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Calibration of the image distortion induced error of plane strain field measured using digital image correlation technique

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Abstract

With the help of the rapid increment of the resolution of the digital camera and the calculation capability of computer, photogrammetry based on the digital image correlation is widely applied in different research fields. However, the optical dispersion gives rise to the image distortion, when a 2D image is recorded from a 3D surface. This image distortion can contribute an additional error at calculating strain tensor. The main purpose of the work is to provide a calibration method in order to reduce the error caused by the image distortion and to raise the precision at the calculation of strain.

In this work the bi-quadratic and bi-cubic interpolation functions are used to establish the relationship between the real coordinate and the distorted image coordinate systems. The results show that the calibration has almost no influence on improving the precision of calculated strain, when the displacement is small. In the case of large displacement, the deviation of strain and displacement can be reduced with the help of the bi-cubic calibration model.

Keywords: calibration, digital image correlation, image distortion, measurement

technique.

1 Introduction

The measurement and analysis of the strain distribution is very important in civil and mechanical engineering, especially in the stress concentration problem in plasticity and fracture mechanics. The traditional measurement techniques can be divided into two types: contact-type and non-contact-type. The digital image correlation [1] is a new developed non-contact-type measurement technique. Because of the fast development of digital camera recently, the instrument cost of this method has dropped very rapidly but the precision raise contrarily. The advantage of this technique is that the whole displacement and strain field can be

Paper 138

Calibration of the Image Distortion Induced Error for a Plane Strain

Field Measured Using Digital Image Correlation Techniques

S.H. Tung†, J.C. Kuo‡ and M.H. Shih*

† Department of Civil and Environmental Engineering National University of Kaohsiung, Taiwan

‡ Department of Materials Science and Engineering National Cheng-Kung University, Tainan, Taiwan * Department of Construction Engineering

National Kaohsiung First University of Science and Technology, Taiwan

©Civil-Comp Press, 2006.

Proceedings of the Fifth International Conference on Engineering Computational Technology, B.H.V. Topping, G. Montero and R. Montenegro, (Editors), Civil-Comp Press, Stirlingshire, Scotland.

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analysed and the sample will not be disturbed. Therefore it can also be used to verify the results of numerical analysis.

The digital image correlation has been applied to analyse various problems [2-7]. But the digital image which is taken by digital camera can be distorted. Figure 1 shows the pictures taken by digital camera with different zooms. The image of Figure 1a shows an obvious distortion. The image distortion can be caused by the following reasons:

1. Unparallel of the CCD or CMOS with the specimen plan : The distance between the CCD or CMOS and the specimen plan is not constant. The near part of the specimen will be larger in the image than the far part.

2. Refraction of light through the lens[8-9] : When the light passes through the lens, it will be refracted. Especially the camera lens is composed of many lenses. The rays of light enter the lens at different positions of the lens and they will be refracted many times, therefore the image will be distorted after the refraction.

3. Non-square of the pixel of CCD or CMOS : If the pixel of CCD or CMOS is not perfectly square, the ratio of horizontal and vertical length of the image will not be 1. This can also cause the distortion of image.

Therefore the analysis result of such images will diverge from the truth. The main purpose of this research is to establish a calibration method, with which the error caused by the image distortion can be reduced in order to raise the precision of this measurement technique.

(a) Zoom 18 mm (b) Zoom 55 mm Figure 1: Standard image (Canon EF-S 18-55mm F3.5-5.6)

2 Calibration Method

The calibration method is based on the establishment of the relation between the standard graph and its digital image. This relation will be used to transfer the analysed node coordinates from the image coordinate system into the real world

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coordinate system. And these transferred node coordinates will be used to calculate the strain and displacement field.

2.1 Standard Image

The standard image for calibration is shown in Figure 1. The size of a single grid is 20 mm × 20 mm. The small black squares are used to increase the accuracy of automatic positioning. A picture of this standard image will be taken under the same condition as the other pictures of the sample. Then the mapping relationship between the real world coordinate system and the image coordinate system will be established with the help of interpolation function.

2.2 Interpolation Functions

The bi-linear and bi-cubic interpolation functions are two commonly used interpolation functions. Investigating the standard image at 18 mm focal length in Figure 1a, we can find that the straight line is distorted to a second order polynomial curve. Therefore the bi-linear interpolation function will not be used to establish the mapping relation between two coordinate systems. The bi-quadratic and the bi-cubic interpolation function will be adopted.

