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Digital image correlation for large deformation applied in Ti alloy compression and tension test
Optik 125 (2014) 5316–5322
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Digital image correlation for large deformation applied in Ti alloy compression and tension test Xiang Guo ∗ , Jin Liang, Zhenzhong Xiao, Binggang Cao School of Mechanical Engineering, Xi’an Jiaotong University, No.28 Xianning Road, Xi’an, Shaanxi province, China
a r t i c l e
i n f o
Article history: Received 23 September 2013 Accepted 25 May 2014 Keywords: Speckle Digital image processing Pattern recognition Target tracking
a b s t r a c t In this paper, digital speckle correlation is used in the measurement of Ti alloy compression and tension test. The key technologies applied in the measurement are discussed in detail, including camera calibration with telephoto lens and digital image correlation in large deformation. Single camera self-calibration algorithm based on photogrammetry is proposed. In the algorithm, the interior parameters of camera are estimated without calibrated object, using the bundle adjustment technique, so the 3-D coordinates of calibration target points are not needed in advance to get a reliable camera calibration result. An updating reference image scheme could be employed to deal with large deformation situation. A large deformation measurement scheme, updating reference image scheme, is proposed in this paper. The undeformed image is used as reference in correlation at first. Only for extremely large deformation field, in which iteration of correlation is not convergent, the reference image is updated to the image of previous deformed stage. Using this method, not only extremely large deformation can be measured successfully but also the accumulated error could be controlled. The 75 mm lens is calibrated in the measurement and compared the result with extensometer and un-calibrated image. Experimental results show that up to 150% tensile deformation and 50% compression deformation can be measured successfully. © 2014 Elsevier GmbH. All rights reserved.
1. Introduction Digital image correlation (DIC) method [1] can provide fullfield displacement and surface strain field of measurement object, by comparing two object digital images before and after deformation. With the development of material, mechanics, aerospace, and vehicle, the requirement of displacement and strain measurement is increasing quickly [2–5]. 2D DIC [6] which is used with a single camera, can measure only in-plane displacement/strain fields on plane objects, while 3D DIC technique [7], which combines DIC with stereo vision, can measure the 3D displacement field and surface strain field of 3D object. Due to its advantages of simple equipment, high precision and non-contact measurement, the DIC method has been used widely for deformation measurement these years. Sztefek et al. used the DIC method to determine the surface strain of bone during loading [8]. Stanford et al. made use of the DIC method to measure the wing deformation of micro air vehicles and the measurement result is compared with the numerical result [9]. Miehe et al. applied the DIC method to the surface strain measurement of amorphous glassy polymers [10].
∗ Corresponding author. E-mail address: [email protected] (X. Guo). http://dx.doi.org/10.1016/j.ijleo.2014.06.067 0030-4026/© 2014 Elsevier GmbH. All rights reserved.
The DIC method is employed in Bambach’s research [11] to capture the full-surface transverse buckling deformation of the ranges and webs and the results are used to describe and explain the fundamental behavior of ranges and webs. Experiments based on DIC method were performed by Hargather and Settles to measure the deformation of thin plates in response to explosive blast [12]. Ti is used to be alloyed with iron, aluminum, vanadium, molybdenum, among other elements and produce strong lightweight alloys for aerospace, military, industrial process, automotive, medical prostheses, orthopedic implants, dental and endodontic instruments and files, dental implants, sporting goods, jewelry, mobile phones, and other applications [13,14]. So the method to obtain accurate parameters of Ti alloy is very important. In Ti alloy test, the size of Ti alloy is 3 mm in diameter and 5 mm in height, the focus of lens is 75 mm and the maximum of compression deformation will be up to 50%. There are some problems for traditional image correlation or speckle correlation method: (1) the telephoto lens is hard to be calibrated; (2) when deformation of test to be measured is large, the result of image correlation will be error or unsuccessful. In this paper, a new method for telephoto lens calibration and large deformation image correlation has been presented. In the experiment, the result of Ti alloy in compression and tension is performed accurately and directly.
X. Guo et al. / Optik 125 (2014) 5316–5322
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Fig. 3. Camera calibration used spatial object.
camera lens, Q is the object point, SO is the camera axis which is vertical to the image plane, and O is the focus. The distance between S and O is the focal length, which is expressed as f, and the coordinates of image point are calculated as:
⎧ a (X − X ) + b (Y − Y ) + c (Z − Z ) ⎪ ⎨ x = −f a1 (X − XS ) + b1 (Y − YS ) + c1 (Z − ZS ) 3
Fig. 1. Measurement process.
Fig. 2. Pinhole imaging model.
