Nomenclature

Below are the technical expressions for some of the variables used in Fig. 2

GSD

ground sample distance

Vertices

ordered vertex coordinates of the closed polygon ( $\left[ {N \times 2} \right]$ as $[{x;\;y}]$ )

${p_{Endlap}}$

forward-overlap ratio ( $0 \lt p \lt 1$ )

${q_{Sidelap}}$

side-overlap ratio $0 \lt q \lt 1$

$H$

Flight altitude (meters)

$ImageWidth$

image width in pixels

$ImageHeight$

image height in pixels

$f$

focal length of camera as meter

$ccdWidth$

width of imaging sensor as meter

$azimuth$

direction angle for the flight start (in degrees)

$flightPaths$

cell array containing $x$ and $y$ spatial coordinates for each flight path

$waypoints$

combined waypoints of all flight paths ( $M \times 2$ sized matrix)

$totalTurns$

total number of turns made by the drone

$dp$

vertices of turn locations

The following are the technical expressions for some of the variables used in Fig. 3.

$U\left( {lo{w_j},u{p_j}} \right)$

continuous uniform distribution with parameters $a$ and $b$

$U\left\{ {1,2,3} \right\}$

discrete uniform distribution with parameters $a$ and $b$

$ObjFun$

objective function

$rperm\left( a \right)$

randomly shuffles the positions of integer values between [1, a] according to a uniform

distribution

${\rm{shuffle}}\left( {a,b} \right)$

randomly shuffles integers [1, $a$ ], returns first $b$ values

$rperm\left( {a;\;b} \right)$

randomly shuffles the positions of integer values between [1, a] according to a uniform

distribution and returns the first b of them

$rnd\left( {} \right)$

generates a random real-valued number between [0,1] according to the rules of a

uniform distribution each time it is called

$rnd\left( {a;b} \right)$

generates a 2D matrix with ‘a’ rows and ‘b’ columns each time it is called; he elements

of the matrix are random values within the range [0,1], following the rules of uniform

distribution

${\rm{sign}}\left( \cdot \right)$

returns sign of numeric value ( $ \cdot $ )

⊘

element-wise division operator (Hadamard division operator)

$sign\left( \cdot \right)$

returns the sign of the numeric value (.)

$ \circ $

element-wise multiplication operator (Hadamard multiplication operator)

$LevyDist\left( {\alpha ,\beta ,N,1} \right)$

generates a matrix with uniform random-valued elements having N rows and 1 column,

using a Levy distribution with parameters $\left( {\alpha ,\beta } \right)$

$zeros\left( {a,b} \right)$

generates a 2D matrix of size [a b] with all elements set to zero

$ \lceil\cdot\rceil $

In discrete mathematics, represents the ceiling function

$sortindex\left( {a,{\rm{'}}ascend{\rm{'}}} \right)$

sorts the elements of the vector ‘a’ in ascending order and returns the resulting sorted

vector

$rndi\left( {\left[ {c,d} \right]} \right)$

generates a uniform single integer value between [c d]

$abs\left( \cdot \right)$

absolute value function

2.0 Flight path planning for aerial imaging

Aerial imaging-based topographic mapping methods can only capture aerial images that meet specific geometric requirements through the use of an optimised FMP process. Therefore, FMP is an essential process for aerial imaging-based topographic mapping applications. From the perspective of aerial imaging-based mapping methods, FMP ensures that the captured aerial images achieve the desired numerical overlap limits in both forward and side directions. This process facilitates the mutual orientation of consecutive images and enables the generation of a 3D stereo model of the imaged area. To meet the need for high-accuracy aerial mapping production, it is crucial to pre-plan an adequate GSD [Reference Yang55–Reference Lee and Sull58] value during the initial step of FMP. In the experiments conducted in this paper, TUSAGA-AKTIF was used for the global positioning of the micro-UAV. TUSAGA-AKTIF is a GPS-based real-time global positioning network [59].

Variations in elevation on the topographic surface imaged using micro-UAVs can cause numerical fluctuations in the GSD. Additionally, micro-UAV platforms generally have very limited onboard energy resources. For this reason, optimising the flight plan to minimise fluctuations in the GSD is a necessity for efficient use of the micro-UAV’s onboard energy. The general analytical definition of GSD is provided in Equation (1).

