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Use of rigid-body motion for the investigation and estimation of the measurement errors related to digital image correlation technique
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Optics and Lasers in Engineering 46 (2008) 185–196 www.elsevier.com/locate/optlaseng
Use of rigid-body motion for the investigation and estimation of the measurement errors related to digital image correlation technique H. Haddadi, S. Belhabib CNRS-LPMTM, Universite´ Paris 13, Institut Galile´e-99, av. J.-B. Cle´ment, 93430 Villetaneuse, France Received 20 October 2006; received in revised form 23 May 2007; accepted 30 May 2007 Available online 18 October 2007
Abstract The aim of this work is to investigate the sources of errors related to digital image correlation (DIC) technique applied to strain measurements. The knowledge of such information is important before the measured kinematic fields can be exploited. After recalling the principle of DIC, some sources of errors related to this technique are listed. Both numerical and experimental tests, based on rigid-body motion, are proposed. These tests are simple and easy-to-implement. They permit to quickly assess the errors related to lighting, the optical lens (distortion), the CCD sensor, the out-of-plane displacement, the speckle pattern, the grid pitch, the size of the subset and the correlation algorithm. The errors sources that cannot be uncoupled were estimated by amplifying their contribution to the global error. The obtained results permit to address a classification of the error related to the used equipment. The paper ends by some suggestions proposed in order to minimize the errors. r 2007 Elsevier Ltd. All rights reserved. Keywords: Strain field measurements; Errors measurements; Rigid-body motion; Digital image correlation technique
1. Introduction The measurement of the mechanical response of a sample subjected to a load history is an important data for the identification of the mechanical behavior of materials. Several methods have been developed to measure materials deformation. They can be classified into three categories. The first category consists in measuring the deformation indirectly by assuming that strain field is homogenous in the gauge area. The lengthening of the sample corresponds to the displacement of the crossbar. The second category is contact methods. This is typically the case of gauges or extensometers. These methods are accurate but limited in terms of measurement points. The third category is non-contact methods. This last category encloses: grid method [1], geometric moire´ [2], speckle laser [3,4] and digital image correlation (DIC) technique [5,6]. These last methods permit to measure either real-time strain of only some points in order to monitor a tensile test Corresponding author. Tel.: +33 1 49 40 34 75; fax: +33 1 49 40 39 38.
E-mail address: [email protected] (H. Haddadi). 0143-8166/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2007.05.008
by imposing constant strain rate, or measure the strain field once the mechanical test is performed because of the time needed to process the data. These methods are based on different physical principles and used for different purposes. The choice of a method depends on the following considerations: measurement precision, cost of the experimental set-up, easiness of use and the studied phenomena. Details and attempts of classification of these methods can be found in [7]. The full-field measurement techniques are used in different domains [8]. These methods are very useful for the study of the mechanical behavior of materials especially heterogeneous ones. They are of great help in the case of composite materials for example [9], where the use of classical tensile tests may be not sufficient. Biomechanics is an other domain, where full-field measurement techniques may be of valuable contribution [10–12]. Full-field methods can also be used for the validation and identification of mechanical models [13,14]. These techniques are increasingly used for the study of crack growth and damage in materials [15,16]. Thus, promising industrial applications like control and testing can be developed [17].
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Fig. 1. Illustration of the principle of the digital image correlation technique. The subset in the image B that gives good correlation corresponds to the minimal value of the correlation coefficient.
DIC technique permits to evaluate the displacement and strain over a virtual grid defined on a digital image [5,6]. The main advantages of DIC are: the simplicity of the needed devices, strain field can be measured over a broad area and for large deformation. In addition to this, DIC is a non-contact method namely it permits to measure strain without disturbing the local mechanical response of the material. Theoretically, this technique permits to measure small deformation. Practically, however, the measurement of small strain is strongly related to the quality of the devices used in the experimental test. Thus, it is obvious that the estimation of the errors related to the different devices involved in the measurement procedure is very important [18,19]. The first question one has to answer when dealing with experimental data is: what is the overall error linked to the measurement process? The second question is: what are the sources of errors and their contribution to the overall error? Bringing answers to these questions can help enormously to classify the sources of errors and try to minimize their contribution. The aim of this work is to assess the errors related to DIC measurements. Investigation of the errors related to the size of the subset can be found in [20–22]. The errors related to speckle pattern are reported in [21–23]. The influence of the contrast of the image on the correlation errors can be found in [24]. The systematic errors caused by gray-value interpolation is addressed in [25]. The influence of distortion was studied by [26,27]. The authors tried in this paper to propose some tests that can help to explore the errors related to different sources using rigid-body motion. The second section recalls briefly the principle of DIC, since this work deals with errors related to this measurement method. In the third section, several experimental and numerical tests are presented. The obtained results are discussed in section 4. In the last section, some suggestions are proposed to minimize errors.
2. Digital image correlation technique 2.1. Principle of DIC The principle of DIC is illustrated in Fig. 1. Image correlation consists in associating to a given material point M located in the center of a pixel with ðX ; Y Þ coordinates in the reference image A, acquired before the beginning of the test, the ðx; yÞ coordinates of the same material point in image B acquired during the test. The coordinates ðx; yÞ are fixed by the minimization of a correlation coefficient given by P i2D f ðX i ; Y i Þgðxi ; yi Þ c ¼ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , (1) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P P 2 2 i2D f ðX i ; Y i Þ i2D g ðxi ; yi Þ where functions f and g represent gray levels, respectively, in the reference image A and the one to be correlated B. D is the domain, called also subset, of evaluation of the correlation coefficient, its center is the material point M. The correlation is carried out at the nodes of a virtual grid. The subset in the image B can be deformed to enhance the quality of the correlation. 2.2. Experimental set-up The vision system used in this study is the Olympus E-10 camera with a four mega pixels CCD sensor providing 8 bit grayscale images. The characteristics of the lens given by the manufacturer are: Zoom Lens Olympus 9–36 mm, F2.0–F2.4, 14 elements in 11 groups. In this study, the focal length was fixed to 9 mm. The camera is placed in front of the sample surface (Fig. 2). The lighting source used is a desk lamp. The accuracy of the correlation is closely related to the quality of the speckle. This means that images must have different gray levels so that the new position of a given
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Fig. 2. Experimental set-up. The distance between the sample surface and the camera is about 21 cm. The size of the studied area, located in the central part of the sample, is about 400 200 pixels (45 20 mm2 ).
point may be found thanks to the neighboring pixels. Some materials are naturally textured and the images are contrasted enough so that the correlation is possible. In the case of non-textured materials, one has to create a random speckle pattern using black and white paint (Fig. 3). The sample preparation consists in applying a thin layer of white paint followed by spray jets of black paint in order to obtain a random speckle. In this work, a standard paint for metallic surfaces was used. The choice of paint differs according to the studied material. One should verify, however, that the used paint remains stick to the sample surface during the mechanical test. The speckle granulometry depends on the operator and sometimes many attempts are needed to obtain a good speckle pattern. Images are processed by a graphical user’s interface software called ‘‘Sept-D’’ 1 version 0.6.0.83, based on DIC. This software gives access to displacement and strain at the nodes of a virtual grid. 3. Description of the tests used for the estimation of stain measurement errors In order to quantify and classify some sources of errors, two categories of investigation tests are proposed in this work. The first one is experimental and the second one is numerical. By experimental tests, we refer to tests that need experimental manipulation and real images acquisition. These tests permit a quick assessment of the errors. Numerical tests, however, concern the tests where a real image is used only as a reference image which is processed by a suitable software to generate the other images of the test. These tests were carried out in order to uncouple the 1 Sept-D is an image correlation software developed by Vacher et al. [28] at the University of Savoie, France.
