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An improved method for shape measurement using two-dimensional digital image correlation
Optik 124 (2013) 4097–4099
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
An improved method for shape measurement using two-dimensional digital image correlation Xing Xu, Kaifu Wang ∗ , Guoqing Gu College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, 29 Yudao Street, Nanjing, Jiangsu 210016, PR China
a r t i c l e
i n f o
Article history: Received 17 July 2012 Accepted 10 December 2012
Keywords: Digital image correlation Shape measurement Plane mirror rotation Newton-Raphson iteration Bicubic spline interpolation
a b s t r a c t Shape measurement is a significant application of digital image correlation (DIC). An improved method that combines a rotatable plane mirror is proposed to measure the shape of an immovable object. In this method, two images, one before and the other after rotating the plane mirror, are obtained and then in-plane translation which related to the shape of the detected object can be calculated by the use of two-dimensional digital image correlation (2D DIC). The relationship between the in-plane translation and the shape of the object is described. Experimental results show that the proposed method is feasible for shape and distance measurement with high accuracy. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction Shape measurement is important in many fields of physics and engineering. Various optical methods have been employed for accurate shape measurement of an object, such as interferometric profilometry [1,2] and digital image correlation [3,4]. Interferometry is a well-established, whole-field method for surface profiling. The classical 2D DIC method, which uses only one camera, has been used for shape measurement according to the relationship between surface height variation and in-plane displacement in the image plane [5,6]. Meanwhile, 3D DIC method, which uses two cameras, can determines the shape of an object based on the principle of binocular stereovision. However, all of these methods have some limitations. Interferometric profilometry often needs complicated phase analysis. Some 2D DIC methods demands for speckle projection [5]. 3D DIC needs a laborious stereovision calibration. Huang et al. developed a simple method by using 2D DIC which does not need speckle projection and also provides accurate results [6]. However, this method needs to translate the determined object which cannot be used to measure the shape of an immovable object. In this paper we have proposed an improved method for the shape measurement of immovable object. It is known that the magnification at different points of an object changes by the surface height variation. Therefore a rigid-body in-plane translation of the object will introduce an inequable displacement in the
image plane [6]. However, if the determined object is immovable, the method based on giving a rigid-body in-plane translation is unusable. We have added a plane mirror between the object and the camera. An approximate rigid-body in-plane translation can occur by rotating the mirror slightly. Then the shape of the object can be calculated from the displacement in the image plane using DIC. 2. Principles 2.1. Principle of two-dimensional digital image correlation Digital image correlation which initially proposed by Peters and Ranson [7] has been developed for thirty years and now has been widely used in many fields. A brief description of this method is given in this section and more details can be found in [7–9]. The basic principle of 2D DIC is to match the same sub-image of two images recorded before and after object deformation. The matching process is implemented by searching the peak position of the distribution of a correlation coefficient which is predefined and reveals the similarity between two sub-images. Once the correlation coefficient extremum is found, the relative displacement of two sub-images can be determined. In order to improve the accuracy of DIC, Newton–Raphson iteration method and bicubic spline interpolation scheme are often used in DIC. 2.2. Principle for shape measurement
∗ Corresponding author. E-mail address: [email protected] (K. Wang). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2012.12.034
Fig. 1 illustrates the typical pinhole imaging system which had been used for shape measurement in [6]. The detected object is imaged on the image plane through a thin lens. In the experiment,
4098
X. Xu et al. / Optik 124 (2013) 4097–4099
X
According to Eq. (7), the z coordinate can be obtained after measuring the x’ using digital image correlation and determining the image distance a by a calibration process:
Image plane Lens
Object
x Z
x
Y
z≈
P ( x, y , z ) P (x
x, y, z )
1 1 1 = + a z−a f
Fig. 1. Schematic geometry of pinhole camera system.
