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Abstract
Digital Image Correlation (DIC) is a superior optical method to measure the surface deformation with a high accuracy. Currently, most researches on DIC are based on random patterns. In this paper, A DIC/Moiré hybrid method using regular patterns is proposed for deformation measurement. In this method, a Moiré fringe technique based on correlation coefficient is developed to provide accurate initial deformation estimation for DIC. Experimental results indicate a higher computational efficiency by the proposed method than the conventional DIC method. It is also found that the calculation accuracy increases using regular patterns. The advantage of obtaining accurate initial estimation by the DIC/Moiré hybrid method may enable potential application in deformation measurements.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Many non-contact, full-field optical measurement techniques have been developed to determine surface deformation over the past years. Basically, these optical techniques could be divided into two categories according to the patterns used. One category is conducted based on regular patterns such as Moiré method [1], Moiré interferometry [2] and grid method [3], while the other employs images with random patterns, including electronic speckle pattern interferometry (ESPI) [4] and Digital Image Correlation (DIC) [5].
In recent years, DIC, based on random speckle patterns, has become the most active optical measuring technique for deformation analysis for its simple equipment and low test environment demands. It has been widely used in scientific research and engineering measurement [6,7]. At the beginning, DIC was closely related to laser speckle [8]. Currently, random patterns have been greatly used in DIC. Because each subset of the random patterns is unique, DIC essentially transforms the gray information in two patterns into relative motions between them. Researchers revealed that it is the gray gradient that enables the use of random patterns in DIC [9–11]. One interesting fact is that regular patterns also include gray gradient in each subset. In this sense, regular patterns may also be used in DIC. However, limited to the traditional point of view, only a few researchers reported on the possibility and feasibility of using regular patterns in DIC. Among them, Dr. Grédiac has done a great deal of inspiring work on the use of checkerboard images for deformation analysis [12–14].
Moiré method is a representative of the methods based on regular patterns. It uses light and dark fringes to measure deformation by superimposing two sets of translucent specimen printed with dense grid lines [15]. With the development of digital image processing technology, phase analysis technology has been applied to fringe pattern analysis. The phase shifting technology was proposed by Bruning [16] and then widely used in various fields. In recent years, the combination of phase shifting technology and digital Moiré technology has already promoted the automatic processing of Moiré fringe patterns [17].
In this paper, a hybrid optical experimental method is proposed for deformation measurement. In this method, regular patterns are applied in the calculation, while DIC and a Moiré method are combined, though they belong to different categories. This paper is organized as follows. In Section 2, we discuss the feasibility to apply regular patterns in DIC. In Section 3, a new Moiré fringe technique based on correlation coefficients is developed for displacement calculation. Section 4 presents the hybrid DIC/Moiré method, which combines DIC and the new Moiré fringe technique. In Section 5, numerical and physical experiments were conducted to compare DIC results by random and regular patterns, and to present the results by the Moiré fringe technique developed in Section 3 and the proposed hybrid DIC/Moiré method in Section 4. Section 6 presents our results and discussion. Section 7 concludes our work.
2. DIC method based on regular patterns
The principle of DIC is to determine the displacement field on the specimen surface before and after deformation by maximizing the correlation coefficient function. Generally speaking, the analysis process of DIC is divided into two steps: coarse and fine searches. But actually the coarse search is only needed for the first subset. Because according to the continuous deformation assumption, the initial deformation estimation of other subsets can be obtained from those subsets that were previously computed [18].
When random patterns are applied, the initial deformation estimation can be easily obtained because each subset is unique due to the randomly distributed speckles, as shown in Fig. 1(a). The initial deformation estimation is usually obtained by searching the subsets in the deformed image using the maximum correlation coefficient. But for regular patterns in Fig. 1(b), it would be much more difficult to estimate the initial deformation because the subsets are similar in the pattern. A translation beyond one period of regular pattern will result in an identical image and be prone to significant error. There are three possible solutions.
The first solution is to use quasi-regular patterns, as indicated in Fig. 1(c). In the figure, four random subsets are added to the corners of a regular pattern. In the estimation, the deformation of these four random subsets are first calculated. The initial deformation estimation of the other subsets is then obtained by interpolation using the deformation of the four random subsets. The fine search for every subset can therefore be conducted based on the interpolated deformation estimation.
