WANG Ji-wu, YU Peng-fei, LUO Hai-bao, YU Pei-long

(1. School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing 100044, China；2. Gansu Dahua Construction Engineering Co., Ltd, Lanzhou 730030, China)

Abstract： For the linear crack skeleton of railway bridges with irregular strike, it is difficult to accurately express the crack contour feature by using a single smoothing fitting algorithm. In order to improve the measurement accuracy, a polynomial curve fitting was proposed, which used the calibration point of crack contour as the boundary point, and then put them all together to produce a continuous contour curve to achieve the crack length measurement. The method was tested by measuring the linar cracks with different shapes. It is shown that this proposed algorithm can not only solve the jagged problem generated in the crack skeleton extraction process, but also improve the crack length measurement accuracy. The relative deviation is less than 0.15, and the measurement accuracy is over 98.05%, which provides a more effective means for the crack length measurement in railway bridges.

Key words： crack skeleton; length measurement; calibration point; polynomial fitting; railway bridge

0 Introduction

With the rapid development of China’s economic construction, meanwhile the operating mileage of high-speed railway continues to expand, the safety operation of bridge is also put forward with higher requirements. In railway bridges, crack is one of the most common problems, which causes 90% of bridge faults according to engineering practice[1]. Therefore, it is of great practical significance to realize real-time cracks classification monitoring.

The length measurement of crack target is the difficult and key link in the bridge crack detection technology. At present, the artificial non-online crack length measurement method is the mainstream method, which is difficult to find the relative position of the problem when checking bridge cracks. At the same time, due to the limitation of space, it is impossible to measure the length of concrete cracks. Although using bridge inspection vehicle and scaffold can check the problem comprehensively and regularity, they are expensive and have a long period, and mostly need to occupy road, which makes it difficult to meet the real-time requirements of daily inspection[2-4]. Domestic and foreign scholars have carried out in-depth and extensive research on it, and many effective algorithms have been proposed.

With the development of measurement technology, the requirements in on-line measurement are increasing rapidly. Different from contact measurement, non-contact measurement can perform real-time measurement, so it is getting more and more attention in engineering practical applications[5-6]. Fang et al. realized the crack width measurement by pasting the pure color box for distance conversion, but this method is difficult to implement[7]. Zhan et al. proposed a crack length calculation method based on crack skeleton points, but the crack measurement accuracy of this algorithm is low due to the irregularity of crack shape and the interference of non-crack noise, and the algorithm has a large amount of computation and low real-time performance[8].

The shape of cracks in railway bridges is roughly composed of one or several narrow linear contours. And non-contact measurement based on curves mainly applies triangulation measurement principle, which is widely used in high-precision on-line detection[9].

The main principle of the measurement method is to use the linear shape to represent the geometric feature of the surface profile of the measured object[10]. Due to the object skeleton has a certain width, the center line of the object is usually used to represent the geometry of measured object. For the unevenness and jaggedness of the object lines extracted by image processing, which affects the measurement accuracy, many scholars have conducted further research to fit the laser lines. Farin et al. proposed a smoothing method of B-spline curves by node elimination and insertion[11]. Its essence is to make manual modification on the curvature of the curve so that the curvature change uniformly to meet the requirements of smooth curve finally. Mu et al. gave an overall smoothing method for parameterized cubic B-spline curve, which mainly smoothed the curve by adjusting the control vertices of the curve[12]. A method for extracting coordinates of laser light stripe centers based on multiscale analysis was proposed by Li et al., and it got the extraction of center line pixel in laser strip[13]. Qin et al. used curvature-chord ratio composite criterion for filtering and simplifying significant noise points, and the random filtering algorithm was applied to reduce low frequency random noise points[14]. Then the piecewise low-order interpolation method was applied to fit cavity point cloud to make the curve smoother. The smoothing processing of curve point cloud data based on Lagrange multiplier method was proposed by Wang et al., which can get high fitting precision[15].

In practical applications, non-single irregular surfaces are often encountered. After feature extraction of the crack, an irregular curve is generated. If the above smoothing method is adopted simply, the key point may be neglected, resulting in the deviation between the final fitting curve and the real contour feature.

In order to extract the laser line profile efficiently, the curve refinement algorithm is adopted based on Canny method[16]. Moreover, a piecewise polynomial curve fitting method with the calibration point as the boundary point is proposed. Accordingly, the length measurement of cracks can be realized. Experimental results show that it can provide high fitting accuracy and improve measurement accuracy.