Object
Lens
Z
P ( x, y , z )
P (x
x, y, z
z)
z-a-l
l
Fig. 2. Schematic geometry of a pinhole camera system with a plane mirror.
when an in-plane translation x is applied to the object, the translation of the image of the object x’ can be expressed as x = −
a x z−a
(1)
At this end, by doing a calibration of determining the image distance a, the z coordinate which is equal to the shape information can be acquired: z =a−
1/2
(10)
3. Experiments and results 2
Y
z − (z 2 − 4zf ) 2
Virtual image
Plane mirror
x
a
(9)
where f is the focal length. The image distance a can be calculated by a=
X Image plane
(8)
There are many methods for calibration of the image distance. Here we introduce a very simple one. According to the image formula of pinhole camera system:
z-a
a
al sin 2 +a a sin 2 − x
ax x
(2)
Using the above system, the shape of an object can be accurate determined. However, this system may be unusable when the detected object is immovable. In view of this, we design an improved system with a plane mirror, showed in Fig. 2. In the experiment, the plane mirror is rotated with an angle in the XZ plane. According to the principle of plane mirror imaging, the virtual image of the object has a deformation after rotating the plane mirror. A point P(x, y, z) which is random selected in the virtual image of the object moves to point P (x + x, y, z + z). Therefore, there are both in-plane translation x and out-of-plane translation z. These two translations can be expressed as x = x(cos 2 − 1) + (z − a − l) sin 2
(3)
z = (z − a − l)(1 − cos 2) + x sin 2
(4)
The CCD camera used here has a resolution of 2560-by-1920 and the lens here has a focal length of 25 mm. The detected objects are one flat plate and two wedges with different separation angles. The objects are all coated with black–white speckles on their surface. In the experiment, the surface of the object is illuminated by a white light source and the plane mirror is located far from the image plane about 260 mm. By rotating the plane mirror a little angle of 0.025◦ , and correlating the images before and after the rotating, an in-plane translation x’ which is related to the height variation can be acquired. Afterwards the distance z of every point of the object surface which represents the shape of the object is obtained according to Eq. (8). After experiment, a calibration of the image distance is essential. The image distance should be kept identical both in experiment and calibration process. The aperture of the camera should be set to maximum in the calibration process so as to find the object distance with a high accuracy. Then, the image distance can be calculated according to Eq. (10). Fig. 3 shows the in-plane translation x’ of the image of the flat plate. According to Eq. (8), the distance z which directly represents the shape of the plate then can be obtained, as shown in Fig. 4. The results in Fig. 4 show that the distance between the plate plane and the plane mirror is nearly 320-260 = 60 mm. As a reason, this method can also be used for distance measurement. Figs. 5 and 6 show the shapes of the two wedges with different separation angles. In order to calculate these two angles, middle sections of Figs. 5 and 6 are plotted in Fig. 7, in which the height variation of the wedges is clearly viewed. After that, the angles can be obtained through computing the slope of middle sections. The real separation angles of these two wedges can be measured by a protractor and the real angles are 12◦ and 30◦ respectively. It can
The relationship among x’ and x, z is acquired as following:
x = a −
x − x x + z−a z − z − a
Based on Eqs. (3) and (4), x’ can be rewritten as
x = a
−
x 2x − x cos 2 − (z − a − l) sin 2 + z−a z − a − (z − a − l)(1 − cos 2) − x sin 2
(5)
(6)
We assume that the rotating angle is so small that we have cos 2 ≈ 1 and sin 2 (1 − cos 2). Meanwhile, considering that z − a x sin 2. Therefore, x’ can be represented by x ≈
a(z − a − l)sin 2 z−a
(7) Fig. 3. The in-plane translation of the image of the flat plate acquired by DIC.
X. Xu et al. / Optik 124 (2013) 4097–4099
Fig. 4. The distance z of the plat plate.
4099
Fig. 7. Middle sections of two wedges.
spline interpolation and a subset of 50 × 50 were used. The rotating angle should be very small due to the analysis in Section 2.2 and is chose as 0.025◦ in this paper. In order to analyze the sensitivity of this method, the relationship between (x’) and z can be obtained based on Eq. (7) and be expressed as (x ) =
Fig. 5. The distance z of the first wedge with a smaller separation angle.
la sin 2 (z − a)2
z
(11)
According to Eq. (11), the value of (la sin 2)/(z − a)2 must be as large as possible. It is noted that the value of a always close to the value of the focal length of the lens. Based on these, the value of (a + l) must be much larger than the value of (z − a − l), i.e. that the plane mirror should be set far away from the image plane and the detected object should be located as close to the plane mirror as possible. 5. Conclusions An improved method combined a rotatable plane mirror for shape measurement using 2-D DIC is proposed in this paper. The principle of this method is described and then experimental demonstration is presented. Results show that the method has a high accuracy on shape and distance measurement. References
Fig. 6. The distance z of the second wedge with a larger separation angle.
