A Monocular Vision-Based Framework for Power Cable Cross-Section Measurement

Abstract

The measurements of the diameter of different layers, the thickness of different layers and the eccentricity of insulation layer in the cross-section of power cables are important items of power cable test, which currently depend on labor-intensive manual operations. To improve efficiency, automatic measurement methods are in urgent need. In this paper, a monocular vision-based framework for automatic measurement of the diameter, thickness, and eccentricity of interest in the cross-section of power cables is proposed. The proposed framework mainly consists of three steps. In the first step, the images of cable cross-section are captured and undistorted with the camera calibration parameters. In the second step, the contours of each layer are detected in the cable cross-section image. In order to detect the complete and accurate contours of each layer, the structural edges in the cross-section image are firstly detected and divided into individual layers, then unconnected edges are connected by arc-based method, and finally contours are refined by the proposed break detection and grouping (BDG) and linear trend-based correction (LTBC) algorithm. In the third step, the monocular vision-based cross-section dimension measurement is accomplished by placing a chessboard coplanar with the power cable cross-section plane. The homography matrix mapping pixel coordinates to chessboard world coordinates is estimated, and the diameter, thickness and eccentricity of specific layers are calculated by homography matrix-based measurement method. Simulated data and actual cable data are both used to validate the proposed method. The experimental results show that diameter, minimum thickness, mean thickness and insulation eccentricity of simulated image without additive noise are measured with root mean squared error (RMSE) of 0.424, 0.103 and 0.063 mm, and 0.002, respectively, those of simulated image with additive Gaussian noise and salt and pepper noise are measured with RMSE of 0.502, 0.243 and 0.058 mm and 0.001. Diameter, minimum thickness and mean thickness of actual cable images are measured with average RMSE of 0.768, 0.308 and 0.327 mm. The measurement error of insulation eccentricity of actual cable image is comparatively large, and the measurement accuracy should be improved.

1. Introduction

High voltage power cables are used for power transmission and distribution in power systems, and cable performance test is essential to reduce operational accidents and improve power supply reliability. Measurements of electrical properties, thermal properties, mechanical properties and different geometric dimensions in the cross-section are included in the power cable test [1,2,3,4]. The defeat of power cables in geometric dimensions will lead to severe electrical and mechanical performance loss, and the accurate measurement of geometric dimensions are of great importance to power cable defect detection.

The geometric dimension measurements of power cables mainly involve the diameter and thickness of different layers and the eccentricity of insulation layer in the cross-section. The traditional manual measurement methods of cable dimensions are kinds of destructive test, in which each layer of interest is measured according to specifications [1,2,3,4] after specimen preparations such as cutting, stripping layer by layer, cleanout, and slicing. Take the power cable with Cross-Linked Polyethylene (XLPE) insulation for rated voltage of 110 kV as an example, the specimen is first stripped layer by layer, diameters up to 25 mm are directly measured with a micrometer or a vernier caliper reading to 0.01 mm, and diameters more than 25 mm are calculated from perimeter measured with a steel tape reading to 0.1 mm. The thickness of nonmetal and metal sheath is measured with a micrometer reading to 0.01 mm. The three insulation related layers are cleaned and sliced first and the thickness is measured with a digital projector or a microscope reading to 0.01 mm.

Diverse manual measuring operations and specimen preparations are involved in traditional measurement methods, which are labor-intensive and inefficient. Besides, the dimension measured by different operators may be different due to inevitable personal errors. Automatic methods for dimension measurements in power cable cross-section are starved for. Current automatic measurement methods mainly include radiation method [5,6,7,8,9], electromagnetic induction method [10,11], and photoelectricity method (vision-based method) [12,13,14,15]. In radiation methods, diameter and thickness of insulation layer and nonmetal sheath layer can be measured online directly with X-ray penetrating the multilayer specimen. There are mature online X-ray measuring systems [5,6], but they are only designed for one insulation layer or three insulation-related layers, and the X-ray apparatus are of high costs. When used for multilayer modeling, the contours of all layers are theoretically treated as eccentric circles [9], so the measuring accuracy is very limited. For electromagnetic induction methods, Wen and Zhao [11] measured the thickness and eccentricity of insulation layer with an eddy current sensor, and Liang et al. [10] measured the thickness of aluminum sheath with two eddy current sensors of high and low frequency excitation. Although multilayer specimen is allowed in electromagnetic induction method, this metal-sensitive measurement method is designed only for a single layer, and the outmost diameter is generally measured by the photoelectric method. In photoelectric methods, the image of insulation layer [12,13] or nonmetal sheath layer [14,15] slice is first acquired by digital microscope or camera, then contours are detected in the image, and the diameter and thickness are finally calculated according to an amplification factor calibrated by a ruler. The existing photoelectric methods are only suitable for cleaned specimen slices, in which cleaning and slicing of specimen are tedious.

Vision-based methods are more and more used for automatic and efficient defect detection and dimension measurement in engineering. Cheng et al. [16] proposed a detection and localization method for self-shattering defects detection in twin glass insulators based on spatial features. Siddiqui et al. [17] utilized a convolutional neural networks based method to automatic powerline equipment detection, and a novel rotation normalization and ellipse detection algorithm was used for insulator detection. Hao et al. [18] conducted image segmentation of ice-covered insulators using proposed GrabCut algorithm, and then calculated the shed overhang and shed spacing with contour convexity defect recognition for icing condition assessment. Fard et al. [19] monitored the electrical tree (ET) in XLPE cable insulation with a camera attached to the microscope and studied the effect of different voltage types on ET and partial discharge. Kang et al. [20] extracted the boundary of a burning wood pellet using scanning lines and then calculated its volume based on radius and height, which is important to understand the wood pellet burning mechanism. Lee et al. [21] proposed a vision-based method with an adaptive region-of-interest algorithm to locate the marker attached to bridge, and the structural displacements of bridge were measured based on planar homography. Zhang et al. [22] measured the steel bar diameter, spacing, and quantity by sub-pixel boundary location in smartphone image and fast image stitch.

In the geometric dimension measurements of power cable, current vision-based automatic methods are mainly used for the sliced insulation related layers or nonmetal sheath layer, and measuring methods for other layers have been rarely reported. Since the vision-based method can obtain a complete cross-section image of power cable along the axial direction, it has a large potential to simultaneously measure the diameter and thickness of different layers and the eccentricity of insulation layer. This paper tries to develop a monocular vision-based method to measure the diameter, thickness, and eccentricity of interest in the cable cross-section. There are single-core and multi-core power cables, in which core means the innermost solid or stranded conductor. Since single-core structure is the most common type and the basis of the multi-core structure, this paper solely deals with the automatic cross-section measurement of single-core power cable.

This paper proposes a monocular vision-based framework to automatically measure the cross-section dimensions of power cable. In our framework, the camera is first calibrated and the distortion parameters are used to undistort the images of power cable cross-section. The next key step of this vision-based method for power cable cross-section measurement is contour detection of different layers. Circle detection methods such as improved Hough Transform methods [23,24,25], geometric feature methods [26,27,28], template matching method [29,30,31], and optimization method [32,33] are usually used to detect such circular contours. Due to the existence of global deformations and local defects in the multilayer contours of cross-section, current circle detection methods cannot meet the demands of contour detection in power cable cross-section measurement. In order to accurately and completely detect the contour of each layer, contours of metal layers are firstly extracted in this paper and used as a reference for following operations, then the structural edges in the cross-section are detected and divided into individual layers. The arc-based method is used to connect the discontinuous edges in each layer. Finally, the contours are refined by proposed break detection and grouping (BDG) and linear trend-based correction (LTBC) algorithm. In the last step, geometric dimension measurements based on monocular vision are accomplished by placing a chessboard coplanar with the power cable cross-section plane. The homography matrix mapping pixel coordinates to chessboard world coordinates is firstly estimated. With the detected contours of each layer, the diameter and thickness of different layers and the eccentricity of insulation layer are then calculated by mapping the contour pixels to chessboard world coordinates based on the estimated homography matrix, which is more logically rigorous than conversion with an amplification factor.

