**4. Result and Discussion**

#### *4.1. Picture Perspective Transform of Test Beam*

According to the principle of perspective transformation, the image of the test beam obtained under each load grades is processed, and the result of image processing under one of the load grades is shown in Figure 8. It can be seen that through perspective transformation, the image of the test beam changes from oblique projection to orthogonal projection, while all the details of the specimen have been well preserved. It is worth noting that the remain parts apart from the specimen are distorted by the perspective transformation. However, the distortion does not affect the acquisition of structural deformation data since the test aims at the overall deformation data of the specimen only.

 **Figure 8.** Perspective transformation of the tested beam; (**a**) after perspective transform; (**b**) integral drawing of specimen after projection transformation.

#### *4.2. Edge Contour Extraction of Structures*

Several types of operators can be employed for the edge detection, including the Sobel [28] operator, Prewitt operator [29], Roberts operator [30], and log operator [31], in which different methods are used to solve the gradient extremum. In this paper, all the five operators have been applied to detect the edge of the specimen using the image after perspective transformation, as shown in Figure 9. Compared with the Log operator, the edge detection results of the other four operators are not satisfactory due to the lack of edge information, which in turn has a negative impact on the accuracy. The advantage of the Log operator over the other methods is that the Gauss spatial filter is employed to smooth the original image, which minimizes the influence of noise on edge detection. The Log operator is a second-order edge detection operator [32], as shown in Equations (19) and (20):

$$
\nabla^2 f = \frac{\partial^2 f}{\partial \mathbf{x}^2} + \frac{\partial^2 f}{\partial y^2} \tag{19}
$$

$$\begin{cases} \frac{\partial^2 f}{\partial x^2} = f(i, j+1) - 2f(i, j) + f(i, j-1) \\ \frac{\partial^2 f}{\partial y^2} = f(i+1, j) - 2f(i, j) + f(i-1, j) \end{cases} . \tag{20}$$

**Figure 9.** The applied five kinds of edge detection operators; (**a**) Sobel operator edge detection; (**b**) Prewitt operator edge detection; (**c**) Roberts operator edge detection; (**d**) log operator edge detection.

Based on the second-order differential of the image, the extreme points can be generated at the abrupt position of the gray value. According to these extreme points, the edge of the structure can be determined.

As shown in Figure 9, the distribution of the light intensity respecting the specimen is inconsistent, and the gray value of some edges does not change significantly, resulting in discontinuity in the detection of some edges. The discontinuous edges like Figure 9 are very normal in actual structure images. On the other hand, the distribution of edge pixels obtained by various edge algorithms is also different. The distribution density of edge pixels in this paper is shown in Figure 10.

**Figure 10.** Distribution density of edge pixels by various edge detection operators; (**a**) edge width distribution degree of Sobel operator; (**b**) edge width distribution degree of Prewitt operator; (**c**) edge width distribution degree of Robert operator; (**d**) edge width distribution degree of log operator.

As shown in Figure 10, the edge features obtained by the above five operators are entirely different. For instance, the edge distributions in some operators are three pixels wide, while in some operators, e.g., the Log operator, the edges are just concentrated in one pixel. Because of the large scatter in the pixel distribution, it is difficult to determine the exact edge position of the structure. Naturally, an efficient edge detection operator should have the centralized pixel distribution and good pixel continuity. After comparison, it is found that the Log operator can extract the edges of the structure relatively intact and maintain a relatively centralized distribution of edge pixels. Thus, the log operator has been selected in extracting the edge of the structure.

The phenomenon of discontinuous edges and scattered edge distribution of the above-mentioned is similar to the detection issue in real structures, which is induced by the environment of measurement. In dealing with such kind of problem, the data processing and analysis process have been employed, as illustrated in the following. As an important part of the deformation, the edge of the structure is the key content of this paper. For the specimens in this paper, the upper and lower edges of the bridge deck and the lower edges of the specimens can be used as characteristic contours to analyze the overall deformation of the structure. From Figure 2, we can see that the noise caused by the environment will make the edge location confused. Points with continuous gradient change are very rare in the actual

structure image. The presence of image noise can lead to the generation of pseudo edges. On the other hand, the real signal on the edge of the structure may also be smoothed out by the Gauss spatial filter, which will cause the edge of the structure to be discontinuous and the edge information missing as shown in Figure 11. Figure 11a shows that the lower edge contour of the bridge deck is relatively continuous. Therefore, the lower edge of the bridge deck is used as the characteristic contour of the test beam to extract the overall deformation of the structure; the position of edge contour extraction in this paper is shown in Figure 11b.