In the measurement, images of specimen in deformation are captured by camera at first (Fig. 1). Each image is captured as one stage in deformation process. Secondly, camera calibration is used to reduce the effect of camera and lens distortions. It improves the accuracy of result. Thirdly, image correlation algorithm is used to calculate the target’s position in each image. Fourthly, full-field result of strain is calculated by strain calculation. Finally, strain result and force data of deformation is used to analysis mechanical properties of specimen, such as elongation percentage in axis and level direction, young’s modulus and Poisson’s ratio. 2. Camera calibration When image is gathering, the distortion of camera and lens is added in the image. Camera calibration can reduce the distortion and improve the accuracy of measurement result. In the measurement, image correlation is used in the whole deformed area in images. Without camera calibration, distortion at different position will enlarge the error of correlation result. Range close photogrammetry [15] is a 3D measurement technique, which uses the pinhole imaging model as its fundamental mathematical model. As shown in Fig. 2, S is the optical center of
S
3
S
3
S
⎪ ⎩ y = −f a2 (X − XS ) + b2 (Y − YS ) + c2 (Z − ZS )
(1)
a3 (X − XS ) + b3 (Y − YS ) + c3 (Z − ZS )
where, (x, y) are the image point coordinates, (X, Y, Z) are the 3D coordinates of artificial target in the object spatial coordinate system, (XS , YS , ZS ) are the 3D coordinates of camera station in the object spatial coordinate system, ai , bi , ci (i = 1, 2, 3) are the direction cosines between the image space coordinate system and the object spatial coordinate system. Eq. (1) is the collinearity equation in photogrammetry. When the interior and exterior elements are given and if two corresponding image points are known, then 4 equations can be listed according to Eq. (1) and the 3D coordinates of object point Q can be calculated. Calibration will use a series of images to obtain the inner parameters of camera. Ideal model for imaging is pinhole model. In reality, there are various factors, such as lens distortion and eccentric radial distortion, distortion of the image plane and image plane uneven ratio and orthogonal distortion, making the point to imaging plane and theoretical position biases [15,16]. The traditional method of calibration is capture image of calibration plane at several different positions. Because the targets in calibration plane are coplanar, calibration result of telephoto lens usually is not convergent. In this paper, a spatial object is used to place targets and camera capture images from different stations as Fig. 2 shows. The distortion mod-el used in our method is:
dx = A1 x(r 2 − r02 ) + A2 x(r 4 − r04 ) + A3 x(r 6 − r06 ) + B1 (r 2 + 2x2 ) + 2xyB2 + E1 x + E2 y dy = A1 y(r 2 − r02 ) + A2 y(r 4 − r04 ) + A3 y(r 6 − r06 ) + B1 (r 2 + 2y2 ) + 2xyB1
where A1 , A2 , A3 are radial distortion parameters, B1 and B2 are tangential distortion parameters, and E1 and E2 are thin prism distortion parameters, r is the image radius and r0 is the second zero crossing of the distortion curve. It can reduce the RMS error of the re-projection. Camera is calibrated as Fig. 3 shown. The result of camera calibration is shown in Table 1. For accuracy, extensometer is used in a larger scale measurement to compare with the method in this paper. Compare the strain result of extensometer, digital speckle measurement without calibration and digital speckle measurement with calibration. As Fig. 4 shows, the result of digital speckle measurement with calibration is closer to extensometer result than the result without calibration. For calibration, the distortion of camera and lens is reduced. In experiment, speckle trace in calibrated mode has more accurate compared with no calibrated mode.
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Table 1 Result of camera calibration.
0.08
7160.7147 1306.3696 957.3804 −8.7284 −2.4291 −6.4099 1.4303 −1.1190 −9.5002 −5.4702
0.06
Strain
f (pixel) x0 (pixel) y0 (pixel) A1 (10−9 ) A2 (10−15 ) A3 (10−24 ) B1 (10−7 ) B2 (10−7 ) E1 (10−5 ) E2 (10−5 )
0.1
0.04
0.02 Extensometer Measurement with calibration Measurement without calibration
0
-0.02
0
50
100
150
Stage
(a) 0.091
0.09
Strain
0.089
0.088
0.087
0.086
Extensometer Measurement with calibration Measurement without calibration
0.085
141
Fig. 4. Camera calibration.