This paper is organized as follows: Section 2 introduces the framework we developed, including image undistortion, multilayer contour detection in cable cross-section, and cross-section dimension measurement. Section 3 demonstrates and analyzes the cross-section measurement results of both simulated data and actual cable specimen data, and discusses the influence of printed chessboard on measurement accuracy. Section 4 presents conclusions and future works.

2. Methodology

Figure 1 shows a typical single-core power cable cross-section, and each contour is denoted by $C_k\left(k=12, 23, 34, 45, 56, 67, 78, 80\right)$, where the subscript $k$ indicates its adjacent structural layers. For instance, $C_{12}$ is interface between layers 1 and 2, and the special $C_{80}$ is outer contour of layer 8. The innermost layer 1 denotes stranded copper conductor with rich chrominance information due to its material. Layer 2 consists of wrapped semi-conductive tape with partial overlapping, which is used to tighten the conductor and almost invisible to naked eyes in cross-section. In order to express the existence of layer 2, contour $C_{23}$ is exaggeratedly drawn in Figure 1. Layers 3–5 are conductor screen, XLPE insulation, and insulation screen, which are produced by the three-layer co-extrusion method with high roundness and tightness. Layer 6 is wrapped semi-conductive buffer water block, layer 7 is flexible corrugated aluminum sheath, and layer 8 is nonmetal outer sheath. The material and manufacturing techniques of each layer determine that inner layers 1–5 are less deformed and good concentric shapes are maintained. The outer layers 6–8 may have larger deformation under extrusion.

According to standards [1,2,3,4], diameter $D_i$ of layer $\left(i=1,5,6,7,8\right)$, minimum thickness ${(t_i)}_{\min }$ and mean thickness ${\overline{t}}_i$ of layer $i$ $\left(i=3,4,5,7,8\right)$, and eccentricity $e_4$ of insulation layer (layer 4) should be measured. So, contour extraction of all layers as much as possible is required in image processing for the cross-section measurement. Since the wrapped tape is almost invisible in cable cross-section, $C_{23}$ cannot be extracted in this vision-based method and the calculated thickness of conductor screen (layer 3) in actual power cable cross-section actually contains wrapped tape (layer 2) and conductor screen. The detection of $C_{56}$ between insulation screen (layer 5) and wrapped buffer (layer 6) is challenging as it is in the similar background and near $C_{45}$.

The flowchart of the proposed framework is shown in Figure 2. This vision-based framework for power cable cross-section measurement mainly contains three steps: image undistortion, multilayer contour detection in cable cross-section, and cross-section dimension measurement. In the first step, the camera is calibrated and captured images of power cable cross-section are undistorted with the camera calibration parameters. In the second step, structural edges in the cross-section are firstly detected and divided into individual layers, unconnected edges are then connected by arc-based method, and contours are further refined by BDG and LTBC algorithm. In the third step, the diameter and the thickness of different layers and the eccentricity of insulation layer are calculated by mapping the detected contour pixels to chessboard world coordinates based on homography matrix $H$.

2.1. Image Undistortion

The camera calibration is conducted with a chessboard. As shown in Figure 3, the world coordinate system $O_w$-$X_w{Y}_w{Z}_w$ is a left-handed coordinate system with the origin $O_w$ at the upper left chessboard corner, $X_w$ axis to the right and $Y_w$ axis down. The pixel coordinate system is denoted by $o$-$uv$, and image coordinate system $o_c$-$xy$ is established with the origin $o_c$ at the principal point, $x$ axis parallel to the horizontal sampling direction of pixel ($u$ axis), and $y$ axis parallel to the vertical sampling direction ($v$ axis).

When we use the chessboard for calibration, arbitrary point $P$ on the chessboard is in the defined $O_w{X}_w{Y}_w$ plane with $Z_w=0$. Mapping relationship from spatial point $P$ to image point $p$ is shown as Equation (1):

$$s\left[\begin{array}{c}u\\ v\\ 1\end{array}\right]=M\left[\begin{array}{cc}R& T\end{array}\right]\left[\begin{array}{c}{X}_w\\ Y_w\\ 0\\ 1\end{array}\right]=M\left[\begin{array}{ccc}{r}_1& r_2& T\end{array}\right]\left[\begin{array}{c}{X}_w\\ Y_w\\ 1\end{array}\right],$$

(1)

where $s$ is an arbitrary scale factor, $M=\left[\begin{array}{ccc}{f}_u& 0& u_0\\ 0& f_v& v_0\\ 0& 0& 1\end{array}\right]$ is the intrinsic matrix for camera, $R$ is a 3-by-3 rotation matrix which can be broken up into three 3-by-1 columns ($R=\left[\begin{array}{ccc}{r}_1& r_2& r_3\end{array}\right]$), and $T$ is a 3-by-1 translation matrix.

Equations (2) and (3) shows the model [34] considering radial distortion and tangential distortion:

$$u^{\prime }=u+\overline{u}\left(k_1{r}^2+k_2{r}^4+k_3{r}^6\right)+\left[2p_1\overline{u}\overline{v}+p_2\left(r^2+2{\overline{u}}^2\right)\right],$$

(2)

$$v^{\prime }=v+\overline{v}\left(k_1{r}^2+k_2{r}^4+k_3{r}^6\right)+\left[p_1\left(r^2+2{\overline{v}}^2\right)+2p_2\overline{u}\overline{v}\right],$$

(3)

$$\overline{u}=u-u_0,$$

(4)

$$\overline{v}=v-v_0,$$

(5)

$$r^2={\overline{u}}^2+{\overline{v}}^2,$$

(6)

where $\left(u^{\prime },v^{\prime }\right)$ are distorted pixel coordinates, $\left(u,v\right)$ are undistorted pixel coordinates, $k_i\left(i=1,2,3\right)$ denotes radial distortion parameter, and $p_i\left(i=1,2\right)$ denotes tangential distortion parameter.

The distortion is firstly assumed to 0, and the initial solution to intrinsic and extrinsic parameters are obtained by Zhang’s method [35]. Then the radial and tangential distortions are considered on the basis of the pinhole model. The final parameters are acquired by Levenberg-Marquardt iterative algorithm based on maximum likelihood criterion [36,37]. Then undistorted images of power cable cross-section can be obtained by distortion correction based on camera calibration parameters.

2.2. Multilayer Contour Detection in Cable Cross-Section

The aim of multilayer contour detection is to get continuous and accurate contour of each layer in the cross-section. Structural edges in the cross-section are first detected and divided into individual layers. Edges in each layer are denoted by $Edg{e}_k\left(kϵ\left\{12, 23, 34, 45, 56, 67, 78, 80\right\}\right)$, in which $k$ indicates its adjacent structural layers. Since edges in a layer may be discontinuous, the unconnected parts in $Edg{e}_k$ are then connected by arc-based method to get the initial closed contour $C_k$. To solve the problem of possible mutations in initial contour caused by interference edges or inaccurate edge connection, each initial contour $C_k$ is refined to final contour by the proposed BDG and LTBC methods.