Fig. 3. Surface of the sample: (a) as received (b) after the application of the spray paint. The used material is a dual-phase steel sheet of 1 mm thickness.
sources of errors by studying them separately. These tests are very important, for example, to assess the errors linked to the used correlation algorithm. All tests were performed on an undeformed sample subjected to a rigid-body motion. Given that the strain field is theoretically equal to zero, the measured strain represents the measurement error. For the sake of simplicity, it is referred by error in what follows to the longitudinal Green–Lagrange mean strain error. It is calculated by the following expression: E¼
nn 1X jE ðiÞ j, nn i¼1 xx
(2)
where nn is the number of the nodes of the virtual grid and E ðiÞ xx is the value of the longitudinal Green–Lagrange strain measured at the ith node of the virtual grid. 3.1. Sources of errors The sources of errors related to full-field strain measurement by DIC technique are numerous (Fig. 4).
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Fig. 4. Sources of errors in the case of DIC measurements.
They can be classified into two categories. The first one is the errors linked to the quality of the measurement devices and the working environment:
lighting; the optical lens (distortion); the CCD sensor; the presence of out-of-plane displacement; and the arrangement of all these devices in the workspace.
The second category is the errors related to the correlation principle itself:
the quality of the speckle pattern; the grid pitch, the size of the subset; and the correlation algorithm (minimization algorithm, interpolation function).
In this study, the correlations were run with the following parameters of the correlation software Sept-D: cross correlation with bilinear interpolation option, the values of the modified parameters (grid pitch, size of the subset) were given for each test. The other correlation parameters of the software were kept with their default values. 3.2. Description of the experimental tests 3.2.1. Tests related to in-plane rigid-body displacement The overall error can be estimated by taking an image before and another after applying a rigid-body motion to the sample followed by the measurement of strain fields. As these fields are theoretically equal to zero, one can thus accede to the overall error of strain measurement. To estimate this error, we have led three series of manipulation: horizontal translation, vertical translation and inplane rotations with respect to the center of the sample.
The horizontal translation test was also used for the study of errors related to the grid pitch and the subset size as well. It should be pointed out that the distortion is present in almost all real tests excepting the out-of-plane displacement (negligible) and the environment test. Given that this error is systematic, its contribution is then constant when the same displacement for real tests is used. It should be clear, then, that no correction of distortion is performed in this study. (a) Test related to in-plane translation: The first test consists in translating the sample along the horizontal (x) and the vertical (y) axes around a reference position using a multi-axis positioning stage (Fig. 2). Images are acquired for each position of the sample (Fig. 5). The horizontal and vertical translations range is 5 mm to þ5 mm by steps of 1 mm with respect to the reference position taken as the origin. The size of the zone of interest of the used sample is 460 200 pixels with a pixel size of about 0:11 mm. (b) Test related to the grid pitch: Several correlations were run using different grid pitches on the same set of images corresponding to horizontal translations of the sample. The studied grid pitches are: 5, 10, 20, 30, 40 and 50 pixels. (c) Test related to the size of the subset: In order to investigate the influence of the size of the subset pattern, several correlations were performed on the same area, using different sizes for the subset (5, 10, 15, 25, 40, 50 pixels) and keeping the same value for the grid pitch (25 pixels). (d) Test related to in-plane rotation: The sample is subjected to in-plane rotation around the normal axis of the sample (Fig. 6). The range of the tested angles is [01, 301]. 3.2.2. Test related to the speckle pattern Five random speckle patterns were tested (Fig. 7): (a) random speckle; (b) small black spots; (c) big black spots; (d) small black spots þ random speckle; and (e) big black spots þ random speckle. These patterns were deposited on white painted background. The test consists, for each pattern, in translating the sample along the horizontal axis as previously explained. 3.2.3. Test related to out-of-plane displacement The out-of-plane displacement consists in forward or backward movement of the sample surface with respect to the vision system (Fig. 8). To characterize the out-of-plane displacement, we have performed a test consisting of moving backward and forward the sample while the CCD camera is kept motionless. Images are acquired for each position of the sample (Fig. 9). For this test, we use the following expression for the evaluation of the error: E¼
nn 1X E ðiÞ . nn i¼1 xx
(3)
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Fig. 5. Horizontal translation of the sample: (a) negative translation; (b) reference position; (c) positive translation. The sample is translated with respect to the reference position and images were acquired for each translation.
Fig. 6. Images of the rotated sample: (a) reference position; (b) rotated sample. The sample is rotated around the reference position and images were acquired for each rotation.
Fig. 8. Principle of the out-of-plane displacement test. The sample is moved forward and backward to the camera. The forward displacement leads to a positive apparent strain field, whereas the backward displacement leads to a negative one.
was fixed to 1 min. This error includes the signal noise of the CCD sensor as well as the mechanical oscillation of the camera with respect to the sample. 3.2.5. Test related to lighting The last source of errors that we have investigated is lighting. The fact that strain measurement is performed on a set of digital images acquired during a test, means that the quality of the measurement is closely linked to the quality of these images. We propose here a test that permit to quantify the influence of lighting on the global error. The correlation is led between a reference position and the one obtained by 5 mm translation under the same light intensity. This procedure was repeated for several light intensities. Fig. 7. The tested patterns were deposited on white painted background: (a) random speckle; (b) small black spots; (c) big black spots; (d) small black spots þ random speckle; and (e) big black spots þ random speckle.
This expression is more suitable than the expression (2) to take into account the sign of the error which is positive in the case of forward displacement and negative otherwise (Fig. 8). The covered range of the out-of-plane displacement is 5 mm up to þ5 mm with steps of 1 mm. 3.2.4. Test related to the environment The test we have led consists in taking 20 images of the same sample kept motionless at equal time intervals. The elapsed time between two consecutive image acquisitions
3.3. Description of the numerical tests 3.3.1. Test related to perfect in-plane translation To estimate the errors related to the software algorithm, we propose a test consisting in cutting in an image two subimages containing a common zone of interest as depicted in Fig. 10. The first sub-image will be considered as the reference image. The correlation between the two subimages will theoretically give a uniform displacement field. Hence, the strain field is equal to zero, the measured strain is simply equal to the correlation software error. 3.3.2. Test related to lighting We propose here a numerical test to explore the errors related to lighting variation. The test consists in generating
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a set of images from a reference one, by varying the luminosity using the image processing software photoshopr (Fig. 11). The variation of the luminosity consists in reducing or increasing the gray level of each pixel by the same constant value. Two sub-images are then cut, with perfect displacement of 10 pixels, from each generated image. The correlation is led then between each couple of sub-images. 4. Results and discussion 4.1. Experimental tests 4.1.1. Errors related to in-plane rigid-body displacement (a) Error related to in-plane translation: The obtained results related to the horizontal and vertical translation are depicted in Figs. 12–14. One can see that the errors evolve proportionally to the imposed translation of the sample.
