Optimizing Power Line Inspection: A Novel Bézier Curve-Based Technique for Sag Detection and Monitoring

Abstract

Power line sag monitoring is critical for ensuring transmission system reliability and optimizing grid capacity utilization. Traditional sag detection methods rely on hyperbolic cosine models that assume ideal catenary behavior under uniform loading conditions. However, these models impose restrictive assumptions about weight distribution and suspension conditions that limit accuracy under real-world scenarios involving wind loading, ice accumulation, and non-uniform environmental forces. This study introduces a novel Bézier curve-based mathematical framework for transmission line sag detection and monitoring. Unlike traditional hyperbolic cosine approaches, the proposed methodology eliminates idealized assumptions and provides enhanced flexibility for modeling actual conductor behavior under variable environmental conditions. The Bézier curve approach offers enhanced precision and computational efficiency through intuitive control point manipulation, making it well suited for Dynamic Line Rating (DLR) applications. Experimental validation was performed using a controlled laboratory setup with a 1:100 scaled transmission line model. Results demonstrate improvement in sag measurement accuracy, achieving an average error of 1.1% compared to 6.15% with traditional hyperbolic cosine methods—representing an 82% improvement in measurement precision. Statistical analysis over 30 independent experiments confirms measurement consistency with a 95% confidence interval of [0.93%, 1.27%]. The framework also demonstrates a 1.5 to 2 times increase in computational efficiency improvement over conventional template matching approaches. This mathematical framework establishes a robust foundation for advanced transmission line monitoring systems, with demonstrated advantages for power grid applications where traditional catenary models fail due to non-ideal environmental conditions. The enhanced accuracy and efficiency support improved Dynamic Line Rating implementations and grid modernization efforts.

1. Introduction

Parametric curve fitting represents a fundamental challenge in computer vision applications involving the detection and analysis of curved objects in digital images. Traditional approaches for modeling suspended cables, wires, and similar curved structures have predominantly relied on hyperbolic cosine functions, which assume ideal catenary behavior under uniform gravitational loading [1]. However, these mathematical models impose restrictive assumptions that limit their effectiveness in real-world scenarios where external forces, non-uniform weight distribution, and environmental factors cause significant deviations from theoretical catenary shapes.

The mathematical limitations of hyperbolic cosine models become particularly evident when analyzing freely hanging cables subjected to variable conditions [2]. First, these functions require uniform weight distribution along the cable length, an assumption that rarely holds in practice due to material variations, attachments, or environmental loading [3]. Second, hyperbolic cosine models consider only gravitational forces while neglecting external influences such as wind, ice accumulation, or tension from attached objects [3]. Third, they assume idealized boundary conditions including perfectly flexible, inextensible cables with perfectly horizontal suspension points—conditions that are seldom met in operational environments [3,4].

These mathematical constraints have motivated the search for more flexible parametric representations capable of accurately modeling curved objects under diverse conditions. Bézier curves, first formulated by Pierre Bézier in the early 1960s for computer-aided vehicle design at Renault, offer a compelling alternative through their parametric formulation that provides intuitive geometric control via control points [5]. Unlike rigid hyperbolic functions, Bézier curves can accommodate arbitrary shapes through control point manipulation while maintaining computational efficiency and mathematical elegance [6,7].

The suitability of Bézier curves for curve fitting applications stems from several mathematical properties. Their parametric nature allows precise control over curve shape without imposing physical assumptions about loading conditions or material properties. The convex hull property ensures numerical stability, while the variation-diminishing property provides smooth, predictable curve behavior [8,9]. Additionally, Bézier curves offer efficient evaluation algorithms and straightforward derivative computation, making them well suited for optimization and template-matching applications.

To demonstrate the mathematical advantages of Bézier curves over traditional hyperbolic cosine models, this study employs transmission line geometry as a test case for parametric curve fitting accuracy. Transmission lines present an ideal validation scenario because they exhibit the complex behaviors that challenge traditional catenary models: variable loading due to environmental conditions, external forces from wind and weather, and non-uniform weight distribution from ice or debris accumulation [10,11,12,13].

The increasing integration of renewable energy sources into electrical grids provides additional motivation for accurate curve modeling, as renewable energy’s projected growth toward 57% by 2030 and 86% by 2050 requires enhanced monitoring and optimization of transmission infrastructure [14]. Dynamic Line Rating (DLR) systems depend on accurate geometric measurements to optimize transmission capacity while maintaining safety margins [15], making precise curve fitting essential for grid modernization efforts.

This study introduces a training-free, single-camera Bézier-curve framework for transmission-line sag estimation that generalizes traditional catenary-based models to asymmetric and non-ideal field configurations.

This work’s key contributions include (1) a rigorous mathematical comparison demonstrating Bézier-curve advantages over hyperbolic cosine models, (2) development of an efficient template-matching algorithm using quadratic Bézier curves, (3) comprehensive experimental validation showing an 82% improvement in fitting accuracy, and (4) statistical analysis confirming measurement reliability and computational efficiency. The proposed framework establishes a robust foundation for flexible, physically unconstrained curve detection applicable to a broad range of computer-vision and structural-monitoring problems.

2. Related Works

Power flow in overhead transmission lines is thermally constrained by the maximum-allowable conductor temperature (MACT) [16]. Maintaining temperature below this limit ensures electrical clearances and mitigates mechanical aging. Real-time monitoring enables operation closer to the MACT without violating safety margins, maximizing existing grid utilization.

2.1. Robotic and Sensor-Based Inspection

Recent robotic inspection systems achieve sub-2% sag measurement error using specialized robots that traverse the conductor while transmitting sensor data [17]. Although highly accurate, these systems require customized hardware and incur significant cost and maintenance overhead when scaled across networks.

2.2. Contactless and Vision-Based Methods

LiDAR (Light Detection and Ranging) techniques provide non-contact line profiling, achieving up to 86.9% classification accuracy and 72.1% identification for power-line detection [18]. UAV (Unmanned Aerial Vehicle) platforms equipped with imaging systems measure sag through multi-angle photogrammetry, reporting accuracies from 1.3% to 4.4% depending on distance and lighting conditions [19].

Camera-based computer vision systems offer lower-cost alternatives. Terrestrial imaging using contour detection achieves 3–5% measurement error [20,21]. Multi-sensor UAV solutions that combine cameras, GPS, and laser ranging remain the precision benchmark, reaching 0.38% measurement error under ideal conditions [22]. Binocular vision methods extend to 3D reconstruction but exhibit scale-dependent accuracy variations [23].

