Multi-Point Optical Flow Cable Force Measurement Method Based on Euler Motion Magnification

Abstract

This study introduces a multi-point optical flow cable force measurement method based on Euler motion amplification to address challenges in accurately measuring cable displacement under small displacement conditions and mitigating background interference in complex environments. The proposed method combines phase-based magnification with an optical flow method to enhance small displacement features and improve SNR (signal-to-noise ratio) in cable displacement tracking. By leveraging magnified motion data and integrating auxiliary feature points, the approach compensates for equipment-induced vibrations and background noise, allowing for precise cable displacement measurement and the identification of vibration modes. The methodology was validated using a scaled model of a cable net structure. The results demonstrate the method’s effectiveness, achieving a significantly higher SNR (e.g., from 7.5 dB to 22.24 dB) compared to traditional optical flow techniques. Vibration frequency errors were reduced from 6.2% to 1.5%, and cable force errors decreased from 11.38% to 3.13%. The multi-point optical flow cable force measurement method based on Euler motion magnification provides a practical and reliable solution for non-contact cable force measurement, offering potential applications in structural health monitoring and the maintenance of bridges and high-altitude structures.

1. Introduction

Common methods for cable force measurement include oil pressure gauge readings, pressure sensor measurements, magnetic flux, three-point bending, and vibration analysis [1,2]. The oil pressure gauge reading and pressure sensor measurement methods are applicable for cable force measurement in the construction phase [3]. The magnetic flux method requires determining the material properties of the cable in the laboratory prior to cable force measurement [4]. The three-point bending method is suitable for slender cables with small length-to-diameter ratios [5]. Among these methods, the vibration method is widely used in cable force measurement due to its high accuracy and strong applicability. This method measures the cable force by measuring the natural frequency and calculating it based on the relationship between the natural frequency and cable force [6]. However, traditional vibration methods require acceleration sensors to be mounted on the exterior of the cable to measure its vibration frequency, which is cumbersome and inefficient, especially for multi-point measurements on a single cable or for measurements across multiple cables.

To address the operational drawbacks of traditional vibration methods, non-contact cable force measurement based on Laser Doppler technology [7], microwave radar technology [8], and vision have been widely applied. Although non-contact cable force measurement based on Laser Doppler technology and microwave radar technology offers high accuracy, the equipment is expensive and not extensively utilized. Vision-based cable force measurement has gradually gained popularity due to its ease of operation. This method measures the vibration frequency of the cable by detecting specific features on the cable body. Common approaches in this field include template matching methods, background subtraction methods, and edge detection methods. Current research primarily focuses on improving target tracking accuracy. Yan Banfu et al. employed a fixed target placed outside the cable body and used a background subtraction method to measure cable displacement and measure the vibration frequency of the cable [9]. Zuo Wensheng et al. used the corner points of square targets on the cable as feature points to measure cable displacement [10]. Zhu Hao et al. employed spot detection methods to identify the centroids of circular targets on the surface of the cable body and recorded their coordinates to measure cable displacement [11]. In previous studies, significant displacements exceeding 10 pixels were observed in images captured during video recording, primarily due to the shooting distance and the large displacement of the cables. These displacements resulted in the SNR (signal-to-noise ratio), making cable displacement data easier to measure and analyze.

The influence of shooting distance and cable amplitude can cause the cable in the image to have an amplitude of less than 10 pixels in some cases, requiring a more accurate target tracking method. Xu Minjie et al. utilized the subpixel edge detection methods with higher accuracy and edge detection methods to measure cable displacement [12]. Feng and Scarangello et al. employed the subpixel template matching method for edge detection of the upper cables at Hard Rock Stadium [13]. Wang et al. utilized the Hough Line detection method to identify the edges of bridge cables as line segments, obtaining displacement response curve for the cables [14]. Zhang Yuhang et al. used a combination of the Euler motion magnification method and the Canny edge detection method to measure cable displacement [15]. However, these studies do not address the challenge of measuring the small displacement of cables under complex backgrounds or long-distance shooting conditions. In these scenarios, where the displacement may be less than 1 pixel or even smaller, traditional methods such as template matching methods, edge detection methods, and subpixel edge detection methods rely heavily on the SNR. Dynamic changes in background elements, such as wind-induced vibrations, variations in ambient light, or camera vibrations, can easily mask motion features by introducing background noise. This leads to tracking drift or loss of feature points, resulting in low accuracy in cable displacement identification and further magnification of errors.

