Effective Strategies for Mitigating the “Bowl” Effect and Optimising Accuracy: A Case Study of UAV Photogrammetry in Corridor Projects

Abstract

UAV-Enabled Corridor Photogrammetry is applied to survey linear transport infrastructure projects’ sites. The corridor flight missions cause a misalignment of the point cloud called the “bowl” effect. The purpose of this study is to offer a methodology based on statistical compensation methods to mitigate this effect and to improve the accuracy and density of the generated point cloud. The aerial images’ post-processing was carried out by varying the aerotriangulation methods. Subsequently, the accuracy improvement was completed by integrating the coordinates of the ground control points (GCPs) through different spatial distributions. Finally, Mean and RANSAC compensations were proposed to address the errors induced by the “bowl” effect on the coordinates of the images’ perspective centres (PCs). The findings indicate that the optimised aerotriangulation using Post-Processed Kinematic (PPK) data significantly contribute to reducing the “bowl” effect. Moreover, the GCP pyramidal spatial distribution allows accuracy improvement to a centimetre level. The Mean compensation method yields optimal outcomes in accuracy. It also helps to optimise on-site survey time and computing resources. RANSAC compensation optimises the accuracy and allows the retrieval of a 5-times-denser point cloud. Furthermore, the results give better accuracy compared to some current approaches.

1. Introduction

Geospatial data acquisition is facing increasing demands in terms of accuracy, speed and efficiency. Given the scope of these growing needs, aerial photogrammetry is considered as an efficient solution to collect data over large areas in a limited timeframe while ensuring optimal accuracy [1].

The use of UAVs in photogrammetry is becoming an effective and practical method for generating cartographic deliverables, especially in linear projects, such as point clouds and Digital Terrain Models (DTMs). UAV application is increasing due to its ability to generate survey outputs quickly and at low cost, thus providing an advantageous alternative to traditional methods of mapping and surveying [2]. Thanks to its low flight heights, this technology offers high spatial resolution and good radiometric homogeneity compared to images taken by photogrammetric aeroplanes or satellites [3].

In photogrammetry, a “corridor project” or “linear project” refers to a long and narrow geographic area where detailed mapping and analyses occur. Such a project is characterised by a width that ranges from hundreds to thousands of metres and a length of several kilometres. Corridor project area is usually linked to infrastructure such as roads, railways, pipelines or utility lines [4,5,6,7,8]. The linearity and continuity of these elements require special methodologies for mapping and analysis. The spatial distribution and number of ground control points (GCPs) are essential components of these methodologies that should be considered to achieve high-accuracy measurements [2].

To execute an infrastructure project in a corridor, the process unfolds in several distinct stages. The routing and preliminary conceptual design stages are the second prime phase of these projects and precede detailed design [9,10]. These stages’ objective is to provide a basic spatial layout of the primary alignments and assess whether the sites’ topography and characteristics respect the engineering standards [11]. When the accuracy of the topographical survey of these projects’ sites is high, the stakeholders can produce more accurate technical calculations, such as excavation and embankment. For the stage of routing and preliminary conceptual design, the surveyed 3D points’ accuracy must be better than 50 cm [12]. However, the corridor projects, when surveyed by UAVs, face many challenges caused by their linearity. These challenges generally lead to non-compliance with the accuracy requirements for the routing and conceptual design purposes.

UAVs substantially reduce the required time to obtain accurate and detailed data, even for difficult-to-access areas. This is crucial to cover linear projects’ spanning and inaccessible areas [13]. According to a study conducted by [14], UAVs minimise human intervention while maximising effective data collection for corridor projects. UAVs also offer beneficial technological synergy for modelling both urban and natural corridors. They offer valuable perspectives for the design and maintenance of infrastructures as well as for the management and protection of natural resources [15]. Indeed, by integrating LiDAR technology with UAV photogrammetry, it is possible to produce DTMs and digital surface models (DSMs) with high accuracy. Numerous studies have illustrated the application of UAVs in linear projects. Ruzgienė et al. [16] use UAV photogrammetry for road surface modelling purposes to demonstrate the significant advantages of this technology in the case of linear infrastructure. The accuracy of the measured 3D points on the road was 2.94 cm. Nahon et al. [15] map the coastal dunes in France by deploying UAVs equipped with both photogrammetry and LiDAR technology at Cap Ferret. The objective of the study is to monitor environmental impacts in this touristic and ecologically sensitive area. The study allows the measurement of erosion with a 15 cm value for vertical accuracy. In a study conducted by Abd Mukti et al. [17] in urban environments, UAVs proved their effectiveness by mapping a wide road reserve. The accuracy of measurement reached 2.19 cm for points’ altitude. The UAV also allows great manoeuvrability in congested areas. Singh et al. [18] use a set of semi-autonomous UAVs to generate precise geospatial results in a linear project. The accuracy of their 3D results reached 2.3 cm, demonstrating significant effectiveness compared to traditional methods.