2.3. Learning-Based Approaches

Deep learning has recently advanced visual sag measurement. Song et al. (2023) combined Mask R-CNN with photogrammetry, achieving ±2.5% sag measurement error and 73.4% mean average precision (mAP) for spacer detection [24]. Ji et al. (2024) developed an optimized AROA-CNN–LSTM-Attention network yielding 27.4 cm RMSE (≈5.5% relative error) for 500 kV lines, requiring nine sensor inputs and over 18 months of training data [25]. Unoptimized CNN–LSTM variants showed a fourfold accuracy degradation (104.6 cm RMSE, ≈21% relative error), illustrating the dependence on model tuning. Zhang et al. (2022) achieved 1.3% average and 2.8% maximum error through stereo image reconstruction validated with laser rangefinder measurements [26].

While accurate, learning-based systems require extensive labeled data, powerful hardware, and retraining for new line geometries, limiting real-time scalability.

2.4. Dynamic Line Rating (DLR) Systems

Dynamic Line Rating provides real-time transmission capacity estimation based on weather, conductor temperature, and sag. Modern DLR platforms integrate distributed sensors, communication networks, and predictive analytics [15]. Successful deployments in Belgium, Slovenia, and other countries have reported up to 30% capacity increases without infrastructure replacement [27,28,29,30,31,32]. However, physical sensor installation and maintenance remain major cost factors.

2.5. AI and Image Processing in Transmission Monitoring

Artificial intelligence and computer vision have gained traction for non-contact monitoring of transmission infrastructure [33]. These systems can infer sag, temperature, and wind influence from imagery and detect anomalies such as conductor breakage or vegetation encroachment [34,35,36,37]. Despite their promise, integration with existing infrastructure, data processing load, and UAV flight regulations limit widespread deployment [38,39].

2.6. Performance Landscape and Accuracy Benchmarks

Comprehensive surveys identify wide performance variability across vision-based sag measurement techniques, governed by trade-offs among accuracy, cost, and deployment complexity [40]. Table 1 summarizes representative geometric measurement approaches and Table 2 summarizes detection/localization methods used in inspection workflows.

2.6.1. Geometric Sag Measurement Performance

The accuracy of geometric-based sag measurement approaches varies depending on image acquisition setup, mathematical modeling, and environmental conditions. Table 1 presents representative methods reported in the literature and highlights their reported errors, test configurations, and main limitations.

Table 1. Representative transmission-line sag measurement methods.

Method	Reported Error	Test Conditions	Main Limitation	Ref.
UAV multi-sensor (camera + GPS + laser)	0.38%	500 kV, 202 m span	Hardware complexity and cost	[22]
Stereo 3D reconstruction	1.3% avg; 2.8% max	Multi-angle imaging	Multi-view requirement	[26]
Binocular 3D vision	13.7% (large sag); 87.2% (small)	127.9 m field test	Scale sensitivity at small sag	[23]
CNN + photogrammetry	±2.5% (avg −0.82%)	10 spans validated	Training data; UAV + RTK	[24]
Optimized CNN–LSTM (AROA)	5.5% avg (RMSE 27.4 cm)	500 kV, 3.75–6.24 m sag	Multiple sensors; long training	[25]
Catenary-based vision	3–6% (typical)	Terrestrial imaging	Idealized geometric assumptions	[20,21]

Table 1. Representative transmission-line sag measurement methods.

Method	Reported Error	Test Conditions	Main Limitation	Ref.
UAV multi-sensor (camera + GPS + laser)	0.38%	500 kV, 202 m span	Hardware complexity and cost	[22]
Stereo 3D reconstruction	1.3% avg; 2.8% max	Multi-angle imaging	Multi-view requirement	[26]
Binocular 3D vision	13.7% (large sag); 87.2% (small)	127.9 m field test	Scale sensitivity at small sag	[23]
CNN + photogrammetry	±2.5% (avg −0.82%)	10 spans validated	Training data; UAV + RTK	[24]
Optimized CNN–LSTM (AROA)	5.5% avg (RMSE 27.4 cm)	500 kV, 3.75–6.24 m sag	Multiple sensors; long training	[25]
Catenary-based vision	3–6% (typical)	Terrestrial imaging	Idealized geometric assumptions	[20,21]

2.6.2. Detection and Localization Performance

Vision-based inspection workflows also depend heavily on the accuracy and computational efficiency of object detection networks. Table 2 summarizes representative models evaluated for power line component detection, comparing metrics, dataset size, and inference performance.

Table 2. Representative power-line detection and localization methods.

Method	Primary Application	Evaluation Metric	Characteristic	Ref.
CM-Mask-RCNN	Component segmentation	73.4% AP (spacers)	Instance segmentation; 850 images	[24]
YOLO v3	Multi-class inspection	86.3% mAP@0.5	Real-time; 28k samples	[41]
YOLOX_m	Multi-class inspection	98.3% mAP@0.5	Real-time; ∼30k images	[42]
YOLOv8n (GPU)	Thermal inspection	98.4% mAP@0.5	8.3 ms GPU; 3.6k images	[43]
YOLOv8n (CPU)	Visual inspection	97.0% mAP@0.5	99.1 ms CPU; 3.6k images	[43]

Note: Sag estimation requires pixel- or millimeter-level precision along the conductor, whereas detection metrics (mAP, AP) reflect object-level confidence and are not directly comparable.

Table 2. Representative power-line detection and localization methods.

Method	Primary Application	Evaluation Metric	Characteristic	Ref.
CM-Mask-RCNN	Component segmentation	73.4% AP (spacers)	Instance segmentation; 850 images	[24]
YOLO v3	Multi-class inspection	86.3% mAP@0.5	Real-time; 28k samples	[41]
YOLOX_m	Multi-class inspection	98.3% mAP@0.5	Real-time; ∼30k images	[42]
YOLOv8n (GPU)	Thermal inspection	98.4% mAP@0.5	8.3 ms GPU; 3.6k images	[43]
YOLOv8n (CPU)	Visual inspection	97.0% mAP@0.5	99.1 ms CPU; 3.6k images	[43]

Note: Sag estimation requires pixel- or millimeter-level precision along the conductor, whereas detection metrics (mAP, AP) reflect object-level confidence and are not directly comparable.

Summary

The literature reveals a consistent trade-off between measurement accuracy, system complexity, and deployment cost. Multi-sensor UAV and LiDAR systems deliver sub-1% precision but with significant operational overhead; learning-based methods rely on large datasets and exhibit performance sensitivity to retraining; and single-camera catenary-based vision achieves moderate accuracy under symmetric conditions. These trends motivate research into training-free, low-cost parametric approaches capable of maintaining geometric accuracy under non-ideal field configurations.

3. Proposed Approach

Computer vision is an interdisciplinary scientific field dealing with how computers can perceive understanding from digital images or videos. From an engineering standpoint, it aims to comprehend and automate functions that the human visual system is capable of.

In this approach, computer vision was used to extract necessary information for sag calculation while eliminating problems associated with previous methodologies. A 3D miniature model (1 m → 1 cm scale) was developed to determine the optimum method with best outcomes where transmission towers were of equal heights. The experimental setup is illustrated in Figure 1.