The errors lay in the interval ½0:521:7 103 . Where the displacement is in the range ½5 mm; þ5 mm. (b) Error related to the grid pitch: Fig. 15 shows that, for a rigid-body motion, the greater is the grid pitch the smaller are the errors. For a grid pitch of 5 pixels the errors range in the interval ½121:9 103 , whereas for a grid pitch of 50 pixels the obtained errors lay in the interval ½0:321:6 103 . The mean error, corresponding to different grid pitches, evolves in the interval ½0:921:4 103 . For 1 mm translation, the errors obtained for a grid pitch of 5 pixels are approximately 3 times greater than those obtained for a grid pitch of 20 pixels. (c) Error related to the size of the subset: The obtained results are depicted in Fig. 16. For a rigid-body motion, one can notice that the smaller is the subset, the greater are the errors. For a subset of 5 pixels the errors evolve in the interval ½222:4 103 , whereas the obtained errors for a subset of 50 pixels lay in the interval ½0:321:6 103 . The
Fig. 9. Images obtained for different values of the sample out-of-plane displacement with respect to the CDD camera: (a) backward displacement; (b) reference position; (c) forward displacement. The dashed rectangles cover the same material area.
Fig. 10. Principle of the ‘‘perfect displacement’’. The sub-images 1 and 2 are chosen in the original image so that the zone of interest belongs to both of them. The displacement field is not equal to zero since each pixel in the zone of interest has different coordinates in both sub-images 1 and 2. The strain field, however, is equal to zero because the zone of interest was not deformed.
Fig. 11. Obtained images with different luminosity variations: (a) 50 levels; (b) 0 level (reference); (c) þ50 levels. The luminosity variation performed using the image processing software photoshopr , consists in reducing or increasing the gray level of each pixel by the same constant value.
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mean error, corresponding to different subsets, evolves in the interval ½0:922:1 103 . Figs. 17(a) and (b) represent the errors fields obtained using subsets of 10 and 50 pixels, respectively. (d) Error related to in-plane rotation: The obtained results are depicted in Figs. 18 and 19. The errors lay in the interval ½0:421:5 103 . These manipulations permit to estimate the overall error. Thus, strain less than 0:5 103 could not be measured correctly. This means that it is quite difficult to measure the strain field accurately in the elastic domain with the devices used in this study.
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4.1.2. Error related to the speckle pattern The obtained results are represented in Fig. 20. The speckle pattern (c) presents the greatest errors. The patterns (d) and (e), however, give smaller errors than patterns (b) and (c) respectively. We can conclude that the addition of a random spray increases the accuracy of the strain measurement. The pattern (a) seems to give good results and as it is easy to paint, we used it for the other tests. For the pattern (c) the software was not able to correlate at some nodes of the virtual grid. This pattern
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Fig. 15. Errors related to in-plane translation obtained for different grid pitches with a subset size of 25 pixels.
Fig. 14. Cartographies of errors related to a horizontal translation of: (a) 5 mm; (b) þ5 mm. The fields (a) and (b) have negative and positive values (grid pitch ¼ 20 pixels, subset size ¼ 25 pixels).
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leads to the greatest errors as shown in Fig. 20. The obtained errors corresponding to this speckle pattern lay in the interval ½223:7 103 , whereas the pattern (e) presents errors ranging in the interval ½122 103 . These tests permit to show the influence of the speckle pattern on the accuracy of the measurements on the one hand. On the other hand, they show that bad speckle pattern is likely to make the correlation impossible at some nodes of the virtual grid and thus limits the number of measuring points and makes the strain map incomplete.
plane displacement according to the following relation: E ¼ a d,
(4)
4.1.3. Error related to the lighting The variation of light induces the variation of the gray levels, that influences on the accuracy of the correlation results. Fig. 21 shows the obtained results for this test. The errors lay in the interval ½2:224 103 . We can see that DIC technique works successfully down to very low values of the mean gray level.
where E is the mean value of the error, d is the out-of-plane displacement and a is a proportionality coefficient depending on the distance between the specimen and the camera. As shown in Fig. 23, the errors are positive in the case of forward out-of-plane displacement and negative in the other case. In our case a 2 103 mm1 , this means that an out-of-plane displacement of 1 mm introduces a measurement error of about 2 103 . The out-of-plane displacement seems to be an important source of errors. It should be pointed out that the obtained value of a in this study corresponds to a distance of about 21 cm between the sample surface and camera. The evolution of the coefficient a vs. the distance between the sample surface and camera was investigated (Fig. 24). The figure shows that greater is the distance, smaller is the value of a and thus smaller are the errors related to the out-of-plane displacement.
4.1.4. Error related to the out-of-plane displacement The obtained results are depicted in Fig. 22. We can notice a linear evolution of the errors against the out-of-
4.1.5. Error related to the environment The results are presented in Fig. 25. The errors are fluctuating in the interval ½228 104 .
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degree [°] Fig. 16. Errors obtained for different sizes of the subset with a grid pitch of 20 pixels. The mean error obtained for a subset of 5 pixels is approximately 2 times greater than the one obtained for a subset of 25 pixels.
Fig. 18. Errors related to in-plane rotation. The errors are of the same magnitude as the ones obtained in the case of in-plane translation (grid pitch ¼ 20 pixels, subset size ¼ 25 pixels).
Fig. 17. Cartographies of errors related to a translation of 5 mm. They are obtained with a grid pitch of 20 pixels and subset sizes of: (a) 5 pixels; (b) 50 pixels. Notice that the error field in (b) is smoother than in (a).
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Fig. 19. Cartographies of errors related to an in-plane rotation of: (a) þ4 ; (b) þ33 . The fields present randomly distributed negative and positive values (grid pitch ¼ 20 pixels, subset size ¼ 25 pixels).
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Fig. 22. Linear evolution of the errors related to out-of-plane displacement: E ¼ a d with a 2 103 mm1 (grid pitch ¼ 20 pixels, subset size ¼ 25 pixels).
4.2. Numerical tests 4.2.1. Error related to perfect in-plane translation The software gives apparent strain that represents the inherent errors of the correlation algorithm. The obtained results show that the errors lay in the interval ½0:522:4 105 for a zone of interest of 120 100 pixels (Fig. 26).
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4.2.2. Error related to lighting The obtained results related to the variation of luminosity are reported in Fig. 27. The results show that the increase of luminosity leads to the increase of errors but the correlation remains possible over all the nodes for mean gray levels smaller than 250. The decrease of luminosity leads first to a small decrease of errors before the correlation starts to be impossible over all the nodes for mean gray levels smaller than 32. These results are obtained for a reference image of 116 mean gray level . Figs. 28(a) and (b) represent the errors fields obtained for luminosity variation corresponding to mean gray levels of 60, 116 (reference) and 170.
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4.3. Attempt to classify the sources of errors The rigid-body in-plane and out-of-plane displacement tests permit to estimate the overall error. These tests are a
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Fig. 23. Cartographies of errors related to an out-of-plane displacement of: (a) 5 mm; (b) þ5 mm. As expected the errors are positive in the case of forward out-of-plane displacement and negative in the other case (grid pitch ¼ 20 pixels, subset size ¼ 25 pixels).
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Fig. 26. Errors related to a perfect translation. The chosen images are 350 250 pixels. The zone of interest, belonging to the chosen images, is 120 100 pixels. The obtained errors permit to estimate the inherent error of the correlation software (grid pitch ¼ 20 pixels, subset size ¼ 25 pixels).
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Fig. 25. Errors related to environment. The errors fluctuate randomly around the value 3:7 104 (grid pitch ¼ 20 pixels, subset size ¼ 25 pixels). Twenty images were acquired of the same motionless sample at equal time intervals of 1 min.
Fig. 27. Errors related to the variation of luminosity. (grid pitch ¼ 10 pixels, subset size ¼ 5 pixels).
quick way to have an idea about the measurement precision of the devices used in DIC technique and their arrangement in the workspace. The estimation of the
contribution of the different sources of errors is summarized in Table 1. The analysis of the obtained results shows that the presence of out-of-plane displacement is a major
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Fig. 28. Cartographies of errors related to luminosity variation. Mean gray level of: (a) 60; (b) 116 (reference); (c) 170 (grid pitch ¼ 10 pixels, subset size ¼ 5 pixels).