The second solution is to use real time computation. Owing to the development of various effective algorithms to accelerate DIC computation, real time DIC computation has become available [19]. In this case, if the relative displacements between two successive patterns are less than the period of the regular pattern, the current initial deformation estimation can be obtained from the last step calculation.
The third solution is to use Moiré method to determine the initial deformation estimation of each subset. This solution will be elaborated in Section 4.
In DIC calculation, the measured displacement is not equal to the actual displacement but contains a measurement error which consists of two components: random error and systematic error. Random error is caused by sensor noise and defined as the standard deviation of the measured displacement. Many studies showed that patterns with a larger mean intensity gradient would produce a smaller random error [9–11,20]. Systematic error is caused by imperfect interpolation and image noise. It is defined as the difference between the actual displacement and the mean of the measured displacement [21]. Schreier et al. [22] recommended patterns with low-frequency content from the view of decreasing the systematic error. These patterns have small mean intensity gradient in spatial domain. But actually the root mean-squared error (RMSE) is a more comprehensive parameter to evaluate the total errors of DIC, which is a combination of random error and systematic error [23–25]. RMSE is formulated as
where u' and u0 denote the measured displacement and the actual displacement respectively, and Var(u') and E(u') represent the variance and the mean of the measured displacement. Thus, there is an optimal mean intensity gradient range to lower RMSE. Some optimal pattern parameters, including mean intensity gradient, speckle radius, and coverage ratio, are presented to obtain optimal speckle pattern [20,25–27]. However, for random patterns, these parameters are usually global indicators to assess the quality of the entire speckle pattern, but are hard to be satisfied in each local individual subset. Because the speckles are randomly scattered over a subset, the speckle positions could be regarded as random variables. Hence, even though the generation parameters are identical, each subset of the random pattern is still different. Compared with random patterns, regular patterns can be designed to meet the recommended optimal pattern parameters in each subset.3. Moiré fringe based on correlation coefficient
In traditional DIC method based on random patterns, only the maximum correlation coefficient is utilized to track the corresponding positions of subsets in the un-deformed and deformed patterns. No useful information can be obtained from other correlation coefficients, though much computing time has to be consumed. However, if regular patterns are used in DIC, all the calculated correlation coefficients contain the information of deformation. Thus, we develop a technique, Moiré fringe based on correlation coefficients (MFCC), for deformation measurement. This method is similar to traditional geometric Moiré methods. The detailed interpretation is presented as follows.
When two periodic translucent structures are superposed, there appears a new periodic structure with a larger spatial period, which is called Moiré patterns. Figure 2 schematically shows the forming process of the fringes in our method. To interpret the forming process of the Moiré fringe theoretically, the regular pattern used is approximately expressed as cosine wave with single frequency for simplicity, as indicated in Eq. (2)
where f1 and f2 denote the regular patterns before and after deformation, T is the period of the one-dimensional cosine wave, and u represents the displacement in the deformed pattern. Assuming a uniform illumination field, as shown in Fig. 2, the transmitted field through the superimposed patterns can be expressed as f1 × f2. Since the Moiré fringes observed by human eyes can be interpreted as fluctuations of average intensity in one subset of the two superposed patterns [28], an integration is conducted over one subset to simulate the low-pass filtering characteristic of human eyes. The final observed Moiré fringes can be expressed aswhere C is exactly the correlation coefficient of DIC.Supposing that the displacement u is constant in one subset and setting the period T to the half-height of the subset, Eq. (3) can be deduced as
Equation (4) presents the relationship between deformation and correlation coefficient. In MFCC method, correlation coefficient constitutes the low-frequency intensity of Moiré fringes. The high-frequency component of the regular pattern is filtered out in the process of cross-correlation operation which acts as a filter. It is apparent that for regular patterns, the correlation coefficient of non-maximum values also contains information of deformation. The displacement distribution can be derived from correlation coefficient by MFCC method.4. The DIC /Moiré hybrid method for deformation measurement
A DIC/Moiré hybrid method is proposed in this section by combining DIC method and MFCC method to measure in-plane deformation based on regular patterns. In this method, MFCC method is performed to provide accurate initial deformation estimation for following DIC calculation.