Method 1: bi-quadratic interpolation function

Using the bi-quadratic interpolation function to describe the mapping relation between two coordinate systems, the image coordinate ( , )x y and the real world coordinate ( , )x y′ ′ can be expressed as Equation (1), where p1~ p and 9 q1 ~q are unknown coefficients. Therefore at least 9 9 points are required to determine the unknown coefficients in Equation (1). 2 2 2 2 2 2 1 2 3 4 5 6 7 8 9 2 2 2 2 2 2 1 2 3 4 5 6 7 8 9 x p x p y p xy p x p y p x y p xy p x y p y q x q y q xy q x q y q x y q xy q x y q ′ = + + + + + + + + ′ = + + + + + + + + (1)

Method 2: bi-cubic interpolation function

Considering the distorted image is simultaneously symmetric and anti-symmetric to the x, y axes with the origin at the central point of the image. Therefore the calibration effect of the bi-cubic interpolation function will also be investigated. The mapping relation is similar to Equation (1) except that there are more unknown coefficients. At least 16 points are required to solve the unknown coefficients.

There are 18 and 32 unknown coefficients in bi-quadratic and bi-cubic interpolation functions, respectively. Therefore the amount of known real world coordinates should be more than half of the unknown coefficients in order to solve these coefficients. However in usual cases the amount of known real world coordinates is much more than the necessary amount. In this situation the least square regression can be used to solve the unknown coefficients. The advantage of

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this method is that the noises by taking picture and the errors by analyzing can be filtered and the precision of measurement can increased.

3 Results and Discussions

In this study a sample is fixed on an x-y sample stage, which can horizontally move with a spatial resolution of 0.02 mm in the x and y directions, respectively. The combination system consisting of digital SLR Camera“ Canon EOS 300D” and the zoom lens “Canon EF-S 18-55mm f/3.5-5.6” is used to take the digital image of the sample surface. The focal length is set to 55 mm, the x and y axes of the sample stage are parallel to the x and y axes of the CMOS of the camera, respectively. The surfaces of the tested sample and the sample with a grid as a reference sample are recorded at different displacement in the x/y directions. Then the images will be analysed with the help of digital image correlation and the effect of two calibration methods will be compared.

Figure 2 shows the strain field, which was calculated without calibration, in x-direction and its average value and corresponding standard deviation. The results using bi-quadratic interpolation function and bi-cubic interpolation were shown in Figure 3 and Figure 4, respectively. Comparing Figure 2 with Figure 3 it is observed that the strain fields in x-direction without calibration and using calibration with bi-quadratic interpolation function make no difference. The deviation between the theoretical strain and the calculated strain with interpolation functions is smaller than that without interpolation functions. The theoretical strain after rigid body movement is zero. However, the standard deviations at these two cases are almost the same. In comparison of Figure 2 with Figure 4 it is shown that the strain field calibrated with bi-cubic interpolation function in x-direction is smaller than that without calibration. The average value and standard deviation of the strain in x-direction is also obviously reduced as shown in Figure 4.

(a) (b) (a) Strain field in x-direction (b) Average value and standard deviation

Figure 2: Results in the case of 2.56 mm displacement in the x-direction without calibration

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(a) (b) (a) Strain field in x-direction (b) Average value and standard deviation

Figure 3: Results in the case of 2.56 mm displacement in the x-direction calibrated using bi-quadratic interpolation function

(a) (b) (a) Strain field in x-direction (b) Average value and standard deviation

Figure 4: Results in the case of 2.56 mm displacement in the x-direction calibrated using bi-cubic interpolation function

The average values and standard deviations of the strain in x-direction of the sample under different displacement are shown in Table 1. Investigating Table 1 we can find that the difference between the results without and with calibration is very small in case of small displacement. But as the displacement increases we can find that the calibration with bi-quadratic interpolation function can reduce the errors of average values of strain but the standard deviations are still similar to the results without calibration. In contrary the calibration with bi-cubic interpolation function can simultaneously reduce the errors of average values and the standard deviations of the strain in x-direction. Therefore the results with calibration are better than the result without calibration and the effect of bi-cubic interpolation function is better than that of bi-quadratic interpolation function.