142
143
144
145
146
147
148
149
150
Stage
(b) 3. Image correlation Image correlation is used to obtain matched position of the random speckle pattern in different deformation stages by optimal correlation. As shown in Fig. 4, in the reference image, a square reference subset of (2M + 1) × (2M + 1) pixel centered at point C is picked. The matching procedure is to find the corresponding subset centered at point C in the deformed image, which has the maximum similarity with the reference subset. Consequently, the two center points C and C are a couple of corresponding points of the two images. A predefined cross correlation (CC) criterion or sum of squared differences (SSD) criterion is used to evaluate the similarity between the reference subset and the target subset. So the corresponding subset location can be determined by searching the maximum or minimum (according to the used correlation criterion) correlation criterion in the specified searching area. Due to its simplicity, the SSD correlation is adopted in this work:
CSSD (p) =
x=M y=M
Fig. 5. Strain result of extensometer, digital speckle measurement without calibration and digital speckle measurement with calibration. (a) Result of extensometer, speckle with calibration and speckle without calibration. (b) Zoom in of (a).
Fig. 6. Image for correlation (a) before deformation. (b) After deformation.
[f (x, y) − r0 − r1 × g(x , y )]
2
x=−M y=−M
where f(x,y) is the gray value of point (x,y) in the reference subset of reference image, g(x ,y ) is the gray value of point (x ,y ) in the corresponding subset of deformed image, r0 , r1 are used to compensate the gray value difference caused by illumination diversity, p = [u,ux,uy,v,vx,vy,r0 ,r1 ] represents the vector of the correlation parameters that depends on the chosen mapping function. The first order mapping function is used in this work:
subset center in x and y directions, and ux,uy,vx,vy are the first-order displacement gradients of the reference subset. To get the minimum of CSSD is a nonlinear minimization problem, which can be solved by using the ILS algorithm. Still it needs to be noticed that a bi-cubic spline interpolation is adopted in the algorithm realization because the coordinates of points in the deformed image are not integer pixel. Moreover, to evaluate the accuracy of ILS results, the standard deviation can be employed:
xi = x0 + x + u + ux x + uy y yi = y0 + y + v + vx x + vy y where x, y are the distances from the subset center to point (xi , yi ), u and v are the displacement components of the reference
s0 =
v2
n−1
where n is the number of pixels in the reference subset, v represents the gray value residual of one pixel after correlation calculation.
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Fig. 7. Pattern correlation. Fig. 10. One strain point.
Fig. 11. Image of compression test.
Fig. 8. Image correlation.
Fig. 12. Image of tension test. Fig. 9. Correlation result of original method and method in paper.
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Cauchy stress(MPa)
1500
1000
500
0 -0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
ε11
Fig. 15. Stress–strain curve of compression.
Fig. 13. Epsilon X of compression.
reference image, are calculated by previous image to obtain full field result of deformation. As Fig. 8 shows, correlation result in big correlation is obtained better by the method in this paper. It can be seen that the result of correlation with reference image is same as the result of original method. 4. Strain calculation
Fig. 14. Epsilon Y of compression.
As Fig. 5 shows, in large deformation, the pattern in image will be big difference with it in reference image [17]. As Fig. 6 shows, in large deformation, each correlation of image will use the reference image and previous image. When pattern correlate, the reference image will be used first. In large deformation image, the pattern has a big difference with reference pattern. If iteration of correlation is not convergence, previous pattern will be used to correlate in current image (Fig. 7). Correlation in this method use the reference image for accurate and the previous image for the best correlation result. Compared with original method (only use reference image), the previous image is used and the result of correlation will be better in large deformation. Compared with the method (only use previous image), the reference image is used for the best accuracy of correlation result. For previous image correlation, the distortion of previous image correlation will be summed in the current image correlation with previous image. Usually, the deformation in object field is not same at each area. In this paper, correlation result of the fields, in which the deformation is not too big, are calculated by reference image to make the result accurate; correlation result of the fields, in which the deformation is too big to correlate with
Digital speckle measurement can calculate the strain field of objects by dense patterns. As Fig. 9 shows, calculate the deformation gradient matrix F of the strain point (Fig. 10). According to the deformation of the point Po and eight points nearest in neighborhood, set the reference position of points ⎡P ⎤ 0x P0y P0z ⎢ P1x P1y P1z ⎥ are matrix P = ⎢ P2x P2y P2z ⎥ and set the deformed posi⎣ ⎦ ··· P8x P8y P8z ⎡ ⎤ P0x P0y P0z P1y P1z ⎢ P1x ⎥ ⎢ ⎥. Then project tion of points are matrix P = ⎢ P2x P2y P2z ⎥ ⎣ ⎦ ··· P8x P8y P8z un-deformed and deformed points to normal planes. One of normal planes is calculated by un-deformed points and another by deformed points. Obtain the projection⎡ points of un-deformed and ⎤ ⎡ ⎤ ¯ P¯0x P0y ¯ P¯0x P0y ⎢ P¯ P¯ ⎥ ¯ ⎥ ⎢ P¯1x P1y ⎢ 1x 1y ⎥ ⎢ ⎥ ¯ ¯ ¯ ⎥. If the number ¯ ¯ deformed as P = ⎢ P2x P2y ⎥ and P = ⎢ P¯2x P2y ⎢ ⎥ ⎣ ··· ··· ⎦ ⎣ ··· ··· ⎦ ¯ P¯8x P8y P¯ P¯ 8x
8y
of points in neighborhood is less than 8, the strain of the point Po can also be calculated. However, for the acceptable accurate, the minimum number of points in neighborhood should be not less than 2. Calculate the deformation gradient matrix F by the equation: ¯ P¯ = F P. εG =
1 (FF T − I) 2
Based on equation, calculate the strain of the strain point. The strain field in each deform stage is made up of strain of dense patterns. In 2D measurement, Z value of P and P is zero. So the projection points can be seen as the X and Y value of P and P .