There are two steps to implement this hybrid method. The flow-process diagram of the procedure is shown in Fig. 3. In the first step, MFCC method is used to obtain the initial deformation estimation, where cross-correlation operation is carried out between un-deformed and deformed patterns to filter out the high-frequency components and form low-frequency Moiré fringe expressed by the correlation coefficient, while phase analysis is conducted to obtain continuous phase distribution of the Moiré fringes and displacement distribution can be thereby derived. The second step is basically a DIC calculation on the regular un-deformed and deformed patterns, but using the displacement distribution obtained from the first step as the initial estimation.
In general, the MFCC method can efficiently determine the initial deformation of the whole field without any iteration process. Based on the accurate initial deformation estimation, the following DIC calculation can achieve fast convergence.
5. Experiments
In this section, one numerical experiment and two physical experiments were conducted. Section 5.1 presents the numerical experiment on performance of random and regular patterns in DIC calculation. Section 5.2 presents a uniaxial tensile test on an aluminum alloy specimen. Based on the experiment, the displacement field was computed using conventional DIC with random and regular patterns, and the MFCC method respectively. Section 5.3 presents a rubber tensile experiment. The displacement field was computed using the DIC /Moiré hybrid method.
5.1. Numerical experiment on regular patterns
One numerical experiment was carried out to compare the performance of random and regular patterns on computational errors. In our experiment, Gaussian speckles were adopted because they have the advantages of slow intensity changes and small interpolation errors [23]. The speckle size used in the experiment is 4 pixels, which was designed to be within the optimal speckle size range suggested by Chen et al. [29]. In random patterns, speckles are randomly distributed, as shown in Fig. 1(a). To avoid speckle overlap, the distances between speckle centers were set as larger than 7 pixels. In regular patterns, speckles are regularly arranged, as shown in Fig. 1(b) and the distance between two neighboring speckle centers is 7 pixels. The image resolution is 800☓800 pixels. Twenty-one translated images were generated by applying sub-pixel displacements from 0 to 1 pixel with a 0.05-pixel step in the horizontal direction. To simulate the influence of noise in actual situations, a random Gaussian white noise with a mean of 0 and a variance of 2 was added into each image. A subset size is 41☓41 pixels. The DIC grid step is 10 pixels and the total number of Point of Interest (POI) is 3600. To reduce systematic error, a random offset following Gaussian distribution with a mean value of 0-pixel and a variance of 1-pixel was imposed on subset points at integer locations in the experiment [30]. In the DIC algorithm, the Newton-Raphson iteration method with zero-order shape function and cubic B-spline interpolation were employed.
5.2. Aluminum alloy uniaxial tensile test
Figure 4 demonstrates a uniaxial tensile test of aluminum alloy specimen. In the experiment, Gaussian speckles proposed by Zhou et al. [26] was adopted in speckle pattern generation. The surface of the specimen was prefabricated with the speckle pattern using the water transfer printing (WTP) method [31]. Every speckle size is 5 pixels. The coverage ratio is 50%. The speckle pattern has two parts. The left part is a random pattern and the right one is a regular pattern. Such arrangement ensures that both patterns are under the same uniform tensile deformation state. A telecentric lens (HX014-5M178) was used to eliminate the influence of the out-of-plane displacement. Images with a 1600☓1000 resolution and 256 gray levels were captured by a CCD camera (Basler acA2040-25gm). One image captured in the experiment is shown in Fig. 4. Real time computation strategy was adopted in the experiment. The u displacement fields were calculated at 5,500 locations using the N-R method with a DIC grid step of 10 pixels.
5.3. Rubber uniaxial tensile test
A uniaxial tensile test of a rubber specimen is performed to verify the DIC/Moiré hybrid method proposed in Section 4. The experimental setup is shown in Fig. 5. The dumbbell-shaped specimen was made from molded rubber. The surface of the rubber specimen was prefabricated with the same regular pattern as detailed in Section 5.2. It was revealed that such prefabrication could ensure well adhesive performance and stability even under large deformation state [31]. On the left side of the regular pattern, a small random subset was added to compensate the translation of the specimen in tensile deformation. In the experiment, two circular cold lights were used to provide enough uniform lighting.
6. Results and discussion
6.1. Regular pattern vs. random pattern
Performance of regular and random patterns can be evaluated by the computational errors of DIC calculation. Figure 6 shows the random error, systematic error, and the total error (RMSE) obtained from the numerical experiment. The results are displayed as boxplots to show the detailed statistical results. In each boxplot, the top and bottom of the box are the third and first quartiles, and the band inside the box is the second quartile. The ends of the whiskers represent the minimum and maximum of each error data group.