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Without calibration Bi-quadratic Bi-cubic Displacement (mm) Average Value Standard deviation Average value Standard deviation Average value Standard deviation 0.02 -6 22 -6 22 -6 22 0.08 -8 33 -9 33 -9 33 0.32 -3 33 -7 33 -5 25 1.28 11 89 -7 90 5 39 2.56 22 222 -10 223 7 61 5.12 45 359 -13 367 12 111

Table 1: Average values and standard deviations of strain in x-direction ( ×10-6)

Figure 5 ~ Figure 7 show the displacement fields in x-direction, the average displacement values and the standard deviations without and with calibration when the sample moves 2.56 mm in the x-direction. The deviation of displacement in Figure 5 and Figure 6 is larger than that in Figure 7. The same conclusion can also be obtained by investigating the statistic standard deviations. Theoretically the displacements of every point should be the same. Therefore the calibration with bi-cubic interpolation function can provide a better analysed displacement result than the cases without calibration and calibrated with bi-quadratic interpolation function.

(a) (b) (a) Displacement field in x-direction (b) Average value and standard deviation

Figure 5: Results in the case of 2.56 mm displacement in the x-direction without calibration

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(a) (b) (a) Displacement field in x-direction (b) Average value and standard deviation

Figure 6: Results in the case of 2.56 mm displacement in the x-direction calibrated using bi-quadratic interpolation function

(a) (b) (a) Displacement field in x-direction (b) Average value and standard deviation

Figure 7: Results in the case of 2.56 mm displacement in the x-direction calibrated using bi-cubic interpolation function

The average displacement values and standard deviations under different displacements are shown in Table 2. Comparing the average displacement values without and with calibration we can find that the average displacement values with calibration are closer to the real displacement values than that without calibration but the improvement is not very obviously. Investigating the standard deviations we can find that the standard deviations in these three cases are similar to each other under small displacement. As the displacement increases the standard deviations calibrated with bi-quadratic interpolation function is almost equal to the standard deviations without calibration. The with bi-cubic interpolation function calibrated standard deviations are clearly better than the standard deviations in the other two cases.

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Without calibration Bi-quadratic Bi-cubic Displacement (mm) Average value Standard deviation Average value Standard deviation Average value Standard deviation 0.02 0.742 0.0600 0.745 0.0606 0.744 0.0596 0.08 5.812 0.0922 5.841 0.0943 5.836 0.0848 0.32 28.95 0.0902 29.09 0.0978 29.07 0.0642 1.28 125.6 0.3115 126.2 0.3011 126.1 0.1265 2.56 255.0 0.7694 256.1 0.7348 255.9 0.1931 5.12 509.0 1.342 511.2 1.286 510.8 0.3832

Table 2: Average values and standard deviations of displacement in x-direction ( ×10-6)

As the sample moves along the y-direction the analysed strains and displacements under different displacements are shown in Table 3 and Table 4. In Figure 3 we can find that the average strains and standard deviations show only slight variation in the cases of small displacement. In case the displacement is larger, the with bi-quadratic interpolation function calibrated average strains are better than the average strains in the other two cases. But investigating the standard deviation we can find that the calibration using bi-cubic interpolation function can achieve smaller standard deviations. This means that the variation of the with bi-cubic interpolation function calibrated strain field is smaller than the variations of strain field in the other two cases. Comparing the average displacement and standard deviation in Table 4 similar results as the sample moves along x-direction can be obtained. In case the displacement is small, the calibrations have almost no influence on the analysed results. The calibration using bi-cubic interpolation function can improve the analysed results in case the displacement is large.

Without calibration Bi-quadratic Bi-cubic Displacement (mm) Average value Standard deviation Average value Average value Standard deviation Average value 0.02 7 26 7 26 7 26 0.08 -1 23 -1 23 -1 23 0.32 9 23 7 23 9 24 1.28 12 59 8 58 15 26 2.56 15 114 4 113 19 48 5.12 33 225 1 229 31 51 Table 3: Average values and standard deviations of strain in y-direction ( ×10-6)

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Without calibration Bi-quadratic Bi-cubic Displacement (mm) Average value Standard deviation Average value Average value Standard deviation Average value 0.02 1.895 0.0927 1.898 0.0925 1.897 0.0932 0.08 6.955 0.0494 6.976 0.0508 6.973 0.0509 0.32 30.38 0.0837 30.49 0.085 30.48 0.0955 1.28 125.9 0.1900 126.4 0.1648 126.4 0.1343 2.56 254.9 0.3397 255.8 0.2806 255.7 0.1848 5.12 508.5 0.7173 510.4 0.5851 510.2 0.3225

Table 4: Average values and standard deviations of displacement in y-direction ( ×10-6)

Investigating the analysed strains and displacements we find that there are still errors between the real values and the calibrated average values. The errors can be induced by the following reasons:

1. The standard image is drawn with the help of AutoCAD. Then this file will be printed out using printer. The ratio of the length in x-direction to the length in y-direction can diverge from 1.