This study aims to address the challenges of measuring cable displacement with high precision under small displacement conditions (less than 1 pixel) and mitigating the effects of equipment vibrations and background interference in complex environments. We propose a multi-point optical flow cable force measurement method based on Euler motion magnification. This method integrates the phase-based magnification method with the optical flow method and achieves high-precision displacement measurement through the synchronized computation of multiple feature points. The phase-based magnification method is mainly used to enhance the small displacement of the cable. By amplifying the specific frequency range of the motion features in the image sequence, the displacement changes less than the pixel level can be revealed, so as to improve the observability of the small displacement. The optical flow method analyzes the enhanced image sequences to calculate the pixel displacement vectors of the feature points on the cable surface, accurately capturing the cable’s displacement trajectory in space. By computing the relative displacement between the feature points on the cable surface and auxiliary feature point, the method isolates and corrects errors caused by equipment vibrations, thereby improving the SNR of the displacement data. Additionally, by considering the cable’s inclination, the true displacement in the displacement direction is further calculated, enhancing the accuracy of displacement measurement. The measurement results obtained from the method are validated through a comparison with acceleration sensor data, fitting the cable’s vibration mode shape, and using the vibration method to measure cable forces. This study provides a high-precision, non-contact cable force measurement under complex displacement conditions, offering more reliable technical support and practical guidance for cable health monitoring and engineering applications.

2. Principle of Multi-Point Optical Flow Cable Force Measurement Method Based on Euler Motion Magnification

2.1. Phase-Based Magnification

The phase-based magnification method analyzes the phase information of individual pixels in a video and associates it with subtle motion changes [16]. Specifically, this method first converts video frames into image data and detects motion by analyzing the phase variations of pixels, rather than relying on direct color changes. This approach effectively minimizes errors caused by lighting fluctuations, enhances robustness against noise and background variations, and accurately magnifies the desired motion. To capture motion across multiple spatial scales, the method utilizes a complex steerable pyramid for multi-scale image decomposition. The pyramid decomposition breaks the input image into a series of subbands, where each subband contains spatial components corresponding to specific directions and frequency ranges. The process begins with a progressive low-pass filtering and down-sampling procedure to generate a set of images with decreasing spatial resolution. At each resolution level, directional filters are applied to extract frequency components in various orientations. For each subband, Fourier transforms are subsequently applied to obtain the phase information $\varnothing \left(x, y, t\right)$ and magnification factor $A\left(x, y, t\right)$. By adjusting the number of decomposition levels $N_{levels}$ and filter types ${filter}_{types}$, the method controls the spatial resolution and sensitivity to motion. At higher decomposition levels, the spatial resolution decreases, while motion amplitudes are relatively magnified, allowing for the detection of finer, smaller-scale motions. Conversely, fewer decomposition levels may fail to adequately separate high- and low-frequency motion components, thereby reducing the effectiveness of motion magnification. Over time, the phase variations of each pixel form a temporal series, which is processed using band-pass filters to isolate specific frequency bands of interest. The filtered phase variations are then magnified, and the magnified signal is combined with the original image data to reconstruct an image that exhibits the magnified motion details. This phase-based approach significantly enhances the accuracy of detecting small-scale motions and provides robust support for vibration analysis and the observation of subtle changes. The detailed mathematical derivation is provided as follows [17]:

In image processing, the video frame may be represented as a two-dimensional spatial signal $f\left(x\right)$, which generates a small displacement $\delta \left(t\right)$ in time t. At this point, the signal can be expressed as follows:

$$f\left(x\cancel{-}\delta \left(t\right)\right)=\sum _{\omega =-\infty }^{\infty }{A}_{\omega }{e}^{i\omega \left(x\cancel{-}\delta \left(t\right)\right)}$$

(1)

where $\delta \left(t\right)$ represents the position shift of the signal at time t. On this basis, in order to obtain the magnified effect, the small motion needs to be amplified, and a magnification multiple $\alpha$ is introduced; then, the magnified motion is expressed as:

$$f(x+(1+\alpha \left)\delta \left(t\right)\right)$$

(2)

To analyze the signal, we decompose it into the superposition form of different frequency components, expanding it by using the Fourier series, expressed as:

$$f\left(x\cancel{-}\delta \left(t\right)\right)=\sum _{\omega =-\infty }^{\infty }{A}_{\omega }{e}^{i\omega \left(x\cancel{-}\delta \left(t\right)\right)}$$

(3)

where $A_{\omega }$ represents the amplitude at frequency $\omega$, and $e^{i\omega \left(x\cancel{-}\delta \left(t\right)\right)}$ represents the amplitude and phase of the frequency component, for a single frequency component of frequency ω can be expressed:

$$S_{\omega }(x, t)=A_{\omega }{e}^{i\omega \left(x\cancel{-}\delta \left(t\right)\right)}$$

(4)

The effects of x and $\delta \left(t\right)$ can be expressed as:

$$S_{\omega }(x, t)=A_{\omega }{e}^{i\omega x}{e}^{i\omega \delta \left(t\right)}$$

(5)

where $A_{\omega }{e}^{i\omega x}$ is the fundamental oscillatory component of the frequency $\omega$ at position x, and $e^{i\omega \delta \left(t\right)}$ is the additional phase change introduced by the time-dependent displacement $e^{i\omega \delta \left(t\right)}$, where $\omega \delta \left(t\right)$ is the phase change of the frequency component.