2.2.1. Edge Detection and Layering

The contours of metal layers are much easier to extract, so the coarse contours of metal layers, including copper conductor (layer 1) and aluminum sheath layer (layer 7), are firstly extracted based on thresholding. Similar to contour $C_k$, notation $r{C}_k\left(k=12, 67, 78\right)$ is used to denote the coarse contours of metal layers, and the pixel length of $r{C}_k$ is denoted by $r{L}_k$. The copper conductor is segmented based on chrominance information in $U^{\prime }$ component, and the coarse contour $r{C}_{12}$ is extracted with the length of $r{L}_{12}$. In YUV color space, $U$ is blue component offset from brightness [38], so $U^{\prime }=-U$ can be seen as yellow component offset from brightness. According to the definitions, Equation (7) can be derived, where constant $U_{\max }$ determines $U^{\prime }\in \left[-U_{\max },U_{\max }\right]$. The coefficients of $R$, $G$, and $B$ sum to 0, which ensures $U^{\prime }\approx 0$ when $R\approx G\approx B$ and can be used to weaken the background in $U^{\prime }$ component:

$$U^{\prime }=U_{\max }\left(\alpha R+\left(1-\alpha \right)G-B\right).$$

(7)

After the measurement image is converted to grayscale image $IG$, the aluminum sheath is segmented based on brightness information in $IG$, and coarse contours $r{C}_{78}$ and $r{C}_{67}$ are then extracted. The coarse contours conform to the actual structural contours but not accurate enough, and they are used as a reference for following line edge detection.

Candidate edges are acquired through Canny edge detection [39] in grayscale image $IG$, including actual structural edge, details inside each structural layer, stain, and cutting texture, which should be further filtered based on the conditions of length and linearity features.

As shown in Figure 4, candidate edges can be firstly divided into three categories based on length. (1) Taking the length of innermost coarse contour $r{C}_{12}$ as reference, the long edges with length no less than $t_l·r{L}_{12}$ are considered as reliable structural edge (Figure 4a). $t_l$ is a proportion of length threshold to the coarse contour length $r{L}_{12}$ of copper conductor. (2) The very short edges with length no more than $t_s$ pixels are regarded as discrete interference details (Figure 4b) and directly removed. (3) The left edges in Figure 4c contain both structural edges and interference details, pending to further linearity judgment. To describe the edge linearity, the eccentricity of ellipse that has the same second moment as $i$-th pending edge is calculated and denoted by $e_i^{\prime }$. The value of $e^{\prime }$ ranges from 0 to 1: the closer $e^{\prime }$ is to 0, the closer the shape is to a circle; the closer $e^{\prime }$ is to 1, the closer the shape is to a line. Short structural edges with fairly good linearity (Figure 4d) are then extracted from pending edges based on the condition of $e_i^{\prime }\ge e_{\min }^{\prime }$. The whole structural edges in cable cross-section are made up of short structural edges with fairly good linearity and the previous long structural edges.

The structural edges are then divided into individual layers with result of $Edg{e}_k$, in which $kϵ\left\{12, 23, 34, 45, 56, 67, 78, 80\right\}$ indicates its adjacent structural layers. The outermost edge $Edg{e}_{80}$ is firstly obtained by hole filling of structural edges. With the previous extracted location of coarse contours $r{C}_k$, $Edg{e}_k\left(k=12, 67, 78\right)$ is then successively extracted from structural edges. The left structural edges involve layers of wrapped tape (layer 2), conductor screen (layer 3), XLPE insulation (layer 4), insulation screen (layer 5), and wrapped buffer (layer 6), which should be further divided into specific layers. Considering that contours of inner layers 1–5 are less deformed and maintain a good concentric shape, edge points in an individual layer are of nearly equal distances from the approximate center. Therefore, the edge points can be divided into specific layers in polar coordinate system by annulus division according to the polar radius. The annulus with structural edge $Edg{e}_k\left(kϵ\left\{23, 34, 45, 56\right\}\right)$ can be extracted based on two characteristics: (1) points in this annulus are far more than those in other annuli with a few useless details; (2) polar angles of points in this annulus are in a larger range than those in other annuli with local useless details. Points in each extracted $Edg{e}_k$ are fitted to an approximate circular contour $C_{ck}$ for further use. The center, radius, and maximum fitting error of fitted circle $C_{ck}$ are denoted by $O_{ck}$, $R_{ck}$, and ${\sigma }_k$ respectively.

2.2.2. Arc-Based Edge Connection

Affected by local interferences, the detected structural edges in a layer may be discontinuous, and unconnected edges in each $Edg{e}_k$ should be connected to get a closed contour. Structural edges in each $Edg{e}_k$ are connected by the proposed arc-based method of direct connection or prediction-and-search connection (PSC) with local constraints of fitted circular contour $C_{ck}$. All structural edges in each $Edg{e}_k$ are firstly clockwise organized as $Ar{c}_p\left(p=1,2,3\dots \right)$, and points in each arc are also clockwise arranged with start point of $P_{p\mathrm{S}}$ and end point $P_{p\mathrm{E}}$. The arc-based edge connection starts with $P_{1\mathrm{E}}$ to the next $P_{2\mathrm{S}}$ as follows.

As shown in Figure 5a, red dots are edge points to be connected, green line denotes fitted circular contour $C_{ck}$, and direct connection is first tried through creating a connecting line $P_{1\mathrm{E}}{P}_{2\mathrm{S}}$. Then distance ${\mathsf{\epsilon }}_i$ between each point in $P_{1\mathrm{E}}{P}_{2\mathrm{S}}$ and $C_{ck}$ is calculated, and root mean squared error (RMSE) is calculated by Equation (8) and used to measure the agreement degree of $P_{1\mathrm{E}}{P}_{2\mathrm{S}}$ and $C_{ck}$:

$$s_c=\sqrt{{\displaystyle \sum }_{i=1}^N{\mathsf{\epsilon }}_i{}^2/N}.$$

(8)

In Equation (8), $N$ is the number of points in line $P_{1\mathrm{E}}{P}_{2\mathrm{S}}$. $s_c\le \max \left({\mathsf{\epsilon }}_{1E},{\mathsf{\epsilon }}_{2S}\right)$ means the distance of $P_{1\mathrm{E}}{P}_{2\mathrm{S}}$ from $C_{ck}$ does not exceed that of endpoints, $P_{1\mathrm{E}}{P}_{2\mathrm{S}}$ agrees well with $C_{ck}$, and $Ar{c}_1$ is updated with connected $Ar{c}_1$, $P_{1\mathrm{E}}{P}_{2\mathrm{S}}$, and $Ar{c}_2$. Otherwise $s_c>\max \left({\mathsf{\epsilon }}_{1E},{\mathsf{\epsilon }}_{2S}\right)$ indicates a large deviation of line $P_{1\mathrm{E}}{P}_{2\mathrm{S}}$ from $C_{ck}$, and the proposed PSC method will be considered.

The idea of PSC is to find appending edge points in local search range with maximum gradient projection in the normal direction. As shown in Figure 5b, red arrow shows the tangential direction of $C_{ck}$ at $P_{1\mathrm{E}}$, and a prediction point $P^{\prime }$ is set to the next point in this tangential direction. Magenta arrow shows the normal (radial) direction of $C_{ck}$ at $P^{\prime }$, and the appending point is then searched in this normal direction around $P^{\prime }$. Gradient and its direction of image $IG$ in search range are calculated, and the point with maximum gradient projection in the normal direction is appended to initial $Ar{c}_1$. Repeat the steps of appending point determination until $Ar{c}_1$ is connected to $Ar{c}_2$, and $Ar{c}_1$ is updated with the connected $Ar{c}_1$ and $Ar{c}_2$.

After $Ar{c}_2$ is brought into $Ar{c}_1$, the next arc connection is continued with above-mentioned direct connection or PSC, until all arcs are contained in $Ar{c}_1$ and $Ar{c}_1$ is closed. The closed $Ar{c}_1$ after edge connection is denoted by initial contour $C_k\left(kϵ\left\{12, 23, 34, 45, 56, 67, 78, 80\right\}\right)$.