(1) \begin{align}{\rm{GS}}{{\rm{D}}_{{\rm{meter}}/{\rm{pixel}}}} = \frac{{{H_{meter}} \cdot ccdWidt{h_{meter/pixel}}}}{{{f_{meter}}}} = \frac{{{H_{meter}} \cdot SensorWidt{h_{mm}}}}{{{f_{mm}} \cdot ImageWidt{h_{pixels}}}},\end{align}

where, $H$ represents the flight altitude (i.e. elevation hight), $SensorWidth$ denotes the physical size of the sensor in millimetres, and $f$ corresponds to the focal length. $ImageWidth$ and $ccdWidt{h_{meter/pixel}}$ denote the width of the image, measured in pixels, and the width of a single pixel in the camera’s CCD, measured in meters, respectively. In this paper, altitude measurements were obtained using barometric methods.

The height of a single image footprint corresponds to the flight direction of the micro-UAV. The width of a single image footprint on the ground ( $GSD{W_{{\rm{meter}}}}$ ) and the height of a single image footprint on the ground ( $GSD{H_{{\rm{meter}}}}$ ) are calculated using Equations (2) and (3), respectively.

(2) \begin{align}GSD{W_{{\rm{meter}}}} = GS{D_{meter/pixels}} \cdot ImageWidt{h_{pixels}},\end{align}

(3) \begin{align}GSD{H_{{\rm{meter}}}} = GS{D_{meter/pixels}} \cdot ImageHeig{h_{pixels}}.\end{align}

Aerial imaging based mapping methods necessitates a sufficient degree of image overlap to ensure accurate processing and reconstruction. Specifically, forward overlap, which refers to the overlapping area along the flight path, is typically maintained within a range of 60 ${\rm{\% }}$ to 80 ${\rm{\% }}$ . Similarly, side overlap, representing the overlapping region between adjacent flight lines, generally falls within the range of 40 ${\rm{\% }}$ to 70 ${\rm{\% }}$ . These overlap parameters are critical for achieving high-quality results in photogrammetric analyses.

The distance between adjacent images, $Endla{p_{meter}}$ , and distance between side-flight lines, $Sidela{p_{meter}}$ , are calculated by using Equations (4) and (5):

(4) \begin{align}Endla{p_{meter}} = GSD{W_{meter}} \cdot \left( {1 - {p_{Endlap}}} \right),\end{align}

(5) \begin{align}Sidela{p_{meter}} = GSD{H_{meter}} \cdot \left( {1 - {q_{Sidelap}}} \right),\end{align}

where ${p_{Endlap}}$ and ${q_{Sidelap}}$ denote forward-overlap and side-overlap values of aerial images as ${\rm{\% }}$ , respectively.

In this paper, the 2D vertex coordinates of the polygon for which the relevant flight plan is to be created have been defined using the WGS84 datum and the EPSG:3395 (WGS84/World Mercator) projection system. The Endlap and Sidelap values were used to completely cover the 2D polygon surface, where the corresponding FMP is to be performed, with a point-mesh grid of size $\left[ {{\rm{Endlap\;\;Sidelap}}} \right]$ . The central position of the generated point-mesh coincides with the centre of the corresponding 2D polygon. Each point in the point-mesh represents a waypoint. Each waypoint corresponds to an aerial image capture location. By optimising the $azimuth$ of the Point-Mesh relative to the centre of the 2D polygon, the number of waypoints remaining inside the polygon is maximised, thereby achieving maximum coverage that satisfies the flight parameters over the corresponding 2D polygon. In this paper, the relevant $azimuth$ value is calculated using an evolutionary optimisation algorithm called the colony-based search algorithm (CSA) [Reference Civicioglu and Besdok51]. The waypoints are organised as an interlacing-path to generate the flight path. The number of turns on the interlacing-path corresponds to the total number of turns.

An FMP with an optimal flight path and conditions designed for a micro-UAV facilitates the efficient use of on-board energy. In Fig. 1, two different non-optimal FMPs, calculated for $azimuth = {30^0}$ and $azimuth = - {30^0}$ , are shown in red and green, respectively, over the test polygon used in the experiments presented in this paper. The additional flying mission parameters for these flights are provided as follows: ${p_{Endlap}}$ = 0.80, ${q_{Sidelap}}$ = 0.70, H = 200 m, ImageWidth = 5472 pixels, ImageHeight = 3648 pixels, $f = 8.8 \cdot {10^{ - 3}}$ m, and $ccdWidth = 2.41 \cdot {10^{ - 6}}$ m. For an FMP that ensures optimal conditions, the $azimuth$ angle should be optimised according to the objective function.

