}/({\mathrm{h}}_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}}-{\Delta \mathrm{h}}_{\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\_\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{l}\mathrm{a}\mathrm{p}})),$$

(6)

Step 7. The condition ${\mathrm{t}}_{\mathrm{U}\mathrm{A}\mathrm{V}\_\mathrm{f}\mathrm{l}\_\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}\le {\mathrm{T}}_{\mathrm{r}\mathrm{e}\mathrm{q}}$ is checked. If the condition is not met, the number of UAVs (required to fulfil it) is determined (Step 8):

$${\mathrm{N}}_{\mathrm{U}\mathrm{A}\mathrm{V}}=⌈{\mathrm{t}}_{\mathrm{U}\mathrm{A}\mathrm{V}\_\mathrm{f}\mathrm{l}\_\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}/{\mathrm{T}}_{\mathrm{r}\mathrm{e}\mathrm{q}}⌉.$$

(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.

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]:

$$\mathit{c}\left[n\right]=\mathit{c}[n-1]-\gamma \left[n\right]{WT}_k\left(\mathit{Q}\right(\mathit{x}\left[n\right],\mathit{c}[n-1]\left)\right),$$

(8)

where $\mathit{Q}(\mathit{x}, \mathit{c})$ is a functional that depends on the vector of parameters $\mathit{c}=(c_1,\dots ,c_N)$ and $\mathit{x}=(x_1, \dots ,x_M)$; $\gamma \left[n\right]$ is a step; $n$ is the iteration number; $k$ is the start number;

$${WT}_k\left(\mathit{Q}\left(\mathit{x}\left[n\right],\mathit{c}\left[n-1\right]\right)\right)=\left\{G_{1k},G_{2k},\dots ,G_{Nk}\right\},$$

(9)

where ${WT}_k\left(\mathit{Q}\left(\mathit{x}\left[n\right],\mathit{c}\left[n-1\right]\right)\right)$ $\mathrm{i}\mathrm{s} \mathrm{a}$ WT, which determines the movement direction toward the extremum;

$$G_{jk}=\sum _{i={\displaystyle \frac{s_k}{2}},i\ne 0}^{{\displaystyle \frac{s_k}{2}}}{\displaystyle \frac{\left(\mathit{Q}\left(\mathit{x}\left[n\right],{\mathit{c}}_j+ia\right)\right)·{\psi }_k\left(i\right)}{s_k}},$$

(10)

where $s_k$ is the length of the WF carrier at the $k$-th start ($s_k$ is an even number); $a$ is the discretisation step of the WF; ${\psi }_k\left(i\right)$ is the WF at the $k$-th start; $j=1,\dots ,N$ is the dimension of the parameter vector.

The step $\gamma \left[n\right]$ is selected to find the optimum with the gradient estimate: $\gamma \left[1\right]$ = 0.4, …, 0.6. If the sign of ${WT}_k\left(\mathit{Q}\left(\mathit{x}\left[n\right],\mathit{c}\left[n-1\right]\right)\right)$ is changing when passing through the optimum at the $n-1$ step, then $\gamma \left[n\right]=0.5· \gamma \left[n-1\right]$.

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 $s_1=12$:

$${\psi }_1\left(i\right)=\left\{\begin{array}{c}1,i=1,\dots ,{\displaystyle \frac{s_1}{2}}\\ -1,i=-1,\dots ,{\displaystyle \frac{s_1}{2}}\end{array}\right..$$

(11)

Then the hyperbolic wavelet function

$${\psi }_k\left(i\right)=\left\{\begin{array}{c}{\displaystyle \frac{1}{a_k}}\left(\left|i\right|+1\right),i>0,\\ {\displaystyle \frac{-1}{a_k}},\left(\left|i\right|+1\right),i<0\end{array}\right.,iϵ\left[-{\displaystyle \frac{s_k}{2}},+{\displaystyle \frac{s_k}{2}}\right],i\ne 0$$

(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 $a_k$ 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):

$$f_1\left(x\right)=100(x_2-x_1^2)+(1-x_1{)}^2.$$

(13)

In the function (13) at $x\in (-2.048; 2.048),$ the global minimum $f_1\left(x\right)=0$ at $x_1=1,x_2=1$. During the study, the sampling step of the WF $a=0.0005$ and the carrier length $s_k=10$.