2. The x/y axes of the translating instrument and the x/y axes of the CMOS are not perfectly parallel. Therefore the analysed displacement in x- or y-direction is only a component. It is smaller than the real displacement.

3. The minimum scale of the translating instrument is 0.02 mm. Thus, the so-called real value can be also divergent from the real displacement.

Therefore we can increase the precision of the analysed average values in the future by improving the above three points.

4 Conclusions

The main purpose of this work is to eliminate the effect of the image distortion and to improve the precision of digital image correlation technique. The bi-quadratic and bi-cubic interpolation functions are used to establish the relationship between the real coordinate and the distorted image coordinate systems. From the results it is observed that the calibration doesn’t have an influence on the improvement of the precision of calculated strain in the case of the small displacement. In the case of large displacement, the calibration using bi-quadratic interpolation function can slightly improve the average values of strain and displacement but the deviations of strain field and displacement field are almost the same as compared with the experiment without calculation. In contrast to the bi-quadratic interpolation model the application of the bi-cubic calibration model leads to reduce dramatically the

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deviation of strain and displacement. Therefore, the calibration methods described in this work provide a method to eliminate the errors induced by the image distortion. In respect to effect the bi-cubic interpolation model is better than the bi-quadratic interpolation model.

References

[1] T.C. Chu, W.F. Ranson, M.A. Sutton and W.H. Peters, 1985, “Application of Digital-Image-Correlation Techniques to Experimental Mechanics”, Experimental Mechanics, 25(3), 232-244.

[2] G. Vendroux and W.G. Knauss, 1998, “Submicron Deformation Field Measurements: Part 2. Improved Digital Image Correlation”, Experimental Mechanics, 38(2), 86-92.

[3] D. Raffard, P. Ienny and J.-P. Henry, “Displacement and Strain Fields at a Stone/Mortar Interface by Digital Image Processing”, Journal of Testing and Evaluation. 29, 2001, 115-122.

[4] J.C. Kuo, S. Zaefferer, Z. Zhao, M. Winning and D. Raabe, “Deformation Behavior of Aluminum Bicrystals”, Advanced Engineering Materials. 5, 2003, 563-566

[5] S. Zaefferer, J.C. Kuo, Z. Zhao, M. Winning and D. Raabe, “On the influence of the grain boundary misorientation on the plastic deformation of aluminum bicrystals”, Acta Materialia. 51, 2003, 4719-4735

[6] S.H. Tung, J.C. Kuo and M.H. Shih, “Strain Distribution Analysis Using Digital-Image-Correlation Techniques”, 18th KKCNN Symposium on Civil Engineering, Taiwan, December 19-21, 2005, 213-218.

[7] J.C. Kuo, S.H. Tung, M.H. Shih and D. Chen, “Application of Digital-Image-Correlation Techniques to crystal plasticity”, Proceedings of the Conference on Computer Applications in Civil & Hydraulic Engineering, Tainan, Taiwan, 2005. (in Chinese)

[8] R. I. Hartley. “An algorithm for self calibration from several views.” In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Seattle, WA, June 1994, 908-912,.

[9] R. Bowden, T.A. Mitchell and M. Sarhadi,”Non-linear statistical models for the 3D reconstruction of human pose and motion from monocular image sequences”, Image And Vision Computing. 18, 2000, 729-737.

數據

Figure 2 shows the strain field, which was calculated without calibration, in x- x-direction and its average value and corresponding standard deviation
Figure 3: Results in the case of 2.56 mm displacement in the x-direction calibrated  using bi-quadratic interpolation function
Table 1: Average values and standard deviations of strain in x-direction ( ×10 -6 )
Figure 6: Results in the case of 2.56 mm displacement in the x-direction calibrated  using bi-quadratic interpolation function
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