Figure 1. Blue-coloured convex polygon is used for FMP optimisation tests. The spatial coordinates are defined in the datum of WGS84 and projection system of EPSG:3395 (WGS84/World Mercator). The red-coloured FMP is associated with an $azimuth = {30^0}$ , whereas the green-coloured FMP corresponds to an $azimuth = - {30^0}$ .

The pseudo-code of the FMP method proposed in this paper is provided in Fig. 2, where the lines #1–4 compactly express the calculation processes related to the GSD value. The process for calculating Endlap and Sidelap given in lines #5–6 and lines #14–28 of Fig. 2 analytically and compactly describe the processes for constructing the interlacing path, calculating waypoint positions, and determining the total number of turns.

Figure 2. The pseudo-code of the FMP algorithm, proposed for computing waypoints and total turns.

In this paper, the relevant FMP problem has been solved separately for four different scenarios.

The endurance of micro-UAVs in the air is generally limited due to the restricted capacity of their on-board energy sources. Since the flight endurance is a function of the total flight path length, the classical approach for FMP aims to minimise the total flight path length. The first FMP problem is based on minimising the total flight path length. This optimisation was achieved by applying Equation (6).

(6) \begin{align}ObjFun\;:{\rm{\;}}\underbrace {{\rm{arg\;min}}}_{{\rm{azimuth}}}{\rm{\;\;TotalDistance}} = \mathop \sum \limits_{i = 1}^{n - 1} \sqrt {{{({x_{i + 1}} - {x_i})}^2} + {{({y_{i + 1}} - {y_i})}^2} + {{({z_{i + 1}} - {z_i})}^2}}, \end{align}

where, the flight path calculated as a result of the FMP is represented by a list of waypoints, ${(x,y,z)_i}$ indicates the 3D spatial position of the ${i^{th}}$ waypoint.

In aerial imaging-based a-SfM applications utilising micro-UAVs, the accuracy levels of the calculated topographic surface points may exhibit undesirable local fluctuations due to variations in GSD values. Therefore, it may be necessary to prepare the FMP in a manner that ensures relatively stable GSD values along the waypoints. The second FMP planning problem aims to find the narrowest 95 ${\rm{\% }}$ confidence interval for the GSD values calculated for the image capture points along the flight path. This ensures that the GSD values for the image capture points, and consequently the flight path, have the minimum average and standard deviation. This problem is modelled by Equation (7).

(7) \begin{align}ObjFun\;:\underbrace {\arg {\rm{min}}}_{azimuth}{\rm{\;}}\left( {{\mu _{GSD}} + 3 \cdot {\sigma _{GSD}}} \right)\end{align}

where ${\left[ {\mu ,\sigma } \right]_{GSD}} = \left[ {mean\left( {GSD} \right),std\left( {GSD} \right)} \right]{\rm{\;\;\;\;\;\;\;}}|{\rm{\;\;\;\;\;\;}}GSD = \left[ {GS{D_{{{\left( {x,y,z} \right)}_i}}}} \right]$ denote the mean and standard deviation of GSD values.

The turning manoeuvres of micro-UAVs involve complex processes, and the efficiency of on-board energy usage increases as the number of turns decreases. In a-SfM applications based on micro-UAV-supported aerial imaging, the accuracy levels of the calculated topographic surface points may exhibit undesirable numerical fluctuations due to the influence of GSD values by local topographic height variations. Therefore, it may be necessary to prepare an FMP that ensures a relatively stable GSD value along the waypoints while minimising the number of turns. The third FMP problem aims to minimise the 95 ${\rm{\% }}$ confidence interval for the GSD values calculated for the image capture points along the flight path and to minimise the number of required turns. This problem is modelled using Equation (8).