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 optimisation 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):

$$f_2\left(x\right)=418.9829+(-x\cdot \mathit{sin}\sqrt{\left|x\right|}).$$

(14)

Here $x\in (-500; 500),$ the global minimum $f_2\left(x\right)=0$ 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 $\delta \le 10^{-2}$ in 123 out of 150 cases. The global minimum was not found when starting point values were selected outside the interval $x\in (-420; 470)$.

The noise immunity of the method using the De Jong function with the addition of noise has been studied (at $x\in (-205;205)$) (Figure 7c):

$$f_3\left(x\right)=x^2.$$

(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 $f_3\left(x\right)=\mathrm{42,000}$. 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 $\delta \le 10^{-2}$. 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 $\lambda$—space including the following procedures:

• Constructing a complete graph in Euclidean space;
• Calculating the normalised distance between all pairs of its vertices $d_i={\displaystyle \frac{{\alpha }_i}{{\beta }_{imax}}}$;
• Calculating the characteristic of the local density of the set in the neighbourhood of the $i$-th edge:

  $${\tau }_i={\displaystyle \frac{{\alpha }_i}{{\beta }_{imin}{\tau }_{max}}},$$

  where ${\alpha }_i$ is the distance between the $i$—th pair of vertices; ${\beta }_{imax}$ and ${\beta }_{imin}$ are the lengths of the longest and shortest edges, respectively; ${\tau }_{max}$ is the maximum value of ${\tau }_i^{*}={\displaystyle \frac{{\alpha }_i}{{\beta }_{imin}}}$;
• Calculating the length of edges in the graph in λ-space [60]:

  $${\lambda }_i={\tau }_i^2\times d_i.$$

Step 2. Construction of the $\lambda$-graph for the shortest open (non-circular) path according to the $\lambda p$ algorithm (see Figure 6) and considering the probability of breaking its edge [62]:

$$p_{iz}={\displaystyle \frac{{\lambda }_{iz}}{\sum _{j=1}^k{\lambda }_{iz}}},$$

where $p_{iz}$ is the probability of breaking the $i$-th edge connected to the vertex $z$; ${\lambda }_{iz}$ is the $\mathsf{\lambda }$ distance corresponding to the $i$-th edge connected to the vertex $z$ in the $\lambda$-graph of the shortest open (non-circular) path.

Step 3. Calculation of the parameter (see Figure 6)

$$q_i={\displaystyle \frac{f(i+1)}{f\left(i\right)}},i\in [1,n-1],$$

(16)

where $f_i$ is the average $\lambda$—distance when adding the $i$-th value to the data cluster.

Step 4. Search for the global maximum of the multi-extremal $Q=\mathrm{m}\mathrm{a}\mathrm{x} \left(q_i\right)$ 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 $X_{30}$ 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 $Log\left(q_i\right)$ 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 $Q=\mathit{max}(q_i)$ successively, excluding from consideration the data assigned to the previous cluster when evaluating $q_i$.

Thus, based on finding optima with WT, the number of clusters is determined after transformation to the $\lambda$–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 $\mathit{c}=c_{opt}$. That provides the extreme value of the quality functional $Q(\mathit{x},\mathit{c})$ (see Figure 6, block 7) relative to the vector of variables $\mathit{c}=(c_1,\dots ,c_N)$, depending on the vector of random sequences $\mathit{x}=(x_1,\dots ,x_M)$. 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):

$$\left.\begin{array}{c}{c}_1\left[n\right]=c_1[n-1]-{\gamma }_1\left[n\right]{\tilde{\nabla }}_{c1+}Q\left(\mathit{x}\right[n],c_1[n-1],c_2[n-1\left]\right)\\ c_2\left[n\right]=c_2[n-1]-{\gamma }_2\left[n\right]{\tilde{\nabla }}_{c2+}Q\left(\mathit{x}\right[n],c_1[n-1],c_2[n-1\left]\right)\end{array}\right\},$$

where ${\gamma }_1\left[n\right], {\gamma }_2\left[n\right]$ are the step sizes;

$n$ is the iteration number;

${\tilde{\nabla }}_{c1+}Q\left(x\left[n\right],c_1\left[n-1\right],c_2\left[n-1\right]\right)$ is the assessment of the movement direction towards the extremum with WT for the first cluster (see Figure 6, block 9);

${\tilde{\nabla }}_{c2+}Q\left(x\right[n],c_1[n-1],c_2[n-1\left]\right)$ is the assessment of the movement direction towards the extremum with WP for the second cluster (see Figure 6, block 9);