be seen that the experimental results and the real values agree very well. 4. Discussion The most important advantage of this method is that the detected object can be immovable and the setup is very simple. However, there are many factors influencing the accuracy and the sensitivity of this method, such as the accuracy of DIC, the rotating angle of the plane mirror, the distance (a + l) between the CCD plane and the plane mirror and the distance (z − a − l) between the plane mirror and the detected object. The accuracy of DIC lies on the algorithm, sub-pixel interpolation, the speckle size, and the subset size. In this study, Newton–Raphson iteration method, bicubic
[1] A. Anand, V.K. Chhaniwal, P. Almoro, G. Pedrini, W. Osten, Shape and deformation measurements of 3D objects using volume speckle field and phase retrieval, Opt. Lett. 34 (2009) 1522–1524. [2] L.C. Chen, S.L. Yeh, A.M. Tapilouw, J.C. Chang, 3-D surface profilometry using simultaneous phase-shifting interferometry, Opt. Commun. 283 (2010) 3376–3382. [3] J.X. Gao, W. Xu, J.P. Geng, Use of shadow-speckle correlation method for 3D tooth model reconstruction, Opt. Las. Eng. 44 (2006) 455–465. [4] B. Pan, H.M. Xie, L.H. Yang, Z.Y. Wang, Accurate measurement of satellite antenna surface using three-dimensional digital image correlation technique, Strain 45 (2009) 194–200. [5] B. Pan, H.M. Xie, J.X. Gao, A. Asundi, Improved speckle profection profilometry for out-of-plane shape measurement, Appl. Opt. 47 (2008) 5527–5533. [6] Y.H. Huang, C. Quan, C.J. Tay, L.J. Chen, Shape measurement by the use of digital image correlation, Opt. Eng. 44 (2005) 087011. [7] W.H. Peters, W.F. Ranson, Digital imaging techniques in experimental stress analysis, Opt. Eng. 21 (1982) 427–431. [8] H.A. Bruck, S.R. McNeill, M.A. Sutton, W.H. Peters, Digital image correlation using Newton–Raphson method of partial differential correction, Exp. Mech. 46 (1989) 261–267. [9] B. Pan, K.M. Qian, H.M. Xie, A. Asundi, Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review, Meas. Sci. Technol. 20 (2009) 062001.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
An improved method for shape measurement using two-dimensional digital image correlation Xing Xu, Kaifu Wang ∗ , Guoqing Gu College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, 29 Yudao Street, Nanjing, Jiangsu 210016, PR China
a r t i c l e
i n f o
Article history: Received 17 July 2012 Accepted 10 December 2012
Keywords: Digital image correlation Shape measurement Plane mirror rotation Newton-Raphson iteration Bicubic spline interpolation
a b s t r a c t Shape measurement is a significant application of digital image correlation (DIC). An improved method that combines a rotatable plane mirror is proposed to measure the shape of an immovable object. In this method, two images, one before and the other after rotating the plane mirror, are obtained and then in-plane translation which related to the shape of the detected object can be calculated by the use of two-dimensional digital image correlation (2D DIC). The relationship between the in-plane translation and the shape of the object is described. Experimental results show that the proposed method is feasible for shape and distance measurement with high accuracy. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction Shape measurement is important in many fields of physics and engineering. Various optical methods have been employed for accurate shape measurement of an object, such as interferometric profilometry [1,2] and digital image correlation [3,4]. Interferometry is a well-established, whole-field method for surface profiling. The classical 2D DIC method, which uses only one camera, has been used for shape measurement according to the relationship between surface height variation and in-plane displacement in the image plane [5,6]. Meanwhile, 3D DIC method, which uses two cameras, can determines the shape of an object based on the principle of binocular stereovision. However, all of these methods have some limitations. Interferometric profilometry often needs complicated phase analysis. Some 2D DIC methods demands for speckle projection [5]. 3D DIC needs a laborious stereovision calibration. Huang et al. developed a simple method by using 2D DIC which does not need speckle projection and also provides accurate results [6]. However, this method needs to translate the determined object which cannot be used to measure the shape of an immovable object. In this paper we have proposed an improved method for the shape measurement of immovable object. It is known that the magnification at different points of an object changes by the surface height variation. Therefore a rigid-body in-plane translation of the object will introduce an inequable displacement in the