1 Method and principle

1.1 Camera calibration

Camera calibration is an important prerequisite for crack measurement, and the quality of camera calibration results will directly determine the subsequent crack length measurement accuracy. Therefore, before crack measurement, it is necessary to determine the mathematical correspondence between the real stereo environment and the plane image through the camera calibration principle.

Fig.1 shows the projection relationship of the camera. The projection pointP(x,y) of any pointP(xc,yc,zc) on the image plane is the intersection of line between the center of lightOcandP(xc,yc,zc) and the imaging plane.

Fig.1 Projection relations of cameras

The transformation relations of world coordinates, camera coordinates and image coordinates are

(1)

whereβx=f/aandβy=f/arepresent the equivalent focal length inXandYdirections respectively;arepresents the actual image size corresponding to the single pixel range;dis the scale transformation factor;W1is the external camera parameter matrix, which is determined by parametersfx，fy，Cx，Cy; (Cx,Cy) is the main point coordinate;W2is the camera external parameter matrix of 4×4;Wis the projection transformation matrix of 3×4;Xwis the corrdinate matrix of the world corrdinate system;RandTrepresent the rotation and translation matrices corresponding to the world coordinate system and the camera coordinate system respectively, andRis an orthogonal unit matrix of 3×3.

In practical work, the actual imaging of industrial cameras is not ideal transmission imaging. The deviation between actual imaging and ideal projection imaging is called as camera distortion, which is mainly divided into radial distortion, eccentric distortion and thin edge distortion[17]. In the crack calibration process of railway bridges, the eccentricity distortion of thin prism and lens are mainly caused by the assembly error of optical system, which has less influence than the radial distortion. Therefore, we focus on the radial distortion of lens. In engineering applications, the first and second order radial distortion can meet the actual needs. The distortion model is

(2)

whereδxandδyare distortion variables in the direction of two axes;k1andk2are radial distortion coefficients;x′ andy′ are distortion coordinates.

In this paper, the calibration method in Ref.[18] is used. The main calibration process is as follows：

1) Take photos of the manufactured chessboard at different angles and directions to collect calibration images as shown in Fig.2.

Fig.2 Calibration images collected in different directions

2) The corners of 15 calibration images are extracted one by one. The corner extraction process of the first calibration image is shown in Fig.3.

Fig.3 Corner extraction

3) According to the perspective model and camera distortion model, the camera internal and external parameter matrices are solved. The final calibration results are

k1=0.096 8,k2=0.179 8.

(3)

4) Projection of the calibration results are carried out as shown in Fig.4.

According to the results of visual calibration projection, the average calibration error is 0.122 3 pixel, and the calibration effect is ideal.

Fig.4 Projection of calibration results

1.2 Line extraction method

Using line triangulation technology, the center line of the crack strip is extracted. The steps are as follows.

1) Extract the crack line in the captured image;

2) Refine the previous extracted line to obtain the center line with a single pixel width;

3) Remove the noise out of the laser line.

For the captured images of the measured object with the crack skeleton, the point cloud data of the crack line is extracted[19]. Then the crack line refinement is performed based on the Canny algorithm. Finally, the center line with the single pixel of the crack strip is calculated. Fig.5 is the extraction result of a practical crack line based on above procedure.

Fig.5 Illustration of crack skeleton extraction process

1.3 Model with different shapes

For a single smooth convex surface, the crack line is a continuous curve. When the surface has concave and convex feature, the crack line becomes a piecewise smooth curve, and the cut-off point of each part is the key point of the measured surface. If the whole line is simply smoothed, the distortion of actual object contour fitting will be caused and the error will be large. Here the key point is defined as calibration point.

According to the curvature change of the measured object skeleton, it can be divided into two categories：

1) Single smooth surface, as shown in Fig.6(a).AandBare the start and end of the laser line, and they are also the mark nodes.

2) Irregular surface, as shown in Fig.6(b).AandDare the starting point and end point of the crack line;A,D,B,Care the calibration points. If the calibration points are neglected in the smoothing process, it will produce measurement errors in the final results.

Fig.6 Laser line with different calibration points

In this paper, the mark nodes are considered as the key points, and a polynomial curve fitting method with the least squares method is based on the above mark nodes.

1.4 Constraint control

In the actual measurement, the number of calibration points of crack skeleton is not large, which is easy to judge. Before fitting the extracted curve, specify the approximate interval of each calibration point on the curve by human eyes. At the same time, in order to obtain the precise calibration point position, it is necessary to check the segment where the calibration point exists, and judge whether the corner errorθstaticand the bow height errorδstaticmeet the requirements.