2.2.3. Contour Refinement

The initial contour $C_k$ is not smooth enough and fluctuations exist in initial contour $C_k$. As shown in Figure 6, red dots are extracted initial contour, green line denotes the fitted circular contour $C_{ck}$, and fluctuations exist in the blue ellipse. The fluctuations in initial $C_k$ include structure-related gentle changes (Figure 6a) and mutations caused by interference edges and inaccurate edge connection (Figure 6b), in which mutations with steep changes should be distinguished and refined. Since mutations occur at locations with a large radial change in a very small tangential range, the mutations can be detected in the polar coordinate system with pole at the center $O_{ck}$ of fitted circle.

The $i$-th point $P_i$ in contour $C_k$ is firstly converted to polar coordinates $\left({\rho }_i,{\theta }_i\right)$, and the radial distance from this point to the fitted circle $C_{ck}$ is denoted as ${\left(V_{\rho }\right)}_i$ and calculated by Equation (9):

$${\left(V_{\rho }\right)}_i={\rho }_i-R_{ck}, i=1, 2, 3\dots ,$$

(9)

where $R_{ck}$ is radius of the fitted circle $C_{ck}$.

In Figure 7, the vertical axis denotes radial distance $V_{\rho }$ to the fitted circle, horizontal axis denotes polar angle $\theta$ in degrees, and the $V_{\rho }\left(\theta \right)$ sequence is drawn in green line. Similar to a pulse in signal processing, the mutation in $V_{\rho }\left(\theta \right)$ is defined as a $B_1-Ste{p}_{\mathrm{b}}-B_2$ (B-S-B) structure in Figure 7, in which break $B_1$ indicates the beginning segment of steep change away from the overall trend, break $B_2$ indicates the ending segment of steep changes back to the overall trend, and $Ste{p}_{\mathrm{b}}$ denotes the sequence between $B_1$ and $B_2$. If $Ste{p}_{\mathrm{b}}$ contains several points, this B-S-B structure is an interval type mutation (I-Mutation) (magenta arrow in Figure 7a); otherwise, $Ste{p}_{\mathrm{b}}$ is the intersection point of $B_1$ and $B_2$, and this B-S-B structure is a point type mutation (P-Mutation) (orange arrow in Figure 7b). $Ste{p}_0$ denotes the point sequence between two adjacent mutations. Mutations in $V_{\rho }\left(\theta \right)$ are detected and corrected by BDG and LTBC algorithm. The BDG algorithm is designed for breaks ($B_1$ and $B_2$) detection, breaks pairing and mutation grouping. To detect the breaks, $V_{\rho }\left(\theta \right)$ sequence is firstly smoothed and simplified by Douglas-Peuker (DP) decimation [40] to keep only major fluctuations, and the result is denoted by$DP\left(\theta \right)$ and drawn with red dashed line in Figure 7. The absolute radial difference of two consecutive points in simplified $DP\left(\theta \right)$ is then calculated by Equation (10):

$$\left|\mathrm{d}DP\left({\theta }_j\right)\right|=\left|DP\left({\theta }_j\right)-DP\left({\theta }_{j-1}\right)\right|, j\in i.$$

(10)

The $\left|\mathrm{d}DP\left(\theta \right)\right|$ sequence is used to quantize the radial change of points in $DP\left(\theta \right)$ and drawn with blue line in Figure 7. As shown in Figure 7, a break in $V_{\rho }\left(\theta \right)$ is a sequence segment with steep changes, and it appears in the form of a peak point (local maxima) in $\left|\mathrm{d}DP\left(\theta \right)\right|$ sequence. So, all points in $\left|\mathrm{d}DP\left(\theta \right)\right|$ sequence that are larger than their two neighboring points are obtained as peaks and breaks are further detected from these peaks. As shown in Figure 8, breaks can be detected based on $\left|\mathrm{d}DP\left(\theta \right)\right|$ and the angle $\mathsf{\alpha }$ between two consecutive points and radial direction. In Figure 8, red dots are continuous contour points in the original $V_{\rho }\left(\theta \right)$ sequence, point $P_{j-1}$ and $P_j$ denote two consecutive points in the simplified $DP\left(\theta \right)$ sequence, $\mathrm{d}DP\left({\theta }_j\right)$ and $\mathrm{d}{\theta }_j$ are the radial and angular difference between point $P_j$ and $P_{j-1}$, and $\alpha$ is the angle between line $P_{j-1}{P}_j$ and radial direction at point $P_j$. The $V_{\rho }\left(\theta \right)$ sequence between point $P_{j-1}$ and $P_j$ is determined as a break if $\left|\mathrm{d}DP\left({\theta }_j\right)\right|\ge t_{\rho }$ and $\mathsf{\alpha }<t_{\alpha }$. Peaks in $\left|\mathrm{d}DP\left(\theta \right)\right|$ meeting the above two conditions are extracted as major peaks (red circle in Figure 7), and the corresponding breaks in $V_{\rho }\left(\theta \right)$ are obtained. After breaks are detected, each detected break should be paired with another break to make up a group of B-S-B structure. Since the breaks $B_1$ and $B_2$ in a pair are steep changes away from and back to the overall trend respectively, the differences $\mathrm{d}DP\left(\theta \right)$ corresponding $B_1$ and $B_2$ must be in opposite signs. Paired P-breaks are first detected from the points adjacent to major peaks in $\left|\mathrm{d}DP\left(\theta \right)\right|$ sequence. For a major peak at point $P_j$, check whether point $P_{j-1}$ or $P_{j+1}$ meets the condition of $\left|\mathrm{d}DP\left(\theta \right)\right|\ge t_p·\left|\mathrm{d}DP\left({\theta }_j\right)\right|$ and $\mathrm{d}DP\left(\theta \right)·\mathrm{d}DP\left({\theta }_j\right)<0$. If so, major peak at $P_j$ (circle in Figure 7b) and its adjacent point $P_{j-1}$ or $P_{j+1}$ (square in Figure 7b) indicate a pair of P-Breaks. Otherwise, each major peak in Figure 7a indicates a pending I-Break. The possible isolated I-Break should be supplemented by another break or directly removed to ensure an even number of final I-Breaks. After paired breaks are detected, the P-Mutation and I-Mutation are finally grouped and the $Ste{p}_0$ sequences can be obtained.

The LTBC algorithm is designed for $V_{\rho }$ correction to points in mutations according to the linear trends of adjacent $Ste{p}_0$ sequences. In Figure 9, magenta and green solid lines denote detected I-Mutation to be corrected, and the blue lines denote the last and next $Ste{p}_0$ sequence. As shown in Figure 9, the $B_1$, $Ste{p}_{\mathrm{b}}$, and $B_2$ segments in a mutation are corrected separately. For an I-Mutation in small angular range, the last $Ste{p}_0$ and next $Ste{p}_0$ are fitted to line $f_l$ and $f_n$ respectively (red lines), and the short line $f_{\mathrm{c}}$ (red line) connecting $f_l$ to $f_n$ is used as a reference for $Ste{p}_{\mathrm{b}}$ correction. $V_{\rho }$ of $Ste{p}_{\mathrm{b}}$ segment is firstly corrected by translation, and the translation $t$ is calculated by least square estimation according to Equation (11):

$${\left(c{V}_{\rho }\right)}_i={\left(V_{\rho }\right)}_i+t,$$

(11)

where ${\left(V_{\rho }\right)}_i$ denotes $V_{\rho }$ of each point in $Ste{p}_{\mathrm{b}}$ to be corrected and ${\left(c{V}_{\rho }\right)}_i$ denotes $V_{\rho }$ of each point in the referential $f_{\mathrm{c}}$ sequence. After $V_{\rho }$ of $Ste{p}_{\mathrm{b}}$ segment is corrected (green dashed line), the $V_{\rho }$ of $B_1$ segment is then corrected by linear transformation (magenta dashed line) based on Equation (12):

$${\left(V_{\rho }\right)}_i^{\prime }=V_{lE}+\frac{V_{bS}^{\prime }-V_{lE}}{V_{bS}-V_{lE}}\times \left({\left(V_{\rho }\right)}_i-V_{lE}\right),$$