X. Guo et al. / Optik 125 (2014) 5316–5322
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0.04 0.035 0.03
ε22
0.025 0.02 0.015 0.01 0.005 0 -0.005 -0.1
-0.08
-0.06
-0.04 ε11
-0.02
0
0.02
Fig. 16. Evolution of transverse strain with longitudinal strain.
Fig. 18. Epsilon Y of tension.
500 450
Cauchy stress(MPa)
400 350 300 250 200 150 100 50 0 -0.05
0
0.05
0.1
0.15
0.2
ε11
Fig. 17. Epsilon X of tension.
Fig. 19. Stress–strain curve of tension.
0.02
5. Experiment
0 -0.02
ε22
Use extensometer to compare the accuracy of digital speckle with calibration and without calibration. Then the method is applied in Ti alloy compression and tension test. Finally, compare the result of measurement with simulation. CMOS sensor (resolution is 2868 × 1926 and pixel size is 2.2 m) and 75 mm lens are used in experiment. The speed of compression and tensile is 0.18 mm/min (Figs. 11–20). Using the method in this paper, the stain field information of compression and tension can be obtained accurately and showed directly. Using the method, it can use in material test for getting the properties of each material. Using the data of each facet, the modulus of elasticity, Poisson’s ratio and other properties can be calculated.
-0.04 -0.06 -0.08 -0.1 -0.12 -0.05
0
0.05 ε11
0.1
0.15
Fig. 20. Evolution of transverse strain with longitudinal strain.
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6. Conclusion This paper presents a new method for small-scale and large deformed material test measurement. Using single camera calibration can improve digital speckle measurement accuracy. A new method of image correlation is used to obtain the full-field deformation result. According to the experiment of Ti alloy in compression and tension, the result of measurement is accurate and effective. The full-field strain result is obtained and showed directly. Using the method, the properties of each material can be obtained easily. Especially, for some anisotropy materials, this method will have a wide application future. Acknowledgments The authors acknowledge the support of the National Natural Science Foundation of China (Grant no. 51275378 and 51275389) and the Science and Technology Innovation Project of Jiangsu Province (SBC201210069). References [1] T. Chu, W. Ranson, M. Sutton, Applications of digital-image-correlation techniques to experimental mechanics, Exp. Mech. 25 (3) (1985) 232–244. [2] H. Dai, X. Su, Shape measurement by digital speckle temporal sequence correlation with digital light projector, Opt. Eng. 40 (5) (2001) 793–800. [3] J.D. Helm, S.R. McNeill, M.A. Sutton, Improved three-dimensional image correlation for surface displacement measurement, Opt. Eng. 35 (7) (1996) 1911–1920. [4] M.A. Sutton, S.R. McNeill, J. Sengjang, Effects of Subpixel image restoration on digital correlation error estimates, Opt. Eng. 27 (3) (1988) 173–175.