(8) \begin{align}ObjFun\;:{\rm{\;\;}}\underbrace {\arg {\rm{min}}}_{azimuth}{\rm{\;}}\left[{\left( {{\mu _{GSD}} + 3 \cdot {\sigma _{GSD}}} \right) + totalTurns} \right].\end{align}

Micro-UAVs commonly utilise the Stop-Turn-Go strategy to complete their turning manoeuvres. The number of turning manoeuvres is a result of the FMP. Therefore, it may be necessary to prepare an FMP that minimises the number of turns for energy conservation. The fourth FMP planning problem aims to minimise the total number of turns, $totalTurns$ (see lines #25-#31 of the Fig. 2). This problem is modelled using Equation (9).

(9) \begin{align}ObjFun\;:{\rm{\;}}\underbrace {\arg {\rm{min}}}_{azimuth}{\rm{\;}}totalTurns\end{align}

In this paper, CSA has been employed to address the planning problems formulated in Equations (6)–(9) for determining the optimal $azimuth$ value. Fundamentally, CSA is responsible for computing the $azimuth$ angle of the flight paths as defined in line #8 of Fig. 2. Once this $azimuth$ angle is determined, the flight plan parameters that satisfy the corresponding geometric constraints can be calculated using the algorithm presented in Fig. 2.

3.0 Colony-based search algorithm (CSA)

The CSA [Reference Civicioglu and Besdok51] was developed as an evolutionary-based global minimiser tailored for addressing single-objective, bounded or unbounded real-valued numerical problems. Within the current iteration, CSA focuses solely on evolving the patterns within the clan matrix, which is formed by randomly chosen patterns from a parent population referred to as the colony matrix. CSA operates as a non-recursive, iterative and reasonably robust approach. Its framework comprises initialisation, selection, numerical evolution and updating phases. The mutation mechanism in CSA is integrated with the random crossover process, collectively termed the morphogenesis process. The algorithmic design of CSA is thoroughly elaborated, step-by-step, in the subsequent descriptions.

The primary population of CSA, termed the Colony Matrix, $p0$ , comprises a random solution vector with dimensions equivalent to the Clan Matrix, $p$ , multiplied by $T$ time. The initial Colony Matrix, $p0$ , along with the objective function values of the Colony patterns, $fitp0$ , are determined through the methodology outlined in Equation (10).

(10) \begin{align}\left\{ {\begin{array}{l}{p{0_{i = 1:T \cdot N,j = 1:D}}\sim U( {lo{w_j},u{p_j}} )}\\[5pt] {fitp0 = {\rm{ObjFun}}\left( {p0} \right)}\end{array},} \right.\end{align}

where $ObjFun$ , $low$ , and $up$ denote the objective function, the lower and upper bounds of the search space, respectively. In this study, the Mersenne Twister uniform pseudo-random number generator is employed to produce random numbers utilised across all experimental methods. In CSA, the initial values of $moment$ and $initindex$ are set to $momen{t_{initindex}} = 0$ and $initindex = \left\{ {1:N} \right\}$ , respectively.

Equation (11) can be used to express the clan, $p$ , and objective function values of the $p$ , $fitp$ , patterns.

(11) \begin{align}[ {p;{\rm{\;\;}}fitp} ] = [ {p0( {index,:} );{\rm{\;\;}}fitp0( {index} )} ]\end{align}

where $index = rprm\left( {T \cdot N;\;N} \right)|\left( {initindex\sim = inex} \right)$ and updated $initindex = index$ . Equation (12) is used to generate the scale factor, or $scale$ , that the CSA uses to regulate the direction vector’s amplitude:

(12) \begin{align}scale = \left\{ {\begin{array}{l@{\quad}l}{\left( {rand\left( {N;\;c} \right) - 0.50} \right)\oslash\left( {rand\left( {N,c} \right) - 0.50} \right)} & {}{If{\rm{\;\;}}rand() \lt rand()}\\[5pt] {sign\left( {rand\left( {N;\;1} \right) - 0.50} \right) \circ {\rm{\Phi }}} {}& {else}\end{array}} \right.\end{align}

where ${\rm{\Phi }}\sim LevyDist\left( {\alpha ,\beta ,N,1} \right)$ and $\oslash$ denote scaling vector, and Hadamard division operator, respectively. The individuals of ${\rm{\Phi }}$ have Levy distribution with $\alpha ,\beta $ location and shape parameters. ${\rm{\Phi }}$ supplies Levy flight ability to CSA. In the CSA framework, Levy flights provide the essential exploration capability needed to overcome local optima and identify a variety of high-quality solutions within intricate search spaces. By incorporating sporadic long jumps, Levy flights strengthen the global search process, enhancing algorithm robustness and reducing the likelihood of stagnation. The Levy distribution, distinguished by its heavy-tailed nature, is a probability distribution that exhibits a greater tendency to generate extreme values than the normal distribution. It is characterised by its probability density function (PDF), as specified in Equation (13).