(12)

where ${\left(V_{\rho }\right)}_i$ and ${\left(V_{\rho }\right)}_i^{\prime }$ denote $V_{\rho }$ of each point in $B_1$ segment before and after correction, $V_{lE}$ is $V_{\rho }$ of the ending point in last $Ste{p}_0$, and $V_{bS}$ and $V_{bS}^{\prime }$ denote $V_{\rho }$ of the starting point in $Ste{p}_{\mathrm{b}}$ before and after correction. This linear transformation method of $B_1$ correction ensures smooth transition from last $Ste{p}_0$ to corrected $Ste{p}_{\mathrm{b}}$. Finally, $V_{\rho }$ of $B_2$ segment is corrected similarly to $B_1$ for smooth connection between corrected $Ste{p}_{\mathrm{b}}$ and the next $Ste{p}_0$. The LTBC method also applies to P-Break group with $Ste{p}_{\mathrm{b}}$ of one point.

Repeat the above-mentioned steps of BDG, LTBC, and $V_{\rho }\left(\theta \right)$ sequence update until less than three break groups are detected. After correction to $V_{\rho }$, the corrected points are converted from polar coordinates $C_k\left(R_{ck}+V_{\rho }\left({\theta }_i\right),{\theta }_i\right)$ to pixel coordinates $C_k\left(u_i,v_i\right)$.

2.3. Cross-Section Dimension Measurement

In our framework, a chessboard is placed near the power cable cross-section and the chessboard plane is coplanar with the power cable cross-section plane. Using the chessboard, the homography matrix mapping chessboard world coordinates of chessboard corners to the pixel coordinates can be estimated. According to Equation (1), mapping from spatial point $P$ on the chessboard plane to image point $p$ can be described by a 3-by-3 homography matrix $H$ as Equations (13) and (14):

$$\tilde{p}=H\tilde{P},$$

(13)

$$H=\left[\begin{array}{ccc}{H}_{11}& H_{12}& H_{13}\\ H_{21}& H_{22}& H_{23}\\ H_{31}& H_{32}& H_{33}\end{array}\right]=\mathsf{\lambda }M\left[\begin{array}{ccc}{r}_1& r_2& T\end{array}\right],$$

(14)

where $\mathsf{\lambda }$ is an arbitrary scale factor.

The homography matrix $H$ has only 8 free elements and can be normalized by $H_{33}=1$. For measurement image after undistortion, we can calculate $H$ with chessboard world coordinates $\left(X_w,Y_w\right)$ and pixel coordinates $\left(u,v\right)$ of chessboard corners according to Equation (15):

$$\left[\begin{array}{cccccccc}{X}_w& Y_w& 1& 0& 0& 0& -u{X}_w& -u{Y}_w\\ 0& 0& 0& X_w& Y_w& 1& -v{X}_w& -v{Y}_w\end{array}\right]\left[\begin{array}{c}{H}_{11}\\ H_{12}\\ H_{13}\\ H_{21}\\ H_{22}\\ H_{23}\\ H_{31}\\ H_{32}\end{array}\right]=\left[\begin{array}{c}u\\ v\end{array}\right].$$

(15)

Since the chessboard is coplanar with the power cable cross-section, the estimated homography matrix$H$ can also be used to map the pixel coordinates of cross-section contours to chessboard world coordinates. The chessboard world coordinates $C_k\left(X_w,Y_w\right)$ of contour points can be calculated with pixel coordinates $C_k\left(u,v\right)$ and $H$ according to Equation (16) after multilayer contour detection:

$$\left[\begin{array}{cc}{H}_{11}-u{H}_{31}& H_{12}-u{H}_{32}\\ H_{21}-v{H}_{31}& H_{22}-v{H}_{32}\end{array}\right]\left[\begin{array}{c}{X}_w\\ Y_w\end{array}\right]=\left[\begin{array}{c}u-H_{13}\\ v-H_{23}\end{array}\right].$$

(16)

After $C_k\left(X_w,Y_w\right)$ is obtained, the diameter $D_i$ is measured based on Equation (17) of the perimeter tape method. As shown in Figure 10a, we first extract the convex hull [41] of the dense contour points in a layer, and the sum of convex vertex distance is used as perimeter $L$ to simulate the manual measurement of the perimeter tape:

$$D_i=L_i/\pi ,i=1,5,6,7,8.$$

(17)

To measure the thickness of layer $i\left(i=3,4,5,7,8\right)$, each point $P_{\left(i-,j\right)}$ of its inner contour is traversed to calculate the thickness $t_{\left(i-,j\right)}$ at the $j$-th point. As shown in Figure 10b, the minimum thickness ${(t_i)}_{\min }$ of layer $i$ is the minimum of all $t_{\left(i-,j\right)}$, and the mean thickness ${\overline{t}}_i$ is the mean value of six $t_{\left(i-,j\right)}$ measured at angle interval of 60 degrees from ${(t_i)}_{\min }$.

The eccentricity of insulation layer is defined as Equation (18):

$$e_4=\left(t_4{}_{\max }-t_4{}_{\min }\right)/t_4{}_{\max },$$

(18)

where $t_4{}_{\min }$ and $t_4{}_{\max }$ are the minimum and maximum thickness of insulation layer respectively.

3. Results and Discussion

3.1. Cross-Section Measurement of Simulated Cable

A chessboard containing $7\times 6$ squares is designed in AutoCAD and the side length of each square is 10 mm. In order to verify the effectiveness of the proposed method, we firstly simulate a cable cross-section with concentric circles in AutoCAD, in which wrapped tape (layer 2) is exaggeratedly drawn. The simulated pattern and chessboard are printed on an A4 paper, and this printed paper is captured by a CMOS camera (Nikon, Tokyo, Japan) with image resolution of $6000\times 4000$ as simulated image $I_1$ (Figure 11a). Gaussian noise with variance of 0.01 and salt and pepper noise with density of 0.01 are successively added to the $R$, $G$ and $B$ channels of $I_1$, and then we can get the simulated image $I_2$ shown in Figure 11b.

3.1.1. Multilayer Contours Detection

Coarse contour of copper conductor is firstly extracted based on chrominance information, and those of aluminum sheath are then obtained from grayscale image based on brightness information. The candidate edges are obtained by Canny detection and result of $I_2$ with mixed noise is shown in Figure 12a. In Canny edge detection, the standard deviation of Gaussian filter is set to $\sqrt{2}$ for noise reduction, the high threshold $t_{high}$ is dynamically 70th percentile of gradients of whole image, and the low threshold is $0.4t_{high}$. The proportional threshold $t_l$ is set to 0.25 for long structural edges, and $t_s$ is set to 64 pixels for very short interference details. Since long structural edges with $e^{\prime }$ more than 0.9 are close to lines, the threshold $e_{\min }^{\prime }$ for short structural edges is loosely set to the minimum $e^{\prime }$ of these long linear edges. The whole structural edges of $I_1$ and $I_2$ are obtained by filtering with length and linearity features, and the result of $I_2$ is shown in Figure 12b.