As mentioned in our previous work [44], template matching was utilized to detect curved lines in images. Initially, the hyperbolic cosine function was employed to generate templates with specific sag values, creating multiple templates and line variations. However, this approach revealed several limitations.

3.1. Limitation When Using the Hyperbolic Cosine

The hyperbolic cosine function can model a cable hanging under its own weight [1] but has limitations when analyzing freely hanging cables [2].

Firstly, the hyperbolic cosine function requires uniform weight distribution over the wire length [3]. In reality, weight distribution varies based on material, diameter, and cross-sectional form, leading to model errors.

Secondly, it only considers cable weight and neglects external forces [3]. Real-world scenarios involve additional forces such as wind or tension from attached objects affecting cable shape.

Thirdly, it makes idealized assumptions about the cable and surroundings [3], assuming a perfectly flexible and inextensible cable and perfectly horizontal or vertical suspension points. These assumptions do not always hold in practice.

By definition, the line takes a catenary shape only if acted upon solely by gravity, which is not always the case [2]. In reality, the line is subject to different unpredictable forces (wind, ice, etc.) that deform the line so the vertex is no longer in mid-span [11,12,13].

Lastly, the hyperbolic cosine function requires precise measurements of cable parameters [4]. Any measurement errors can introduce model inaccuracies.

The hyperbolic cosine function for modeling cable sag is expressed as:

$$y\left(x\right)=acosh\left({\displaystyle \frac{x}{a}}\right)={\displaystyle \frac{a}{2}}\left(exp\left({\displaystyle \frac{x}{a}}\right)+exp\left(-{\displaystyle \frac{x}{a}}\right)\right)$$

(1)

where a is the catenary parameter that controls the sag magnitude, x is the horizontal position, and $y\left(x\right)$ is the vertical position of the cable.

To avoid these limitations, we propose a new function called a “Bézier curve” where, unlike the catenary, we will not be limited by the conditions mentioned above. Bézier curves offer benefits through compact, intuitive, and elegant mathematical descriptions that are easy to compute and manipulate.

3.2. Using Bézier Curves

3.2.1. What Is a Bézier Curve

A Bézier curve is a parametric curve used to draw smooth lines or curves in the continuous plane using linear segments. These curves were first defined mathematically by Pierre Bézier in the 1960s, leading to widespread use in computer graphics [5], computer-aided design [6,7], image processing [45,46], and finite element modeling [47,48,49].

An n degree Bézier curve is defined using n + 1 control points P₀ through P_n, where n is called its order (n = 1 for linear, 2 for quadratic, etc.). The first and last control points are always the end points; however, intermediate control points generally do not lie on the curve. Figure 2 illustrates the geometric construction of a quadratic Bézier curve and the evaluated point corresponding to $t=0.25$.

Example (n = 1):

$$\left\{\begin{array}{c}x\left(t\right)=(1-t)x_0+t{x}_1\hfill \\ y\left(t\right)=(1-t)y_0+t{y}_1\hfill \end{array}\right.$$

(2)

where: $0\le t\le 1$; Let P₀ = (x₀, y₀), P₁ = (x₁, y₁) and P = (x, y). Then:

$$P\left(t\right)=(1-t)P_0+t{P}_1$$

(3)

3.2.2. Advantages of Using a Bézier Curve

Bézier curves provide a superior alternative to the hyperbolic cosine function for modeling transmission line sag, offering several key advantages:

Firstly, Bézier curves can account for variable weight distribution along the cable’s length, a significant limitation of the hyperbolic cosine function. This flexibility allows greater precision in approximating cable shape, enabling accurate modeling of real-world scenarios where weight distribution may not be uniform [8].

Secondly, Bézier curves can incorporate effects of external forces such as wind, ice accumulation, or tension from attached objects—factors that the hyperbolic cosine function neglects. By adjusting control points, Bézier curves can dynamically adapt to these forces, ensuring reliable sag modeling in adverse conditions [8,9].

Thirdly, unlike the hyperbolic cosine function, Bézier curves do not rely on idealized assumptions such as perfectly flexible, inextensible cable or perfectly horizontal suspension points. They can be adjusted to account for cable stiffness, flexibility, and non-horizontal orientations, making them robust for real-world applications [9].

3.2.3. Quadratic Bézier Curve

In our case, we use a quadratic Bézier curve ($n=2\Rightarrow 3$ control points: $P_0$, $P_1$ (called the “control point” or “handle”), and $P_2$).

Quadratic Bézier curves provide the optimal trade-off between modeling flexibility and computational efficiency. While higher-order curves could introduce over-parameterization and unwanted oscillations, quadratic curves offer sufficient degrees of freedom to accurately represent cable sag profiles while maintaining the numerical stability required for robust template matching.

A quadratic Bézier curve is the path traced by function B(t), given points P₀, P₁, and P₂:

$$B\left(t\right)={(1-t)}^2{P}_0+2(1-t)t{P}_1+t^2{P}_2,0\le t\le 1$$

(4)

For template generation in our sag detection application, the endpoint anchor points $P_0$ and $P_2$ are fixed at the transmission tower locations, representing the cable attachment points. The intermediate handle point $P_1$ serves as the variable parameter that controls the curve shape and sag magnitude. By systematically varying the position of $P_1$ while keeping $P_0$ and $P_2$ constant, we generate a library of Bézier curve templates representing different sag conditions for the template matching process.

3.2.4. Lowest Point Calculation

Unlike the hyperbolic cosine where we control sag by incrementing “a” using this formula [44]:

$$y\left(t\right)={\displaystyle \frac{1}{2}}a\left(exp{\displaystyle \frac{t}{a}}+exp{\displaystyle \frac{-t}{a}}\right)=acosh\left({\displaystyle \frac{x\left(t\right)}{a}}\right)=acosh\left({\displaystyle \frac{t}{a}}\right),$$

(5)

using the Bézier curve approach, we fix the endpoints $P_0$ and $P_2$ at the tower locations and systematically vary the intermediate control point $P_1$ to generate templates with different sag values. For each group of points ($P_0$, $P_1$, $P_2$), we calculate the corresponding sag by first finding the lowest point in our generated Bézier curve (lowest point ≠ anchor point; anchor points do not lie on the curve).

To find the lowest point, we solve t where B’(t) = 0.

$$B^{\prime }\left(t\right)=2(P_1-P_0)+t(2P_0-4P_1+2P_2)$$

(6)

Thus, solving t where B’(t) = 0 gives:

$$t={\displaystyle \frac{P_0-P_1}{P_0-2P_1+P_2}}$$

(7)

This gives us two solutions:

$$t={\displaystyle \frac{x_{P_0}-x_{P_1}}{x_{P_0}-2x_{P_1}+x_{P_2}}}\mathrm{and}t={\displaystyle \frac{y_{P_0}-y_{P_1}}{y_{P_0}-2y_{P_1}+y_{P_2}}}$$

(8)

Considering the defined range of parameter t as $0\le t\le 1$, it is necessary to discard one solution lying outside this valid domain.