Table 1 Summary of the estimated errors related to different sources according to the proposed tests Error source
Mean error
Out-of-plane displacement Lighting Speckle pattern Subset size In-plane translation In-plane rotation Grid pitch Environment Perfect displacement
Displacement (mm) 2 103 ðmm1 Þ ½2:224 103 ½0:523:7 103 ½0:922:1 103 ½0:421:7 103 ½0:421:5 103 ½0:921:4 103 ½228 104 ½0:522:4 105
source of errors. The test shows that the errors evolve rapidly against the out-of-plane displacement. The errors related to light variation seems to be important as lighting variation leads to changes in the gray levels of the acquired images which affects the correlation accuracy. The obtained results show that the errors linked to the speckle pattern can be important if a bad speckle pattern is used. This can be seen when the correlation is not achieved all over the virtual grid. The measurement errors depend also on the grid pitch, and they are inversely proportional to the size of the grid pitch. After that come the errors related to the in-plane displacement. These errors are mainly related to the lens distortion and the quality of parallelism of the lens and the sample surfaces. Then, come the errors linked to the CCD sensor and the environment stability that shows the real limits of the used devices. The minor errors are those linked to the software algorithm. These errors depend on the interpolation and the optimization methods implemented in the correlation software. 5. Conclusion DIC technique is very attractive as it permits mapping displacement and strain fields for different types of applications (control, identification, motion detectiony). However, its setting-up needs special attention because of the length of the measure chain which introduces several
errors. In this study, the errors of strain measurement related to different sources were investigated and classified (Table 1). We propose hereafter some suggestions to improve the accuracy of DIC technique. Minimization of the out-of-plane displacement: The first thing to do is to use adequate solution to limit the out-ofplane displacement. This can be achieved by using special clumps for instance. The use of a telecentric lens, when possible, is a good solution to minimize the effect of the out-of-plane displacement [29]. An other solution consists in using stereo-correlation so that one can accede to the three components of the displacement. Lighting source: As shown by the test, the errors related to the variation of the lighting source may be important. Thus, one has to ensure a uniform and sufficient lighting covering the whole zone of interest of the sample surface during the mechanical test. Choice of the speckle pattern: The quality of the applied random pattern is very important. It has an influence on the correlation results. A bad speckle pattern leads to no correlation at some nodes of the virtual grid. It has also an influence on the accuracy of the correlation. Using qan optimized pattern that can be printed directly on the surface of the sample is an attractive solution. Thus, the obtained speckle pattern will be independent of the operator and; hence the results are more reproducible. Choice of the size of the subset: The main role of the subset is to characterize the resemblance between two regions and it depends on the speckle pattern. A subset too small is likely to give wrong correlation due to the increase of the number of local minima of the correlation coefficient defined by Eq. (1). The increase of the size of the subset enhance the quality of the correlation. The increase of the size of the subset over a given size do not present a tangible enhancement of measurements quality. One should keep in mind that the increase of the size of the subset is synonymous of the increase of computation time. So the subset must belong to a given range and one may try several sizes for the subset to find a suitable one that presents a good compromise between accuracy and computation time. It should be noticed that the size of the subset is closely linked to the granulometry of the speckle pattern.
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Choice of the grid pitch: A great grid pitch gives accurate strain field but a limited number of measurement points. A small pitch, however, gives access to a large number of measurement points but the obtained strain field is more noisy. Thus an optimized grid pitch has to be chosen according to the aimed application. For a given displacement pffiffiffi resolution su , the strain resolution s is given by s ¼ 2su =Dx [30]. Filtering the raw data: The use of a filtration technique may significantly decrease strain measurements noise and hence make them easily exploitable. The analysis of these errors in the case of large deformation is the next objective. This will be achieved necessarily by generating synthetic images using a chosen kinematic field. Acknowledgments The authors would like to thank Pr. P. Vacher from the University of Savoie-France for providing free of charge licences of the image correlation software ‘‘Sept-D’’ to LPMTM Laboratory. References [1] Avril S, Vautrin A, Hamelin P, Surrel Y. A full-field optical method for the experimental analysis of reinforced concrete beams repaired with composites. Compos Part A—Appl S 2004;35(7–8):873–84. [2] Boone PM, Vinckier AG, Denys RM, Sys WM, Deleu EN. Application of specimen-grid moire´ techniques in large scale steel testing. Opt Eng 1982;21(4):615–25. [3] Sjo¨dahl M. Some recent advances in electronic speckle photography. Opt Laser Eng 1998;29(2–3):125–44. [4] Yamaguchi I. Holography, speckle, and computers. Opt Laser Eng 2003;39(4):411–29. [5] Peters WH, Ranson WF. Digital imaging techniques in experimental stress analysis. Opt Eng 1982;21(3):427–31. [6] Sutton MA, Wolters WJ, Peters WH, McNeill SR. Digital image processing using and improved digital correlation method. Image Vision Comput 1983;1(3):133–9. [7] Surrel Y. Les techniques de mesure de champ: essai de classification. In: Orteu J-J, Balageas D, editors, Nume´ro spe´cial de la revue RSInstrumentation, Mesure, Me´trologie, vol. 4(3–4), 2004. p. 11–42. [8] Gre´diac M. The use of full field measurement methods in composite material characterization: interest and limitations. Compos Part A— Appl Sci 2004;35(7–8):751–61. [9] Mathias J-D, Balandraud X, Grediac M. Experimental investigation of composite patches with a full-field measurement method. Compos Part A—Appl Sci 2006;37(2):177–90. [10] Zhang D, Eggleton CD, Dwayne DA. Evaluating the mechanical behavior of arterial tissue using digital image correlation. Exp Mech 2002;42(4):409–16.