For edge layering, $Edg{e}_k\left(k=12, 67, 78, 80\right)$ is firstly extracted from structural edges based on coarse contours, and $Edg{e}_k\left(k=23, 34, 45\right)$ is then extracted by annulus division. As shown in Figure 12c, nearly all edges (including $Edg{e}_{23}$) can be extracted from $I_2$ except that $Edg{e}_{56}$ in $I_2$ is covered by the mixed noise. Discontinuous structural edges in $I_1$ and $I_2$ are then connected by arc-based method with search radius of 1. As shown in Figure 13, there are several edge discontinuities in $Edg{e}_{12}$ (yellow dots) in $I_1$ and $I_2$, which are connected with magenta dots and enclosed in this step. Since $Edg{e}_{56}$ between insulation screen (layer 5) and wrapped buffer (layer 6) with close grayscale in $I_2$ is missing after edge layering, we try to detect the $Edg{e}_{56}$ and $C_{56}$ from enhanced image. As shown in Figure 14a, grayscale $\left[0,0.2\right]$ in the original grayscale image of $I_2$ is mapped to $\left[0,1\right]$, $Edg{e}_{56}$ (yellow dots) is then detected in the enhanced grayscale image, and closed contour $C_{56}$ can be obtained by arc-based edge connection (magenta dots). The arc-based edge connection relies on the previously detected edge, so connection result would absorb the deviations of inaccurate edge (yellow dots in red circle in Figure 14b) caused by busy background.

Each initial $C_k$ is fitted to circular contours $C_{ck}$ with max fitting error of ${\sigma }_k$. Major peaks with corresponding breaks are extracted by thresholds of $t_{\rho }=0.9{\sigma }_k$ and $t_{\alpha }=45$, and $t_p$ is set to 0.7 for paired P-Breaks determination in adjacent location of major peaks. As shown in Figure 14c, one P-Mutation in initial $C_{56}$ (yellow dots in red circle) of $I_2$ is detected by BDG, LTBC is then made to the mutation, and the refined results are plotted with red dots, which reduces the bias to some extent.

3.1.2. Cross-Section Dimension Measurement

Extracted contour points are drawn with red dots in Figure 15, and their vertexes of convex hull are plotted with blue asterisks. Figure 15 shows that all contours required (including exaggeratedly drawn $C_{23}$) can be extracted from simulated images. Locations of minimum thickness ${(t_i)}_{\min }$ are drawn with magenta lines in Figure 15, and the other five thicknesses measured clockwise are drawn with green lines. Influenced by noise, ${(t_i)}_{\min }$ locations in $I_1$ and $I_2$ may be inconsistent, so the six thicknesses measured from ${(t_i)}_{\min }$ location for mean thickness calculation are all different.

The diameter measurement results are listed in

Table 1. Diameter measurement results for $I_1$ and $I_2$. (unit: mm).

Layer	${\mathit{D}}_0$	${\mathit{D}}_1$	$\left|{\mathit{V}}_{\mathit{D}1}\right|$	${\mathit{D}}_2$	$\left|{\mathit{V}}_{\mathit{D}2}\right|$
8	115.470	115.958	0.488	115.772	0.302
7	105.600	105.694	0.094	105.499	0.101
6	99.910	100.557	0.647	100.390	0.480
5	85.330	85.791	0.461	86.292	0.962
1	46.752	46.901	0.149	46.681	0.071
RMSE	-	-	0.424	-	0.502

, in which $D_1$ and $D_2$ are diameter measurements of $I_1$ and $I_2$, and $\left|V_{D1}\right|$ and $\left|V_{D2}\right|$ are corresponding errors between vision-based measurements and manual measurements $D_0$. Diameters are measured with RMSE of 0.424 mm for image $I_1$, and the mixed noise brings about a slight increase in diameter measurement errors for image $I_2$ with RMSE of 0.502 mm. Compared with $I_1$, the salt and pepper noise will cover the contour $C_{56}$ to some extent and leading to a larger diameter measurement error for layer 5 in $I_2$. Since the RMSE of image $I_1$ and $I_2$ is almost equal, the diameter measurement is robust to noise.

The thickness measurement results are listed in

Table 2. Thickness measurement results for $I_1$ and $I_2$. (unit: mm).

Layer	Manual		${\mathit{I}}_1$				${\mathit{I}}_2$
	${\mathit{t}}_{\min 0}$	${\overline{\mathit{t}}}_0$	${\mathit{t}}_{\min 1}$	$\left|{\mathit{V}}_{\mathit{t}\mathrm{m}1}\right|$	${\overline{\mathit{t}}}_1$	$\left|{\mathit{V}}_{\overline{\mathit{t}}1}\right|$	${\mathit{t}}_{\min 2}$	$\left|{\mathit{V}}_{\mathit{t}\mathrm{m}2}\right|$	${\overline{\mathit{t}}}_2$	$\left|{\mathit{V}}_{\overline{\mathit{t}}2}\right|$
8	4.820	4.995	4.941	0.121	5.051	0.099	4.936	0.116	5.074	0.079
7	2.450	2.552	2.446	0.004	2.476	0.048	2.411	0.039	2.516	0.036
5	1.520	1.552	1.328	0.192	1.510	0.073	0.999	0.521	1.646	0.094
4	14.980	15.045	14.944	0.036	14.896	0.036	14.894	0.086	15.041	0.004
3	1.900	2.020	1.891	0.009	2.045	0.032	1.861	0.039	2.033	0.013
RMSE	-	-	-	0.103		0.063		0.243		0.058

, in which $t_{\min 1}$, ${\overline{t}}_1$, $t_{\min 2}$ and ${\overline{t}}_2$ are thickness measurements for $I_1$ and $I_2$ respectively. $t_{\min }$ and $\overline{t}$ in a layer are theoretically equal for concentric circles, while it’s not true in manual and vision-based measurements due to errors caused by printing, imaging, and noise. Mean thickness for $I_1$ and $I_2$ are measured with RMSE of 0.063 mm and 0.058 mm, which reflects to some extent the overall measurement accuracy, and minimum thickness measurement errors are with RMSE of 0.103 mm and 0.243 mm, which shows the accuracy in worst cases. The mixed noise in image $I_2$ may cover local edges, so the minimum thicknesses especially that in layer 5 are measured with a relatively larger RMSE. The insulation eccentricity is theoretically 0 and the manual measurement is 0.013. Influenced by both the measurement errors of maximum and minimum thickness, the $e_4$ measurements in

Table 3. Eccentricity measurement of insulation layer in $I_1$ and $I_2$.

Method	${\mathit{t}}_4{}_{\max }\left(\mathbf{mm}\right)$	${\mathit{t}}_4{}_{\min }\left(\mathbf{mm}\right)$	Insulation Eccentricity ${\mathit{e}}_4$
Manual	15.180	14.980	0.013
$I_1$	15.171	14.944	0.015
$I_2$	15.106	14.894	0.014

are 0.015 for $I_1$ and 0.014 for $I_2$ with very small measurement errors.

3.2. Cross-Section Measurement of Actual Cable

The chessboard pattern is printed on an A4 image and used for homography matrix estimation. Actual cable specimen is cut with an axial length of about 100 mm, cable cross-section and chessboard are simply pressed by the glass for coplanar condition, and the cross-section image is captured and shown in Figure 16. Note that the wrapped semi-conductive tape (Layer 2) is invisible to naked eyes in this cross-section image. All parameters in multilayer contour detection of actual specimen image are selected in the same way as those of simulated images above.