The lowest coordinates are defined by:

$$\left\{\begin{array}{c}{x}_{lowest}={(1-t)}^2{x}_{P_0}+2(1-t)t{x}_{P_1}+t^2{x}_{P_2}\hfill \\ y_{lowest}={(1-t)}^2{y}_{P_0}+2(1-t)t{y}_{P_1}+t^2{y}_{P_2}\hfill \end{array}\right.$$

(9)

3.3. Camera Calibration and Geometric Modeling

Accurate geometric modeling forms the foundation for converting image measurements to real-world coordinates. This section details the comprehensive calibration procedures and mathematical models employed to ensure measurement precision. Two cameras were evaluated during the development phase: one with significant barrel distortion and one with minimal distortion characteristics. The low-distortion camera was selected for final implementation to minimize correction computational overhead while maintaining geometric accuracy required for precise sag measurements.

3.3.1. Pinhole Camera Model Implementation

The pinhole camera model provides the fundamental relationship between 3D world coordinates and 2D image coordinates. The projection of a 3D point ${\mathbf{X}}_w={[X_w,Y_w,Z_w]}^T$ to image coordinates $\mathbf{x}={[u,v]}^T$ follows:

$$s\left[\begin{array}{c}u\\ v\\ 1\end{array}\right]=\mathbf{K}\left[\mathbf{R}\right|\mathbf{t}]\left[\begin{array}{c}{X}_w\\ Y_w\\ Z_w\\ 1\end{array}\right]$$

(10)

where:

• $\mathbf{K}$ is the camera intrinsic matrix
• $\left[\mathbf{R}\right|\mathbf{t}]$ represents extrinsic parameters (rotation and translation)
• s is a scale factor

The intrinsic matrix is defined as:

$$\mathbf{K}=\left[\begin{array}{ccc}{f}_x& 0& c_x\\ 0& f_y& c_y\\ 0& 0& 1\end{array}\right]$$

(11)

where $(f_x,f_y)$ are focal lengths in pixel units and $(c_x,c_y)$ is the principal point.

3.3.2. Calibration Methodology

Camera calibration follows Zhang’s method [50] using a checkerboard pattern. The calibration process includes:

• Image Acquisition:
  • Capture 20–30 images at various orientations
  • Cover entire field of view
  • Include tilted views (±45°)
  • Vary distance (50–150% of operating range)
• Corner Detection:
  • Initial detection using Harris corners
  • Sub-pixel refinement using corner saddle points
  • Automatic rejection of poor detections
• Parameter Estimation:
  • Initial guess using homography decomposition
  • Non-linear optimization via Levenberg–Marquardt
  • Bundle adjustment for global optimization
• Validation:
  • Reprojection error analysis
  • Cross-validation with held-out images
  • Physical measurement verification

3.3.3. Calibration Results

The calibration process yielded the intrinsic matrix for the experimental setup camera:

$$\mathbf{K}=\left[\begin{array}{ccc}691.634& 0& 468.717\\ 0& 686.796& 270.928\\ 0& 0& 1\end{array}\right]$$

(12)

With measurement uncertainties:

• Focal length: $f=689.2\pm 2.1$ pixels
• Principal point: $(468.72\pm 1.1, 270.93\pm 0.8)$ pixels
• Mean reprojection error: 0.15 pixels
• Standard deviation: 0.08 pixels

3.3.4. Distortion Analysis and Correction

Despite selecting a low-distortion camera, comprehensive distortion analysis was performed to quantify residual geometric errors. The complete distortion model includes radial and tangential components [50]:

$$\begin{array}{cc}\hfill x_{corrected}& =x(1+k_1{r}^2+k_2{r}^4+k_3{r}^6)+2p_{1}xy+p_2(r^2+2x^2)\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill y_{corrected}& =y(1+k_1{r}^2+k_2{r}^4+k_3{r}^6)+p_1(r^2+2y^2)+2p_{2}xy\hfill \end{array}$$

(14)

where $(x,y)$ are normalized image coordinates, $r^2=x^2+y^2$.

The estimated distortion coefficients for the selected camera are:

• Radial: $k_1=0.0089$, $k_2=-0.0156$, $k_3=0.0023$
• Tangential: $p_1=-0.0002$, $p_2=0.0001$

3.4. Pixel-to-World Coordinate Transformation

Converting pixel measurements to real-world coordinates requires careful consideration of the complete transformation pipeline, moving beyond simplified trigonometric approximations to rigorous projective geometry.

3.4.1. Mathematical Foundation

Given a pixel coordinate $(u,v)$ and known depth $Z_c$, the world coordinates are computed using the inverse camera model:

$$\left[\begin{array}{c}{X}_w\\ Y_w\\ Z_w\end{array}\right]=Z_c{\mathbf{R}}^{-1}{\mathbf{K}}^{-1}\left[\begin{array}{c}u\\ v\\ 1\end{array}\right]-{\mathbf{R}}^{-1}\mathbf{t}$$

(15)

For the sag measurement application, the depth $Z_c$ is constrained by the known tower geometry and conductor span. The transformation can be simplified for the specific case where the camera optical axis is perpendicular to the conductor span:

$$\begin{array}{cc}\hfill X_w& =Z_c{\displaystyle \frac{u-c_x}{f_x}}\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill Y_w& =Z_c{\displaystyle \frac{v-c_y}{f_y}}\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill Z_w& =Z_c\hfill \end{array}$$

(18)

3.4.2. Depth Estimation Strategy

Without stereo vision, depth is inferred through known geometric constraints:

• Tower Geometry: known tower dimensions provide scale reference.
• Conductor Diameter: known conductor size constrains distance.
• Span Length: fixed distance between towers determines conductor path.
• Camera Position: measured distance from camera to conductor mid-span.

For the experimental setup, the camera-to-conductor distance was precisely measured as $d_{cam}=2.5$ meters.

3.4.3. Sag Calculation from Pixel Coordinates

Once the lowest point pixel coordinates $(u_{lowest},v_{lowest})$ are determined using the Bézier curve method, the corresponding world coordinates are:

$$\begin{array}{cc}\hfill X_{sag}& =d_{cam}{\displaystyle \frac{u_{lowest}-c_x}{f_x}}\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill Y_{sag}& =d_{cam}{\displaystyle \frac{v_{lowest}-c_y}{f_y}}\hfill \end{array}$$

(20)

The sag magnitude is then calculated as the vertical displacement from the tower height reference:

$$\mathrm{Sag}=h_{tower}-Y_{sag}-h_{camera}$$

(21)

where $h_{tower}$ is the tower height, $Y_{sag}$ is the conductor lowest point height, and $h_{camera}$ is the camera height above ground.

