## Abstract

Digital Image Correlation (DIC) has been established as a flexible and effective technique to measure the displacements on specimen surface by matching the reference subsets in the undeformed image with the target subsets in the deformed image. With the existing DIC techniques, the user must rely on experience and intuition to manually define the size of the reference subset, which is found to be critical to the accuracy of measured displacements. In this paper, the problem of subset size selection in the DIC technique is investigated. Based on the Sum of Squared Differences (SSD) correlation criterion as well as the assumption that the gray intensity gradients of image noise are much lower than that of speckle image, a theoretical model of the displacement measurement accuracy of DIC is derived. The theoretical model indicates that the displacement measurement accuracy of DIC can be accurately predicted based on the variance of image noise and Sum of Square of Subset Intensity Gradients (SSSIG). The model further leads to a simple criterion for choosing a proper subset size for the DIC analysis. Numerical experiments have been performed to validate the proposed concepts, and the calculated results show good agreements with the theoretical predictions.

© 2008 Optical Society of America

## 1. Introduction

Digital Image Correlation (DIC) [1] technique has been established as a flexible and effective optical metrology tool for deformation measurement in the field of experimental mechanics. With the DIC technique, the full-field deformation of a planar specimen surface can be obtained through processing the grayscale digital images of the specimen surface captured before and after deformation. The DIC analysis program or software is generally developed based on the well-established algorithms that use either subset-based cross-correlation (CC) criteria or sum-squared difference (SSD) correlation criteria [2].

In practice, the measurement accuracy of DIC can be affected by many factors, such as sub-pixel optimization algorithm, subset shape function, subset size, sub-pixel intensity interpolation scheme, image noise as well as camera lens distortion. The effects of some of these factors on displacement measurements have been thoroughly investigated by many researchers. For example, Schreier and Sutton, *et al.*, studied the systematic errors in DIC caused by undermatched shape functions [3] and intensity interpolation [4], and the Pan, *et al.*, studied the performance of sub-pixel registration algorithms on displacement estimations [5]. Recently, Zhang, *et al.*, [6] and Yaneyama, *et al.*, [7] investigated the effect of lens distortion on displacement measurements in DIC.

In addition to the numerous custom DIC program or software developed in various laboratories, a few powerful commercial DIC software packages with easy-to-use interface and high processing speed are available in the market, such as the Vic-2D/3D system [8]. Nevertheless, whether custom or commercial DIC software packages are used, the users must manually select a subset size varying from several pixels to even more than a hundred pixels before the DIC analysis. Since the subset size directly determines the area of the subset (sub-image) being used to track the displacements between the reference and target subsets, it is found to be critical to the accuracy of the measured displacements.

To achieve a reliable correlation analysis in DIC, the size of a subset should be large enough so that there is a sufficiently distinctive intensity pattern contained in the subset to distinguish itself from other subsets. This means that a large subset size is desirable. On the other hand, however, it is noted that the underlying deformation field of a small subset can be readily and accurately approximated by a first-order or second-order subset shape function, whereas a larger subset size normally leads to larger errors in the approximation of the underlying deformations. For this reason, to guarantee a reliable displacement measurement, a small subset size is preferable in DIC. The above two conflicting demands imply that there is a trade-off between using large and small subset sizes.

In actual applications, the issue of subset size selection becomes more complicated. For instance, Sun, *et al.*, pointed out that the subset size should have a lower limit in order to suppress the influence of random noise [10]. Another notable example is that since the speckle patterns prepared on the same object through spraying paints by different people may demonstrate different grayscale distribution characteristics such as different image contrasts and speckle sizes, the DIC measurements conducted by different people may yield substantially different results even though the specimen, the actual deformation state and the subset size are all identical. From above descriptions, it can be seen that how to select a proper subset size for various speckle patterns in practice is important but confusing.

Recently, Wang, *et al.*, [9] investigated the effect of intensity pattern noise on the displacement measurement precision of DIC using self-correlation experiments. In their study, a concise theoretical model is deduced from a one-dimensional (1D) cross-correlation function, and the model is capable of predicting the achievable measurement accuracy of DIC. In spite of the advances, the mathematical derivation involved in their work is extremely complicated and cumbersome, and the work is limited to 1D case only.