[11] Wang Y, Cuitin˜o AM. Full-field measurements of heterogeneous deformation patterns on polymeric foams using digital image correlation. Int J Solids Struct 2002;39(13–14):3777–96. [12] Gao J, Xu W, Geng J. 3d shape reconstruction of teeth by shadow speckle correlation method. Opt Laser Eng 2006;44(5):455–65. [13] Meuwissen MHH, Oomens CWJ, Baaijens FPT, Petterson R, Janssen JD. Determination of the elasto-plastic properties of aluminium using a mixed numerical-experimental method. J Mater Process Technol 1998;75(1–3):204–11. [14] Haddadi H, Belhabib S, Gaspe´rini M, Vacher P. Identification of the parameters of swift law using strain field measurements. In: Eighth European mechanics of materials conference, Cachan, France; 2005. [15] Claire D, Hild F, Roux S. Identification of damage fields using kinematic measurements. C R Mec 2002;330(11):729–34. [16] Avril S, Vautrin A, Surrel Y. Grid method: application to the characterization of cracks. Exp Mech 2004;44(1):37–43. [17] Siebert T, El-Ratal W, Wegner R, Ettemeyer A. Combine simulation and experiment in automotive testing with ESPI measurement. Exp Techn 2002;26(3):42. [18] Mistou S, Surrel Y. Caracte´risation me´trologique des syste`mes optiques de mesure de champs cine´matiques. In: Congre`s Franc- ais de Me´canique Troyes, 2005. [19] Patterson EA, Hack E, Brailly P, Burguete RL, Saleem Q, Siebert T, et al. Calibration and evaluation of optical systems for full-field strain measurement. Opt Laser Eng 2007;45(5):550–64. [20] Knauss WG, Chasiotis I, Huang Y. Mechanical measurements at the micron and nanometer scales. Mech Mater 2003;35(3–6): 217–31. [21] Lecompte D, Smits A, Bossuyt S, Sol H, Vantomme J, Van Hemelrijck D, et al. Quality assessment of speckle patterns for digital image correlation. Opt Laser Eng 2006;44(11):1132–45. [22] Bornert M. Workgroup 1 Gdr2519. Resolution and spatial resolution of digital image correlation techniques. In: Photomechanics 2006. France: Clermont Ferrand; 2006. [23] Hung PC, Voloshin AS. In-plane strain measurement by digital image correlation. J Brazil Soc Mech Sci Eng 2003;25(3):215–21. [24] Sun Z, Lyons JS, McNeil SR. Measuring microscopic deformation with digital image correlation. Opt Laser Eng 1997;27:409–28. [25] Schreier HW, Braasch JR, Sutton MA. Systematic errors in digital image correlation caused by intensity interpolation. Opt Eng 2000; 39(11):2915–21. [26] Schreier HW, Garcia D, Sutton MA. Advances in light microscope stereo vision. Exp Mech 2004;44(3):278–88. [27] Yoneyama S, Kikuta H, Kitagawa A, Kitamura K. Lens distortion correction for digital image correlation by measuring rigid body displacement. Opt Eng 2006;45(2):1–9. [28] Vacher P, Dumoulin S, Morestin F, Mguil-Touchal S. Bidimensional deformation measurement using digital images. Proc Inst Mech Eng 1999;213(Part C):811–7. [29] Pannier Y. Identification de parame`tres e´lastoplastiques par des essais statiquement inde´termine´s: mise en oeuvre expe´rimentale et validation de la me´thode des champs virtuels. PhD thesis, E´cole Nationale Supe´rieure d’Arts et Me´tiers, Centre Chaˆlons-enChampagne, France; 2006. [30] Surrel Y. Me´thodes optiques de mesure de champ: choix actuel et crite`res de choix. Se´minaire formation doctorale GMGC, E´gletons, France; 2001.
Optics and Lasers in Engineering 46 (2008) 185–196 www.elsevier.com/locate/optlaseng
Use of rigid-body motion for the investigation and estimation of the measurement errors related to digital image correlation technique H. Haddadi, S. Belhabib CNRS-LPMTM, Universite´ Paris 13, Institut Galile´e-99, av. J.-B. Cle´ment, 93430 Villetaneuse, France Received 20 October 2006; received in revised form 23 May 2007; accepted 30 May 2007 Available online 18 October 2007
Abstract The aim of this work is to investigate the sources of errors related to digital image correlation (DIC) technique applied to strain measurements. The knowledge of such information is important before the measured kinematic fields can be exploited. After recalling the principle of DIC, some sources of errors related to this technique are listed. Both numerical and experimental tests, based on rigid-body motion, are proposed. These tests are simple and easy-to-implement. They permit to quickly assess the errors related to lighting, the optical lens (distortion), the CCD sensor, the out-of-plane displacement, the speckle pattern, the grid pitch, the size of the subset and the correlation algorithm. The errors sources that cannot be uncoupled were estimated by amplifying their contribution to the global error. The obtained results permit to address a classification of the error related to the used equipment. The paper ends by some suggestions proposed in order to minimize the errors. r 2007 Elsevier Ltd. All rights reserved. Keywords: Strain field measurements; Errors measurements; Rigid-body motion; Digital image correlation technique
1. Introduction The measurement of the mechanical response of a sample subjected to a load history is an important data for the identification of the mechanical behavior of materials. Several methods have been developed to measure materials deformation. They can be classified into three categories. The first category consists in measuring the deformation indirectly by assuming that strain field is homogenous in the gauge area. The lengthening of the sample corresponds to the displacement of the crossbar. The second category is contact methods. This is typically the case of gauges or extensometers. These methods are accurate but limited in terms of measurement points. The third category is non-contact methods. This last category encloses: grid method [1], geometric moire´ [2], speckle laser [3,4] and digital image correlation (DIC) technique [5,6]. These last methods permit to measure either real-time strain of only some points in order to monitor a tensile test Corresponding author. Tel.: +33 1 49 40 34 75; fax: +33 1 49 40 39 38.
E-mail address: [email protected] (H. Haddadi). 0143-8166/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2007.05.008
by imposing constant strain rate, or measure the strain field once the mechanical test is performed because of the time needed to process the data. These methods are based on different physical principles and used for different purposes. The choice of a method depends on the following considerations: measurement precision, cost of the experimental set-up, easiness of use and the studied phenomena. Details and attempts of classification of these methods can be found in [7]. The full-field measurement techniques are used in different domains [8]. These methods are very useful for the study of the mechanical behavior of materials especially heterogeneous ones. They are of great help in the case of composite materials for example [9], where the use of classical tensile tests may be not sufficient. Biomechanics is an other domain, where full-field measurement techniques may be of valuable contribution [10–12]. Full-field methods can also be used for the validation and identification of mechanical models [13,14]. These techniques are increasingly used for the study of crack growth and damage in materials [15,16]. Thus, promising industrial applications like control and testing can be developed [17].
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Fig. 1. Illustration of the principle of the digital image correlation technique. The subset in the image B that gives good correlation corresponds to the minimal value of the correlation coefficient.
DIC technique permits to evaluate the displacement and strain over a virtual grid defined on a digital image [5,6]. The main advantages of DIC are: the simplicity of the needed devices, strain field can be measured over a broad area and for large deformation. In addition to this, DIC is a non-contact method namely it permits to measure strain without disturbing the local mechanical response of the material. Theoretically, this technique permits to measure small deformation. Practically, however, the measurement of small strain is strongly related to the quality of the devices used in the experimental test. Thus, it is obvious that the estimation of the errors related to the different devices involved in the measurement procedure is very important [18,19]. The first question one has to answer when dealing with experimental data is: what is the overall error linked to the measurement process? The second question is: what are the sources of errors and their contribution to the overall error? Bringing answers to these questions can help enormously to classify the sources of errors and try to minimize their contribution. The aim of this work is to assess the errors related to DIC measurements. Investigation of the errors related to the size of the subset can be found in [20–22]. The errors related to speckle pattern are reported in [21–23]. The influence of the contrast of the image on the correlation errors can be found in [24]. The systematic errors caused by gray-value interpolation is addressed in [25]. The influence of distortion was studied by [26,27]. The authors tried in this paper to propose some tests that can help to explore the errors related to different sources using rigid-body motion. The second section recalls briefly the principle of DIC, since this work deals with errors related to this measurement method. In the third section, several experimental and numerical tests are presented. The obtained results are discussed in section 4. In the last section, some suggestions are proposed to minimize errors.