[5] I. Yamaguchi, K. Kobayashi, L. Yaroslavsky, Measurement of surface roughness by speckle correlation, Opt. Eng. 43 (11) (2004) 2753–2761. [6] B. Pan, et al., Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review, Meas. Sci. Technol. 20 (6) (2009) 062001. [7] P. Luo, et al., Accurate measurement of three-dimensional deformations in deformable and rigid bodies using computer vision, Exp. Mech. 33 (2) (1993) 123–132. [8] P. Sztefek, et al., Using digital image correlation to determine bone surface strains during loading and after adaptation of the mouse tibia, J. Biomech. 43 (4) (2010) 599–605. [9] B. Stanford, et al., Fixed membrane wings for micro air vehicles: experimental characterization, numerical modeling, and tailoring, Prog. Aerosp. Sci. 44 (4) (2008) 258–294. [10] C. Miehe, S. Göktepe, J. Méndez Diez, Finite viscoplasticity of amorphous glassy polymers in the logarithmic strain space, Int. J. Solids Struct. 46 (1) (2009) 181–202. [11] M.R. Bambach, Photogrammetry measurements of buckling modes and interactions in channels with edge-stiffened flanges, Thin-Walled Struct. 47 (5) (2009) 485–504. [12] M.J. Hargather, G.S. Settles, Laboratory-scale techniques for the measurement of a material response to an explosive blast, Int. J. Impact Eng. 36 (7) (2009) 940–947. [13] V.K. Chandravanshi, et al., Effect of boron on microstructure and mechanical properties of thermomechanically processed near alpha titanium alloy Ti-1100, J. Alloys Compd. 509 (18) (2011) 5506–5514. [14] A.H.M. Dom, et al., Embedment of HA/Ti composite on superplastic titanium alloy (Ti–6Al–4V), Mater. Sci. Eng. A 527 (21–22) (2010) 5831–5836. [15] X. Guo, et al., Precision Control of Scale Using in Industrial Close-range Photogrammetry, SPIE, Hangzhou, China, 2010. [16] C. Fraser (Ed.), Photogrammetric Camera Component Calibration – A Review of Analytical Techniques, Springer Berlin Heidelberg, Washington, DC, USA, 1992. [17] C. Zhou, et al., Digital speckle correlation method improved by hybrid method, in: ICO20: Optical Information Processing, SPIE, 2006.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Digital image correlation for large deformation applied in Ti alloy compression and tension test Xiang Guo ∗ , Jin Liang, Zhenzhong Xiao, Binggang Cao School of Mechanical Engineering, Xi’an Jiaotong University, No.28 Xianning Road, Xi’an, Shaanxi province, China
a r t i c l e
i n f o
Article history: Received 23 September 2013 Accepted 25 May 2014 Keywords: Speckle Digital image processing Pattern recognition Target tracking
a b s t r a c t In this paper, digital speckle correlation is used in the measurement of Ti alloy compression and tension test. The key technologies applied in the measurement are discussed in detail, including camera calibration with telephoto lens and digital image correlation in large deformation. Single camera self-calibration algorithm based on photogrammetry is proposed. In the algorithm, the interior parameters of camera are estimated without calibrated object, using the bundle adjustment technique, so the 3-D coordinates of calibration target points are not needed in advance to get a reliable camera calibration result. An updating reference image scheme could be employed to deal with large deformation situation. A large deformation measurement scheme, updating reference image scheme, is proposed in this paper. The undeformed image is used as reference in correlation at first. Only for extremely large deformation field, in which iteration of correlation is not convergent, the reference image is updated to the image of previous deformed stage. Using this method, not only extremely large deformation can be measured successfully but also the accumulated error could be controlled. The 75 mm lens is calibrated in the measurement and compared the result with extensometer and un-calibrated image. Experimental results show that up to 150% tensile deformation and 50% compression deformation can be measured successfully. © 2014 Elsevier GmbH. All rights reserved.
1. Introduction Digital image correlation (DIC) method [1] can provide fullfield displacement and surface strain field of measurement object, by comparing two object digital images before and after deformation. With the development of material, mechanics, aerospace, and vehicle, the requirement of displacement and strain measurement is increasing quickly [2–5]. 2D DIC [6] which is used with a single camera, can measure only in-plane displacement/strain fields on plane objects, while 3D DIC technique [7], which combines DIC with stereo vision, can measure the 3D displacement field and surface strain field of 3D object. Due to its advantages of simple equipment, high precision and non-contact measurement, the DIC method has been used widely for deformation measurement these years. Sztefek et al. used the DIC method to determine the surface strain of bone during loading [8]. Stanford et al. made use of the DIC method to measure the wing deformation of micro air vehicles and the measurement result is compared with the numerical result [9]. Miehe et al. applied the DIC method to the surface strain measurement of amorphous glassy polymers [10].
∗ Corresponding author. E-mail address: [email protected] (X. Guo). http://dx.doi.org/10.1016/j.ijleo.2014.06.067 0030-4026/© 2014 Elsevier GmbH. All rights reserved.
The DIC method is employed in Bambach’s research [11] to capture the full-surface transverse buckling deformation of the ranges and webs and the results are used to describe and explain the fundamental behavior of ranges and webs. Experiments based on DIC method were performed by Hargather and Settles to measure the deformation of thin plates in response to explosive blast [12]. Ti is used to be alloyed with iron, aluminum, vanadium, molybdenum, among other elements and produce strong lightweight alloys for aerospace, military, industrial process, automotive, medical prostheses, orthopedic implants, dental and endodontic instruments and files, dental implants, sporting goods, jewelry, mobile phones, and other applications [13,14]. So the method to obtain accurate parameters of Ti alloy is very important. In Ti alloy test, the size of Ti alloy is 3 mm in diameter and 5 mm in height, the focus of lens is 75 mm and the maximum of compression deformation will be up to 50%. There are some problems for traditional image correlation or speckle correlation method: (1) the telephoto lens is hard to be calibrated; (2) when deformation of test to be measured is large, the result of image correlation will be error or unsuccessful. In this paper, a new method for telephoto lens calibration and large deformation image correlation has been presented. In the experiment, the result of Ti alloy in compression and tension is performed accurately and directly.