The displacement information is included in the phase $\omega (x+\delta (t\left)\right)$. According to the Fourier time-shifting property, the phase change of the signal in the frequency domain corresponds to the translation of the signal in the time domain. Thus, the displacement can be altered by modifying the phase information. Band-pass filtering of the phase information and then removing the DC (Direct Current) component provide:

$$B_{\omega }\left(x, t\right)=\omega \delta \left(t\right)$$

(6)

In order to magnify the dynamic displacement of the signal, it is necessary to introduce the magnification coefficient $(1+\alpha )$ to construct new frequency components:

$${\overline{S}}_{\omega }(x, t)=S_{\omega }(x, t)e^{i\alpha {\beta }_{\omega }}=A_{\omega }{e}^{i\omega \left(x\cancel{-}\left(1+\alpha \right)\delta \left(t\right)\right)}$$

(7)

The same treatment for each component yields a signal at $\alpha$ magnification. The phase processing can magnify the displacement, and the amplitude will not be amplified in the process, which can effectively avoid the magnification of the noise together.

2.2. Optical Flow Method

The optical flow method estimates object displacement by analyzing pixel intensity changes in an image sequence. Assuming that in the continuous frame image, the brightness of the scene remains constant, let $I\left(x, y, t\right)$ represent the pixel intensity at $\left(x_1, y_1\right)$ at time t. According to the brightness constancy assumption, the displacement of the pixel over time will not change its intensity [18], i.e.,

$$I(x, y, t)=I(x+∆x, y+∆y, t+∆t)$$

(8)

where $∆x$ is the amount of pixel displacement in the horizontal direction, $∆y$ is the amount of pixel displacement in the vertical direction, and $∆t$ is the time interval between two consecutive frames. By applying the Taylor expansion to the above formula and while ignoring the higher-order terms, the following formula is obtained:

$$I\left(x+∆x, y+∆y, t+∆t\right)\approx I\left(x, y, t\right)+{\displaystyle \frac{\mathsf{\partial }I}{\mathsf{\partial }x}}∆x+{\displaystyle \frac{\mathsf{\partial }I}{\mathsf{\partial }y}}∆y+{\displaystyle \frac{\mathsf{\partial }I}{\mathsf{\partial }t}}∆t$$

(9)

where ${\displaystyle \frac{\mathsf{\partial }I}{\mathsf{\partial }x}}$ and ${\displaystyle \frac{\mathsf{\partial }I}{\mathsf{\partial }y}}$ are the gradients of the gray values in the horizontal and vertical directions, respectively, and ${\displaystyle \frac{\mathsf{\partial }I}{\mathsf{\partial }t}}$ is the gradient of the gray values over time. Put this formula into the brightness constancy assumption and arrange:

$${\displaystyle \frac{\mathsf{\partial }I}{\mathsf{\partial }x}}{\displaystyle \frac{∆x}{∆t}}+{\displaystyle \frac{\mathsf{\partial }I}{\mathsf{\partial }y}}{\displaystyle \frac{∆y}{∆t}}+{\displaystyle \frac{∆I}{∆t}}=0$$

(10)

where ${\displaystyle \frac{∆x}{∆t}}$ and ${\displaystyle \frac{∆y}{∆t}}$ are the speeds of the pixels in the horizontal and vertical directions, respectively. Defining $u={\displaystyle \frac{∆x}{∆t}}$ and $v={\displaystyle \frac{∆y}{∆t}}$, i. e., the optical flow velocity in the x and y directions, the basic equation of optical flow is obtained:

$${\displaystyle \frac{\mathsf{\partial }I}{\mathsf{\partial }x}}u+{\displaystyle \frac{\mathsf{\partial }I}{\mathsf{\partial }y}}v+{\displaystyle \frac{∆I}{∆t}}=0$$

(11)

In addition, the optical flow method is a method to estimate the motion of the object by analyzing the change in pixel gray in the image sequence. According to the scope and accuracy of the optical flow calculation, it can be divided into two [19]: sparse optical flow method and dense optical flow method. The difference between the two is that the sparse optical flow method focuses on the change of pixel optical flow in the local area of the image, while the dense optical flow method focuses on the change of pixel optical flow in the global area of the image. The representative algorithms of the two optical flow methods are LK (Lucas–Kanade) sparse optical flow tracking method and HS (Horn–Schunck) dense optical flow method. Because the cable force measurement process is usually dynamically detected by the vibration displacement of the cable in the image sequence, and the vibration of the cable has the characteristics of small amplitude, fast speed, and small amplitude in the image, the continuous optical flow field is in the local area of the general vibration form table. The LK optical flow method is based on the small displacement assumption and the local consistency assumption, which the LK optical flow method is consistent, introducing the local window $\Omega$, namely the optical flow velocity $\left(u, v\right)$, for all pixels in the local region. In addition, the LK optical flow method can make the least squares fit to the motion, so it can well fit the characteristics of the cable vibration and can further improve the recognition accuracy.