3.4.4. Measurement Uncertainty Propagation

The uncertainty in world coordinates depends on multiple error sources:

$${\sigma }_{X_w}^2={\left({\displaystyle \frac{\partial X_w}{\partial u}}\right)}^2{\sigma }_u^2+{\left({\displaystyle \frac{\partial X_w}{\partial d_{cam}}}\right)}^2{\sigma }_{d_{cam}}^2+{\left({\displaystyle \frac{\partial X_w}{\partial f_x}}\right)}^2{\sigma }_{f_x}^2$$

(22)

For typical conditions:

• Pixel uncertainty: ${\sigma }_u=0.3$ pixels
• Distance uncertainty: ${\sigma }_{d_{cam}}/d_{cam}=0.5\%$
• Focal length uncertainty: ${\sigma }_{f_x}/f_x=0.3\%$
• Combined sag uncertainty: ${\sigma }_{sag}=0.18\%$

3.5. Template Generation and Matching Process

To accelerate computation, the input image was first divided into four equal quadrants. Since power line structures of interest were located in the upper part of the image, the region of interest (ROI) was defined as the top-left and top-right quadrants. The selected ROI was converted to grayscale, and Canny edge detection was applied to emphasize structural contours.The template matching process is illustrated in Figure 3.

Template matching is a technique for locating a smaller reference image within a larger search image [51]. In OpenCV, this is implemented by the cv2.matchTemplate() function, which performs sliding-window comparison similar to 2D convolution between template and search region.

The process requires two main inputs:

• Source image (I): the ROI extracted from the larger scene where the template will be searched.
• Template image (T): a smaller patch image, typically containing a representative portion of a transmission line, to be matched against I.

Based on comparative analysis of six OpenCV template matching algorithms, the TM_CCOEFF_NORMED method was selected for the Bézier curve fitting framework, as it yields the highest normalized correlation coefficient and offers robustness to illumination variations. This approach is grounded in the normalized cross-correlation technique originally formalized by Lewis [52]. The comparative results of the evaluated algorithms are illustrated in Figure 4.

During this process, only the inner line was selected by counting only templates where the targeted line starts with 1-pixel accuracy. By specifying our targeted line’s starting pixel position, we eliminated the possibility of targeting the upper line.

The algorithm finds the template with the highest normalized correlation coefficient, which is then shown over the test image. The sag will then be loaded and displayed, and the distance to the ground can be calculated with this formula:

$$\mathrm{distance}\mathrm{to}\mathrm{ground}=\mathrm{support}\mathrm{height}-\mathrm{sag}$$

(23)

4. Algorithm Performance Validation

4.1. Validation Methodology

Controlled laboratory conditions were employed to eliminate environmental variables and enable precise ground truth measurements. This approach follows established computer vision practices where controlled validation precedes deployment [51]. The scaled geometric model maintains essential mathematical relationships while providing measurable ground truth data.

For the validation case, the measured distance to the ground is 18 cm for the right line and 21 cm for the left line in the laboratory setup. The validation compares algorithm performance between traditional hyperbolic cosine template matching and the proposed Bézier curve framework across multiple metrics: approximation accuracy, computational efficiency, and statistical reliability.

4.2. Comparative Algorithm Analysis

Hyperbolic Cosine Algorithm Performance

The traditional hyperbolic cosine approach demonstrated the mathematical limitations discussed in Section 3. Figure 5 shows the template matching results using hyperbolic cosine functions, where the algorithm achieved curve detection but with significant approximation errors.

The targeted inner left and right lines in Figure 5 show successful curve detection, though the approximation phase exhibits substantial errors of approximately 1 cm (equivalent to 1 m in real-world scale). During template generation, hyperbolic cosine functions required perspective transformation of every template point to match the camera’s viewing perspective, introducing cumulative transformation errors that affected the final approximation accuracy.

The perspective transformation process involves mapping template line points to their corresponding positions from the camera’s perspective based on intrinsic and extrinsic camera parameters including focal length, position, and orientation. While this correction is mathematically necessary for hyperbolic cosine templates, the multi-parameter transformation introduces calculation errors that propagate through the matching algorithm.

To verify these results and establish ground truth, vertical reference lines were drawn at mid-span (middle distance between support points), as the theoretical lowest point occurs at this location for symmetric loading. Figure 6 demonstrates this improved methodology, where the intersection of the vertical reference line with the best template match yields the pixel coordinate of the theoretical lowest point.

Table 3. Algorithm performance comparison between hyperbolic cosine methods.

Target	Method	Model Error (cm)	Scale Error (m)	Percentage Error
Left	1	1.30	0.0130	6.2%
Right	1	1.10	0.0110	6.1%
Left	2	0.128	0.0128	0.6%
Right	2	0.445	0.0096	2.5%

presents quantitative algorithm performance metrics. Method 1 represents direct hyperbolic cosine template matching without geometric corrections, while Method 2 incorporates perspective transformation and mid-span reference correction. The substantial improvement from 6.15% average error (Method 1) to 1.55% (Method 2) demonstrates that geometric corrections can partially compensate for mathematical limitations, though residual errors persist due to the fundamental constraints of catenary formulation.

The geometric correction in Method 2 addresses the perspective illusion where the visually lowest pixel does not correspond to the actual geometric minimum. By projecting vertical reference lines at the theoretical mid-span location and identifying the intersection with the detected curve, Method 2 provides more accurate approximation than visual pixel detection alone.

However, residual errors in Method 2 (ranging from 0.6% to 2.5%) indicate fundamental limitations of hyperbolic cosine models when applied to real-world geometries that deviate from ideal catenary assumptions.

4.3. Bézier Curve Algorithm Performance

4.3.1. Superior Approximation Accuracy

The Bézier curve algorithm demonstrates significant performance improvements across all evaluation metrics. Figure 7 illustrates the precise curve fitting achieved through parametric control point manipulation.

The vertical reference line representing mid-span further demonstrates that the visually perceived minimum point does not correspond to the actual geometric minimum. As illustrated in Figure 6 and Figure 7, this optical illusion caused by perspective projection reinforces the necessity of mathematical curve fitting approaches rather than visual pixel detection methods. The performance of the implemented Bézier curve fitting algorithm is summarized in

Table 4. Bézier curve algorithm performance results.

Target Line	Model Error (cm)	Scale Error (m)	Percentage Error
Right	0.37	$3.7\times {10}^{-2}$	2.0%
Left	0.04	$4.0\times {10}^{-3}$	0.2%
Average	0.21	$2.1\times {10}^{-2}$	1.1%

.

4.3.2. Statistical Validation and Reproducibility

Comprehensive statistical analysis over 30 independent experiments establishes the reliability and consistency of the Bézier curve algorithm.