Corridor UAV surveys are generally considered problematic due to many challenges caused by their linearity. First, the continuity of linear structures does not allow homogeneous coverage to be obtained along the project. This increases the risk of having gaps in the collected data and compromises their reliability [19]. Second, linear and parallel flight configurations can restrict camera angles and flight orientations, reducing image redundancy, which is essential for having cartography that reflects real-world conditions [20]. Then, having a long and thin geometry of the corridor is not optimal for matching aerial images because they can only be taken in a linear sequence [21]. In several cases, this leads to recommending oblique images for the survey to avoid any decline in accuracy of photogrammetric surveys [20]. Indeed, the integration of this type of image improves the quality of generated 3D models [22]. Also, the linear aspect of corridor projects can complicate georeferencing due to the difficulty in obtaining more stable and homogeneously distributed GCPs along wide structures [23]. Additionally, the limitations of IMUs, based on micro-electromechanical systems (MEMSs) technology, degrade the quality of height data, notably in relation to yaw angle in the case of corridors flights [24]. Finally, an accumulation of lengthwise deformations is observed in corridor projects surveyed using UAVs. This accumulation of errors, which mainly concerns the longest dimension, is caused by camera calibration inconsistencies. This may induce salient deformations, particularly because of erroneous and variable focal lengths, the effect of the rolling shutter, and the influence of the rotational movement [25].

Among the length distortions that are observed in UAV shots in corridor projects, the “bowl” effect emerges. It refers to a distortion that appears on the surface of the produced data, such as point clouds or DTMs, characterised by a bowl-shaped deformation in the output point cloud [26]. Several products derived from vertical UAV images taken in corridors show large-scale systematic deformations, shown in the form of a central bowl [27,28]. The flatter the area, the greater the accumulation of errors, introducing strong correlations between parameters, which consequently decreases the accuracy of results in corridor configurations compared to traditional configurations [29].

The “bowl” effect is a hindrance in the photogrammetric workflow. It is caused by the accumulation of camera calibration errors or a combination of near-parallel imaging directions and inaccurate correction of the lens radial distortion [30]. These systematic errors can be caused by an unstable rolling shutter affecting georeferencing accuracy and inaccurate self-calibration of the software based on Structure from Motion (SFM) without the use of GCPs [31]. Wackrow and Chandler [32] demonstrate how such deformation is associated with the processing of image sets with predominantly parallel viewing directions. They also demonstrate that this deformation relates to inaccuracies in the modelling of the camera lens’s radial distortion. Their study analyses this correlation and shows how variations in distortion parameters can introduce errors. As an example, Wackrow and Chandler [33] examine how a biased lens model or wrong georeferencing can cause systematic errors in DTMs. Errors are caused by the inability of the adjustment process to correctly model the real shape of the field. The study of Wackrow and Chandler [33] shows that the radial distortion parameter k₁ can be adjusted to simulate the effects of a distorted model using a virtual simulator.

The “bowl” deformation appears because the variation in points’ altitudes is symmetrical and co-centred. It results in an underestimation of altitudes in the centre of the studied area and an overestimation on its edges, or vice versa. Barazzetti et al. [34] estimate the impact of the “bowl” effect on a DTM by adding random offsets to the altitudes calculated for each 3D point in order to represent the measurement noise. The introduced offsets present the “bowl” effect. James and Robson [30] implemented simulations without any control measurements using an internal constraint method so that the “bowl” effect and other systematic errors can directly affect the accuracy of the coordinate’s measurements.

The variation in points’ altitudes due to the “bowl” effect is proportional to the cube of the radial distance. It affects the adjusted altitude of the points, as explained by Equation (1) [35]. In this equation, $h_0$ is the initial altitude at point (x, y), $k_1$ the radial distortion parameter, and $\mathrm{r}$ the radial distance from this point to the centre of the image.

$$h\left(x,y\right)=h_0+k_1·{\mathrm{r}}^3$$

(1)

Wackrow and Chandler [33] studied how a biased lens model or wrong georeferencing workflow can cause the “bowl” effect at the DTM level. Similarly, they showed that the radial distortion parameter $k_1$ can be adjusted to simulate the effects of a distorted model using a virtual simulator. Thus, radial distortion incorporated in the “bowl” effect can be described by Equation (2) [36]. In this equation, $\mathrm{r}$ is the radial distance to the image centre and $k_1$ is the radial distortion parameter.

$$\mathsf{\Delta }\mathrm{r}=k_1·{\mathrm{r}}^3$$

(2)

Several solutions exist to resolve the “bowl” effect. These solutions provide approaches that can be categorised as follows: 1/ procedural approaches based on the optimisation of flight parameters; 2/ geometric approaches based on camera calibration methods and the improvement of camera distortions; or 3/ mathematical approaches based on the error’s compensation at the level of image coordinates.

Table 1. Recap of solution options to resolve the “bowl” effect.

Study Reference	Approach Type	Description
Abd Mukti and Tahar (2021) [17]	Procedural	Urban road mapping. Optimisation of flight parameters and camera settings. Spatial distribution of GCPs.
Elsheshtawy and Gavrilova [24]	Mathematical	Coordinates of images’ PC obtained by GNSS. Compensation of images’ PC coordinates by averaging. Use of checkpoints for error evaluation.
Huang et al. [26]	Geometric	Analysis of distortion patterns. Adjustment using GNSS observations.
Ferrer-González et al. [2]	Procedural	Evaluating the impact of GCP number and distribution on the accuracy.
Jaud et al. [19]	Procedural and Geometric	Survey of linear coastal reliefs. Variation in flight plans. Use of different distortion models. Comparison between two software programs.
Molina et al. [37]	Procedural	Improved measurement accuracy with optimised GCPs and air traffic control parameters.
James and Robson [30]	Procedural	Simulation of multi-image networks. Use of the k1 distortion parameter to reduce systematic errors. Added oblique images.

summarises examples from the literature review carried out as part of this study with regard to these different solutions.