X. Guo et al. / Optik 125 (2014) 5316–5322
5317
Fig. 3. Camera calibration used spatial object.
camera lens, Q is the object point, SO is the camera axis which is vertical to the image plane, and O is the focus. The distance between S and O is the focal length, which is expressed as f, and the coordinates of image point are calculated as:
⎧ a (X − X ) + b (Y − Y ) + c (Z − Z ) ⎪ ⎨ x = −f a1 (X − XS ) + b1 (Y − YS ) + c1 (Z − ZS ) 3
Fig. 1. Measurement process.
Fig. 2. Pinhole imaging model.
In the measurement, images of specimen in deformation are captured by camera at first (Fig. 1). Each image is captured as one stage in deformation process. Secondly, camera calibration is used to reduce the effect of camera and lens distortions. It improves the accuracy of result. Thirdly, image correlation algorithm is used to calculate the target’s position in each image. Fourthly, full-field result of strain is calculated by strain calculation. Finally, strain result and force data of deformation is used to analysis mechanical properties of specimen, such as elongation percentage in axis and level direction, young’s modulus and Poisson’s ratio. 2. Camera calibration When image is gathering, the distortion of camera and lens is added in the image. Camera calibration can reduce the distortion and improve the accuracy of measurement result. In the measurement, image correlation is used in the whole deformed area in images. Without camera calibration, distortion at different position will enlarge the error of correlation result. Range close photogrammetry [15] is a 3D measurement technique, which uses the pinhole imaging model as its fundamental mathematical model. As shown in Fig. 2, S is the optical center of
S
3
S
3
S
⎪ ⎩ y = −f a2 (X − XS ) + b2 (Y − YS ) + c2 (Z − ZS )
(1)
a3 (X − XS ) + b3 (Y − YS ) + c3 (Z − ZS )
where, (x, y) are the image point coordinates, (X, Y, Z) are the 3D coordinates of artificial target in the object spatial coordinate system, (XS , YS , ZS ) are the 3D coordinates of camera station in the object spatial coordinate system, ai , bi , ci (i = 1, 2, 3) are the direction cosines between the image space coordinate system and the object spatial coordinate system. Eq. (1) is the collinearity equation in photogrammetry. When the interior and exterior elements are given and if two corresponding image points are known, then 4 equations can be listed according to Eq. (1) and the 3D coordinates of object point Q can be calculated. Calibration will use a series of images to obtain the inner parameters of camera. Ideal model for imaging is pinhole model. In reality, there are various factors, such as lens distortion and eccentric radial distortion, distortion of the image plane and image plane uneven ratio and orthogonal distortion, making the point to imaging plane and theoretical position biases [15,16]. The traditional method of calibration is capture image of calibration plane at several different positions. Because the targets in calibration plane are coplanar, calibration result of telephoto lens usually is not convergent. In this paper, a spatial object is used to place targets and camera capture images from different stations as Fig. 2 shows. The distortion mod-el used in our method is:
dx = A1 x(r 2 − r02 ) + A2 x(r 4 − r04 ) + A3 x(r 6 − r06 ) + B1 (r 2 + 2x2 ) + 2xyB2 + E1 x + E2 y dy = A1 y(r 2 − r02 ) + A2 y(r 4 − r04 ) + A3 y(r 6 − r06 ) + B1 (r 2 + 2y2 ) + 2xyB1
where A1 , A2 , A3 are radial distortion parameters, B1 and B2 are tangential distortion parameters, and E1 and E2 are thin prism distortion parameters, r is the image radius and r0 is the second zero crossing of the distortion curve. It can reduce the RMS error of the re-projection. Camera is calibrated as Fig. 3 shown. The result of camera calibration is shown in Table 1. For accuracy, extensometer is used in a larger scale measurement to compare with the method in this paper. Compare the strain result of extensometer, digital speckle measurement without calibration and digital speckle measurement with calibration. As Fig. 4 shows, the result of digital speckle measurement with calibration is closer to extensometer result than the result without calibration. For calibration, the distortion of camera and lens is reduced. In experiment, speckle trace in calibrated mode has more accurate compared with no calibrated mode.
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Table 1 Result of camera calibration.