2. Digital image correlation technique 2.1. Principle of DIC The principle of DIC is illustrated in Fig. 1. Image correlation consists in associating to a given material point M located in the center of a pixel with ðX ; Y Þ coordinates in the reference image A, acquired before the beginning of the test, the ðx; yÞ coordinates of the same material point in image B acquired during the test. The coordinates ðx; yÞ are fixed by the minimization of a correlation coefficient given by P i2D f ðX i ; Y i Þgðxi ; yi Þ c ¼ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , (1) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P P 2 2 i2D f ðX i ; Y i Þ i2D g ðxi ; yi Þ where functions f and g represent gray levels, respectively, in the reference image A and the one to be correlated B. D is the domain, called also subset, of evaluation of the correlation coefficient, its center is the material point M. The correlation is carried out at the nodes of a virtual grid. The subset in the image B can be deformed to enhance the quality of the correlation. 2.2. Experimental set-up The vision system used in this study is the Olympus E-10 camera with a four mega pixels CCD sensor providing 8 bit grayscale images. The characteristics of the lens given by the manufacturer are: Zoom Lens Olympus 9–36 mm, F2.0–F2.4, 14 elements in 11 groups. In this study, the focal length was fixed to 9 mm. The camera is placed in front of the sample surface (Fig. 2). The lighting source used is a desk lamp. The accuracy of the correlation is closely related to the quality of the speckle. This means that images must have different gray levels so that the new position of a given
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Fig. 2. Experimental set-up. The distance between the sample surface and the camera is about 21 cm. The size of the studied area, located in the central part of the sample, is about 400 200 pixels (45 20 mm2 ).
point may be found thanks to the neighboring pixels. Some materials are naturally textured and the images are contrasted enough so that the correlation is possible. In the case of non-textured materials, one has to create a random speckle pattern using black and white paint (Fig. 3). The sample preparation consists in applying a thin layer of white paint followed by spray jets of black paint in order to obtain a random speckle. In this work, a standard paint for metallic surfaces was used. The choice of paint differs according to the studied material. One should verify, however, that the used paint remains stick to the sample surface during the mechanical test. The speckle granulometry depends on the operator and sometimes many attempts are needed to obtain a good speckle pattern. Images are processed by a graphical user’s interface software called ‘‘Sept-D’’ 1 version 0.6.0.83, based on DIC. This software gives access to displacement and strain at the nodes of a virtual grid. 3. Description of the tests used for the estimation of stain measurement errors In order to quantify and classify some sources of errors, two categories of investigation tests are proposed in this work. The first one is experimental and the second one is numerical. By experimental tests, we refer to tests that need experimental manipulation and real images acquisition. These tests permit a quick assessment of the errors. Numerical tests, however, concern the tests where a real image is used only as a reference image which is processed by a suitable software to generate the other images of the test. These tests were carried out in order to uncouple the 1 Sept-D is an image correlation software developed by Vacher et al. [28] at the University of Savoie, France.
Fig. 3. Surface of the sample: (a) as received (b) after the application of the spray paint. The used material is a dual-phase steel sheet of 1 mm thickness.
sources of errors by studying them separately. These tests are very important, for example, to assess the errors linked to the used correlation algorithm. All tests were performed on an undeformed sample subjected to a rigid-body motion. Given that the strain field is theoretically equal to zero, the measured strain represents the measurement error. For the sake of simplicity, it is referred by error in what follows to the longitudinal Green–Lagrange mean strain error. It is calculated by the following expression: E¼
nn 1X jE ðiÞ j, nn i¼1 xx
(2)
where nn is the number of the nodes of the virtual grid and E ðiÞ xx is the value of the longitudinal Green–Lagrange strain measured at the ith node of the virtual grid. 3.1. Sources of errors The sources of errors related to full-field strain measurement by DIC technique are numerous (Fig. 4).
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Fig. 4. Sources of errors in the case of DIC measurements.
They can be classified into two categories. The first one is the errors linked to the quality of the measurement devices and the working environment:
lighting; the optical lens (distortion); the CCD sensor; the presence of out-of-plane displacement; and the arrangement of all these devices in the workspace.
The second category is the errors related to the correlation principle itself:
the quality of the speckle pattern; the grid pitch, the size of the subset; and the correlation algorithm (minimization algorithm, interpolation function).
In this study, the correlations were run with the following parameters of the correlation software Sept-D: cross correlation with bilinear interpolation option, the values of the modified parameters (grid pitch, size of the subset) were given for each test. The other correlation parameters of the software were kept with their default values. 3.2. Description of the experimental tests 3.2.1. Tests related to in-plane rigid-body displacement The overall error can be estimated by taking an image before and another after applying a rigid-body motion to the sample followed by the measurement of strain fields. As these fields are theoretically equal to zero, one can thus accede to the overall error of strain measurement. To estimate this error, we have led three series of manipulation: horizontal translation, vertical translation and inplane rotations with respect to the center of the sample.
The horizontal translation test was also used for the study of errors related to the grid pitch and the subset size as well. It should be pointed out that the distortion is present in almost all real tests excepting the out-of-plane displacement (negligible) and the environment test. Given that this error is systematic, its contribution is then constant when the same displacement for real tests is used. It should be clear, then, that no correction of distortion is performed in this study. (a) Test related to in-plane translation: The first test consists in translating the sample along the horizontal (x) and the vertical (y) axes around a reference position using a multi-axis positioning stage (Fig. 2). Images are acquired for each position of the sample (Fig. 5). The horizontal and vertical translations range is 5 mm to þ5 mm by steps of 1 mm with respect to the reference position taken as the origin. The size of the zone of interest of the used sample is 460 200 pixels with a pixel size of about 0:11 mm. (b) Test related to the grid pitch: Several correlations were run using different grid pitches on the same set of images corresponding to horizontal translations of the sample. The studied grid pitches are: 5, 10, 20, 30, 40 and 50 pixels. (c) Test related to the size of the subset: In order to investigate the influence of the size of the subset pattern, several correlations were performed on the same area, using different sizes for the subset (5, 10, 15, 25, 40, 50 pixels) and keeping the same value for the grid pitch (25 pixels). (d) Test related to in-plane rotation: The sample is subjected to in-plane rotation around the normal axis of the sample (Fig. 6). The range of the tested angles is [01, 301]. 3.2.2. Test related to the speckle pattern Five random speckle patterns were tested (Fig. 7): (a) random speckle; (b) small black spots; (c) big black spots; (d) small black spots þ random speckle; and (e) big black spots þ random speckle. These patterns were deposited on white painted background. The test consists, for each pattern, in translating the sample along the horizontal axis as previously explained. 3.2.3. Test related to out-of-plane displacement The out-of-plane displacement consists in forward or backward movement of the sample surface with respect to the vision system (Fig. 8). To characterize the out-of-plane displacement, we have performed a test consisting of moving backward and forward the sample while the CCD camera is kept motionless. Images are acquired for each position of the sample (Fig. 9). For this test, we use the following expression for the evaluation of the error: E¼
nn 1X E ðiÞ . nn i¼1 xx
(3)
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Fig. 5. Horizontal translation of the sample: (a) negative translation; (b) reference position; (c) positive translation. The sample is translated with respect to the reference position and images were acquired for each translation.
Fig. 6. Images of the rotated sample: (a) reference position; (b) rotated sample. The sample is rotated around the reference position and images were acquired for each rotation.
Fig. 8. Principle of the out-of-plane displacement test. The sample is moved forward and backward to the camera. The forward displacement leads to a positive apparent strain field, whereas the backward displacement leads to a negative one.
was fixed to 1 min. This error includes the signal noise of the CCD sensor as well as the mechanical oscillation of the camera with respect to the sample. 3.2.5. Test related to lighting The last source of errors that we have investigated is lighting. The fact that strain measurement is performed on a set of digital images acquired during a test, means that the quality of the measurement is closely linked to the quality of these images. We propose here a test that permit to quantify the influence of lighting on the global error. The correlation is led between a reference position and the one obtained by 5 mm translation under the same light intensity. This procedure was repeated for several light intensities. Fig. 7. The tested patterns were deposited on white painted background: (a) random speckle; (b) small black spots; (c) big black spots; (d) small black spots þ random speckle; and (e) big black spots þ random speckle.