The studies based on procedural and/or geometric approaches show that it is necessary to determine the optimal GCP number and distribution. This leads to the optimisation of accuracy as well as the field work timeframe. However, the deployment of some GCP spatial distributions is strongly contingent on the available resources and on environmental constraints. At the same time, the self-calibration method with a single GCP, using high-accuracy GNSS observations, helps to lessen field effort but can occasionally decrease accuracy due to the low number of GCPs. In addition, flight plans optimised with oblique images greatly reduce systematic errors. This solution is quite practical for improving the accuracy but requires additional captures, which increases the time of field missions as well as the processing time. Thereby, existing procedural and/or geometric approaches in the literature vary in terms of application and technical requirements. On the one hand, some require fewer GCPs but additional technologies. On the other hand, some are based on the optimisation of camera distortion models. Alternatively, others are working on optimising flight parameters and varying camera angles.

Studies that use a mathematical approach to resolve the “bowl” effect are infrequent. Further, even fewer studies that combine all three approaches to resolve this problem exist. Accordingly, this study seeks to propose a practical methodology based on statistical compensation methods to mitigate the “bowl” effect and improve the accuracy and density of photogrammetric results for UAV flights on corridor infrastructure projects. The methodology is based on a combination of procedural, geometric and mathematical approaches.

In the following sections of this paper, the methodology of this study is presented as well as the results obtained. Then, the results are discussed to establish effective practices for aerial photogrammetry by UAVs in corridor projects with the aim of mitigating the “bowl” effect on photogrammetric deliverables. The methodology proposed aims to respond to accuracy requirements, focusing solely on routing and preliminary conceptual design stages for corridor projects. The main contributions of this paper are as follows: 1/ to propose RANSAC compensation as a mathematically grounded yet underutilised approach to mitigate the “bowl” effect; 2/ to test the combination of procedural, geometric and mathematical approaches to reach better accuracies for similar projects in comparison with current approaches; and 3/ to suggest practical methods to improve fieldwork efficiency while mitigating the “bowl” effect for corridor projects using UAVs.

2. Materials and Methods

2.1. Methodology

To mitigate the “bowl” effect in UAV photogrammetry for corridor projects, a four-phase methodological framework is proposed. Its main steps are presented in Figure 1. The first phase is an aerial mission in both Real-Time Kinematic (RTK) and Post-Processed Kinematic (PPK) modes. In this phase, the aerial images are captured, and the coordinates of their perspective centres (PCs) are observed using the two modes RTK then PPK. A GNSS station on the ground is used for surveys.

The second phase pertains to surveying the GCPs on the ground. Four GCP spatial distributions are carried out to meet the comparison objectives of this research. Zigzag, pyramid and pair distributions are conducted to identify the best spatial distribution for corridor projects. These spatial distributions demanded significant time spent on field surveying. For this reason, a fourth spatial distribution is established. It is a 5-GCP distribution combined with a coordinate compensation method. It aims to study the effect of reducing the number of GCPs, when paired with a compensation method, on the accuracy for this kind of project. Opting for 5 GCPs is based on two criteria: 1/ 5 is the minimum number of GCPs (when combined with accurate calibration of the camera) used in the reviewed literature to obtain accurate results [38,39]; 2/ to allow for comparison with some existing works that also use a 5-GCP configuration in their experiments. These works were conducted by Amr et al. [23] and Elsheshtawy and Gavrilova [24].

The third phase involves the photogrammetric calculations. Aerial images are processed by aerotriangulation. This workflow is based on the mathematical principle of Bundle Adjustment (BA) [40]. Then, the tie points are extracted using the Scale-Invariant Feature Transform (SIFT) algorithm [41]. Finally, the matching of images is carried out by the Structure from Motion (SFM) algorithm [42]. This step aims to obtain a block of georeferenced and scaled images. During this phase, a comparative approach is adopted, involving the simultaneous implementation of two sub-steps. First, the standard aerotriangulation consists of matching aerial images using the RTK coordinates of the PC. Then, optimised aerotriangulation involves both using the PPK coordinates of the PC and re-calculating the camera’s auto-calibration parameters repetitively using the initial parameters as pseudo-observations. The optimised aerotriangulation applied to the 5-GCP project is the reference calculation that is used to compare results.

The fourth phase consists of compensation for the PC coordinates. Figure 2 gives an overview of the specific steps of this phase. This phase is conducted using a mathematical approach where two statistical methods are used. The first method is the Mean compensation presented in the literature by Elsheshtawy and Gavrilova [24], while the second method is that of RANdom SAmple Consensus (RANSAC). This method is specifically used to overcome common problems in image matching algorithms in photogrammetry and computer vision [43]. However, its use in the literature is limited for PC coordinate compensation for errors. This finding prompted further investigation into the potential of this statistical method to resolve or mitigate the “bowl” effect. After the compensation step, the results are compared to the reference calculation achieved using the 5-GCP configuration project.