In this paper, a two-dimensional (2D) theoretical model of displacement measurement accuracy of DIC is proposed. The model is established on the SSD correlation criterion and it is assumed that the adopted shape function can accurately describe the underlying local deformation field and the gray intensity gradients of image noise are much lower than that of speckle image. The theoretical model clearly indicates that the displacement measurement accuracy of DIC can be quantitatively predicted based on the known image noise variations and Sum of Square of Subset Intensity Gradients (SSSIG). It is also noteworthy that the 1D theoretical model presented in Ref. [9] can be readily deduced from the proposed 2D theoretical model. In addition, the 2D model further enlightens the authors of this paper to propose an algorithm of how to choose a proper subset size for different speckle patterns in DIC analysis. In the proposed algorithm, a novel parameter named SSSIG is adopted as a key parameter for the proper selection of a subset size. In order to validate the correctness of the theoretical model and the effectiveness of the proposed algorithm, numerical experiments using actual speckle patterns with different grayscale characteristics have been performed, and the results show good agreements with the theoretical predictions.

## 2. Fundamental principle of DIC

In general, DIC uses the random intensity distributions on the specimen surface to obtain the full-field surface displacements by matching the subsets of interest before and after deformation. As shown schematically in Fig. 1, a subset of (2M+1)×(2M+1) pixels centered at point (*x*
_{0}, *y*
_{0}) from the reference image is chosen and used to determine its corresponding location in the deformed image. In routine practice of DIC, a CC criterion or SSD correlation criterion [2] is predefined to evaluate the similarity between the reference subset and the target subset. The matching procedure is carried out through searching the peak position of the distribution of correlation coefficients. Once the maximum or minimum (dependent on the correlation function used) correlation coefficient is detected, the target subset can then be directly determined. The differences of the positions of the reference subset center and the target subset center yield the in-plane displacements *u* and *v* at point (*x*
_{0}, *y*
_{0}). More details of the DIC technique can be found in Ref. [11].

In existing DIC techniques, a subset of (2M+1)×(2M+1) pixels is chosen manually. As stated in the previous section, the subset size is critical to the displacement measurement accuracy of DIC, so it should be properly selected according to the actual random intensity distributions of the speckle patterns as will discussed in the following sections.

## 3. Displacement measurement accuracy of DIC

For the convenience of analysis, the following SSD correlation function is used for deriving a theoretical model of the displacement measurement accuracy of DIC. The SSD correlation function is defined as

where *f*(*x*, *y*) is the grayscale intensity value at point (*x*, *y*) of the reference image, and *g*(*x*′,*y*′) is the grayscale intensity value at point (*x*′, *y*′) of the deformed image. The coordinates (*x*, *y*) and (*x*′, *y*′) are related to the so-called subset shape function [3], which is utilized to approximate the underlying displacement field.

Although several kinds of shape functions are available, a zero-order shape function (pure in-plane translation) is employed here for the following mathematical derivation due to its simplicity and convenience.

where *u*
_{0} and *v*
_{0} are the integral pixel displacements in *x* and *y* directions, which can be accurately determined with ease and are assumed as known values; ^{Δ}u and ^{Δ}v are the corresponding sub-pixel displacements in *x* and *y* directions, respectively.

After neglecting the high-order terms, the first-order Taylor expansions of *g*(*x*′,*y*′) yields

$$=g(x+{u}_{0},y+{v}_{0})+\Delta u\xb7{g}_{x}(x+{u}_{0},y+{v}_{0})+\Delta v\xb7{g}_{y}(x+{u}_{0},y+{v}_{0})$$

where *g _{x}* and

*g*are the first-order derivatives of grayscale intensities, and in this study they are calculated by central difference of neighboring points in

_{y}*x*and

*y*directions, respectively.