(13) \begin{align}f\!\left( {x;\;\alpha ,\beta } \right) = \frac{\beta }{{2\pi }}{\rm{exp}}\left( { - \frac{\beta }{{2\left( {x - \alpha } \right)}}} \right)\left( { - \frac{1}{{{{\left( {x - \alpha } \right)}^{3/2}}}}} \right)\end{align}

where, $x$ , $\alpha $ and $\beta $ denote the random variable, the location parameter, and the scale parameter of the distribution, respectively. In CSA, the ${\rm{\Phi }}$ used in Equation (12) is generated using Equation (14):

(14) \begin{align}{\rm{\Phi }} = \beta \cdot \left( {rand() + 1} \right)\oslash{\omega ^{\frac{1}{\alpha }}}{\rm{\;\;\;}}|{\rm{\;\;\;\;}}\omega \sim {\rm{\Gamma }}\left( {\alpha ;\;\kappa } \right),{\rm{\;\;}}\kappa \sim \left\{ {2\;:\;5} \right\}\end{align}

where ${\rm{\Gamma }}\left( {\alpha ;\;\kappa } \right)$ denotes Gamma distribution with the parameters of $\left( {\alpha ;\;\kappa } \right)$ .

The value of $c$ used in Equation (12) is computed using Equation (15):

(15) \begin{align}If{\rm{\;\;}}rand() \lt rand(){\rm{\;\;}},{\rm{\;\;}}then{\rm{\;\;}}c = 1{\rm{\;\;\;}}else{\rm{\;\;\;}}c = D\end{align}

Using the steps outlined in Equation (16), the binary valued mutation control matrix, m, of CSA is produced.

(16) \begin{align}\left\{ {\begin{array}{*{20}{l}}{m = {0_{N \times D}}}\\[5pt] {for{\rm{\;\;\;}}j = {\rm{\;}}1{\rm{\;\;\;}}to{\rm{\;\;\;}}N}\\[5pt] {ind = rperm\left( D \right)}\\[5pt] {{k_{index}} = abs(randi( {[ {0,1} ]} ) - rand {{()^{ \bullet randi( {[ {1,10} ]} )}}} )}\\[5pt] {e = ind\left( {1\;:\;\lceil{k_{index}} \cdot D}\rceil \right)}\\[5pt] {m\left( {k,e} \right): = 1}\\{end}\end{array}} \right.\end{align}

where $ \bullet $ denotes piecewise-power operator. CSA uses Equation (17) to generate evolutionary direction, $dx$ .

(17) \begin{align}dx = \left\{\begin{array}{l@{\quad}l} {p\!\left[ {v2\left] { - p} \right[v1} \right]} & {}{v = 1} \\[5pt] {p\!\left[ {v1} \right] - p}& {}{v = 2} \\[5pt] {p\!\left[ {index0\left[ {randi\left( {1,N/5} \right)} \right],:} \right] - p} & {}{v = 3} \end{array} \right.,\end{align}

where $\left[ {v1,v2} \right] = \left[ {rperm\left( {N;\;N} \right),rperm\left( {N;\;N} \right)} \right]$ , $v1 \ne v2$ and $index0 = sortindex\left( {fit,{\rm{'}}ascend{\rm{'}}} \right){\rm{\;\;}} \wedge {\rm{\;\;\;}}v\sim U\left\{ {1,2,3} \right\}$ .

The morphogenesis matrix, $px$ , of CSA is generated using Equation (18):

(18) \begin{align}px = p + scale \circ m \circ dx + s\end{align}

where $s = \left( {rand\left( {N;\;1} \right) - 0.50} \right) \circ rand{(N;\;1)^{ \bullet randi\left( {\left[ {2;10} \right]} \right)}}$ .