Mean compensation detailed steps are illustrated in Figure 3. Its algorithm is developed within this study using the Python programming language in its version 3.12.0. The algorithm table is presented in Algorithm 1. The calculation is carried out considering the means (${D\phi }_{av}$, ${D\lambda }_{av}$, and ${DH}_{av}$) of the differences ${DX}^{(i,j)}$ obtained between the coordinates ${XR}_i$ of the PC calculated through the standard aerotriangulation and ${XG}^{(i,j)}$ obtained through optimised aerotriangulation. The results of the Mean compensation are the coordinates (${\phi N}_i$, ${\lambda N}_i$, ${HN}_i$) of the images’ PC.

Algorithm 1. Algorithm table for the Mean compensation used

Require: Images’ PC coordinates calculated after standard aerotriangulation XRi; images’ PC coordinates calculated after optimised aerotriangulation XGi (in several aero iterations j)

$1: \mathbf{Input}\mathbf{:} {XR}_{i=1,\dots ,n}=\left[\begin{array}{c}\begin{array}{ccc}{\phi }_i& {\lambda }_1& H_1\\ {\phi }_2& {\lambda }_2& H_2\\ \dots & \dots & \dots \end{array} \\ \begin{array}{ccc}{\phi }_n& {\lambda }_n& H_n\end{array}\end{array}\right]$, ${XG}_{\begin{array}{c}i=1,\dots ,n\\ j=1,\dots ,p\end{array}}^{(i,j)}={\left[\begin{array}{c}\begin{array}{ccc}{\phi \prime }_i& {\lambda \prime }_1& {H\prime }_1\\ {\phi \prime }_2& {\lambda \prime }_2& {H\prime }_2\\ \dots & \dots & \dots \end{array} \\ \begin{array}{ccc}{\phi \prime }_n& {\lambda \prime }_n& H_n\prime \end{array}\end{array}\right]}_j$ $\mathrm{Number} \mathrm{of} \mathrm{images} nϵN$ $\mathrm{Number} \mathrm{of} \mathrm{iterations} \mathrm{of} \mathrm{optimised} \mathrm{aerotriangulation} p ϵ N$ $2: \mathbf{Output}: \mathrm{Deviations} {\mathrm{D}\mathrm{X}}_i$, Mean Deviations (${\mathrm{D}\mathsf{\phi }}_{i,av}$, ${\mathrm{D}\mathsf{\lambda }}_{i,av}$, ${\mathrm{D}\mathrm{H}}_{i,av}$), Compensated coordinates (${\mathsf{\phi }\mathrm{N}}_i, {\mathsf{\lambda }\mathrm{N}}_i, {\mathrm{H}\mathrm{N}}_i$) 3: for i= 1 to n do                  // Compute offsets                   ${DX}_{\begin{array}{c}i=1,\dots ,n\\ j=1,\dots ,p\end{array}}^{(i,j)}={\left[\begin{array}{c}\begin{array}{ccc}{\phi \prime }_1-{\phi }_1& {\lambda \prime }_1-{\lambda }_1& {H\prime }_1-H_1\\ {\phi \prime }_2-{\phi }_2& {\lambda \prime }_2-{\lambda }_2& {H\prime }_2-H_2\\ \dots & \dots & \dots \end{array} \\ \begin{array}{ccc}{\phi \prime }_n-{\phi }_n& {\lambda \prime }_n-{\lambda }_n& {H\prime }_n-H_n\end{array}\end{array}\right]}_j$ 4: for i=1 to n and k=1 to j do             // Compute average offsets                      ${\mathrm{D}\mathsf{\phi }}_{i,av}={\displaystyle \frac{1}{j}}\sum _{i=1, k=1}^{n,j}{({{\phi }^{\prime }}_i-{\phi }_i)}_k$                      ${\mathrm{D}\mathsf{\lambda }}_{i,av}={\displaystyle \frac{1}{j}}\sum _{i=1, k=1}^{n,j}{({{\lambda }^{\prime }}_i-{\lambda }_i)}_k$                      ${\mathrm{D}\mathrm{H}}_{i,av}={\displaystyle \frac{1}{j}}\sum _{i=1, k=1}^{n,j}{({H^{\prime }}_i-H_i)}_k$ 5: for i=1 to n do                 // Apply Compensation                       ${{\mathsf{\phi }\mathrm{N}}_i=\phi }_i+{\mathrm{D}\mathsf{\phi }}_{i,av}$                       ${{\mathsf{\lambda }\mathrm{N}}_i=\mathsf{\lambda }}_i+{\mathrm{D}\mathsf{\lambda }}_{i,av}$                       ${{\mathrm{H}\mathrm{N}}_i=H}_i+{\mathrm{D}\mathrm{H}}_{i,av}$ 6: end

RANSAC compensation is a predictive statistical method that provides an estimate of the probability of obtaining reliable predictions [44]. The RANSAC modelling used is the regression fitting. The algorithm applied is the one proposed by [45]. The steps of the algorithm are detailed in Figure 4. Then, Algorithm 2 displays an extract of the algorithm table. This algorithm focuses on reducing mis-matched points gradually. RANSAC first selects an initial set of random samples (points) and solves the model parameters. The inlier threshold t value is fixed to the median absolute deviation (MAD) of the target values ${\phi \prime }_i$. Then, the algorithm checks the number of inliers and calculates the number of iterations k compared to the maximum allowed M. If the inlier ratio is greater than the one from the last iteration, the number of inliers e is updated. This procedure continues until the number of iterations reaches M.