**Figure 11.** Feature contour extraction; (**a**) edge of the test beam affected by environment; (**b**) sketch map of edge extraction position.

Because the space position of the camera is fixed and the same perspective transformation method is used in each load grade, the edge contour before and after deformation can be directly extracted and compared without looking for fixed points in each load grade. The pixel coordinates of the lower edge of the bridge deck are extracted, and the original edge contour of the structure is obtained under each load grade, as shown in Figure 12.

**Figure 12.** Edge line of lower edge of bridge deck.

The result shows that the deflection of the specimen increases with the load grade. Since the edge line of the specimen is obscured near the right bearing by the reaction frame, the edge information is partially missed. By contrast, the information of the left edge is well established. Considering the parallel change of the contour line near the bearing on the left, it can be inferred that the rigid displacement of the specimen exists under each load condition. When loaded from 0 to 100 kN, the rigid displacement reaches its largest. This large displacement is due to the fact that the bearings are not in close contact with the reaction frame before loading. After the load of 100 kN, the bearings will be tightly contacted with the reaction frame. However, due to the deformation of the reaction frame since it is not ideally stiff, a small amount of rigid body displacement still exists in all the load conditions. Besides, it can be found that the edge contour of the specimen is piecewise continuous under every single load condition, with the step change occurred at the four connections between the segments. During the fabrication, the specimen is first divided into five segments and each of the segments is manufactured independently. Then, the separated segments are assembled at the four points together, resulting in the inevitable assembly error. As a result, the shape of the specimen will change abruptly at those assembly points, which in turn lead to the step change as reflected in the edge contour.

#### *4.3. Deformation Curve Obtained by Overlapping Di*ff*erence of Contour Line*

By calibrating each pixel in the image, the size of each pixel can be obtained. The size of each pixel is the theoretical limit of the accuracy in the proposed photogrammetric method. In the steel truss bridge tested, the vertical member is orthogonal to the pixels, as shown in Figure 13. Thus, the calibration of the vertical member is relatively simple. As shown in Table 3, by calibrating 13 visible vertical members, the measurement accuracy of this experiment is 1.12 mm. According to the calibration value, the contour of the pixel in Figure 12 can be transformed into the actual deformation value.

**Figure 13.** Pixel calibration schematic.

**Table 3.** Pixel size calibration table of vertical members.


The edge line shown in Figure 12 is notably discontinuous at the connecting points due to assembly error as analyzed before. However, the discontinuity is not caused by the structural deformation and will not change with the applied load. Therefore, the influence of the discontinuity can be eliminated by the overlapping difference method. Since the bearings contact the reaction frame closely after 100 kN, the edge line of the initial working condition can be overlap differenced by the edge line of the other load grades. Thus, the load-displacement curve of 100–600 kN can be derived, as shown in Figure 14.

**Figure 14.** Load-displacement curves under various load grades.

It can be found that the load-displacement curve presents a zigzag feature. The reason for this problem is that the edge of the structure in the image is composed of one or more pixels, as shown in the red pixel of Figure 15. These pixels are arranged side by side to form a bandwidth. The final edge position (shown in deep red in Figure 15) is determined by the gradient change rate of the gray value of all pixels in the bandwidth. Under the influence of illumination conditions, the final edge and the actual edge will have errors, as shown in the dotted line of Figure 15. As a result, the zigzag phenomenon occurs in the load-displacement curve shown in Figure 14. Based on the above analysis, it can be indicated that the deformation extracted from the image will be distributed around the actual deformation of the structure. On this end, the load-deformation curve is polynomial fitted to approximate the actual deformation value of the structure, as shown in Figure 16. The measured comparison of three dial meters is shown in Figure 17.

between pixel edge and actual edge.

Difference

**Figure 15.**

**Figure 17.** The load-displacement curve measured by dial meters.

#### *4.4. Error Analysis of Photogrammetry*

The overall deformation of the structure edge extracted by photogrammetry is compared with the dial gauge as shown in Figure 18. The numerical comparison results are shown in Table 4. From Figure 18 and Table 4, it can be seen that the structural deformation data obtained by photogrammetry are consistent with the data measured by the dial meters, and the maximum error is less than 5%. Compared with the traditional methods, the photogrammetry method has a wider observation range. Thus, the proposed method can obtain the displacement of any section without extensive efforts, which in turn can better reflect the overall displacement and deformation of the structure.

**Figure 18.** Error comparison chart.

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**Table 4.** Comparison of displacement measurement error.
