Photogrammetry-based two-dimensional digital image correlation with nonperpendicular camera alignmen

Abstract. 2-D digital image correlation (DIC) is traditionally used with the camera’s axis perpendicular to the object surface. The nonperpendicular camera alignment will inevitably introduce measurement error. An obliqueaxis 2-D DIC method based on photogrammetry compensation is proposed in this paper. Not less than three reference points are pasted on the specimen surface. The photogrammetry technique is first used to get the 3D coordinates of these reference points. A reference plane is obtained by fitting these reference points. Then the oblique-axis camera is oriented by using the space resection method. During deformation measurement, for each image points matched by DIC, its 3-D point can be acquired by simply intersecting the point ray with the reference plane.

Using this method, not only is the measurement error caused by oblique axis compensated, but also the displacement in millimeters rather than in pixels can be determined directly. Experiments results indicate that, for in-plane deformation in oblique-axis measurement situation,our method has comparable accuracy with 3-D DIC. © 2012 Society of Photo-Optical Instrumentation Engineers (SPIE). [DOI: 10.1117/1.OE.51.2.023602]

1 Introduction

Digital image correlation (DIC) has been extensively applied to experimental mechanics testing of different materials since the early eighties.1–6 In the last thirty years, much study has been carried out on DIC method for improving its accuracy,robustness, and computational efficiency. Different factors affecting the accuracy of DIC, including subpixel registration algorithms,7 subset size,8 interpolation algorithm,9 correlation criterion,10 and shape function,11 have been discussed by different researchers.

There is also much work providing detailed analysis of the influence of different error sources.12,13 A so-called reliability-guided DIC (RG-DIC)was proposed by Pan14 that makes the correlation calculation more robust. Chen et al.15 developed a two-step extended DIC (X-DIC) that can be used to measure the full-field displacements across the crack domain. Recently, a fast DIC method that is 120 to 200 times much faster than traditional DIC has been proposed by Pan and Li.16

2-D DIC, which uses a single camera, can measure inplane deformation of plane object. 3-D DIC,17,18 which combines stereo-vision with DIC, can measure 3-D deformation of a 3-D object. Compared to 3-D DIC, 2-D DIC is used more widely,19 because it is more convenient and is sufficient to meet the measurement needs in most cases. When using 2-D DIC to measure object deformation, three assumptions are commonly employed.20 First, the specimen is assumed to be flat. Second, the camera axis is perpendicular to the specimen surface.

Third, the specimen remains in the same plane during the load process, which means the out-of-plane motion is small enough. However, these assumptions can’t be fully met in the actual measurement process, especially the second assumption.

In fact, the misalignment between the camera axis and object surface always exists in actual measurement, and for some special cases, for example, when measuring deformation of a large object (e.g., bridge or power tower), the camera axis oblique angle may be quite large. A nonperpendicular camera alignment will inevitably introduce measurement errors. Meng et al.21 theoretically analyzed the relationship between measurement errors and the oblique angle, and the results show that the oblique angle must be less than 5 deg to reach a measuring sensitivity of 0.01 pixels. Some research has been done to compensate the errors brought by the nonperpendicular camera setup. A calibration method using a regular gird was developed by Helm et al.22 to compensate for the off-axis angles, and in the calibration process, a bicubic correction function is gained to map the off-axis displacement

data back to its original coordinate system. Lava et al.23,24 proposed a so-called “rectified 2D-DIC” setup, in which
the nonperpendicular alignment images are rectified with the obtained camera parameters, simulating a perpendicular
camera alignment. In both methods, a regular gird pattern with known coordinates is employed.

In this work, an oblique-axis 2-D DIC based on photogrammetry compensation is developed. In our proposed
method, at least three reference points are pasted on the object surface. The 3-D coordinates of these reference points
are measured using the photogrammetry technique. During deformation measurement, these reference points are used to get the exterior orientation of the oblique-axis camera and reconstruct the 3-D points of measured image points.
Using this method, not only is the measurement error caused by oblique-axis compensated, but also the displacement in 0091-3286/2012/$25.00 © 2012 SPIE millimeters rather than in pixels can be determined directly.The outline of this paper is as follows. In Sec. 2, the basic principle of DIC is briefly described, and the effect of oblique-axis camera alignment on 2-D DIC is analyzed.