Algorithm 2. Extract from the RANSAC compensation algorithm table used to compensate images’ PC latitudes

Require: Images’ PC coordinates calculated after standard aerotriangulation XRi; images’ PC coordinates calculated after optimised aerotriangulation XGi

1: Input: ${XR}_i=\left[\begin{array}{c}\begin{array}{ccc}{\phi }_1& {\lambda }_1& H_1\\ {\phi }_2& {\lambda }_2& H_2\\ \dots & \dots & \dots \end{array} \\ \begin{array}{ccc}{\phi }_n& {\lambda }_n& H_n\end{array}\end{array}\right]$, ${XG}_{\begin{array}{c}i=1,\dots ,n\\ j=1,\dots ,p\end{array}}^{(i,j)}={\left[\begin{array}{c}\begin{array}{ccc}{\phi \prime }_1& {\lambda \prime }_1& {H\prime }_1\\ {\phi \prime }_2& {\lambda \prime }_2& {H\prime }_2\\ \dots & \dots & \dots \end{array} \\ \begin{array}{ccc}{\phi \prime }_n& {\lambda \prime }_n& H_n\prime \end{array}\end{array}\right]}_j$ $n$: Number of images $p$: Number of iterations of optimised aerotriangulation $t^{\left(\phi \right)}$: Distance thresholds to the model $M$: Max iterations $e$: number of inliers 2: Output:  ${\mathrm{D}\mathsf{\phi }}_{i,av}:$ Mean Deviations on images’ PC latitudes ${\mathsf{\phi }\mathrm{N}}_i$: Compensated latitudes $3: \mathbf{Initialisation} k=0$, $e=0$   $\mathbf{while} k<=M$ do    $\mathbf{for} i=1 to n$$\mathbf{and} j=1 to p$ do     $\mathbf{Read} {XR}_i$$\mathrm{and} {XG}^{(i,j)}$             // Define feature set and targets     $\mathbf{Fit} \mathbf{Ransac}. \mathbf{Regressor} (t^{\left(\phi \right) }$$, e, M)$$\mathbf{to} {XG}^{(i,j)}$ points  // Train RANSAC model for latitude     ${Median}_{{\phi \prime }_i}=median \left(\left|{\phi \prime }_i-median \left({\phi \prime }_i\right)\right|\right)$       $\mathbf{If} {Median}_{{\phi \prime }_i}<=t^{\left(\phi \right) }$ then           // Define inliers and outliers       $inlier{\phi }_i\leftarrow {\phi \prime }_i$$\mathbf{and} k=k+1$$\mathbf{and} \mathrm{e}=e+1$       Else       $outlier\leftarrow {\phi \prime }_i$$\mathbf{and} k=k+1$ $4: {\mathrm{D}\mathsf{\phi }}_{i,av}={\displaystyle \frac{1}{j}}\sum _{i=1, e=1}^{n,j}{(inlier{\phi }_i-{\phi }_i)}_e$            // Compute average offsets on inliers $5: \mathbf{For} i=1 to n$ do   ${{\mathsf{\phi }\mathrm{N}}_i=\phi }_i+{\mathrm{D}\mathsf{\phi }}_{i,av}$                     // Apply Compensation 12: end

To validate the research results, a visual identification of the “bowl” effect correction is performed first. Then, an analysis of the Random Mean Square Error (RMSE) based on the checkpoints is realised. Afterwards, a comparison of the two compensation method results is performed.

Table 2. Tests realised to mitigate the “bowl” effect.

Test	Test Type	Test Description
Test 1	Procedural	Standard aerotriangulation only GCP Integration: 3 spatial distributions Images’ PC coordinates: RTK mode
Test 2	Procedural and Geometric	Optimised aerotriangulation GCP Integration: 3 spatial distributions Images’ PC coordinates: PPK mode
Test 3	Mathematical	Mean compensation and Ransac compensation for images’ PC. Optimised aerotriangulation with 5 GCPs as reference calculation GCP Integration: 5 GCPs only after Mean and RANSAC compensation.

synthesises the three tests conducted to obtain and validate the results. Test 1 refers to the integration of GCPs into the standard aerotriangulation calculations. It aims to visually assess the impact of GCPs only on the “bowl” effect result. Test 2 refers to applying optimised aerotriangulation with the use of PPK coordinates of the images’ PC. It allows the exploration of the impact of changing the aerotriangulation approach on the “bowl” effect results. Test 3 is built upon a mathematical model, which, in addition to the two previous procedural and geometric tests, aims to increase the accuracy of the results obtained. Thereby, error compensation on the PPK coordinates of the images’ PC is carried out by two statistical methods: Mean and RANSAC compensations. This last test also aims to explore whether it is possible to reduce the GCP number by keeping an accuracy threshold that meets the accuracy requirements of the routing and preliminary conceptual design stages for corridor projects.

2.2. Study Site

The suggested methodology is applied to the High-Speed Railway project in Morocco at the section linking the two cities of Kenitra and Rabat. The project is in its routing and preliminary conceptual design stages. The project site is characterised by a curved shape and a mildly uneven terrain in some zones. The site’s altitudes vary between 360 m and 395 m. Two UAV corridor missions were executed. The missions’ technical characteristics are described in

Table 3. Technical characteristics of the missions.