This expression is more suitable than the expression (2) to take into account the sign of the error which is positive in the case of forward displacement and negative otherwise (Fig. 8). The covered range of the out-of-plane displacement is 5 mm up to þ5 mm with steps of 1 mm. 3.2.4. Test related to the environment The test we have led consists in taking 20 images of the same sample kept motionless at equal time intervals. The elapsed time between two consecutive image acquisitions
3.3. Description of the numerical tests 3.3.1. Test related to perfect in-plane translation To estimate the errors related to the software algorithm, we propose a test consisting in cutting in an image two subimages containing a common zone of interest as depicted in Fig. 10. The first sub-image will be considered as the reference image. The correlation between the two subimages will theoretically give a uniform displacement field. Hence, the strain field is equal to zero, the measured strain is simply equal to the correlation software error. 3.3.2. Test related to lighting We propose here a numerical test to explore the errors related to lighting variation. The test consists in generating
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a set of images from a reference one, by varying the luminosity using the image processing software photoshopr (Fig. 11). The variation of the luminosity consists in reducing or increasing the gray level of each pixel by the same constant value. Two sub-images are then cut, with perfect displacement of 10 pixels, from each generated image. The correlation is led then between each couple of sub-images. 4. Results and discussion 4.1. Experimental tests 4.1.1. Errors related to in-plane rigid-body displacement (a) Error related to in-plane translation: The obtained results related to the horizontal and vertical translation are depicted in Figs. 12–14. One can see that the errors evolve proportionally to the imposed translation of the sample.
The errors lay in the interval ½0:521:7 103 . Where the displacement is in the range ½5 mm; þ5 mm. (b) Error related to the grid pitch: Fig. 15 shows that, for a rigid-body motion, the greater is the grid pitch the smaller are the errors. For a grid pitch of 5 pixels the errors range in the interval ½121:9 103 , whereas for a grid pitch of 50 pixels the obtained errors lay in the interval ½0:321:6 103 . The mean error, corresponding to different grid pitches, evolves in the interval ½0:921:4 103 . For 1 mm translation, the errors obtained for a grid pitch of 5 pixels are approximately 3 times greater than those obtained for a grid pitch of 20 pixels. (c) Error related to the size of the subset: The obtained results are depicted in Fig. 16. For a rigid-body motion, one can notice that the smaller is the subset, the greater are the errors. For a subset of 5 pixels the errors evolve in the interval ½222:4 103 , whereas the obtained errors for a subset of 50 pixels lay in the interval ½0:321:6 103 . The
Fig. 9. Images obtained for different values of the sample out-of-plane displacement with respect to the CDD camera: (a) backward displacement; (b) reference position; (c) forward displacement. The dashed rectangles cover the same material area.
Fig. 10. Principle of the ‘‘perfect displacement’’. The sub-images 1 and 2 are chosen in the original image so that the zone of interest belongs to both of them. The displacement field is not equal to zero since each pixel in the zone of interest has different coordinates in both sub-images 1 and 2. The strain field, however, is equal to zero because the zone of interest was not deformed.
Fig. 11. Obtained images with different luminosity variations: (a) 50 levels; (b) 0 level (reference); (c) þ50 levels. The luminosity variation performed using the image processing software photoshopr , consists in reducing or increasing the gray level of each pixel by the same constant value.
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mean error, corresponding to different subsets, evolves in the interval ½0:922:1 103 . Figs. 17(a) and (b) represent the errors fields obtained using subsets of 10 and 50 pixels, respectively. (d) Error related to in-plane rotation: The obtained results are depicted in Figs. 18 and 19. The errors lay in the interval ½0:421:5 103 . These manipulations permit to estimate the overall error. Thus, strain less than 0:5 103 could not be measured correctly. This means that it is quite difficult to measure the strain field accurately in the elastic domain with the devices used in this study.
x 10-3 1.6 1.4
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1
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4.1.2. Error related to the speckle pattern The obtained results are represented in Fig. 20. The speckle pattern (c) presents the greatest errors. The patterns (d) and (e), however, give smaller errors than patterns (b) and (c) respectively. We can conclude that the addition of a random spray increases the accuracy of the strain measurement. The pattern (a) seems to give good results and as it is easy to paint, we used it for the other tests. For the pattern (c) the software was not able to correlate at some nodes of the virtual grid. This pattern
x 10-3 x 10-3
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grid pitch [pixels] Fig. 13. Errors related to in-plane vertical translation. The errors increase against the vertical displacement (grid pitch ¼ 20 pixels, subset size ¼ 25 pixels).
Fig. 15. Errors related to in-plane translation obtained for different grid pitches with a subset size of 25 pixels.
Fig. 14. Cartographies of errors related to a horizontal translation of: (a) 5 mm; (b) þ5 mm. The fields (a) and (b) have negative and positive values (grid pitch ¼ 20 pixels, subset size ¼ 25 pixels).
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leads to the greatest errors as shown in Fig. 20. The obtained errors corresponding to this speckle pattern lay in the interval ½223:7 103 , whereas the pattern (e) presents errors ranging in the interval ½122 103 . These tests permit to show the influence of the speckle pattern on the accuracy of the measurements on the one hand. On the other hand, they show that bad speckle pattern is likely to make the correlation impossible at some nodes of the virtual grid and thus limits the number of measuring points and makes the strain map incomplete.
plane displacement according to the following relation: E ¼ a d,
(4)
4.1.3. Error related to the lighting The variation of light induces the variation of the gray levels, that influences on the accuracy of the correlation results. Fig. 21 shows the obtained results for this test. The errors lay in the interval ½2:224 103 . We can see that DIC technique works successfully down to very low values of the mean gray level.
where E is the mean value of the error, d is the out-of-plane displacement and a is a proportionality coefficient depending on the distance between the specimen and the camera. As shown in Fig. 23, the errors are positive in the case of forward out-of-plane displacement and negative in the other case. In our case a 2 103 mm1 , this means that an out-of-plane displacement of 1 mm introduces a measurement error of about 2 103 . The out-of-plane displacement seems to be an important source of errors. It should be pointed out that the obtained value of a in this study corresponds to a distance of about 21 cm between the sample surface and camera. The evolution of the coefficient a vs. the distance between the sample surface and camera was investigated (Fig. 24). The figure shows that greater is the distance, smaller is the value of a and thus smaller are the errors related to the out-of-plane displacement.
4.1.4. Error related to the out-of-plane displacement The obtained results are depicted in Fig. 22. We can notice a linear evolution of the errors against the out-of-
4.1.5. Error related to the environment The results are presented in Fig. 25. The errors are fluctuating in the interval ½228 104 .
x 10-3 2.5 1.6 1 mm translation 5 mm translation mean error
2
x 10-3
1.4 1.2
1.5 Exx [ ]
Exx [ ]
1
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0.8 0.6
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0.4 0.2
0 5
10
20
30
40
50
subset size [pixel]
0 0
5
10
15
20
25
30
35
degree [°] Fig. 16. Errors obtained for different sizes of the subset with a grid pitch of 20 pixels. The mean error obtained for a subset of 5 pixels is approximately 2 times greater than the one obtained for a subset of 25 pixels.
Fig. 18. Errors related to in-plane rotation. The errors are of the same magnitude as the ones obtained in the case of in-plane translation (grid pitch ¼ 20 pixels, subset size ¼ 25 pixels).
Fig. 17. Cartographies of errors related to a translation of 5 mm. They are obtained with a grid pitch of 20 pixels and subset sizes of: (a) 5 pixels; (b) 50 pixels. Notice that the error field in (b) is smoother than in (a).