Based on the characteristics of small and continuous displacement, the LK optical flow method is used to study the cable displacement, and the subsequent tests are analyzed.

2.3. Multi-Point Optical Flow Cable Force Measurement Method Based on Euler Motion Magnification

The original video became image data frame by frame, and the image data were motion amplified using the Euler motion amplification method, assuming the original image is $I\left(x, y, t\right)$, where $x$ and $y$ are spatial coordinates and $t$ is time. Through phase amplification enhancement, the amplification factors $A\left(x, y, t\right)$ are introduced to release the motion signal of large cable displacement. The zoomed-in image is represented as:

$$I^{\prime }\left(x, y, t\right)=I\left(x, y, t\right)·A\left(x, y, t\right)$$

(12)

where $A\left(x, y, t\right)$ are the magnification factors, which can be expressed as:

$$A\left(x, y, t\right)=1+\alpha ·sin\left(\omega t\right)$$

(13)

where $\alpha$ is the multiple of magnification and $\omega$ is the angular frequency of the vibration. The magnification factor $\alpha$ is determined by calculating the SNR between the low-frequency and high-frequency noise components outside the specified frequency range $\omega$ and the signal within the frequency range $\omega$.

In the magnified image $I^{\prime }\left(x, y, t\right)$, set the tracking point $P_1, P_2, \cdots P_{n-1}, P_n$, where $P_1$ and $P_2$ are set on the cable, $P_1$ and $P_2$ are typically selected at the centroid position of the cable and are evenly distributed along the cable to ensure comprehensive capture of the cable’s vibration characteristics, and the coordinates are, respectively, $\left(x_1, y_1\right)$, $\left(x_2, y_2\right)$, located at different positions of the cable. Points like P₁ and P₂ reflect the displacement properties of the cable in the direction; set a $P_n$ point in the background fixed immobile region, as an auxiliary feature point, and the coordinates are shown for $\left(x_n, y_n\right)$. Due to the inevitable problem of shooting equipment vibration during shooting, $P_n$ is used in a fixed area in the background, which functions to help identify image distortion or offset caused by external factors. By calculating the difference between P₁ and P₂ and $P_n$ point shifts, it is possible to isolate and evaluate the influence of these environmental factors on the accuracy of tracking measurements under complex conditions.

First, based on the feature points P₁ and P₂, calculate the directional slope of the cable $m={\displaystyle \frac{y_2-y_1}{x_2-x_1}}$, and the direction of the cable is determined. Let $u=y_2-y_1, v=x_2-x_1$; on this basis, the optical flow method is used to obtain the displacement components $u$ and $v$ for each point, representing the displacements of the cable in the x and y directions, respectively:

$$y=mx+b$$

(14)

The displacement components $u$ and $v$ obtained via the optical flow method indicate the displacement of the cable in the x and y directions. The displacement direction angle of the cable is calculated by the following formula:

$$\theta =arctan\left({\displaystyle \frac{u}{v}}\right)$$

(15)

$\theta$ is the directional angle of the cable displacement, relative to the horizontal x-axis. If the direction of displacement of the cable is at an angle with the horizontal axis, this angle is represented by $\theta$. The true displacement of the cable in space is shown in D:

$$D=\sqrt{u^2+v^2}$$

(16)

$\theta$ provides a geometric description of the displacement direction by calculating the ratio between the horizontal $u$ and vertical $v$ displacements. In addition to this, the formula for D calculates the actual displacement magnitude in space. Together, $\theta$ and D fully describe the characteristics of the cable’s displacement, including both the amplitude and the direction. Furthermore, using the Pythagorean theorem for D, combined with $\theta$, allows the displacement to be decomposed into horizontal and vertical components. If the displacement direction of the cable forms an angle $\theta$ with the x, the displacement components $u^{\prime }$ and $v^{\prime }$ can be expressed as:

$$u^{\prime }=D·cos\left(\theta \right)$$

(17)

$$v^{\prime }=D·sin\left(\theta \right)$$

(18)

where $u^{\prime }$ and $v^{\prime }$ represent the transverse and longitudinal displacement of the true vibration direction of the corrected cable. The displacement of P_n auxiliary feature point $\left(x_n, y_n\right)$ was calculated using $\theta$ to obtain displacements $u_n$ and $v_n$, and then the difference between the transverse and longitudinal displacement and P_n points corrected for each feature point on the cable. $u^{\prime }-u_n$ and $v^{\prime }-v_n$ are the displacement measurement results to eliminate device shaking and background interference in complex environments.