In CSA, Equation (19) is used to update the individuals that exceed the search boundaries:

(19) \begin{align}\begin{array}{*{20}{l}}{for{\rm{\;\;}}k = 1{\rm{\;\;}}to{\rm{\;\;}}N}\\[5pt]{for{\rm{\;\;}}l = 1{\rm{\;\;}}to{\rm{\;\;}}D}\\[5pt]{If{\rm{\;\;}}px\!\left( {k,l} \right) \lt low\!\left( l \right)\!,{\rm{\;\;}}px\!\left( {k,l} \right) = low\!\left( l \right) + rand{{()}^{randi\left( {\left[ {1;5} \right]} \right)}} \cdot \left( {up\!\left( l \right) - low\!\left( l \right)} \right)}\\[5pt]{If{\rm{\;\;}}px\!\left( {k,l} \right) \gt up\!\left( l \right)\!,{\rm{\;\;}}px\!\left( {k,l} \right) = up\!\left( l \right) + rand{{()}^{randi\left( {\left[ {1;5} \right]} \right)}} \cdot \left( {low\!\left( l \right) - up\!\left( l \right)} \right)}\\{end}\\{end}\end{array}\end{align}

The objective function values of $px$ patterns are computed using Equation (20):

(20) \begin{align}fitpx = {\rm{ObjFun}}\!\left( {px} \right)\end{align}

The orders, $ind$ of the $px$ patterns supply better solutions then the corresponding patterns are obtained using Equation (21):

(21) \begin{align}ind \leftarrow fitp{x_{\left\{ {1:N} \right\}}}\lt {fit{p_{\left\{ {1:N} \right\}}}{\rm{\;\;\;}}} |{\rm{\;\;\;\;}}ind \in \left\{ {1\;:\;N} \right\}\end{align}

In the concluding stage of each iteration, CSA modifies the associated colony patterns by incorporating the clan patterns as outlined in Equation (22).

(22) \begin{align}\left\{ {\begin{array}{*{20}{l}}{\begin{array}{*{20}{l}}{\left[ {p\!\left( {ind,:} \right),fitp\!\left( {ind} \right)} \right] = \left[ {px\!\left( {ind,:} \right),fitpx\!\left( {ind} \right)} \right]}\\[5pt] [p0(index,:), fit0(index)]= {\left[ {p,fitp} \right]}\\[5pt] {indbest \leftarrow fitp{0_{indbest}} = {\rm{min}}\!\left( {fitp0} \right)}\\[5pt] [gbest,gmin] = {\left[ {p0\!\left( {indbest,:} \right),fit0\!\left( {indbest} \right)} \right]}\end{array}}\end{array}.} \right.\end{align}

Figure 3. Pseudo-code of CSA with the sub-function $levy\_dist$ .

At the conclusion of the present iteration, the numerical value of is revised according to Equation (23):

(23) \begin{align}moment = \left( {abs\!\left( {randi\!\left( {\left[ {0,1} \right];\;N,1} \right)} \right) - m} \right) \circ dx\end{align}

The pseudo algorithm of the CSA has been illustrated in Fig. 3.

CSA is an iterative, non-recursive, stochastic evolutionary search method. CSA is developed for solving single-objective, real valued numerical problems. The version of CSA presented in this paper is designed as a global minimiser. CSA scales the amplitude of the direction vectors using a unique scaling method.

4.0 Experiments

In the experiments conducted in this section, four distinct optimal FMP were calculated using the following parameter values: $p = 0.80$ , $q = 0.70$ , $H = 200$ m, $ImageWidth = 5472$ pixels, $ImageHeight = 3648$ pixels, $F = 8.8$ mm, and $ccdWidth = 2.41{\rm{\;}}\mu $ m. The flight plans were designed over a test polygon that partially covers Mt. Ali, with geographical coordinates (lat= ${38^ \circ }.6621$ , Lon= ${35^ \circ }.5533$ , WGS84). Figure 4 visualises the general view of test area, Mt. Ali.

Figure 4. Mt. Ali and Mt. Erciyes.