image plane [6]. However, if the determined object is immovable, the method based on giving a rigid-body in-plane translation is unusable. We have added a plane mirror between the object and the camera. An approximate rigid-body in-plane translation can occur by rotating the mirror slightly. Then the shape of the object can be calculated from the displacement in the image plane using DIC. 2. Principles 2.1. Principle of two-dimensional digital image correlation Digital image correlation which initially proposed by Peters and Ranson [7] has been developed for thirty years and now has been widely used in many fields. A brief description of this method is given in this section and more details can be found in [7–9]. The basic principle of 2D DIC is to match the same sub-image of two images recorded before and after object deformation. The matching process is implemented by searching the peak position of the distribution of a correlation coefficient which is predefined and reveals the similarity between two sub-images. Once the correlation coefficient extremum is found, the relative displacement of two sub-images can be determined. In order to improve the accuracy of DIC, Newton–Raphson iteration method and bicubic spline interpolation scheme are often used in DIC. 2.2. Principle for shape measurement
∗ Corresponding author. E-mail address: [email protected] (K. Wang). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2012.12.034
Fig. 1 illustrates the typical pinhole imaging system which had been used for shape measurement in [6]. The detected object is imaged on the image plane through a thin lens. In the experiment,
4098
X. Xu et al. / Optik 124 (2013) 4097–4099
X
According to Eq. (7), the z coordinate can be obtained after measuring the x’ using digital image correlation and determining the image distance a by a calibration process:
Image plane Lens
Object
x Z
x
Y
z≈
P ( x, y , z ) P (x
x, y, z )
1 1 1 = + a z−a f
Fig. 1. Schematic geometry of pinhole camera system.
Object
Lens
Z
P ( x, y , z )
P (x
x, y, z
z)
z-a-l
l
Fig. 2. Schematic geometry of a pinhole camera system with a plane mirror.
when an in-plane translation x is applied to the object, the translation of the image of the object x’ can be expressed as x = −
a x z−a
(1)
At this end, by doing a calibration of determining the image distance a, the z coordinate which is equal to the shape information can be acquired: z =a−
1/2
(10)
3. Experiments and results 2
Y
z − (z 2 − 4zf ) 2
Virtual image
Plane mirror
x
a
(9)
where f is the focal length. The image distance a can be calculated by a=
X Image plane
(8)
There are many methods for calibration of the image distance. Here we introduce a very simple one. According to the image formula of pinhole camera system:
z-a
a
al sin 2 +a a sin 2 − x
ax x
(2)
Using the above system, the shape of an object can be accurate determined. However, this system may be unusable when the detected object is immovable. In view of this, we design an improved system with a plane mirror, showed in Fig. 2. In the experiment, the plane mirror is rotated with an angle in the XZ plane. According to the principle of plane mirror imaging, the virtual image of the object has a deformation after rotating the plane mirror. A point P(x, y, z) which is random selected in the virtual image of the object moves to point P (x + x, y, z + z). Therefore, there are both in-plane translation x and out-of-plane translation z. These two translations can be expressed as x = x(cos 2 − 1) + (z − a − l) sin 2
(3)
z = (z − a − l)(1 − cos 2) + x sin 2
(4)
The CCD camera used here has a resolution of 2560-by-1920 and the lens here has a focal length of 25 mm. The detected objects are one flat plate and two wedges with different separation angles. The objects are all coated with black–white speckles on their surface. In the experiment, the surface of the object is illuminated by a white light source and the plane mirror is located far from the image plane about 260 mm. By rotating the plane mirror a little angle of 0.025◦ , and correlating the images before and after the rotating, an in-plane translation x’ which is related to the height variation can be acquired. Afterwards the distance z of every point of the object surface which represents the shape of the object is obtained according to Eq. (8). After experiment, a calibration of the image distance is essential. The image distance should be kept identical both in experiment and calibration process. The aperture of the camera should be set to maximum in the calibration process so as to find the object distance with a high accuracy. Then, the image distance can be calculated according to Eq. (10). Fig. 3 shows the in-plane translation x’ of the image of the flat plate. According to Eq. (8), the distance z which directly represents the shape of the plate then can be obtained, as shown in Fig. 4. The results in Fig. 4 show that the distance between the plate plane and the plane mirror is nearly 320-260 = 60 mm. As a reason, this method can also be used for distance measurement. Figs. 5 and 6 show the shapes of the two wedges with different separation angles. In order to calculate these two angles, middle sections of Figs. 5 and 6 are plotted in Fig. 7, in which the height variation of the wedges is clearly viewed. After that, the angles can be obtained through computing the slope of middle sections. The real separation angles of these two wedges can be measured by a protractor and the real angles are 12◦ and 30◦ respectively. It can
The relationship among x’ and x, z is acquired as following:
x = a −
x − x x + z−a z − z − a
Based on Eqs. (3) and (4), x’ can be rewritten as
x = a
−
x 2x − x cos 2 − (z − a − l) sin 2 + z−a z − a − (z − a − l)(1 − cos 2) − x sin 2
(5)
(6)
We assume that the rotating angle is so small that we have cos 2 ≈ 1 and sin 2 (1 − cos 2). Meanwhile, considering that z − a x sin 2. Therefore, x’ can be represented by x ≈
a(z − a − l)sin 2 z−a
(7) Fig. 3. The in-plane translation of the image of the flat plate acquired by DIC.