In Sec. 3, we present the photogrammetry technique used in our method, and attention is paid to the space resection

algorithm based on a three-point pyramid. In Sec. 4, our proposed oblique-axis 2-D DIC is described in details. Results
and discussion of two experiments are included in Sec. 5. We conclude in Sec. 6.

5.2 Tensile Test

A specimen (size 160 mm × 30 mm ×1 mm) made of sheet steel is used in the tensile test. Figure 10 shows the experimental site. After spraying a random speckle pattern and pasting reference points on the specimen, the XTDP photogrammetry system is first used to measure the 3-D coordinates of the reference points. Then the specimen is mounted on the universal test machine and the stereo vision system is placed for image acquisition. Before the test, the space resection method is employed to calculate the exterior orientation of the left camera. During the test, the images of 50 stages are captured.

The results of 3-D DIC and 2-D DIC with photogrammetry compensation are compared. For the two methods, the same calculation area is chosen in the reference image (the left image of the first stage), and the same subset size (15 × 15 pixels) and point step (13 pixels) are used.

There are 435 points altogether in the calculation area. For a stage with n points, we define the average deviation of displacement calculated by the two methods as 1n P
ni
¼0
jd1i − d2ij, in which d1 and d2 are the displacement of one point calculated by 3-D DIC and our method respectively.
And we also define the average deviation of major strain calculated by the two methods as 1n
P
ni
¼0
jS1i − S2ij,

in which S1 and S2 are the major strain of one point calculated by 3-D DIC and our method respectively. Figure 11
displays the average deviation curve of (a) displacement and (b) major strain between the two methods. As can be seen from Fig. 11, the maximum deviation of average displacement is less than 0.04 mm, and the maximum deviation of average major strain is less than 0.2%. Figure 12 illustrates the (a) displacement and (b) major strain deviation color map of the 50th stage between 3-D DIC and our method. As indicated

in Fig. 12, for the 50th stage, the maximum displacement deviation is less than 0.04 mm, and the maximum major strain deviation is less than 0.5%.

Experiment results indicate that our 2-D DIC with photogrammetry compensation has comparable measurement
accuracy with 3-D DIC. However, an important issue should be pointed out. For 2D DIC, including our method, the measurement is restricted to planar specimens subject to in-plane deformations. The effect of out-of-plane deformation on 2-D

5.2 Tensile Test

A specimen (size 160 mm × 30 mm ×1 mm) made of sheet steel is used in the tensile test. Figure 10 shows the experimental site. After spraying a random speckle pattern and pasting reference points on the specimen, the XTDP photogrammetry system is first used to measure the 3-D coordinates of the reference points. Then the specimen is mounted on the universal test machine and the stereo vision system is placed for image acquisition. Before the test, the space resection method is employed to calculate the exterior orientation of the left camera. During the test, the images of 50 stages are captured.

The results of 3-D DIC and 2-D DIC with photogrammetry compensation are compared. For the two methods, the same calculation area is chosen in the reference image (the left image of the first stage), and the same subset size (15 × 15 pixels) and point step (13 pixels) are used.

There are 435 points altogether in the calculation area. For a stage with n points, we define the average deviation of displacement calculated by the two methods as
1n
P
ni
¼0

jd1i − d2ij, in which d1 and d2 are the displacement of one point calculated by 3-D DIC and our method respectively. And we also define the average deviation of major strain calculated by the two methods as 1n
P
ni
¼0
jS1i − S2ij,

in which S1 and S2 are the major strain of one point calculated by 3-D DIC and our method respectively. Figure 11
displays the average deviation curve of (a) displacement and (b) major strain between the two methods. As can be seen from Fig. 11, the maximum deviation of average displacement is less than 0.04 mm, and the maximum deviation of average major strain is less than 0.2%. Figure 12 illustrates the (a) displacement and (b) major strain deviation color map of the 50th stage between 3-D DIC and our method. As indicated in Fig. 12, for the 50th stage, the maximum displacement
deviation is less than 0.04 mm, and the maximum major strain deviation is less than 0.5%.

Experiment results indicate that our 2-D DIC with photogrammetry compensation has comparable measurement accuracy with 3-D DIC. However, an important issue should be pointed out. For 2-D DIC, including our method, the measurement
is restricted to planar specimens subject to in-plane deformations. The effect of out-of-plane deformation on 2-D