The EPSG:3395 (WGS84/World Mercator) projection system was used for visualising the flight plans obtained from the conducted experiments. The EPSG:3395, also known as the WGS84/World Mercator projection, is a widely used mapping system that provides a conformal representation of the Earth’s surface. This projection system preserves angles and shapes over small areas, making it particularly suitable for applications requiring accurate directional relationships, such as navigation and flight path visualisation. One of its key advantages is its ability to represent large-scale maps with minimal distortion near the equator, although distortions increase at higher latitudes. The uniformity of scale along the equator and its compatibility with global datasets make EPSG:3395 a preferred choice for tasks involving geospatial analysis and cartographic rendering. Additionally, the projection’s grid-based structure simplifies calculations for mapping purposes, enhancing computational efficiency. Its widespread adoption in geographic information systems (GIS) ensures seamless integration with existing tools and frameworks. Another notable benefit is its consistent representation of longitudinal lines as vertical and latitudinal lines as horizontal, which aids in the interpretation of spatial data. These characteristics contribute to its reliability and usability in applications where precise spatial alignment is critical.

The local altitude, $H$ , of the micro-UAV for the ${(i,j)^{{\rm{th}}}}$ pixel location of the digital terrain model (DTM) has been fixed as $H = 200{\rm{\;}}m.$ The highest geodesic peak within the flight polygon is ${h_0} = {\rm{max}}\!\left( {DTM} \right) = 1851.44m.$ Thus, the micro-UAV flies at a steady elevation of ${h_{{\rm{max}}}} = {h_0} + 200 = 2051.44{\rm{\;}}m$ throughout the flight.

In this paper, the digital elevation model (DEM) data of Mt. Ali, obtained from the ASTER GDEM v3 global elevation dataset [60] with a spatial resolution of 1 arc-second, was used to generate the GSD map presented in Fig. 5.

Figure 5. GSD map for ${h_{{\rm{max}}}} = 2051.44{\rm{\;m}}.$ and H = 200 m.

In this paper, each of the four distinct FMP problems addressed contains a single unknown corresponding to the $azimuth$ value. To determine the optimal value of $\left( {azimuth} \right)$ , the CSA was employed. The control parameters of CSA were configured as follows: clan size $N = 30$ , problem dimension $D = 1$ , maximum number of iterations $MaxCycle = 100,000$ , lower bounds ${\rm{low}} = \left[ {0, - 100, - 100} \right]$ and upper bounds ${\rm{up}} = \left[ {360,200,200} \right]$ . These values were selected empirically. CSA is not overly sensitive to the initial parameter settings.

Figure 6. Visualisation of experimental results of : (a) Experiment #1, (b) Experiment #2, (c) Experiment #3 and (d) Experiment #4.

The first FMP problem (Experiment #1), which aims to minimise the total flight distance, is defined using Equation (6). In this experiment, a global solution value of $azimuth = {91^0}.06024$ was obtained. According to the obtained global solution, the total flight distance is 33,432.55 m. In this scenario, a total of 29 turns were made. A total of 789 image capture point locations were calculated. The average GSD is 14.80, and the standard deviation value is 3.28.

The second FMP problem (Experiment #2), as given in Equation (7), focuses on achieving the narrowest 95 ${\rm{\% }}$ confidence interval for the GSD values at the image capture points along the flight path. In this experiment, a global solution value of $azimuth = {268^0}.6806$ was obtained. According to the obtained global solution, the total flight distance is 34051.11 m. In this scenario, a total of 30 turns were made. A total of 800 image capture point locations were calculated. The average GSD is 14.79, and the standard deviation value is 3.26.

The third FMP problem (Experiment #3), as given in Equation (8), aims to minimise both the 95 ${\rm{\% }}$ confidence interval for the GSD values calculated at the image capture points along the flight path and the number of required turns. In this experiment, a global solution value of $azimuth = {199^0}.1012$ was obtained. According to the obtained global solution, the total flight distance is 35574.54 m. In this scenario, a total of 30 turns were made. A total of 819 image capture point locations were calculated. The average GSD is 14.93, and the standard deviation value is 3.37.

The fourth FMP problem (Experiment #4), as defined in Equation (9), aims to minimise the total number of turns. In this experiment, a global solution value of $azimuth = {47^0}.8804$ was obtained. According to the obtained global solution, the total flight distance is 36232.08 m. In this scenario, a total of 35 turns were made. A total of 819 image capture point locations were calculated. The average GSD is 14.95, and the standard deviation value is 3.40. The results obtained from Experiments #1–4 are illustrated in Fig. 6.