Minimization of the SSD correlation function would provide the best estimate of the desired sub-pixel displacements ^{Δ}u and^{Δ}v. Thus we have

The solution of Eq. (4) can be expressed in the following closed form:

By rearranging Eq. (5), we get

Because the images captured from the image acquisition system inevitably contain various kinds of noise such as thermal noise, readout noise, and shot noise, the sub-pixel displacement calculated from Eqs. (6) and (7) will deviate from their actual values. To incorporate the effects of noise into the theoretical derivation, the image noise, *η*
_{1} and *η*
_{2}, are considered as random and are additively added to the images. Thus, the grayscale intensity distribution of the noisy images can be written as:

Then, by replacing *f* and *g* in Eqs. (6) and (7) with *f*′ and *g*′, respectively, the following equations can be obtained,

With *η*(*x*, *y*)=*η*
_{2}(*x*, *y*)-*η*
_{1}(*x*, *y*). Similar to the Ref. [9], two assumptions are made about the random noise *η*(*x*, *y*): First, the mean value of the noise is a constant, that is, *E*(*η*)=*const*; Second, the noise values at different pixels are independent.

Normally, the gray intensity gradient of the speckle image is much larger than that of the image noise; consequently, the effect of random image noise on the value of subset gradients can be neglected. With this assumption, we have

Since the noise of each pixel is random, the computed sub-pixel of each point is also a random variable. If the mathematical expectation of the noise is a constant, the mathematical expectations of Δ*u*′ and Δ*v*′ will be identical to those of Δ*u* and Δ*v*. Hence, we get

Equation (13) denotes that the mathematical expectations of the measured displacements Δ*u*′ and Δ*v*′ are not affected by the image noise, and this conclusion has been verified in the work of Wang, *et al.*, [9]. Further, with both numerators and denominators of Eqs. (11) and (12) divided by (∑∑(*g _{x}*)

^{2}∑∑(

*g*)

_{y}^{2})

^{2}, the variances of Δ

*u*′ and Δ

*v*′ can be derived and expressed as

where *D*(*η*) is the variance of image noise, ∑∑(*g _{x}*)

^{2}and ∑∑(

*g*)

_{y}^{2}are the Sum of Square of Subset Intensity Gradients (SSSIG) in

*x*and

*y*direction, respectively.

*H*is of form:

Here,
${C}_{{g}_{x},{g}_{y}}^{2}$
is the square of cross-correlation coefficient between *g _{x}* and

*g*, defined as

_{y}For random speckle patterns,
${C}_{{g}_{x},{g}_{y}}^{2}$
approximates to zero, as will be shown in the following section. That is to say, *H* approximates to the value of 1, so it can be neglected in Eqs. (14) and (15).

Besides, by assuming *g _{x}*(

*x*+

*u*

_{0}),

*y*+

*v*

_{0})≈

*f*(

_{x}*x*,

*y*),

*g*(

_{y}*x*+

*u*

_{0},

*y*+

*v*

_{0})≈

*f*(

_{y}*x*,

*y*) in Eqs. (14) and (15), the Standard deviation (SD) error of the displacement measurement of DIC can be further expressed as:

It is noted that the governing equations, i.e., Eqs. (18) and (19), are very similar to the Eq. (25) presented in the work of Wang, *et al.*, [9]. The differences are as follows: First, unlike the previous work which is valid for 1D only [9], Eqs. (18) and (19) in this paper represent a 2D description. Second, it can be seen from the denominators in Eqs. (18) and (19) that a novel parameter, namely, SSSIG rather than the sum of square of the grayscale level differences appears. Actually, if a forward difference algorithm is used for computing the intensity gradients, Eqs. (18) and (19) will have identical form to the Eq. (25) of Ref. [9].

It is evident from Eqs. (18) and (19) that the displacement measurement accuracy of DIC is determined by the variance of image noise and SSSIG. Figure 2 shows a plot of the SD errors for the displacements as a function of SSSIG according to Eqs. (18) and (19).