3.2.1. Multilayer Contours Detection

Linear combination $U^{\prime }$ is firstly constructed to express the yellow component offset from brightness, and coarse contour of copper conductor is segmented from $U^{\prime }$. Let $U_{\max }=1$ to enhance the $U^{\prime }$ perception of color difference, and Equation (7) can be simplified as $U^{\prime }=\alpha R+\left(1-\alpha \right)G-B$. Considering the color of copper conductor approximates $RGB\left(173,113,68\right)$, the linear combination $U^{\prime }=0.6R+0.4G-B$ is constructed with $\alpha :\left(1-\alpha \right)=173:113$. As shown in Figure 17a, coarse contour $r{C}_{12}$ conforms well to the actual edges of copper conductor except for a few deviations caused by specular reflection point on contour segmented to the background. Coarse contours of aluminum sheath (layer 7) are then segmented from grayscale image and $r{C}_{67}$ (blue dots) and $r{C}_{78}$ (green dots) are plotted in Figure 17b. To illustrate the details of coarse aluminum contours, partial enlargement of Figure 17b is drawn in Figure 17c, from which we can see $r{C}_{78}$ and $r{C}_{67}$ conform well to actual contours except for a few deviations in $r{C}_{67}$ (blue dots in magenta ellipses) caused by wrapped buffer (layer 6) peeling off and aluminum sheath cutting burr.

The candidate edges are obtained by Canny detection and shown in Figure 18a, including structural edge, details inside each layer, stain and cutting texture. Thresholds for filtering based on length and linearity features are selected in the same way as those of simulated images. After edge filtering, long edges and short edges with good linearity are reserved, and the whole structural edges are shown in Figure 18b.

For edge points layering, cable region is firstly obtained by hole filling of structural edges, and the outermost boundary is extracted as $Edg{e}_{80}$ and plotted in Figure 19a. Structural edges that are near the previously extracted coarse contours are then extracted as $Edg{e}_k\left(k=12, 67, 78\right)$ and drawn in Figure 19a. For the left structural edges related to layers with good concentric shapes, edge points are divided into different annuli and rendered with different colors (Figure 19b), and the annuli with structural edges of $Edg{e}_{34}$ and $Edg{e}_{45}$ are detected (Figure 19c) based on the point count and polar angle span. Since insulation screen (layer 5) and wrapped buffer (layer 6) are in close grayscale and location, few edge points of the contour between them exist in previously detected structural edges, and $Edg{e}_{56}$ cannot be extracted by annulus division. All layered edge points $Edg{e}_k\left(k=12, 34, 45, 67, 78, 80\right)$ are plotted in Figure 20a. To illustrate the details of result, partial edges in Figure 20a are redrawn with red dots in Figure 20b, and the green lines denote the fitted circular contour of each layer. The distances between layered edges and fitted circles confirm the good concentric shape of inner layers and larger deformation of outer layers. Besides, it can be seen from Figure 20b that interference edge points (dots in blue ellipses) and discontinuity (dots in magenta ellipses) exist in the edge layering result.

After layered $Edg{e}_k$ is obtained, discontinuous edges in each layer are connected by arc-based edge connection and the connections are shown in Figure 21a, in which yellow dots are previous $Edg{e}_k$ and magenta dots are connections. For this specimen with average stain, the discontinuities between arcs in extracted $Edg{e}_k$ are not long, so direct connection can meet the requirements. The only one closed contour in each layer after connection is the initial contour $C_k$, and the result without $C_{56}$ is shown in Figure 21b. The contrast of Figure 20a and Figure 21b shows that all unconnected structural edges are successfully connected, and the unconnected interferences are removed. Since $Edg{e}_{56}$ is not extracted in previous edge layering, we try to detect $Edg{e}_{56}$ in enhanced grayscale image and closed $C_{56}$ is then obtained by edge connection of $Edg{e}_{56}$ (Figure 22a). To illustrate the details of edge connection, three local positions in (Figure 22a is enlarged and shown in Figure 22b–d. The arc-based edge connection relies on the previously layered edges, connection result is reliable if $Edg{e}_{56}$ is accurate (Figure 22b), and the result would absorb the deviations of inaccurate $Edg{e}_{56}$ caused by busy background (Figure 22c,d). As shown in Figure 22c,d, there are several textures inside the wrapped buffer (layer 6) mistaken for $Edg{e}_{56}$, and this problem remains in the arc-based edge connection result without improvement.

For contour refinement, one P-Mutation in initial $C_{45}$ and two P-Mutations in initial $C_{56}$ are detected by BDG algorithm and LTBC is made to these mutations. As shown in Figure 23a, yellow dots denote initial contour $C_{45}$, red dots are refined contour after LTBC, and the inaccurate edges caused by cutting texture of XLPE insulation are detected as mutation and corrected. As shown in Figure 23b, yellow dots denote initial contour $C_{56}$ and the steep edges in busy background are detected as mutation and corrected. The inaccurate edges of $C_{56}$ are caused by inaccurate arc-based edge connection in busy background (Figure 22c,d), LTBC algorithm can fix mutations in Figure 22c to Figure 23b with circle characteristics and sequence continuity, but inaccurate edges with gentle changes in radial direction (Figure 22d) are failed to be corrected. For other layers without break detected, it means good consistency of points in the contour.

3.2.2. Cross-Section Dimension Measurement

Extracted contour points are drawn with red dots in Figure 24, and vertexes of their convex hull are plotted with blue asterisks. Similar to cross-section measurement of simulated images, it can be seen from Figure 24 that all contours required can be detected by this method except for invisible $C_{23}$.

Table 4. Measurement results of actual specimen cross-section. (unit: mm).

Layer	${\mathit{D}}_0$	$\mathit{D}$	$\left|{\mathit{V}}_{\mathit{D}}\right|$	${\mathit{t}}_{\min 0}$	${\mathit{t}}_{\min }$	$\left|{\mathit{V}}_{\mathit{t}\min }\right|$	${\overline{\mathit{t}}}_0$	$\overline{\mathit{t}}$	$\left|{\mathit{V}}_{\overline{\mathit{t}}}\right|$
8	118.729	119.207	0.478	4.940	4.839	0.101	7.628	7.903	0.275
7	103.387	103.098	0.289	2.340	2.209	0.131	2.747	2.798	0.051
6	96.784	97.580	0.796	-	-	-	-	-	-
5	86.803	87.984	1.181	0.930	0.803	0.127	1.062	1.303	0.241
4	-	-	-	16.140	15.438	0.702	16.325	16.175	0.150
3	-	-	-	1.440	1.477	0.037	1.567	2.042	0.475
1	48.000	48.176	0.176	-	-	-	-	-	-
RMSE	-	-	0.689	-	-	0.328	-	-	0.281

is the measurement results of actual specimen cross-section. The $D_0$, $t_{\min 0}$ and ${\overline{t}}_0$ are manually measured diameter, minimum thickness and mean thickness, and the $D$, $t_{\min }$ and $\overline{t}$ are vision-based measurement results.

The terms $\left|{\mathrm{V}}_D\right|$, $\left|V_{t\min }\right|$, and $\left|V_{\overline{t}}\right|$ are the corresponding measurement errors between vision-based and manual methods. Note that the outermost $D_0$ is measured with perimeter tape, and $D_0$ of other layers is the mean of multiple diameters nondestructively measured by a vernier caliper. $t_{\min 0}$ and ${\overline{t}}_0$ are nondestructive measurements on the cable cross-section. In Table 4, the RMSE of diameter measurement is 0.689 mm, and the diameter error of layer 5 reaches up to 1.181 mm. Though closed $C_{56}$ can be detected from the enhanced image, there are still several textures inside layer 6 mistaken for $C_{56}$ (Figure 22d), which bring about the local expansion of $C_{56}$ and larger diameter and mean thickness measurements of layer 5. For minimum thickness measurement, vision-based measurement results are all smaller than manual measurement results except for layer 3, and this consistency may be because point traverse in vision-based measurement is more sensitive to minimum value determination than naked-eye observation. $C_{23}$ is invisible and not extracted from image of actual cable cross-section, so slightly larger minimum thickness and mean thickness of layer 3 in Table 4 are calculated based on contours $C_{13}$ and $C_{34}$. The minimum thickness measurement error of insulation layer reaches 0.702 mm, and no clear influence factors for this relatively large error have been found. The larger mean thickness and diameter measurements of the outermost layer may be related to the local expansion of $C_{80}$ due to cutting burrs shown in Figure 25. A different minimum thickness location between vision-based and manual measurement means six thickness measurement locations for mean thickness calculation are all different, and the overall minimum and mean thickness errors are with RMSE of 0.328 mm and 0.281 mm. As shown in

Table 5. Eccentricity measurement of insulation layer.