Fig. 6 Boxplots of random error (RE)、systematic error (SE) and total error (TE) of random pattern and regular pattern in DIC method.
As shown in Fig. 6, the random error of the regular pattern is apparently smaller than the random pattern, which is in consistent with our analysis in Section 2. Even though the reduction in the systematic error of regular pattern is not apparent, a significant improvement in the total error could still be observed for the regular patterns. This result indicates that regular patterns are superior to random patterns in the respect of computation errors.
Comparison between random and regular patterns is also conducted on the displacement calculation in the aluminum alloy uniaxial tensile test. Figure 7 shows the displacement curves of the random and regular patterns. The displacement curves are generated by averaging the displacement field in the horizontal direction. It is indicated in the figure that both curves are well consistent with the pre-added linear deformation distribution. In Fig. 7, a slight difference is also observed in the slopes of the two curves. This difference could not be totally avoided because it is hard to make the lens and the specimen perfectly parallel. Our experimental result indicates a difference in the slopes of 1.3e-5.
Fig. 7 Displacement curve averaged in horizontal direction and the variance of random and regular pattern.
The two blocks in Fig. 7 demonstrate the standard deviation between the original displacement and the fitted value of the random and regular patterns, respectively. The standard deviation corresponds to the random error in the numerical experiment. Figure 7 shows that the standard deviation of the regular pattern is smaller than the random pattern. That is, the regular pattern provides a more stable displacement distribution than the random pattern.
6.2. MFCC method
The strain of the aluminum alloy specimen in the uniaxial tensile test was also obtained by MFCC. Figure 8(a) illustrates the contour map of the correlation coefficient formed by MFCC. As expected, the fringes are nearly regularly spaced. Under the pre-added tensile deformation, an evolution of correlation coefficient is clearly visible in the figure. Figure 8(b) shows the correlation coefficient curve averaged along the horizontal direction. The curve is approximated as a segment of cosine function, which is in accordance with our conclusion in Eq. (4). The displacement in the field of view was derived by fitting the correlation coefficient curve using Eq. (4) and the strain of the pre-add displacement field was finally obtained as 1.67e-4, which is very close to the result 1.47e-4 by DIC method. It is also exciting to find that the computational time by MFCC is shortened compared with DIC since multiple non-linear iterative optimization and repeated sub-pixel interpolation are no longer needed in MFCC.
Fig. 8 MFCC results: (a) typical contour map and (b) the curve averaged along horizontal direction of correlation coefficient in MFCC.
6.3. DIC/Moiré hybrid method
The DIC/Moiré hybrid method was used in displacement calculation in the rubber tensile test. The calculation result obtained by MFCC method was used as the initial deformation estimate for the following DIC calculation. Figure 9(a) shows the Moiré fringe obtained by MFCC method. Along the horizontal direction, displacements of integral period occur at the maximum correlation coefficient. Figure 9(b) shows the fluctuation of the correlation coefficient in the vertical direction. The fluctuation is resulted from the deformation caused by Poisson effect.
Figure 10 presents the displacement calculation results by the DIC/Moiré hybrid method. Figures 10(a) and 10(b) show the u and v displacement respectively. The initial estimates by MFCC method are also presented in the figures. It can be observed from the figures that the initial deformation estimation by MFCC is very close to the final result. In our calculation, rapid convergence was achieved with only one iteration. The computational time is therefore shortened compared with conventional DIC, where three or more iterations are usually needed before convergence.
Fig. 10 Displacement obtained by the DIC/Moiré hybrid method: (a) averaged u displacement. (b) averaged v displacement.
7. Conclusion
In this paper, a DIC/Moiré hybrid method has been proposed to measure deformation using regular patterns. In this method, a new Moiré fringe technique based on correlation coefficient (MFCC) has been developed to estimate the initial deformation for the following DIC calculation. Displacement calculation for a rubber specimen under a uniaxial tensile loading indicates that the use of MFCC could improve the calculation efficiency of the following DIC. Application of regular patterns in DIC has also been proved to be feasible and advantageous through theoretical analysis and experimental verification.
Funding
National Natural Science Foundation of China (11572217, 11572218, 11602167, 11772222, 11472186, and 11602166); Natural Science Foundation of Tianjin (18JCQNJC03500, 17JCQNJCO4800, and 16JCYBJC40500).
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