0.08
7160.7147 1306.3696 957.3804 −8.7284 −2.4291 −6.4099 1.4303 −1.1190 −9.5002 −5.4702
0.06
Strain
f (pixel) x0 (pixel) y0 (pixel) A1 (10−9 ) A2 (10−15 ) A3 (10−24 ) B1 (10−7 ) B2 (10−7 ) E1 (10−5 ) E2 (10−5 )
0.1
0.04
0.02 Extensometer Measurement with calibration Measurement without calibration
0
-0.02
0
50
100
150
Stage
(a) 0.091
0.09
Strain
0.089
0.088
0.087
0.086
Extensometer Measurement with calibration Measurement without calibration
0.085
141
Fig. 4. Camera calibration.
142
143
144
145
146
147
148
149
150
Stage
(b) 3. Image correlation Image correlation is used to obtain matched position of the random speckle pattern in different deformation stages by optimal correlation. As shown in Fig. 4, in the reference image, a square reference subset of (2M + 1) × (2M + 1) pixel centered at point C is picked. The matching procedure is to find the corresponding subset centered at point C in the deformed image, which has the maximum similarity with the reference subset. Consequently, the two center points C and C are a couple of corresponding points of the two images. A predefined cross correlation (CC) criterion or sum of squared differences (SSD) criterion is used to evaluate the similarity between the reference subset and the target subset. So the corresponding subset location can be determined by searching the maximum or minimum (according to the used correlation criterion) correlation criterion in the specified searching area. Due to its simplicity, the SSD correlation is adopted in this work:
CSSD (p) =
x=M y=M
Fig. 5. Strain result of extensometer, digital speckle measurement without calibration and digital speckle measurement with calibration. (a) Result of extensometer, speckle with calibration and speckle without calibration. (b) Zoom in of (a).
Fig. 6. Image for correlation (a) before deformation. (b) After deformation.
[f (x, y) − r0 − r1 × g(x , y )]
2
x=−M y=−M
where f(x,y) is the gray value of point (x,y) in the reference subset of reference image, g(x ,y ) is the gray value of point (x ,y ) in the corresponding subset of deformed image, r0 , r1 are used to compensate the gray value difference caused by illumination diversity, p = [u,ux,uy,v,vx,vy,r0 ,r1 ] represents the vector of the correlation parameters that depends on the chosen mapping function. The first order mapping function is used in this work:
subset center in x and y directions, and ux,uy,vx,vy are the first-order displacement gradients of the reference subset. To get the minimum of CSSD is a nonlinear minimization problem, which can be solved by using the ILS algorithm. Still it needs to be noticed that a bi-cubic spline interpolation is adopted in the algorithm realization because the coordinates of points in the deformed image are not integer pixel. Moreover, to evaluate the accuracy of ILS results, the standard deviation can be employed:
xi = x0 + x + u + ux x + uy y yi = y0 + y + v + vx x + vy y where x, y are the distances from the subset center to point (xi , yi ), u and v are the displacement components of the reference
s0 =
v2
n−1
where n is the number of pixels in the reference subset, v represents the gray value residual of one pixel after correlation calculation.
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Fig. 7. Pattern correlation. Fig. 10. One strain point.
Fig. 11. Image of compression test.
Fig. 8. Image correlation.
Fig. 12. Image of tension test. Fig. 9. Correlation result of original method and method in paper.
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Cauchy stress(MPa)
1500
1000
500
0 -0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
ε11
Fig. 15. Stress–strain curve of compression.
Fig. 13. Epsilon X of compression.
reference image, are calculated by previous image to obtain full field result of deformation. As Fig. 8 shows, correlation result in big correlation is obtained better by the method in this paper. It can be seen that the result of correlation with reference image is same as the result of original method. 4. Strain calculation
Fig. 14. Epsilon Y of compression.