Method	${\mathit{t}}_4{}_{\max }\left(\mathbf{mm}\right)$	${\mathit{t}}_4{}_{\min }\left(\mathbf{mm}\right)$	Insulation Eccentricity ${\mathit{e}}_4$
Manual	16.560	16.140	0.025
Image	16.631	15.438	0.072

, manually measured $e_4$ is 0.025, and the vision-based measurement influenced by measurement errors of maximum and minimum thicknesses is 0.072. Note that the error of maximum thickness measurement of insulation layer is only 0.071 mm, and the large error of eccentricity is mainly determined by the above-mentioned large error of minimum thickness.

In order to estimate the precision of this method, we performed four similar experiments with an actual specimen. In these experiments, specimen and chessboard are placed with different relative locations and rotations, and images are captured at different object distances. The terms $\left|{\mathrm{V}}_D\right|$, $\left|V_{t\min }\right|$, and $\left|V_{\overline{t}}\right|$ in

Table 6. Measurement errors of diameters and thicknesses in four experiments. (unit: mm).

Layer	8	7	6	5	4	3	1	RMSE	RMSEa
$\left|V_{D1}\right|$	0.478	0.289	0.796	1.181	-	-	0.176	0.689	0.768
$\left|V_{D2}\right|$	0.702	0.073	0.984	1.627	-	-	0.116	0.908
$\left|V_{D3}\right|$	0.568	0.192	0.876	1.132	-	-	0.056	0.694
$\left|V_{D4}\right|$	0.611	0.145	0.893	1.362	-	-	0.078	0.781
$\left|V_{t\mathrm{m}1}\right|$	0.101	0.131	-	0.127	0.702	0.037	-	0.328	0.308
$\left|V_{t\mathrm{m}2}\right|$	0.428	0.038	-	0.113	0.343	0.070	-	0.253
$\left|V_{t\mathrm{m}3}\right|$	0.434	0.028	-	0.110	0.518	0.170	-	0.316
$\left|V_{t\mathrm{m}4}\right|$	0.377	0.012	-	0.090	0.614	0.204	-	0.337
$\left|V_{\overline{t}1}\right|$	0.275	0.051	-	0.241	0.183	0.475	-	0.281	0.327
$\left|V_{\overline{t}2}\right|$	0.326	0.118	-	0.413	0.219	0.475	-	0.336
$\left|V_{\overline{t}3}\right|$	0.304	0.144	-	0.360	0.190	0.394	-	0.295
$\left|V_{\overline{t}4}\right|$	0.289	0.144	-	0.607	0.245	0.507	-	0.397

are the corresponding measurement errors between vision-based and manual methods in four experiments, and these statistical results are arranged for each measured item. Table 6 shows that diameters are measured with average RMSE of 0.768 mm, and it reduces to 0.545 mm if we remove the measurements of layer 5 with inaccurate contour points. The minimum thicknesses are measured with average RMSE of 0.308 mm, and the mean thicknesses are measured with average RMSE of 0.327 mm. It can be seen from

Table 7. Measurement errors of insulation eccentricity in four experiments.

Experiment	${\mathit{t}}_4{}_{\max }\left(\mathbf{mm}\right)$	${\mathit{t}}_4{}_{\min }\left(\mathbf{mm}\right)$	Insulation Eccentricity ${\mathit{e}}_4$	Error	Errora
1	16.631	15.438	0.072	0.047	0.026
2	16.429	15.797	0.038	0.013
3	16.417	15.622	0.048	0.023
4	16.289	15.526	0.047	0.022

that the insulation eccentricities in four experiments are measured with errors of 0.047, 0.013, 0.023 and 0.022, and the average error is 0.026.

3.3. Influence of Printed Chessboard on Measurement Accuracy

The side length of each chessboard square is designed as 10 mm in AutoCAD, and the actual length in printed chessboard is on average 9.600 mm measured by a Vernier caliper. After image undistortion based on the actual size of chessboard, the homography matrix $H$ mapping the chessboard world coordinates of the chessboard corners to the pixel coordinates is calculated. To evaluate the influence of printed chessboard on plane measurement, the coordinates differences $\left(\mathrm{d}{X}_w,\mathrm{d}{Y}_w\right)$ between mapped chessboard world coordinates $\left(X_w^{\prime },Y_w^{\prime }\right)$ and actual coordinates $\left(X_w,Y_w\right)$ of chessboard corners are calculated. The coordinates differences in experiments of simulated cables and actual cables are shown in Figure 26a,b.

It can be seen from Figure 26 that most coordinates differences of chessboard corners in simulated cable images are within 0.05 mm, and those in actual cable images are within 0.1 mm. The placements of chessboard papers in experiments make a difference to chessboard smoothness and then corner point precision. Compared with cross-section measurement of simulated images, there is a loss of accuracy in the measurement results of actual cable images. In this homography matrix-based measurement method, the measurement accuracy is highly dependent on the coplanar precision of cable cross-section and the chessboard. For simulated images, cable cross-sections are strictly coplanar with the chessboard, so the homography matrix-based coordinate mapping is of higher positional accuracy. Since the actual specimen is simply pressed by the glass for coplanar condition, there are deviations between the mapped chessboard world coordinates and the physical positions of contour points, and the cross-section is measured with comparatively lower accuracy.

4. Conclusions and Future Works

To improve the geometric dimension measurement efficiency in power cable testing, a monocular vision-based cross-section measurement framework for single-core power cable was proposed in this paper. This vision-based framework consists of image undistortion, multilayer contour detection, and cross-section dimension measurement. The images of cable cross-section are firstly captured and undistorted with the camera calibration parameters. Then the contour of each layer is detected in the cable cross-section image. Finally, the diameter and thickness of different layers and eccentricity of insulation layer are calculated by the proposed homography matrix-based measurement method. The feasibility of this framework is verified by experiments on simulated images and actual images of power cable specimen cross-section. For multilayer contour detection, continuous contour of each layer can be synchronously extracted by edge detection and layering, arc-based edge connection, and contour refinement. And the proposed contour detection algorithm has good adaptability for structural deformation. Diameter, minimum thickness, mean thickness and insulation eccentricity of simulated image without additive noise are measured with RMSE of 0.424, 0.103 and 0.063 mm, and 0.002, respectively, and those of simulated image with mixed noised are measured with RMSE of 0.502, 0.243 and 0.058 mm and 0.001. Diameter and mean thickness measurements are robust to noise, while the mixed noise will obviously increase the measurement error of minimum thickness due to local interferences. Diameter, minimum thickness and mean thickness of actual cable images are measured with averaged RMSE of 0.768, 0.308 and 0.327 mm, and the insulation eccentricity result is most sensitive since it is influenced by both the maximum and minimum thickness in extreme cases.

There is a large potential for accuracy improvement in dimension measurement based on this monocular vision-based framework. In order to improve the measurement accuracy, the specimen should be first cleaned up before imaging to eliminate the influence of cutting burrs, debris, and stain. For the placement scheme of chessboard, high coplanar precision would contribute to accuracy improvement of this homography matrix-based cross-section measurement on one hand. On the other hand, a hollowed-out chessboard placed outside the cable cross-section would be considered to avoid the extrapolation from chessboard corners in coordinate mapping, however this puts forward a higher coplanar requirement for hardware design. As to contour detection algorithm, methods for accurate contour extraction from regions with close grayscale are needed.