$Q\left(x,c_1,c_2\right)={\sum }_{b=1}^2{\epsilon }_k\left(x,{\mathrm{c}}_1,{\mathrm{c}}_2\right)F_b(x,c_1,c_2)$ is the implementation of the quality functional;

$F_b(x, c_1,c_2)$ is the distance function of elements $\mathrm{x}$ from the set $\mathrm{X}$ to the ‘centres’ of the clusters;

${\epsilon }_b\left(·\right)$ are characteristic functions, where:

$${\epsilon }_b\left(x,c_1,c_2\right)=\left\{\begin{array}{c}1,x\in X_{b,}\\ 0,x\notin X_b.\end{array}\right..$$

After initialising the parameters per each of the $i$ elements for the weighted sum with WP, the values of the characteristic functions ${\epsilon }_b\left(\mathrm{x},c_1,c_2\right)$ at $b=1,2$ are determined.

For this, pairs of values are:

$$\begin{array}{l}{c}_1[n-1],c_2[n-1],\\ c_1[n-1]\pm i{e}_{1}a\left[n\right],c_2[n-1],\\ c_1[n-1],c_2[n-1]\pm i{e}_{2}a\left[n\right] (i=\overline{1,N}),\end{array}$$

where $a\left[n\right]$ is a scalar, $N$ is the length of the wavelet function carrier. Then the scalar for a given $x\left[n\right]$ is substituted in the expression:

$$f\left(\mathrm{x},c_1,c_2\right)={‖\mathit{x}\left[n\right]-c_1‖}^2-{‖\mathit{x}\left[n\right]-c_2‖}^2.$$

The function $f(\mathit{x},c_1,c_2)$ is zero on the boundary between two clusters and it has different signs on either side of the boundary. Therefore, if the value of $f(\mathit{x},c_1,c_2)$ is negative, then ${\epsilon }_1=1,{\epsilon }_2=0$. In the opposite case ${\epsilon }_1=0,{\epsilon }_2=1$.

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 $C_{zj}^{tm}$ and supervised labels are given, indicating that the patterns belong to the $m$-th cluster out of $\mathrm{M}$ clusters. Here $z=\overline{1,Z}$, where $Z$ is the dimension of the feature space; $j=\overline{1,L}$ is the pattern number belonging to the cluster; $L$ is the total number of cluster patterns; $t=\overline{1,T}$ 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:

$$d_{rs}(C_{zr}^{tm},C_{zs}^{0m})=\left\{\begin{array}{c}0,if{C}_{zr}^{tm}=C_{zs}^{0m}\\ 1,if{C}_{zr}^{tm}\ne C_{zs}^{0m}\end{array}\right.,$$

(17)

where $C_{zs}^{0m}$ is the supervised labels indicating that the pattern belongs to the cluster; $C_{zr}^{tm}$ is the result of clustering by the method being studied.

Let us define the normalised distance between vectors, which characterise the difference 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 D for the applied clustering method $t$ using expression:

$$D_{t,0}={\displaystyle \frac{1}{L}}\sum _{j=1}^L\sum _{m=1}^M\sum _{z=1}^Z{d}_{jj}\left(C_{zj}^{tm},C_{zj}^{0m}\right).$$

(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 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:

$$P=P_1+P_2,$$

(19)

where $P_1$ is the probability of correctly assigning a pattern to a cluster, characterising type 1 error; $P_2$ 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):

$$R={\mathit{log}}_2{R}_i-{\mathit{log}}_2{R}_0={\mathit{log}}_2{\displaystyle \frac{R_i}{R_0}},$$

(20)

where $R_i$ and $R_0$ 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 m² 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:

$$t={\displaystyle \frac{S_i}{{0.8W}_{s}V}},$$

(21)

where $S_i$ is the defined area of the inspection region, $W_s$ is the width of the viewing strip considering a 10% frame overlap, $V$ is 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:

$$W_s=2h\ast tg{\displaystyle \frac{\alpha }{2}},$$

(22)

where $\alpha$ 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.

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 significantly 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:

$$N_{UAV}=⌈{\displaystyle \frac{t}{T_{req}}}⌉,$$

(23)

where $T_{req}$ 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 recommended, 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:

$$q={\displaystyle \frac{q_p}{q_0·∆}},$$

where $q_p$ and $q_0$ are the standard deviations of parameters in clusters in the testing and training phase, respectively, Δ 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 6 the 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.