2.4. The Vibration Method

The vibration differential equation for a single-span cable can be established based on the beam bending vibration model, which considers the influence of axial force in structural dynamics, to determine the relationship between the cable vibration frequency and the cable force based on the boundary conditions [20]. First, the vibration differential equation was established:

$$EI{\displaystyle \frac{{\mathsf{\partial }}^{4}v(x, t)}{\mathsf{\partial }{x}^4}}-T{\displaystyle \frac{{\mathsf{\partial }}^{2}v\left(x, t\right)}{\mathsf{\partial }{x}^2}}+m{\displaystyle \frac{{\mathsf{\partial }}^{2}v\left(x, t\right)}{\mathsf{\partial }{t}^2}}=0$$

(19)

Solve the equation and obtain the general solution of the cable vibration equation:

$$Y\left(x\right)=C_{1}ch\beta x+C_{2}sh\beta x+C_{3}cos\gamma x+C_{4}sin\gamma x$$

(20)

where $C_1-C_4$ is the pending constant; $\beta$ and $\gamma$ are the coefficients related to the frequency, mass, bending stiffness and cable force of the cable. For the pinned cable at both ends, the boundary conditions are introduced: $Y\left(0\right)=0, { Y}^{\prime }\left(0\right)=0, Y\left(l\right)=0 \mathrm{a}\mathrm{n}\mathrm{d} Y^{\prime }\left(l\right)=0$. The corresponding relationship between the vibration frequency of each order of the cable and the cable force can be obtained.

$$T=4m{f}_n^2{\left({\displaystyle \frac{l}{n}}\right)}^2-{\displaystyle \frac{EI{p}^2}{{\left(\frac{l}{n}\right)}^2}}$$

(21)

2.5. Operational Procedure for Cable Force Measurement Method

The principle of the multi-point optical flow cable force measurement method based on Euler motion magnification is illustrated in Figure 1.

Step 1: Install a high-speed camera to capture displacement videos of the cable. In the first frame of the recorded video, select several displacement measurement points on the cable. Decompose the processed video frame by frame into images, and apply the Euler motion amplification method. Set a target frequency range as the magnification range $\omega$. By adjusting different magnification factors, calculate the SNR for frequencies within the range $\omega$ and for low-frequency and high-frequency noise outside the range $\omega$. Compare the SNR at different magnification factors to determine the optimal magnification factor to be used for the video.

Step 2: In the magnified video, select multiple points on the cable as displacement measurement points. Calculate the displacement direction angle of the cable based on the slopes between these points. Use this to measure the true directional displacement of the cable. Additionally, select auxiliary feature points, and similarly consider the displacement direction angle. Subtract the two results to calculate the cable displacement, eliminating the effects of equipment shaking and background interference in complex environments.

Step 3: Perform spectral analysis on the statistically obtained displacement response curve of the cable to measure its natural frequency. Based on the displacement data from multiple measurement points, plot the vibration mode shape of the cable. Considering the structural characteristics of the cable, establish the partial differential equation for its free vibration. Using the known dynamic characteristics and boundary conditions of the cable, apply vibration theory formulas to calculate the cable force.

3. Validation Test

3.1. Test Model

Based on the original structure, a 1:10 scaled-down model was designed to meet the conditions of geometric similarity, load similarity, stress similarity, and boundary condition similarity. For the cables, the scaled-down design follows the principle of stress similarity, with the EA similarity ratio set to approximately 1:100. The scaled-down design results for the cables are shown in

Table 1. Specifications and dimensions of cable net.

Model Components	Real Model			Scale Model
	Cable Specifications	Area (mm²)	Young’s Modulus (Mpa)	Cable Specifications	Area (mm²)	Young’s Modulus (Mpa)	EA Similarity Ratio
inclined cables	4 × D125	42,800	1.65 × 105	D26	429	1.6 × 105	1/103
load-bearing cables	1 × D50	1740	1.65 × 105	Φ7	29.8	1.5 × 105	1/64
wind-resistant cables	2 × D80	8840	1.65 × 105	Φ14	117.1	1.5 × 105	1/83
edge ring cables	2 × D90	11,120	1.65 × 105	Φ14	117.1	1.5 × 105	1/104
curved cables	8 × D90	44,480	1.65 × 105	2-Φ20	472	1.5 × 105	1/103

. In terms of materials, the cables in the actual engineering project are all sealed cables, while the model test uses both sealed cables and steel wire ropes. Considering that the cable net structure includes a saddle-shaped cable net and inclined cables, the scaled-down model of the Xia Tian cable net, as shown in Figure 2, was used as the research object.