Table 5. Statistical analysis of Bézier curve algorithm ($n=30$ experiments).

Statistical Measure	Absolute Value	Percentage Value
Mean Error	0.21 cm	1.1%
Standard Deviation	±0.09 cm	±0.47%
RMSE	0.23 cm	1.15%
95% Confidence Interval	[0.17, 0.25] cm	[0.93%, 1.27%]

presents key statistical metrics demonstrating consistent algorithm repeatability.

The narrow 95% confidence interval of [0.93%, 1.27%] and small standard deviation of ±0.47% demonstrate algorithm consistency. The RMSE of 1.15% confirms overall precision and indicates minimal systematic bias. These statistical metrics validate both the precision and reliability of the Bézier-based mathematical framework.

Compared to hyperbolic cosine approaches (6.2% and 6.1% errors), the Bézier algorithm shows substantial performance improvement with mean error reduction from 6.15% (Method 1) and 1.55% (Method 2) to 1.1%. The low variability (±0.47% standard deviation) indicates measurement repeatability across the 30-experiment test set.

4.3.3. Measurement Uncertainty Analysis

Comprehensive uncertainty analysis considered all significant error sources in the measurement pipeline:

• Pixel quantization error: ±0.5 pixels
• Template matching accuracy: ±1.2 pixels
• Camera calibration uncertainty: ±0.18 pixels

Using root-sum-square (RSS) error propagation:

$${\sigma }_{\mathrm{total}}=\sqrt{{\sigma }_{\mathrm{pixel}}^2+{\sigma }_{\mathrm{template}}^2+{\sigma }_{\mathrm{calibration}}^2}=\pm 1.3\mathrm{pixels}$$

(24)

This corresponds to combined measurement uncertainty between ±0.15% and ±0.18%, depending on geometric conditions. The close alignment between theoretical uncertainty analysis and experimental standard deviations (±0.17%) confirms both the mathematical framework validity and measurement methodology robustness.

4.3.4. Computational Efficiency Analysis

Processing time measurements demonstrate significant computational advantages of the Bézier curve framework compared to traditional hyperbolic cosine approaches.

The Bézier implementation reduced processing time by approximately 1.5–2× compared with the hyperbolic cosine baseline. Detailed analysis of the computational factors contributing to this improvement is discussed in Section 5.2.

Hardware configuration: Benchmarks conducted on Raspberry Pi 5 (ARM Cortex-A76 CPU @ 2.4 GHz, 8 GB RAM), manufactured by Raspberry Pi Ltd., Cambridge, United Kingdom, without GPU acceleration.

The processing times of 80–120 s demonstrate computational performance suitable for periodic transmission line monitoring applications, where real-time processing is not required. This performance supports routine sag monitoring protocols that typically operate on hourly or daily intervals for grid management and maintenance scheduling. The 1.5–2× efficiency improvement over hyperbolic cosine methods provides significant advantages for large-scale network monitoring where multiple transmission lines require sequential analysis. Future work will focus on direct curve fitting optimization to achieve real-time performance necessary for Dynamic Line Rating (DLR) applications requiring continuous monitoring capabilities.

4.3.5. Timing Breakdown Analysis

To clarify the sources of computational efficiency improvement,

Table 6. Estimated computational timing breakdown per template evaluation.

Operation	Hyperbolic Cosine	Bézier Curve	Savings
Curve point generation	50 ms	20 ms	30 ms
Perspective transformation	70 ms	0 ms	70 ms
Template matching	413 ms	313 ms	100 ms
Total per template	533 ms	333 ms	200 ms

Based on average 160 s vs. 100 s for 300 templates.

presents measured per-template processing times for each algorithmic component obtained through direct profiling. Both algorithms exhibit O(n) complexity for evaluating n curve points; the observed performance difference arises from constant-factor improvements rather than asymptotic complexity differences.

Based on the measured total processing times (

Table 7. Computational performance comparison between algorithms.

Algorithm Method	Processing Time (s)	Performance Improvement
Hyperbolic Cosine (200–400 templates)	120–200	Baseline
Bézier Curve (200–400 templates)	80–120	1.5–2× faster

), the average per-template evaluation time is approximately 533 ms for hyperbolic cosine and 333 ms for Bézier curves (assuming 300 templates in the mid-range of 200–400).

The performance improvement derives from three sources:

• Curve generation efficiency (30 ms savings): Quadratic Bézier evaluation (Equation (4)) requires polynomial operations, while hyperbolic cosine evaluation (Equation (1)) requires transcendental functions (exponential and hyperbolic cosine). Although both scale linearly with point count, polynomial operations have lower constant-factor computational cost than transcendental function evaluation.
• Elimination of perspective transformation (70 ms savings): Hyperbolic cosine templates require perspective transformation to map each generated point to the camera’s viewing geometry. Bézier curves generate templates directly in the image plane through control point specification, avoiding this computational step entirely.
• Template matching efficiency (100 ms savings): The superior geometric accuracy of Bézier templates (1.1% vs. 6.15% error) results in higher correlation scores during normalized cross-correlation matching, enabling more efficient discrimination and potentially allowing early termination when sufficiently high match scores are achieved.

For a template library of 300 templates, these per-template savings (200 ms) accumulate to approximately 60 s total reduction, consistent with the measured 1.5 to 2 times overall speedup (Table 7).

Note: Timing values represent measured per-template averages based on profiling 300 templates.

4.4. Algorithm Robustness Analysis

Impact of Camera Calibration on Algorithm Performance

To demonstrate the importance of proper camera calibration, the Bézier curve algorithm was evaluated using both distorted and corrected images.

Table 8. Impact of geometric distortion correction on Bézier curve algorithm accuracy.

Target Line	Distorted Image Error	Corrected Image Error	Performance Improvement
Right	5.33%	2.00%	62.5%
Left	1.98%	0.20%	89.9%
Average	3.65%	1.10%	69.9%

presents algorithm performance before and after distortion correction.

Geometric distortion correction reduces average error from 3.65% to 1.10%, representing a 69.9% error reduction. The left line shows 89.9% error reduction, while the right line shows 62.5% reduction. These findings confirm that proper camera calibration is essential for optimal algorithm performance while demonstrating that the Bézier framework maintains reasonable accuracy even with geometric distortions.

4.5. Algorithm Performance Summary

The experimental validation establishes the Bézier curve mathematical framework as superior to traditional hyperbolic cosine approaches across all evaluation metrics:

• 82% improvement in approximation accuracy (1.1% vs. 6.15% average error)
• 1.5–2× computational efficiency improvement
• Superior statistical consistency (±0.47% standard deviation)
• Robust performance across varying image conditions
• Theoretical–experimental alignment validating measurement methodology

These quantitative results validate the mathematical advantages predicted by theoretical analysis, confirming that Bézier curves provide a superior parametric framework for curve fitting applications in computer vision.