Since WT clustering methods are characterised by increased quality and efficiency, they can be recommended for clustering tasks with high levels of noise and/or complex cluster shapes, as well as for a wide range of applied problems solved in OTI that satisfy these criteria. The obtained results enable to recommend the developed clustering methods for unsupervised classification tasks, if necessary to increase the reliability of decisions made by OTI with small sets of training parameters, intersecting clusters, and a high level of noise in diagnostic parameters. The decision to choose a clustering method and its parameters can be made after studying the influence of the method on the overall system efficiency.

The influence of varying the wavelet function parameters on the time to search for the clustering centre coordinates and on the speed of evaluation was investigated too. It was found that when changing the studied initial values parameter of WF $s_k$ from 10 to 20 at the size $\gamma$ = 0.4, then the search time for clustering changing centres coordinates by less than 10%.

In addition, we assessed the influence of step size $\gamma \left[n\right]$ on the evaluation speed. Thus, at the initial value $s_k$ = 12 and changing $\gamma$ from 0.4 to 0.6, the time of defining the coordinates of extrema is increased by 5%. Herewith, for coordinates of the clusters center for the test data set, the relative errors in the determination are no more than 1.8%. In the study of noise immunity, the relative error in determining coordinates for clusters center is less than 4% with a signal-to-noise ratio in amplitude of 20 … 10. These results allow us to recommend the developed method for the proper monitoring of images from UAVs.

4. Discussion

The sustainable development of CSs/SRs is inextricably linked to the sustainable development of TI. The growing workload on TI and the requirements for TI sustainability indicators are leading to increased attention to determining its current state. Traditional approaches to inspection have a few disadvantages due to the limited number of means for performing traditional inspection procedures involving large-sized means, which leads to limitations in the throughput of the TI [34].

In addition, an increase in the number of TIs and the workload on them due to an increase in traffic may lead to the need to reduce the inspection intervals. The number of vehicles and personnel to be involved in this case and the amount of inspection data to be analysed may increase many times over, leading to higher costs, both in terms of resources and time, for the inspection and limiting the use of TI reducing the number of transport accidents caused by the unsatisfactory condition of the TI, as in the following:

• Reducing the average journey time/transportation price and maintenance of TI capacity through timely measures to maintain the proper condition of the TI;
• Reducing the amount of harmful transport emissions.

The authors have identified the stages in the process of assessing the condition of the TI, which are not clearly described in existing works [18,20,25], as follows:

• Determination of the structure and composition of UAV-IS-OTI-SCs and the specifics of UAVs’ use during inspection;
• Cluster analysis of a pre-processed set of collected data and quality assessment using wavelet transform.

As a result, the structure of UAV-IS-OTI-SCs in the form of a multi-agent system using ontologies of different levels is proposed, and it allows:

• The determining of the list of UAVs and their equipment for inspection in the current environmental conditions;
• The calculation of the required number of UAVs to ensure the fulfilment of functional requirements (such as system performance and data resolution), and non-functional requirements (such as dependability and safety) to the system during the inspection.

To address the identified problematic issues, the authors propose to inspect TI using UAVs and improve data processing methods based on the quality assessment of cluster analysis for a pre-processed set of collected data. The developed methods and tools are aimed at maintaining the OTI at a level that ensures sustainable development, as follows:

• Reducing of the number of transport accidents caused by the unsatisfactory condition of the TI;
• Decreasing the average journey time/transportation price and maintenance of TI capacity through timely measures to maintain the proper condition of the TI;
• Reducing the amount of harmful transport emissions.

Authors propose the new integrated approach using the wavelet transform and providing the analysis of the efficiency and quality of clustering data obtained by UAVs. That enables expanding the capabilities of the information approach when choosing options and assessing the parameters of this procedure to adapt the parameters of processing procedures, when a high level of noise and/or complex cluster shapes exists. As a result, that enabled to improve the quality of processing and analysis of images obtained from UAVs, including due to the increased noise immunity through the WT employment.

Furthermore, implementation of the proposed WT clustering method has been demonstrated to enhance the quality of clustering with some improvements, as follows:

• A 10% enhancement in comparison to the k-means method;
• An approximately 8% enhancement in comparison to the c-means method.

The objective of enhancing the quality of clustering is to diminish the occurrence of the first and second types of errors in the determination of the state of OTT, which can potentially give rise to errors of the first and second types. In the case for errors of the first type, an OTI being in a satisfactory condition is erroneously determined to be out of compliance. It results in the implementation of measures, which are not necessary, and that leads ultimately to a deterioration in throughput, average travel time, and emissions of harmful substances. In contrast, errors of the second type may result in the emergency facility being assessed as satisfactory, increasing the probability of accidents and leading, subsequently, to a decline in throughput, average journey time, and emissions.