X. Xu et al. / Optik 124 (2013) 4097–4099
Fig. 4. The distance z of the plat plate.
4099
Fig. 7. Middle sections of two wedges.
spline interpolation and a subset of 50 × 50 were used. The rotating angle should be very small due to the analysis in Section 2.2 and is chose as 0.025◦ in this paper. In order to analyze the sensitivity of this method, the relationship between (x’) and z can be obtained based on Eq. (7) and be expressed as (x ) =
Fig. 5. The distance z of the first wedge with a smaller separation angle.
la sin 2 (z − a)2
z
(11)
According to Eq. (11), the value of (la sin 2)/(z − a)2 must be as large as possible. It is noted that the value of a always close to the value of the focal length of the lens. Based on these, the value of (a + l) must be much larger than the value of (z − a − l), i.e. that the plane mirror should be set far away from the image plane and the detected object should be located as close to the plane mirror as possible. 5. Conclusions An improved method combined a rotatable plane mirror for shape measurement using 2-D DIC is proposed in this paper. The principle of this method is described and then experimental demonstration is presented. Results show that the method has a high accuracy on shape and distance measurement. References
Fig. 6. The distance z of the second wedge with a larger separation angle.
be seen that the experimental results and the real values agree very well. 4. Discussion The most important advantage of this method is that the detected object can be immovable and the setup is very simple. However, there are many factors influencing the accuracy and the sensitivity of this method, such as the accuracy of DIC, the rotating angle of the plane mirror, the distance (a + l) between the CCD plane and the plane mirror and the distance (z − a − l) between the plane mirror and the detected object. The accuracy of DIC lies on the algorithm, sub-pixel interpolation, the speckle size, and the subset size. In this study, Newton–Raphson iteration method, bicubic
[1] A. Anand, V.K. Chhaniwal, P. Almoro, G. Pedrini, W. Osten, Shape and deformation measurements of 3D objects using volume speckle field and phase retrieval, Opt. Lett. 34 (2009) 1522–1524. [2] L.C. Chen, S.L. Yeh, A.M. Tapilouw, J.C. Chang, 3-D surface profilometry using simultaneous phase-shifting interferometry, Opt. Commun. 283 (2010) 3376–3382. [3] J.X. Gao, W. Xu, J.P. Geng, Use of shadow-speckle correlation method for 3D tooth model reconstruction, Opt. Las. Eng. 44 (2006) 455–465. [4] B. Pan, H.M. Xie, L.H. Yang, Z.Y. Wang, Accurate measurement of satellite antenna surface using three-dimensional digital image correlation technique, Strain 45 (2009) 194–200. [5] B. Pan, H.M. Xie, J.X. Gao, A. Asundi, Improved speckle profection profilometry for out-of-plane shape measurement, Appl. Opt. 47 (2008) 5527–5533. [6] Y.H. Huang, C. Quan, C.J. Tay, L.J. Chen, Shape measurement by the use of digital image correlation, Opt. Eng. 44 (2005) 087011. [7] W.H. Peters, W.F. Ranson, Digital imaging techniques in experimental stress analysis, Opt. Eng. 21 (1982) 427–431. [8] H.A. Bruck, S.R. McNeill, M.A. Sutton, W.H. Peters, Digital image correlation using Newton–Raphson method of partial differential correction, Exp. Mech. 46 (1989) 261–267. [9] B. Pan, K.M. Qian, H.M. Xie, A. Asundi, Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review, Meas. Sci. Technol. 20 (2009) 062001.