Mission	Description	Number of Images	Overlaps	Flight Height/Ground Sample Distance (GSD)
Mission 1	Length: 10 km Width: 294 m	653	Forward: 60% Side: 65%	233 m/3 cm
Mission 2	Length: 10 km Width: 300 m	668	Forward: 60% Side: 65%	233 m/3 cm

. Figure 5 illustrates the geographic location of the corridor, the areas covered by the two flight missions and the positions of all GCPs used in the project.

2.3. Ressources

For the aerial missions, the UAV used was the Trinity F90+ [46]. It is a fixed-wing UAV with Vertical Take-off and Landing (VTOL). It is equipped with an integrated Sony RX1RII Camera.

Table 4. Technical characteristics of the camera.

Camera Characteristic	Description
Model	Sony RX1R II (DSC-RX1RM2)
Sensor	CMOS Exmor R BSI
Bands	R, G, B
Lens	Type: Fixed Focal length: 35 mm Max opening: f/2.0 FOV: 63°
ISO range	100–25,600
Shutter speed	1/4000 s to 30 s, mechanic
Frame rate	Up to 1/32,000 s electronic

describes the technical specifications of this camera.

The adopted GCP spatial distributions in this study are illustrated in Figure 6. Each spatial distribution allows a different accuracy of the results. To obtain the PPK coordinates of the images PC, QBase software (version 2.35.40) [47] is used.

For photogrammetric calculations, Pix4D software (version 1.75.0) [48] is deployed. It helps to carry out the two sub-steps of aerotriangulation and the point cloud extraction from the images. Finally, and for the last phase of the methodology, the Google Colab platform [49] is used. It allows applying the two statistical algorithms RANSAC and Mean for the compensation of PC coordinates.

3. Results

The calculation of aerotriangulation using the standard method shows the “bowl” effect. This effect is observed only in mission 2. Figure 7 illustrates these results for the two UAV missions. This is attributable to multiple reasons. First, and in addition to the restricted viewing angles, required by corridor shooting, the field covered by mission 2 is characterised by low contrast in several images, as described in Figure 8. Then, the specific environmental features of the site used in mission 2 can explain this discrepancy. Even with the same flight parameters and lighting conditions, the richness of textures at this site produces repeated patterns, which makes it difficult to detect homologous points while searching for tie points. Additionally, and since the vegetation is dense and in similar hues, the matching process from successive images remains difficult to precisely accomplish in mission 2. Finally, pixels that describe the ground are scarce in mission 2 images. This causes errors in the estimation of flight heights while calibrating the camera parameters in the aerotriangulation process. All these conditions lead to errors at several levels in the case of mission 2: 1/ in the calculation of the camera’s auto-calibration parameters; 2/ in the extraction of tie points; and 3/ in the detection of the tie points’ homologous points.

The first test to mitigate the “bowl” effect is to integrate the different GCP spatial distributions into the standard aerotriangulation. The results of Test 1 are illustrated in Figure 9. The visual inspection of these results shows only a slight improvement in the standard aerotriangulation with the zigzag distribution. However, the “bowl” effect still exists after this test.

The second test to overcome the “bowl” effect is to change the aerotriangulation method by using an iterative approach and integrating the PPK coordinates of the images’ PC. During Test 1, it is the RTK coordinates of the images’ PC that are used. In Test 2, an optimised aerotriangulation is adopted by integrating the PPK coordinates of the PC and iteratively recalculating the auto-calibration parameters of the camera. In this iterative calculation, the initial auto-calibration parameters are used as pseudo-observations. The visual result of Test 2 is illustrated in Figure 10. Visually, the “bowl” effect is mitigated. The results of the planimetric and altimetric RMSE are given comparatively in

Table 5. Comparative RMSE results after GCP spatial distribution integration with optimised aerotriangulation.

GCP Spatial Distribution.	GCP Number	Checkpoint Number	RMSE of GCPs (cm) on x, y and z, Respectively	RMSE of Checkpoints (cm) on x, y and z, Respectively
Pairs	14	6	8.0 3.1 16.0	17.4 15.1 34.5
Zigzag	17	6	3.1 3.2 6.1	7.1 6.1 23.4
Pyramid	18	5	3.3 3.0 1.0	5.7 5.5 9.3

.

All three spatial distributions offer accurate outcomes that respond to the routing and preliminary conceptual design stage requirements. From a practical point of view, and to optimise the duration and costs of field missions for the GCP survey phase, the distribution of GCPs in pairs remains the most optimal. However, the results of Test 2 indicate that it is the latter distribution that provides the lowest level of accuracy compared to the others. The pyramid distribution offers the best accuracy level in our case. Furthermore, the use of UAVs for corridor projects aims to optimise the entire process. To elaborate on the results, a third test is carried out in this research.

In Test 3, the number of GCPs is reduced to a minimum of five GCPs for the entire corridor. The spatial distribution of these GCPs is illustrated above by Figure 6d. Then, an optimised aerotriangulation with the five GCPs is calculated. The result of this calculation is taken as a reference. Afterwards, by retaining the same distribution of the five GCPs, two optimised aerotriangulation calculations are made: The first is completed by applying Mean compensation to the PPK coordinates of the images’ PC. And the second is performed by applying the RANSAC compensation method to the same coordinates.