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Fig. 19. Cartographies of errors related to an in-plane rotation of: (a) þ4 ; (b) þ33 . The fields present randomly distributed negative and positive values (grid pitch ¼ 20 pixels, subset size ¼ 25 pixels).
x 10-3
0.01
5
0.008
4.5
3
0.004 0.002 Exx [ ]
3.5
Exx [ ]
0.006
speckle a speckle b speckle c speckle d speckle e
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0 -0.002
2.5
-0.004
2
-0.006 1.5
-0.008
1
-0.01
0.5
-5 -5
-4
-3
-2
-1
0
1
2
3
4
5
5
displacement [mm] Fig. 20. Errors related to in-plane translation for different speckle patterns: (a) random speckle; (b) small black spots; (c) big black spots; (d) small black spots þ random speckle; and (e) big black spots þ random speckle (grid pitch ¼ 20 pixels, subset size ¼ 25 pixels).
4
0 out-of-plane displacement [mm]
0
Fig. 22. Linear evolution of the errors related to out-of-plane displacement: E ¼ a d with a 2 103 mm1 (grid pitch ¼ 20 pixels, subset size ¼ 25 pixels).
4.2. Numerical tests 4.2.1. Error related to perfect in-plane translation The software gives apparent strain that represents the inherent errors of the correlation algorithm. The obtained results show that the errors lay in the interval ½0:522:4 105 for a zone of interest of 120 100 pixels (Fig. 26).
x 10-3
3.8
4.2.2. Error related to lighting The obtained results related to the variation of luminosity are reported in Fig. 27. The results show that the increase of luminosity leads to the increase of errors but the correlation remains possible over all the nodes for mean gray levels smaller than 250. The decrease of luminosity leads first to a small decrease of errors before the correlation starts to be impossible over all the nodes for mean gray levels smaller than 32. These results are obtained for a reference image of 116 mean gray level . Figs. 28(a) and (b) represent the errors fields obtained for luminosity variation corresponding to mean gray levels of 60, 116 (reference) and 170.
3.6 3.4
Exx [ ]
3.2 3 2.8 2.6 2.4 2.2 2 0
50
100
150
200
Mean gray level Fig. 21. Errors related to light intensity variation. Light variation corresponds to different mean gray levels (grid pitch ¼ 20 pixels, subset size ¼ 25 pixels).
4.3. Attempt to classify the sources of errors The rigid-body in-plane and out-of-plane displacement tests permit to estimate the overall error. These tests are a
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Fig. 23. Cartographies of errors related to an out-of-plane displacement of: (a) 5 mm; (b) þ5 mm. As expected the errors are positive in the case of forward out-of-plane displacement and negative in the other case (grid pitch ¼ 20 pixels, subset size ¼ 25 pixels).
3.4
x 10-3
2.5
x 10-5
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2
2.6
1.5 Exx [ ]
α [mm-1]
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2.4 2.2
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2 1.8
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0 10
15
20
25
30
35
0
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10
distance [cm]
25
30
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40
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50
x 10-4
Fig. 26. Errors related to a perfect translation. The chosen images are 350 250 pixels. The zone of interest, belonging to the chosen images, is 120 100 pixels. The obtained errors permit to estimate the inherent error of the correlation software (grid pitch ¼ 20 pixels, subset size ¼ 25 pixels).
7
x 10-3 5
5
100
4
80 error
Exx [ ]
4 3 2 1
correlated points
3
60
2
40
1
20
Correlated points [%]
6
Exx [ ]
20
displacement [pixel]
Fig. 24. Evolution of the parameter a vs. the distance between the camera and the sample surface (grid pitch ¼ 20 pixels, subset size ¼ 25 pixels).
8
15
0 0
5
10 time [min]
15
20
0
0 0
50
100 150 Mean gray level
200
250
Fig. 25. Errors related to environment. The errors fluctuate randomly around the value 3:7 104 (grid pitch ¼ 20 pixels, subset size ¼ 25 pixels). Twenty images were acquired of the same motionless sample at equal time intervals of 1 min.
Fig. 27. Errors related to the variation of luminosity. (grid pitch ¼ 10 pixels, subset size ¼ 5 pixels).
quick way to have an idea about the measurement precision of the devices used in DIC technique and their arrangement in the workspace. The estimation of the
contribution of the different sources of errors is summarized in Table 1. The analysis of the obtained results shows that the presence of out-of-plane displacement is a major
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Fig. 28. Cartographies of errors related to luminosity variation. Mean gray level of: (a) 60; (b) 116 (reference); (c) 170 (grid pitch ¼ 10 pixels, subset size ¼ 5 pixels).
Table 1 Summary of the estimated errors related to different sources according to the proposed tests Error source
Mean error
Out-of-plane displacement Lighting Speckle pattern Subset size In-plane translation In-plane rotation Grid pitch Environment Perfect displacement
Displacement (mm) 2 103 ðmm1 Þ ½2:224 103 ½0:523:7 103 ½0:922:1 103 ½0:421:7 103 ½0:421:5 103 ½0:921:4 103 ½228 104 ½0:522:4 105
source of errors. The test shows that the errors evolve rapidly against the out-of-plane displacement. The errors related to light variation seems to be important as lighting variation leads to changes in the gray levels of the acquired images which affects the correlation accuracy. The obtained results show that the errors linked to the speckle pattern can be important if a bad speckle pattern is used. This can be seen when the correlation is not achieved all over the virtual grid. The measurement errors depend also on the grid pitch, and they are inversely proportional to the size of the grid pitch. After that come the errors related to the in-plane displacement. These errors are mainly related to the lens distortion and the quality of parallelism of the lens and the sample surfaces. Then, come the errors linked to the CCD sensor and the environment stability that shows the real limits of the used devices. The minor errors are those linked to the software algorithm. These errors depend on the interpolation and the optimization methods implemented in the correlation software. 5. Conclusion DIC technique is very attractive as it permits mapping displacement and strain fields for different types of applications (control, identification, motion detectiony). However, its setting-up needs special attention because of the length of the measure chain which introduces several
errors. In this study, the errors of strain measurement related to different sources were investigated and classified (Table 1). We propose hereafter some suggestions to improve the accuracy of DIC technique. Minimization of the out-of-plane displacement: The first thing to do is to use adequate solution to limit the out-ofplane displacement. This can be achieved by using special clumps for instance. The use of a telecentric lens, when possible, is a good solution to minimize the effect of the out-of-plane displacement [29]. An other solution consists in using stereo-correlation so that one can accede to the three components of the displacement. Lighting source: As shown by the test, the errors related to the variation of the lighting source may be important. Thus, one has to ensure a uniform and sufficient lighting covering the whole zone of interest of the sample surface during the mechanical test. Choice of the speckle pattern: The quality of the applied random pattern is very important. It has an influence on the correlation results. A bad speckle pattern leads to no correlation at some nodes of the virtual grid. It has also an influence on the accuracy of the correlation. Using qan optimized pattern that can be printed directly on the surface of the sample is an attractive solution. Thus, the obtained speckle pattern will be independent of the operator and; hence the results are more reproducible. Choice of the size of the subset: The main role of the subset is to characterize the resemblance between two regions and it depends on the speckle pattern. A subset too small is likely to give wrong correlation due to the increase of the number of local minima of the correlation coefficient defined by Eq. (1). The increase of the size of the subset enhance the quality of the correlation. The increase of the size of the subset over a given size do not present a tangible enhancement of measurements quality. One should keep in mind that the increase of the size of the subset is synonymous of the increase of computation time. So the subset must belong to a given range and one may try several sizes for the subset to find a suitable one that presents a good compromise between accuracy and computation time. It should be noticed that the size of the subset is closely linked to the granulometry of the speckle pattern.
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