1) Corner constraint

(4)

Fig.7 Calculation of angle and chord errors

As seen from Fig.7, in the interval of [X1,Xn],θ′ decreases first and then increases, andθreaches the minimum at the calibration pointXk. In practical work, in order to avoid the influence of noise, it must be satisfied with

θ>θstatic.

(5)

2) Bow height constraint

As shown in Fig.7, the errors of the two-sided bow height areδ1andδ2.

(6)

(7)

δ1<δ1static,

(8)

δ2<δ2static.

(9)

With these two constraints, the feature nodes can be found accurately.

1.5 Piecewise polynomial curve fitting algorithm based on mark nodes

Piecewise polynomial curve fitting is applied in each curve with the feature nodes as the cut-off points, and the corresponding polynomial fitting equations are obtained in each divided curve. The argument interval of each equation is separated by the calibration points, and the neighbor sub-sections curve is connected at the mark nodes. The whole curve is formed by putting all segments of fitting curves together.

Let the starting point coordinate of the crack line be (a0,b0) and the end point be (aN,bN). It hasN-1 middle calibration points with coordinates of (ai,bi),i=1,2,…,N-1. The function of each piecewise curve isφi(t), and the fitting equation set is

(10)

where Λ is a positioning symbol indicating the sequence of the data sets. For example,a0ΛtΛa1means that the variabletis between the left calibration pointsa0and the right calibration pointsa1. Thek-th (k=1,2,…,N) piecewise curveφk(t) is

(11)

whereiis the order of the fitting polynomial. And the polynomial equation set can be obtained by the least-squares polynomial fitting method[20].

Because the curve is continuous, then

φk(ak)=φk+1(ak+1).

(12)

Therefore, the length of linear crack can be obtained by integral curve fitting equation. The length of curve can be calculated as

(13)

2 Experimental results analysis

In order to verify the accuracy of the fitting curve, polynomial fitting is adopted for the extracted crack line to calculate the maximum deviation and the standard deviation of the central pixel, where the standard deviation can reflect the error degree between the fitting point and the actual point.

Four samples with simple surface are tested with crack image as shown in Fig.8.

Fig.8 Crack skeleton images of four different samples

Fig.9 shows the relationship between the polynomial fitting order and the root mean square. When the polynomial fitting order is in the third order to the tenth order, the root mean square error is big. If the fitting order is bigger than the tenth order, the root mean square deviation influence gets smaller. In order to balance the amount of computation and the fittingaccuracy, the 10 order polynomial is used in the smoothing process.

Fig.9 Relationship of polynomial fitting order and fitting deviatie

Fig.10 is the smoothing effect comparison of two algorithms: method of least squares and the algorithm proposed in this paper.

Fig.10 Smoothing effect comparison of two algorithms

As seen from Fig.10, the extracted line has many calibration points for piecewise fitting. Obviously, the result shows that the algorithm in this paper has a great ability to reproduce the actual cracks contour, which is much better than the traditional least-squares smoothing algorithm. Because this method guarantees the fitting accuracy of each segment and can accurately show the true contour of the measured object, the measurement accuracy of the actual cracks length has also been greatly improved.

Then a lot of experiments were done, and three experimental results were extracted. Table 1 shows the comparison of relative deviation of two smoothing algorithms, which further proves that the practical value of the algorithm proposed in this paper has higher efficiency and applicability.

Next, the actual crack length can be measured by high-precison crack measurement instrument. Ten groups of linear fracture samples are randomly selected. Firstly, the artificial measured values of the selected samples are obtained as a reference. Then these samples are measured by the algorithm proposed in this paper. The calculated values are compared with those measured by professional measuring tools. Table 2 gives the comparison of the measured values using these two methods.

Table 1 Comparison of relative deviation of two smoothing algorithms

SampleMethod of least squaresAlgorithm proposed in this paper10.025 50.004 55220.014 80.005 13530.027 80.005 723

Table 2 Comparison of measured values

The accuracy errorpis calculated by

(14)

wherelrepresents the actual measured value;l′ represents the value calculated by the algorithm.

In Table 2, the average accuracy error between calculated and measured values is 1.95%, that is, the accuracy of crack length measurement reaches 98.05%. It shows that the polynomial curve fitting algorithm proposed in this paper has high efficiency and accuracy, and can meet the needs of engineering practice.

3 Conclusion

In the non-contact measurement with a crack skeleton, it is important to extract the crack skeleton accurately. Considering the practical application of crack with various shapes, the piecewise polynomial curve fitting method with calibration points is given. Verified with the measurement experiments, the results show that the proposed method can fit the crack skeleton accurately. For the early treatment of problem, it has prominent practical significance to prevent them and realize the early warning of bridge safety operation.