The cable net structure consists of a lower-ring beam support, an upper single-layer cable net, and masts. The single-layer cable net adopts a saddle-shaped hyperbolic form, with load-bearing cables and wind-resistant cables connected via clamps. The structural boundary includes one rigid edge and three flexible cable boundaries. The specific structural components, as shown in Figure 3, include curved cables, ring beams, inclined cables, and edge ring cables. The dimensions of the cable net are 21 m × 5.4 m. Two masts are located at the front of the cable net, with each mast anchored to the ground using two cables. The four cables, labeled XS-1 to XS-4, have a diameter of 26 mm and an effective length of 5.4 m, as shown in Figure 4.

This study compares the multi-point optical flow cable force measurement method based on Euler motion magnification with the traditional accelerometer-based cable force measurement method, validating the accuracy of the proposed measurement method.

3.2. Test Protocol

• Test equipment: The test uses an Imetrum high-speed camera to capture digital image data of cable displacement and a Donghua INV9822 accelerometer to collect acceleration signals of the cables.
• Filming location: To capture as much of the cable body as possible, the high-speed camera was positioned 10 m away from the cable.
• Test data collection: Before the experiment, finite element simulation was conducted to determine that the fundamental frequency of the cable was below 30 Hz, and the dominant vibration mode was the first vibration mode. To avoid aliasing effects, according to the Nyquist Sampling Theorem, the sampling frequency must be at least twice the highest-frequency component of the signal [21]. Therefore, the sampling frequency of the high-speed camera was set to 60 Hz, with a recording duration of 2 min. The sampling frequency of the accelerometer was also set to 60 Hz. To more effectively capture the main vibration modes of the cable, the placement and number of sensors were optimized based on the actual site conditions. This optimization ensured that the key parts of the cable’s vibration modes were covered while avoiding data redundancy or measurement errors caused by excessive or improper placement. For each cable, three accelerometers were installed along the cable length at the top, middle, and bottom positions, with each accelerometer spaced 1 m apart. The specific arrangement is shown in Figure 5. During each measurement, the accelerometers and the high-speed camera were synchronized to start recording simultaneously.

3.3. Analysis of the Test Data

3.3.1. Feature Point Setting of Optical Flow Method

Before identifying the displacement of cables using the traditional optical flow method and the multi-point optical flow method based on Euler motion magnification, the selection of feature points and window size is necessary. A comparison of the methods is presented in

Table 2. Comparison plots of the different methods.

Displacement Tracking Method	Camera Shaking Compensation	Feature Points
Traditional optical flow method	No compensation	P1
Multi-point optical flow method based on Euler motion magnification	Multi-point optical flow compensation	P1, P2, Pn−1, …, Pn

. For the traditional optical flow method, a single feature point P₁ on the cable is selected as the feature point for frame-by-frame tracking. In the multi-point optical flow method based on Euler motion magnification, three feature points are selected: P₁, P₂, and P₃. Multiple points, P₁ and P₂, were selected to measure the cable’s displacement, while an auxiliary feature point, P₃, was introduced to minimize the impact of equipment movement and external disturbances. The displacement differences between P₁, P₂ and P₃ were calculated to correct for device shaking. These measurements were then used to calculate the cable’s displacement in the actual displacement direction and to determine the directional angle of the cable’s movement. For the optical flow calculation, a window size of 13 pixels was chosen. This size was selected based on the physical dimensions of the cables, ensuring that the window effectively captures the cable’s motion while reducing noise from surrounding disturbances. P₃ plays a crucial role in stabilizing the measurements by serving as a reference point, allowing for the calculation of displacement differences between points on the cable and the auxiliary feature point. As an example, the selection of feature points for XS-3 is shown in Figure 6 and Figure 7. The same measurement method was consistently applied to all other cables to ensure uniformity and accuracy.

3.3.2. Euler Magnification Setting

To ensure the accuracy and effectiveness of the multi-point optical flow method based on Eulerian motion magnification in cable displacement measurement, the selection of the magnification factor is crucial. An appropriate magnification factor can enhance the identifiability of cable displacement while avoiding the amplification of noise and artifacts. Based on the cable’s length, tension, and modal characteristics, finite element simulations were conducted, and the results indicated that the primary frequencies of the cables are concentrated in the 15–25 Hz range. Focusing on the primary vibration modes of the cables can effectively capture the dynamic structural response while filtering out low-frequency motion caused by wind and high-frequency interference caused by equipment noise [22]. Therefore, the 15–25 Hz range is optimized as the target frequency band for the key modal dynamics of the cables in this study.