As Fig. 5 shows, in large deformation, the pattern in image will be big difference with it in reference image [17]. As Fig. 6 shows, in large deformation, each correlation of image will use the reference image and previous image. When pattern correlate, the reference image will be used first. In large deformation image, the pattern has a big difference with reference pattern. If iteration of correlation is not convergence, previous pattern will be used to correlate in current image (Fig. 7). Correlation in this method use the reference image for accurate and the previous image for the best correlation result. Compared with original method (only use reference image), the previous image is used and the result of correlation will be better in large deformation. Compared with the method (only use previous image), the reference image is used for the best accuracy of correlation result. For previous image correlation, the distortion of previous image correlation will be summed in the current image correlation with previous image. Usually, the deformation in object field is not same at each area. In this paper, correlation result of the fields, in which the deformation is not too big, are calculated by reference image to make the result accurate; correlation result of the fields, in which the deformation is too big to correlate with
Digital speckle measurement can calculate the strain field of objects by dense patterns. As Fig. 9 shows, calculate the deformation gradient matrix F of the strain point (Fig. 10). According to the deformation of the point Po and eight points nearest in neighborhood, set the reference position of points ⎡P ⎤ 0x P0y P0z ⎢ P1x P1y P1z ⎥ are matrix P = ⎢ P2x P2y P2z ⎥ and set the deformed posi⎣ ⎦ ··· P8x P8y P8z ⎡ ⎤ P0x P0y P0z P1y P1z ⎢ P1x ⎥ ⎢ ⎥. Then project tion of points are matrix P = ⎢ P2x P2y P2z ⎥ ⎣ ⎦ ··· P8x P8y P8z un-deformed and deformed points to normal planes. One of normal planes is calculated by un-deformed points and another by deformed points. Obtain the projection⎡ points of un-deformed and ⎤ ⎡ ⎤ ¯ P¯0x P0y ¯ P¯0x P0y ⎢ P¯ P¯ ⎥ ¯ ⎥ ⎢ P¯1x P1y ⎢ 1x 1y ⎥ ⎢ ⎥ ¯ ¯ ¯ ⎥. If the number ¯ ¯ deformed as P = ⎢ P2x P2y ⎥ and P = ⎢ P¯2x P2y ⎢ ⎥ ⎣ ··· ··· ⎦ ⎣ ··· ··· ⎦ ¯ P¯8x P8y P¯ P¯ 8x
8y
of points in neighborhood is less than 8, the strain of the point Po can also be calculated. However, for the acceptable accurate, the minimum number of points in neighborhood should be not less than 2. Calculate the deformation gradient matrix F by the equation: ¯ P¯ = F P. εG =
1 (FF T − I) 2
Based on equation, calculate the strain of the strain point. The strain field in each deform stage is made up of strain of dense patterns. In 2D measurement, Z value of P and P is zero. So the projection points can be seen as the X and Y value of P and P .
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0.04 0.035 0.03
ε22
0.025 0.02 0.015 0.01 0.005 0 -0.005 -0.1
-0.08
-0.06
-0.04 ε11
-0.02
0
0.02
Fig. 16. Evolution of transverse strain with longitudinal strain.
Fig. 18. Epsilon Y of tension.
500 450
Cauchy stress(MPa)
400 350 300 250 200 150 100 50 0 -0.05
0
0.05
0.1
0.15
0.2
ε11
Fig. 17. Epsilon X of tension.
Fig. 19. Stress–strain curve of tension.
0.02
5. Experiment
0 -0.02
ε22
Use extensometer to compare the accuracy of digital speckle with calibration and without calibration. Then the method is applied in Ti alloy compression and tension test. Finally, compare the result of measurement with simulation. CMOS sensor (resolution is 2868 × 1926 and pixel size is 2.2 m) and 75 mm lens are used in experiment. The speed of compression and tensile is 0.18 mm/min (Figs. 11–20). Using the method in this paper, the stain field information of compression and tension can be obtained accurately and showed directly. Using the method, it can use in material test for getting the properties of each material. Using the data of each facet, the modulus of elasticity, Poisson’s ratio and other properties can be calculated.
-0.04 -0.06 -0.08 -0.1 -0.12 -0.05
0
0.05 ε11
0.1
0.15
Fig. 20. Evolution of transverse strain with longitudinal strain.
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6. Conclusion This paper presents a new method for small-scale and large deformed material test measurement. Using single camera calibration can improve digital speckle measurement accuracy. A new method of image correlation is used to obtain the full-field deformation result. According to the experiment of Ti alloy in compression and tension, the result of measurement is accurate and effective. The full-field strain result is obtained and showed directly. Using the method, the properties of each material can be obtained easily. Especially, for some anisotropy materials, this method will have a wide application future. Acknowledgments The authors acknowledge the support of the National Natural Science Foundation of China (Grant no. 51275378 and 51275389) and the Science and Technology Innovation Project of Jiangsu Province (SBC201210069). References [1] T. Chu, W. Ranson, M. Sutton, Applications of digital-image-correlation techniques to experimental mechanics, Exp. Mech. 25 (3) (1985) 232–244. [2] H. Dai, X. Su, Shape measurement by digital speckle temporal sequence correlation with digital light projector, Opt. Eng. 40 (5) (2001) 793–800. [3] J.D. Helm, S.R. McNeill, M.A. Sutton, Improved three-dimensional image correlation for surface displacement measurement, Opt. Eng. 35 (7) (1996) 1911–1920. [4] M.A. Sutton, S.R. McNeill, J. Sengjang, Effects of Subpixel image restoration on digital correlation error estimates, Opt. Eng. 27 (3) (1988) 173–175.
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