## 4. Selection of subset size

Equations (18) and (19) further provide a criterion for subset size selection in DIC analysis. As can be seen from Fig. 2, the displacement measurement accuracy can be quantitatively predicted if the variance of image noise and SSSIG are known. For certain types of cameras, the image noise can be accurately quantified by comparing the images captured at different time intervals. Generally, the mean value and the variance of image noise are fixed values, thus, the displacement measurement accuracy can be controlled by adjusting the SSSIG, which can be increased or decreased by adjusting the subset size. This is the fundamental idea of the proposed algorithm for subset size selection in DIC analysis.

A flowchart of the proposed algorithm is illustrated in Fig. 3. First, the threshold of SSSIG can be calculated based on the desired accuracy according to Eqs. (18) and (19). Then, beginning with a small subset size (e.g., 11×11pixels), the algorithm calculates the SSSIG of the subset and compares it with the threshold. If the SSSIG is less than the threshold value, the algorithm will increase the subset size by 2 pixels and obtain an updated SSSIG. The cycle repeats until the SSSIG becomes larger than the threshold value, which indicates that a proper subset size has been detected. After that, a common DIC analysis procedure is performed to calculate the displacements at the current point.

## 5. Experimental validations

#### 5.1 Numerical experiments

To better evaluate the proposed concepts and eliminate the possible errors caused by the image acquisition system (e.g., the camera lens distortion), imperfect loading, fluctuation of illumination during image capture, etc., numerical experiments were utilized in this study to validate the correctness of the proposed theoretical model for accuracy prediction and the effectiveness of the proposed algorithm for subset size selection. In the experiments, three sets of test speckle image pairs with an image size of 768×576 pixels and 256 gray levels are used. The speckle patterns were acquired by randomly spraying black and white paints on flat specimen surfaces. Figure 4 shows the reference images of each test speckle image pair, and it can be seen that the grayscale distributions of various speckle patterns are distinctly different. To simulate the image noise, random Gaussian noise with variance of 4, i.e. *D*(*η*)=4, was added to the speckle images to form the deformed images.

The displacements were calculated at regularly distributed 2400(=60×40) points by optimizing the SSD correlation criteria as illustrated in Eq. (1) using Newton-Raphson method for each test image pair with subset sizes ranging from 17×17 to 71×71 pixels at an increment of 6 pixels. The distance between neighboring points is chosen to be 10 pixels. Since the actual displacement is zero, the SD error of the measured displacement can be defined as:

where *d _{i}* is the displacement component

*u*or

*v*of point

*i*(i=1, 2, …, 2400). The SD error reflects the deviation of the measured displacements from their mean value.

For the above three reference speckle patterns, we calculate the
${C}_{{g}_{x},{g}_{y}}^{2}$
of each point using different subset size. Table 1 presents the calculated value of
${C}_{{g}_{x},{g}_{y}}^{2}$
for six different points of speckle image A, B and C using subset of 11×11 pixels. It is found that
${C}_{{g}_{x},{g}_{y}}^{2}$
always approximates to zero. Thus, it is reasonable to remove *H* from Eqs. (14) and (15), since H approximates to the value of 1.

#### 5.2 Effect of subset size on displacement measurement

Figure 5 shows the SD errors of the *u*-displacement and the *v*-displacement for the three test image pairs with subset sizes ranging from 17×17 pixels to 71×71 pixels. It is obvious that, for all the three test image pairs, the SD errors decrease as the subset size increases. According to Eqs. (18) and (19), the improvement of displacement measurement accuracy can be attributed to the increase of SSSIG, when a larger subset size is used for the calculation.

It can be also observed from Fig. 5 that the SD errors of the displacements calculated from test image pairs C are much larger than those obtained from test image pairs A and test image pairs B, even if subsets of the same size are employed for the three cases. This can be explained by the fact that the image contrast of the speckle images in pairs C is much lower than those of the speckle images in pairs A and pairs B; therefore, even for the same subset size, the subset’s SSSIG from image pair C is much smaller than those from the image pair A and image pair B. For this reason, the results shown in Fig. 5 are reasonable.