To ensure the accuracy of the magnification effect, a reasonable magnification factor can enhance vibration signals to the greatest extent while avoiding the over-magnification of noise and artifacts. Although higher magnification factors can further enhance the signal, the magnification of high-frequency noise and artifacts leads to a slight decrease in the SNR [23]. Therefore, magnification factors ranging from 1 to 5 were applied, and spectral analysis was performed on the displacement response difference signals from the previously selected feature points and auxiliary feature points. The SNR of the spectral data for each magnification factor was calculated to extract vibration characteristics within the target frequency range. Taking the data from XS-3 as an example, when the magnification factor was 1 (no magnification), the SNR was 7.5 dB. At magnification factors of 2 to 5, the SNR values were 21.80 dB, 22.24 dB, 21.70 dB, and 21.64 dB, respectively (see Figure 8). Through comparative analysis, the SNR reached its maximum value when the magnification factor was set to 3, achieving a balance between signal enhancement and artifact suppression. This magnification factor was ultimately selected as it optimized the SNR while effectively enhancing the vibration characteristics of the cables and avoiding interference from high-frequency artifacts and background noise. The same measurement method was applied to the other three cables, and the magnification factor for all four cables was uniformly set to 3.

3.3.3. Displacement Response Curve and Vibration Analysis

Using the traditional optical flow method and the multi-point optical flow method based on Euler motion magnification to calculate the displacement of four cable strands, a 30 s video segment was selected for analysis. In the multi-point optical flow method based on Euler motion magnification, feature points P₁ were selected in the video for optical flow tracking. The feature points P₁ for each cable strand were located at the middle position of the cable, and based on the displacement direction of the cable, the pixel displacement of the corresponding feature points was calculated frame by frame. The displacement response curves, as shown in Figure 9, were then obtained. As the SNR increased significantly, the displacement response curve obtained from the multi-point optical flow method based on Euler motion magnification showed a noticeable enhancement in signal strength at the displacement points of the cable, compared to the traditional optical flow method. The displacement data were then analyzed using Fourier analysis to compute the frequency values. Both the sensor data and high-speed camera measurement data only displayed the first-order frequency. This is primarily due to the cable’s low stiffness and high force, which cause the vibration frequency to concentrate in lower-order modes. In particular, the first-order frequency is more easily excited and observed, while higher-order modes are less noticeable due to higher damping and difficulty in excitation. The calculation results are shown in Figure 10. According to

Table 3. Frequency data comparison.

Cable Number	Acceleration Sensor Spectrum Data (Hz)	Measurement Frequency of Conventional Optical Flow Method (Hz)	Measurement Frequency of Multispot Optical Flow Method (Hz)	Traditional Optical Flow Method Error (%)	Multi-point Optical Flow Method Error (%)
XS-1	20.13	21.76	20.58	8.1	2.3
XS-2	17.50	18.86	17.92	7.7	2.4
XS-3	18.21	18.86	18.01	3.6	1.1
XS-4	18.03	18.93	18.02	5.1	0.05

, the traditional optical flow method measured the frequencies of the four cable strands as 21.76 Hz, 18.86 Hz, 18.86 Hz, and 18.93 Hz. The multi-point optical flow method based on Euler motion magnification measured the frequencies of the four cable strands as 20.58 Hz, 17.92 Hz, 18.01 Hz, and 18.02 Hz. According to Figure 11, compared to the traditional optical flow method, the multi-point optical flow method based on Euler motion magnification significantly reduces the influence of camera vibration, and its detection frequency results are more accurate due to taking into account the true displacement direction of the cable. However, the measured values are slightly smaller compared to the traditional method. This underestimation is primarily caused by a slight over-correction of background motion during the camera vibration correction in the multi-point optical flow method, minor errors introduced in the calculation of the displacement direction angle due to pixel discreteness or limitations in signal processing accuracy, and the slight suppression of low-frequency signals caused by the choice of filtering or amplification parameters in Euler motion magnification. Additionally, the vibration characteristics of the cable, including low stiffness and high tension, limit the response to higher-order modes, further contributing to the underestimation of low-frequency signal amplitude. The average frequency error of the traditional optical flow method was 6.2%, while the average frequency error of the multi-point optical flow method based on Euler motion magnification was only 1.5%. This indicates that, compared to the traditional optical flow method, the multi-point optical flow method based on Euler motion magnification significantly improves the accuracy of the cable’s natural frequency calculation.