After compensation, the analysis of images’ matching shows an increase in the number of tie points.

Table 6. Image matching results following compensation of PC coordinates by Mean and RANSAC method.

Test	Average Match Number Per Image	Point Cloud Density (Points/m²)
Test 2: Optimised aerotriangulation with GCP pyramid distribution	1535.23	0.34
Test 3: Mean compensation with 5 GCPs	7184.66	1.58
Test 3: RANSAC compensation with 5 GCPs	7202.01	1.59

represents the average number of tie points, highlighting the compensations’ role in a remarkable increase in the image’s matching points, particularly for the RANSAC compensation.

Figure 11 gives a comparative overview between the point cloud produced following RANSAC compensation and the one from the reference calculation. These results show that the RANSAC cloud is significantly denser than the reference one. This is justified by the partial correction of image positions through this compensation, which allows better identification of tie points existing on a larger number of images.

The results of calculating the RMSE of these different aerotriangulations are reported in

Table 7. RMSE results after error compensation for images’ PC coordinates.

Calculation	GCP Number	Checkpoint Number	RMSE of GCPs (cm) on x, y and z, Respectively	RMSE of Checkpoints (cm) on x, y and z, Respectively
Reference calculation with 5 GCPs	5	6	2.3 6.7 19.2	20.1 23.3 54.6
Mean compensation with 5 GCPs	5	6	4.8 5.6 14.8	9.9 15.4 23.4
RANSAC compensation with 5 GCPs	5	6	5.1 7.4 14.8	12.1 17.2 32.5

. The reference calculation gives a vertical RMSE exceeding 50 cm (the accuracy threshold for routing and preliminary conceptual design for corridor projects). The RANSAC compensation method offers significant improvement for the three dimensions, particularly for the vertical RMSE, which was improved from 54 cm to 32 cm. Furthermore, the Mean compensation method improves the vertical error by 50% and shows a clear improvement in the vertical RMSE. For this reason, this last method represents a good compromise between acceptable accuracy and the minimum number of GCPs to survey on the field.

For a better understanding of the outcomes of the two compensation methods, we carried out a statistical analysis of the distributions of the deviations between the coordinates of the estimated and initial PC for each of the two missions.

For mission 1, regarding the longitudes and latitudes of the images’ PC, the visualisation of the deviations’ frequencies showed a symmetric and homogeneous distribution with deviations order of 10⁻⁶, as shown in Figure 12. Altitude deviations are slightly asymmetric but remain homogeneous without the presence of noise or outliers. Altitude deviations do not exceed 1 m.

The 3D visualisation presented in Figure 13 for each of the coordinates of the initial and estimated PC validates the results of Test 3. We clearly notice the homogeneous distribution of deviations throughout the cloud without any noise or outlier.

For mission 2, where the “bowl” effect is noticed, the distribution of the deviations in latitudes, longitudes and altitudes is relatively normal and symmetrical, as shown in Figure 14. The differences are more visible for altitudes. However, altitude’s deviation values do not present outlier values. The coordinate plot in a 3D reference frame, presented in Figure 15 for mission 2, shows a significant deviation in altitude; however, this deviation is uniformly distributed over all points of the cloud. The altitudes’ deviations for mission 2 are due to errors in calculating flight heights while calibrating camera parameters for aerotriangulation. The conditions that lead to this issue are explained in the beginning of the Results Section.

4. Discussion

The results of this study show two levels of mitigating the “bowl” effect and increasing the 3D accuracy of results in the case of UAV photogrammetry in corridors. In the first level, procedural and geometric approaches are conducted. The aerotriangulation procedure and the GCP spatial distribution are varied. The “bowl” effect is mitigated and the 3D accuracies are acceptable in relation to the requirement value. In the second level, the goal is to explore the reduction in field work on surveyed GCPs. The number of the latter is reduced and two mathematical compensations are applied to give an optimal approach to conduct UAV flights in corridors.

Aerotriangulation methods play a major role in correcting the “bowl” effect for corridor flights. Their ability to reduce errors is apparent through the calculation of internal and external camera parameters. These parameters are used for image geolocation and distortion correction. In this paper, the variation in the standard aerotriangulation method towards an optimised approach is carried out.

The improvement of the altimetric and planimetric accuracies of the point clouds is achieved by the integration of several GCP spatial distributions while calculating the optimised aerotriangulation. The tests carried out show a pronounced improvement in the RMSE, particularly at the altitude level. Its value decreased from 34.5 cm for the GCP pair distribution to 11.8 cm for the GCP zigzag distribution and finally to 9.3 cm for the GCP pyramid distribution. The improvement in the horizontal RMSE is also notable, ranging from 23 cm for the GCP pair distribution to 7.9 cm for the GCP pyramid distribution. The latter offers the best results’ accuracy among all the tests carried out in our research. The 3D accuracy reached by varying the GCP spatial distributions is sufficient to respond to the accuracy requirements.

However, the pyramid distribution required 18 GCPs distributed over the entire corridor. This demanded an increased investment of time and effort on the surveyed site. In order to propose an optimal process for aerial photogrammetry by UAVs in corridor cases, this study proposes the integration of a mathematical approach. The latter draws upon the compensation of errors at the level of the images’ PC coordinates. The two statistical methods proposed are Mean and RANSAC compensations.