In addition, authors would like to mention some limitations of the study, which is focusing on transportation infrastructures such as bridges and highways. The applicability of the proposed method and its effectiveness to different types of transportation infrastructure and in different environments requires more extensive (in-depth) research and this can be a subject of future works. Furthermore, while WT improves clustering quality, it may require significant computational resources. One way to reduce dependence on computing power is to compress the obtained images with quality control [78,79]. The authors would also like to note that when estimating the number of clusters for the test example, the initial carrier length of the Haar WF was set to a fixed value, which may vary in practice. Therefore, one of the directions for future research could be determining this parameter in applied tasks during the processing of images from UAVs.

Besides, the authors recognise that there are still several challenging issues that require attention in the process of implementing the proposed approach to developing an inspection system and data processing method, which were not addressed in this study, which are as follows:

• Development of planning models for the use of UAVs for inspection of various TDFs;
• Development of control systems for UAV fleets that will autonomously inspect CCAs;
• Implementation of IT infrastructure for transmission, storage, and implementation of inspection data processing methods;
• Development of models for determining quantitative estimates of changes in the indicators of sustainable development of TI as functions with parameters: system performance indicators, data processing quality.

Therefore, the main areas for further research in the deployment of inspection systems as intelligent mobile systems include: an in-depth study of ontologies and models of UAV fleets, the development of procedures for improving the efficiency and quality of data processing, and ways to use them in inspection systems for OTI aimed at improving their sustainability. Among other things, it is promising to study clustering methods to create adaptive systems that can vary their own parameters depending on changes in observation conditions and the impact of interference.

Finally, we should mention, the developed methods can be integrated with other Smart City technologies, for instance, the Internet of Things and big data platforms, to improve existing bridge condition monitoring systems [80], as well as facilitate city operations management.

5. Conclusions

This paper presents the results of research aimed at improving the sustainability of SCs/SRs TI by improving the stages of assessing their technical condition, namely, the use of UAVs for collecting and processing data during the inspection of OTI.

The obtained results are aimed at improving sustainability indicators, such as: the number of transport accidents; average travel time/transportation cost; TI throughput; and the amount of harmful emissions. The improvement of indicators is achieved by increasing the efficiency and quality of assessing the technical condition of transport infrastructure using UAV-IS-OTI-SCs and the integrated application of WT clustering accompanied by an assessment of its quality and efficiency based on the semantic measure [77]. This generally ensures the sustainable development of Smart Regions. An approach to the construction of UAV-IS-OTI-SCs in the form of the multi-agent system is proposed using the ontologies of different levels.

To improve the quality of clustering of the obtained data, it is necessary to consider the specifics of UAV flight planning, depending on the type and size of the defects in the design of OTI. The use of UAVs increases the efficiency of inspections by reducing costs, increasing the volume of data, and improving its quality. However, time constraints on the duration of the inspection may require the use of UAV swarms, as well as the creation and deployment of infrastructure to support long-term OTI inspection missions.

The proposed approach using the wavelet transform provides the analysis of the efficiency and quality of clustering data obtained by UAVs. A comparison of crisp and fuzzy clustering with WT and clustering methods: $c$-means and $k$-means is completed. It has been established that when changing the standard deviation of data within clusters, the quality of WT clustering is 10% higher, on average, compared to $k$-means method, and it is 8% higher than the $c$-means method in fuzzy clustering with WT. The assessment showed that WT clustering has increased quality and efficiency, and reduced sensitivity to the initial (starting) search point. Hence, we may recommend using WT clustering when a high level of noise and/or complex shape of clusters exists.

In general, using the proposed method enables quantitative determination of the efficiency and quality for clustering, any comparative assessments, and the impact of clustering procedures on the overall effectiveness of autonomous transport systems.

More detailed results reveal the stages of using UAVs in performing inspection tasks by describing the planning and features of building a UAV-based system for the inspection of TI Scs and clustering procedures during the processing of the obtained data. They are generally aimed at improving or maintaining the required level of sustainability indicators of TI Scs/SRs.

In the future, the authors are going to study deeper the planning process of UAV utilisation for OTI status assessment, and clustering methods for creating adaptive systems that can vary own parameters depending on changing observation conditions and interference influence.