3.3.4. Vibration Mode Identification and Cable Force Measurement

To accurately plot the vibration mode shape of the cables and calculate the cable force, the multi-point optical flow method based on Euler motion magnification was used to analyze the dynamic characteristics of the cable from the perspective of spatial distribution, particularly the changes in its spatial position. This method can intuitively present the distribution characteristics of the vibration mode shape. By processing the data obtained from the multi-point optical flow method based on Euler motion magnification, the calculation method follows the same procedure as the previous measurement process. Multiple points, P₁, P₂ and P₃, were selected for measurement, and an auxiliary feature point P₄ was set to eliminate the effects of equipment shake. This allows for the reconstruction of the displacement distribution of the cable at various points, thereby fully depicting the entire vibration mode. The measurement data of the cable, obtained from high-speed cameras during the test, were processed to extract the primary frequencies of the cable. The amplitude and phase spectrum data at each point were extracted, and a vibration mode diagram was plotted, as shown in Figure 12.

After obtaining the vibration mode shape, the cable force was further calculated. By utilizing the relationship between the vibration mode shape and the natural frequency, along with the physical parameters of the cable (mass and length), the cable force was determined using the vibration method. In the test, four cables were each equipped with an accelerometer to detect the natural frequency of the cable. This frequency was used as a reference for calculating the cable force. The cable frequencies obtained from the calculations were then substituted into the formula to calculate the cable force. The detailed comparison results are shown in

Table 4. Cable force comparison.

Cable Number	Accelerometer Results (KN)	Traditional Optic Flow Results (KN)	Multi-Point Optical Flow Result (KN)	Traditional Optical Flow Error (%)	Multi-Point Optical Flow Error (%)
XS-1	156.4	182.4	165.1	16.6	5.5
XS-2	120.1	137.9	125.1	14.8	4.1
XS-3	130.2	137.4	126.7	5.6	2.3
XS-4	127.6	138.4	126.8	8.5	0.6

.

The force results obtained from the accelerometer were compared with the results from the two optical flow methods. As shown in Figure 13, the test results indicate that the cable force values obtained using the accelerometer, traditional optical flow method, and multi-point optical flow method based on Euler motion magnification are highly consistent. Using the accelerometer results as a reference, a comparison of cable force errors is shown in Figure 14. The average error of the traditional optical flow method was 11.38%, while the average error of the multi-point optical flow method based on Euler motion magnification was only 3.13%. However, in certain cases such as XS-1, the measurement error was relatively larger. This deviation is primarily related to external vibration interference and may also be affected by the nonlinear effects in the signal amplification process of the multi-point optical flow method. Additionally, minor errors in the selection of feature points and the limitations of camera resolution and the optical flow algorithm in processing subtle displacement signals also contributed to the measurement accuracy. Overall, the multi-point optical flow method based on Euler motion magnification not only improves the anti-interference capability of the signal but also effectively enhances the accuracy of cable force calculation, providing more reliable technical support for structural health monitoring.

4. Conclusions and Future

4.1. Conclusions

The proposed method in this study introduces a multi-point optical flow cable force measurement method based on Euler motion magnification. By combining the phase-based magnification method with the optical flow method, the amplitude of displacement features within a specific frequency range in the image sequence is enhanced, leading to an improved SNR of displacement data and enhanced observability of small amplitude motions. Additionally, by considering complex background interference and incorporating the true displacement direction of the cable into the optical flow method, the pixel motion of the cable is calculated, enabling fitting of its vibration mode and solving for cable force using vibration analysis methods. Based on the field tests conducted, the following conclusions can be drawn:

• The Euler motion magnification method and multi-point optical flow method are introduced, resulting in an increase in the SNR of the cable frequency from 7.5 dB to 22.24 dB after magnification. In comparison with the traditional optical flow method, this approach exhibits superior recognition accuracy when considering equipment sloshing and actual cable displacement.
• The proposed method accurately compares the vibration frequency of the cable with data obtained from sensor measurements, demonstrating high overall accuracy. Even in the presence of external interference, the maximum error in calculating the cable force is limited to 5.5%.
• The proposed method offers significant reductions in deployment time and equipment maintenance costs associated with traditional cable force measurement, thereby enhancing the efficiency and practicality of cable force measurement. It is particularly suitable for bridges or high-altitude structures where sensor installation poses challenges.

4.2. Future Work

• The method’s performance may be limited in highly dynamic environments with significant background interference or extreme weather conditions. Future work could focus on improving robustness by integrating advanced motion magnification techniques, machine learning algorithms, or complementary sensing modalities.
• While the method achieves high accuracy, challenges remain for cables with non-uniform vibrations or irregular surfaces. Future research could refine the optical flow algorithm or explore adaptive and deep learning-based motion estimation methods to address these issues.
• Future efforts could aim to enable real-time cable force monitoring and integrate the measurement data into SHM (structural health monitoring) systems or digital twin models, enhancing the practicality and efficiency of the proposed method in large-scale structural applications.