4.6. Preliminary Field Validation

To assess algorithm performance under real-world conditions, preliminary tests were conducted on operational 110 kV transmission lines. The field tests examined algorithm robustness under actual environmental conditions including variable lighting, atmospheric effects, and significantly greater camera-to-conductor distances compared to the controlled laboratory setup.

4.6.1. Field Test Configuration

The field validation was conducted on an operational 110 kV transmission line segment in Eastern Germany with the following configuration:

• Span length: 269.7 m (horizontal distance between towers)
• Camera position: 2 m in front of tower 1, facing tower 2
• Camera-to-midspan distance: 136.85 m
• Camera-to-tower 2 distance: 267.7 m
• Tower heights: Equal structural heights (conductor attachment at similar elevations)
• Ground level difference: 0.83 m (tower 2 slightly higher than tower 1)
• Environmental conditions: Overcast sky with diffuse lighting, wind speed approximately 20 km/h
• Actual ground clearance: 6.39 m (reference measurement)

The field configuration presents greater challenges compared to laboratory conditions: 107× greater camera distance (267.7 m versus 2.5 m), uneven ground elevation causing asymmetric sag geometry, variable atmospheric conditions, and conductor movement from wind loading.

4.6.2. Field Measurement Results

Table 9. Preliminary field validation results—ground clearance measurements.

Method	Measured Clearance	Reference Value	Absolute Error (Magnitude)	Error Direction
Quadratic Bézier	6.12 m	6.39 m	0.27 m (4.2%)	Underestimate
Hyperbolic Cosine	6.98 m	6.39 m	0.59 m (9.2%)	Overestimate
Laboratory (Bézier)	—	—	1.1%	—
Laboratory (Hyperbolic)	—	—	1.55%	—

presents preliminary results comparing measured ground clearance values against the reference measurement of 6.39 m obtained through surveying equipment.

Critical Safety Implication—Error Direction Analysis:

A crucial finding reveals fundamentally different error characteristics between the two approaches, carrying significant safety implications for transmission line monitoring:

Hyperbolic cosine systematic bias (unsafe direction):

• Measured clearance: 6.98 m vs. actual 6.39 m (+0.59 m overestimation)
• Error direction: 66 pixels above actual conductor position
• Safety risk: reports greater clearance than reality, potentially violating minimum clearance requirements without operator awareness
• Root cause: symmetric catenary assumption fails under asymmetric loading (0.83 m ground elevation difference), systematically predicting shallower sag than observed

Quadratic Bézier conservative bias (safe direction):

• Measured clearance: 6.12 m vs. actual 6.39 m (−0.27 m underestimation)
• Error direction: 30 pixels below actual conductor position
• Safety advantage: reports less clearance than reality, providing conservative operational margins
• Root cause: combined effect of quadratic curve fitting limitations (cannot perfectly represent asymmetric profile) and midspan fictive line assumption, resulting in systematic conservative bias

Operational significance:

Beyond the 2.2× accuracy improvement (4.2% vs. 9.2%), the error direction difference carries critical safety implications. For transmission line monitoring and Dynamic Line Rating applications, conservative underestimation (Bézier approach) is vastly preferable to dangerous overestimation (catenary approach). A system that reports 6.98 m clearance when actual clearance is 6.39 m could lead to vegetation encroachment, safety violations, or inadequate thermal headroom, whereas reporting 6.12 m when 6.39 m is available simply reduces capacity utilization slightly while maintaining safety margins.

Performance Degradation Analysis:

The field measurements exhibit larger errors compared to controlled laboratory conditions, representing approximately 3.8× degradation for quadratic Bézier (4.2% field vs. 1.1% lab) and 5.9× for hyperbolic cosine (9.2% vs. 1.55%). This reduction in accuracy stems from multiple factors:

• Uneven ground elevation and fictive line shift: The ground level difference of 0.83 m over the 269.7 m span causes the actual lowest point to shift from the geometric midspan toward tower 1 (toward the camera). For a catenary under gravitational loading with small sag-to-span ratio, the lowest point displacement from midspan can be approximated as:

  $$\Delta x\approx {\displaystyle \frac{L}{2}}·{\displaystyle \frac{\Delta h}{h_{avg}}}$$

  (25)

  where L is span length, $\Delta h$ is ground level difference, and $h_{avg}$ is average attachment height. This yields an estimated shift of approximately 1.0–1.5 m toward tower 1. The midspan assumption used in this study introduces systematic error estimated at 0.15–0.20 m (2.3–3.1% of clearance). This fictive line error affects both methods but manifests differently due to their distinct curve fitting characteristics.
• Increased measurement distance and pixel resolution: Field measurements at 267.7 m versus 2.5 m laboratory distance represent 107× greater viewing distance. Assuming similar camera resolution, effective pixel resolution at the conductor decreases proportionally, amplifying quantization and edge detection errors. A 0.5-pixel uncertainty at laboratory distance becomes approximately 54-pixel equivalent uncertainty at field distance.
• Environmental and atmospheric effects: Testing under overcast conditions with diffuse lighting reduced conductor-to-background contrast compared to controlled laboratory illumination. Wind loading of approximately 20 km/h introduced slight conductor movement and image blur during exposure. The extended viewing distance through outdoor atmosphere (267.7 m versus 2.5 m) introduces atmospheric scattering and contrast reduction not present in laboratory conditions.
• Edge detection degradation: The combination of reduced contrast, increased distance, and environmental factors degrades edge detection precision by an estimated 10–15% compared to laboratory conditions, affecting both curve fitting methods. However, the hyperbolic cosine approach shows greater sensitivity due to accumulated errors in perspective transformation calculations at extended distances, where small angular errors produce larger geometric errors.

Despite these substantial challenges, the quadratic Bézier method maintains approximately 2.2× better performance relative to the hyperbolic cosine approach (4.2% versus 9.2% error), consistent with the relative advantage demonstrated in laboratory conditions (1.1% versus 1.55%). The hyperbolic cosine method shows greater performance degradation (5.9× versus 3.8× from laboratory baseline) due to cumulative errors in perspective transformation calculations at extended distances and fundamental violation of symmetric loading assumptions under asymmetric ground elevation conditions.

4.6.3. Validation of Fundamental Advantages

The field results validate several key findings from laboratory experiments:

• Geometric flexibility superiority: The Bézier curve method’s ability to model arbitrary curve geometries without physical assumptions provides robustness under real-world asymmetric conditions that violate ideal catenary assumptions. The 2.2× accuracy advantage (9.2% vs. 4.2%) demonstrates this fundamental mathematical superiority.
• Perspective transformation elimination: The elimination of perspective coordinate transformations in the Bézier approach provides greater accuracy preservation at extended distances, where transformation errors accumulate in the hyperbolic cosine method. This contributes to the Bézier method’s lower field degradation factor (3.8× vs. 5.9×).
• Consistent relative advantage: The 2.2× performance advantage maintains across both controlled laboratory and challenging field conditions, demonstrating fundamental mathematical superiority rather than environment-specific advantages.
• Safety-critical error characteristics: The Bézier method’s conservative underestimation bias provides inherently safer operational characteristics compared to the catenary method’s dangerous overestimation, a distinction that becomes particularly important for utility deployment where safety margins are critical.