#### 5.3 Effect of SSSIG on displacement measurement

Figures 6(a)–6(c) show the SD errors of the *u*-displacement and the *v*-displacement for the three test speckle image pairs with subset sizes ranging from 11×11 pixels to 71×71 pixels at an increment of 6 pixels. It is clear that the SD errors decrease as SSSIG increases in all the three cases. In contrast with Fig. 5, it is noted that the relationship between the SD errors and SSSIG is independent of the image contrast. Moreover, the experimental results are in good agreement with the theoretical results. Therefore, the SSSIG, instead of the subset size itself, is the determinant parameter that directly affects the displacement measurement accuracy.

Based on Eqs. (18) and (19), we can also obtain the relationship between the displacement deviation and the SSSIG for each point. From the results plotted in Figs. (7), (8), and (9), it is clear that the deviation of the measured displacement decreases as the SSSIG increases.

#### 5.4 Selection of subset size

In this section, the proposed algorithm for subset size selection is used to calculate the displacements for the three test image pairs. The threshold of SSSIG in the calculation is set to 1×10^{5}. Based on the image noise variance of 4 and Eqs. (18) or (19), the theoretical SD error is determined as 0.007 pixels. The results obtained by the proposed algorithm are listed in Table 2. It can be seen from the table that the calculated displacements are in good agreement with pre-assigned theoretical ones, and the SD errors of the *u*-displacement and *v*-displacement also match well with the theoretical value.

Additionally, Table 2 shows that in order to obtain an identical accuracy, the average subset size is different for each of the three test image pairs. Specifically, with a SSSIG threshold of 1×10^{5}, the average subset size is 12 pixels, 14 pixels, and 36 pixels respectively for test image pairs A, B, and C. This reveals that a small subset can lead to a high accuracy of displacement measurement for images with larger contrast. It is also noteworthy that because the values of SSSIG in *x* direction and *y* direction can not be precisely controlled through increasing the subset size, the average SSSIG of the optimal subset size for each of the six cases is different, as can be seen from Table 2. This explains why the STD errors of the displacements are not exactly equal to each other.

## 6. Discussion and Conclusion

More recently, Sun and Pang [10] introduced a concept of subset entropy for optimal selection of subset size for the DIC technique. In their work, the subset entropy is defined as the sum of absolute difference of eight neighboring pixels, which is related to the SSSIG of the proposed method to a certain extent. However, the subset entropy concept is just based on an intuitive idea and it lacks mathematical support. In comparison with the subset entropy, we believe that the SSSIG proposed in this paper is more straightforward and easy to be accepted by the users of DIC.

In DIC analysis, the subset size specifies the sizes of the local reference and target subsets between which the displacements are sought. The subsets must contain enough unique and identifiable features to achieve a reliable and accurate displacement determination. It is well known that the subset size has significant effects on the accuracy of displacement measurement in DIC. In this paper, the problem of subset size selection for speckle patterns is investigated, and a corresponding simple algorithm is proposed based on the subset SSSIG, a novel parameter defined in this paper. The presented algorithm is capable of detecting the proper subset size for various speckle patterns in DIC analysis, which the authors hope to provide a helpful guideline to the real applications of DIC. However, it should be noted that the theoretical model is derived with the assumption that the gray intensity gradient of the speckle image is much larger that that of the image noise. Fortunately, for normal speckle patterns this assumption is always well satisfied.

Additionally, based on the presented study, it is also clear that the displacement measurement accuracy of DIC can be effectively improved through the following two ways:

- Increasing the subset SSSIG. In addition to using larger subsets when applicable, the subset SSSIG can also be increased by using a high bit-depth (e.g., 12-bit or 16-bit) CCD camera or increasing the contrast of the speckle patterns.
- Decreasing the image noise. Image noise can be reduced or suppressed through the use of high-performance hardware, such as a cooled CCD and a high-quality camera lens. In addition, frame averaging during image acquisition is another commonly used effective scheme to alleviate the influence of image noise.

## Acknowledgment

The work is supported by the National Basic Research Program of China (under grant 2004CB619304), the National Natural Science Foundation of China (under grants 10732080, 10625209 and 10472050), Beijing Natural Sciences Foundation (under grant 3072007), and the Program for New Century Excellent Talents (NCET) in University, Chinese Ministry of Education (under grant NCET-05-0059).

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