Concerning image matching, the implementation of the proposed compensations results in a fivefold improvement in image matching accuracy in the case of the reference calculation. Also, the accurate images’ PC coordinates resulting from the compensation allows a much denser point cloud to be obtained. This induces a surge in the overlap area of the images and therefore a more productive search for homologous points between sequential images.

RANSAC compensation allows a decrease in vertical RMSE of 40% and a refinement of horizontal RMSE of 32%. The vertical RMSE obtained is 32 cm. Mean compensation gives better results. The RMSE is reduced by 57% in altitude and by 43% in planimetry. The RMSE obtained with Mean compensation is 23 cm. The efficiency of Mean compensation, in our case, is explained by the statistical distribution of the deviations between the PC surveyed coordinates and calculated ones. This distribution was homogeneous and quasi-normal. This makes the RANSAC method less robust. In fact, the RANSAC method proves most effective when the number of outliers is important [43], which was not the case in our study.

Compared to existing approaches, our proposed methodological framework gives better results. Amr et al. [23] and Elsheshtawy and Gavrilova [24] used a Linear Relation Model to correct images’ PC coordinates. This model is based on Mean compensation to enhance images’ PC coordinates. When using the same number of GCPs and checkpoints used in our study, Amr et al. [23] obtained an RMSE of 74 cm and Elsheshtawy and Gavrilova [24] obtained an RMSE value of 66 cm. These values can be sufficient for some photogrammetric applications in corridor projects, but they do not respond to the accuracy requirements of the routing and preliminary conceptual design stage of linear infrastructure projects.

Overall, it is undeniable that the GCPs’ pyramid distribution integrated with an optimised aerotriangulation offers the best accuracy compared to other tests of the proposed methodology. This GCPs’ spatial distribution allows a centimetre-level accuracy. However, this approach consumes more time in the field and therefore induces more costs in the overall process. The mathematical approach through compensation for the images’ PC coordinates is an alternative that has proven its worth in our research. The two proposed compensation methods remain an interesting compromise between a remarkable improvement in accuracy and an optimisation of the GCPs’ number to be surveyed on site. Also, applying compensation for the images’ PC coordinates allows very dense point clouds to be obtained in the case of corridor flights. This is a major pre-requisite for DTM calculations on this type of project.

Finally, the accuracy thresholds achieved through these two compensations, with an optimal number of GCPs to be surveyed on the field, allow UAV flights in corridors to respond to the accuracy requirements of the routing and preliminary conceptual design stage of linear infrastructure projects.

5. Conclusions

This study highlights several practices that can correct the “bowl” effect and optimise the accuracy of corridor photogrammetric surveys by UAVs for linear infrastructure projects. It also proposes RANSAC compensation to correct images’ PC coordinates, which is not frequently used in the literature to solve the “bowl” effect issue. The results give better accuracy compared to some existing approaches. This study shows two levels of mitigating the “bowl” effect and increasing the 3D accuracy of UAV photogrammetry in corridors for the routing and preliminary conceptual design stage. This study is based on the following practices: 1/ the integration of ground data while optimising the effort provided; 2/ the optimisation of photogrammetric calculations; and 3/ the correction of the coordinates of images’ PC using statistical compensation.

The “bowl” effect is mitigated by an iterative aerotriangulation approach. This approach integrates the initial parameters of camera auto-calibration as pseudo-observation and the PPK coordinates of the PC. The absolute accuracy of the point cloud produced is improved either by an appropriate spatial distribution of the GCPs or by using one of the proposed statistical compensation methods. For RMSE analysis, the GCPs’ pyramid distribution gives the best results. This is due to the fact that this distribution offers homogeneous coverage of the site and avoids any extrapolation. To improve accuracy while optimising effort, time and costs, a statistical compensation of images’ PC coordinates is proposed. The effectiveness of the compensation method depends on the magnitude of discrepancies between surveyed coordinates and calculated ones after aerotriangulation. This requires careful inspection of the data before deciding on the compensation method to use. In projects similar to this study, where the discrepancies are uniformly distributed, Mean compensation is recommended to optimise accuracy and RANSAC compensation is recommended to optimise accuracy and to have denser point clouds.

To implement the results of this study, the authors recommend first defining the accuracy threshold required from the UAV photogrammetry corridor project. In the case of medium accuracies, the optimisation of GCP number is preferred when combined with a compensation method. RANSAC compensation in the latter case would yield satisfactory outcomes if the terrain is uneven. In case of high accuracies, the pyramid spatial distribution of GCPs is recommended. This distribution can be combined with a compensation method for images’ PC for more accurate results in the case of uneven terrain. Lastly, the site characteristics should be analysed before flight. It is recommended to orient the flight lines in a way that allows for rich and detailed variation on all captured images. This supports the prevention of problems in calculating camera calibration parameters or detecting homologous points which leads to the “bowl” effect.

Finally, the results obtained in this paper should also be interpreted in the light of the limitations encountered. First, the access to site points was limited due to dense vegetation in some areas. Then, the length of the site required consideration in take-off and landing zones. In addition, iterative calculating requires high processing performance and time. It is advisable for future studies to 1/ explore a large number of more complex compensation methods for better geolocation of images and 2/ study the potential of artificial intelligence in predicting corrected coordinates for images’ PC and optimising calculation performances.