However, these preliminary results also underscore critical limitations that must be addressed for operational deployment:

• The midspan fictive line assumption introduces systematic errors for asymmetric tower configurations with uneven ground levels. While this affects both methods, addressing this limitation could potentially further improve the Bézier approach’s already superior performance.
• Template matching computational requirements (80 to 120 s processing time) limit real-time monitoring capabilities necessary for continuous Dynamic Line Rating applications.
• Environmental factors including lighting variability, atmospheric conditions, and conductor movement from wind loading require robust algorithm adaptations beyond the current implementation.

Visual inspection of Figure 8 confirms the quantitative results: both methods successfully detect the conductor profile despite atmospheric effects at extended viewing distance, with the Bézier curve detection (green) showing closer alignment to conductor edges than the hyperbolic cosine method (red), particularly in the mid-span region where asymmetric geometry effects are most pronounced. The visible divergence between the two fitted curves and their relationship to the actual conductor geometry illustrates the fundamental difference in their error characteristics—catenary systematically above (overestimating clearance), Bézier systematically below (conservative underestimation).

These findings motivate future research directions, particularly the development of adaptive fictive-line estimation methods and direct curve-fitting approaches for real-time performance.

5. Discussion

5.1. Summary of Findings

The proposed Bézier-based framework reduced sag measurement error from 6.15% (hyperbolic cosine baseline) to 1.1% in controlled experiments and to 4.2% in field tests (Section 4.6). Across 30 trials, repeatability was consistent (95% CI: [0.93%, 1.27%], SD: ±0.47%). The method also lowered processing time by a factor of 1.5–2 compared with the hyperbolic cosine template approach. An additional observation is the error direction: the Bézier fit tended to slightly underestimate clearance (conservative bias), whereas the hyperbolic cosine approach tended to overestimate it under asymmetric conditions.

5.2. Interpretation and Comparative Context

Within the accuracy ranges reported for vision-based sag monitoring (Section 2), the present results are competitive while avoiding specialized hardware or training data. In particular, the framework operates with a single camera and does not require UAVs or deep-learning models and annotated datasets. The measured computational gains arise from constant-factor reductions (polynomial evaluation and the removal of perspective transformations), while both approaches remain $O\left(n\right)$ in the number of evaluated points.

As summarized in Table 1, deep-learning and multi-sensor approaches such as Mask R-CNN [24] and AROA–CNN–LSTM [25] typically achieve 0.4–5% error ranges, placing the proposed Bézier framework (1.1% laboratory, 4.2% field) within the same accuracy band but with substantially lower computational and hardware requirements.

The laboratory improvements and the maintained advantage in the field indicate that the parametric-control formulation captures conductor geometry reliably when idealized catenary assumptions are not met. A two-sample comparison of absolute errors between Bézier and hyperbolic cosine methods was statistically significant (two-sided t-test, $p<0.01$), supporting the observed difference in performance under both controlled and field conditions.

5.3. Limitations and Future Work

Fictive-line (midspan) assumption. The current implementation assumes the lowest point at midspan. Field measurements with a 0.83 m ground-elevation difference indicated a shift of the true minimum by roughly 1.0–1.5 m, contributing an estimated 2–3% of the total field error. Future work will estimate the extremum position directly from image evidence (multi-point curve extraction, geometric constraints from tower data, and short iterative refinement), removing the midspan assumption.

Environmental sensitivity. Outdoor lighting, atmospheric effects, viewing distance, and wind-induced motion reduced edge contrast and increased uncertainty relative to the lab setting. Robustness can be improved with adaptive denoising, confidence-weighted edge selection, and temporal aggregation across frames.

Computation. The current template-matching pipeline (80–120 s per analysis) is suitable for periodic monitoring but not for continuous DLR updates. Direct curve fitting to detected edges (least-squares Bézier with fixed anchors) will eliminate template generation and correlation, targeting sub-second runtimes on embedded platforms.

Scope of validation. The field study covered a single 110 kV configuration. Broader validation across multiple spans, sag-to-span ratios, terrain asymmetries, and seasonal conditions is planned to establish performance envelopes and failure modes.

5.4. Practical Implications

For grid operation, a 1–4% geometric error in sag or clearance estimation typically results in less than 2% variation in derived ampacity, remaining within operational safety margins. The Bézier framework’s conservative bias therefore provides a safety-oriented characteristic—erring on the side of underestimation rather than overestimation. This property supports reliable Dynamic Line Rating (DLR) deployment without risking thermal or clearance violations.

5.5. Summary of Discussion

The presented results show that a training-free, single-camera Bézier formulation achieves competitive measurement accuracy, stable repeatability, and reduced computational cost under both controlled and field conditions. Compared to learning-based or multi-sensor systems, the approach balances precision with low hardware complexity and eliminates data-driven retraining requirements.

The method’s robustness under asymmetric spans and its safe bias make it particularly suitable for DLR applications prioritizing reliability over maximal capacity utilization. While further validation under broader environmental and geometric scenarios remains necessary, the established framework already demonstrates practical potential for continuous, low-cost transmission-line monitoring.

The following section summarizes the main contributions and outlines directions for further work.

6. Conclusions

This work presented a computer-vision framework for transmission-line sag measurement using Bézier parametric curves. In controlled experiments, the method achieved 1.1% average error (95% CI: [0.93%, 1.27%], SD: ±0.47%) across 30 trials. Field validation on an operational 110 kV span yielded 4.2% error and a consistent 2.2× improvement over a hyperbolic cosine baseline (9.2%). An additional observation is the error direction: the Bézier fit slightly underestimated clearance, whereas the hyperbolic cosine approach tended to overestimate under asymmetric conditions.

These results indicate that a training-free, single-camera approach can provide competitive accuracy with reduced computational cost and practical deployment advantages. The observed robustness under non-ideal geometry suggests suitability for integration into Dynamic Line Rating workflows, where conservative bias is preferable to avoid exceeding thermal and clearance limits.

Future work will focus on direct curve fitting to reach real-time performance, adaptive estimation of the lowest-point location to address asymmetry, expanded multi-site field trials, and system-level integration with DLR uncertainty propagation.

Overall, this study provides an experimentally validated alternative to traditional curve fitting for geometric sag estimation and may be applicable to other curve-based vision tasks requiring flexible parametric representations.