1. Introduction

As a powerful and flexible tool for surface deformation measurement, digital image correlation (DIC) [1,2,3] has been widely accepted and commonly used in the field of experimental mechanics. With the DIC technique, the full-field displacements with sub-pixel accuracies [4,5] and full-field strains can be directly computed through processing the grayscale digital images of the specimen surface recorded before and after deformation. More recently, the methodologies of existing 2D DIC technique for displacement and strain measurements are systematically reviewed in a review paper written by the first author of this paper [3]. It is noteworthy that the same image processing technique has also been used in the three-dimensional digital image correlation (3D DIC) method [6] and speckle projection method [7] to determine the disparity data or the in-plane motions of speckle granules encoded in the digital images recorded at different configurations, and subsequently the detected image motions (in units of pixels) can be converted to the desired 3D full-field displacement field or the 3D shape of the test object surface.

In principle, DIC is a simple and easy-to-implement optical metrology based on digital image processing and numerical computing. The basic principle of the standard subset-based DIC method is to match (or track) the same physical points located in the reference (or source) image and deformed (or target) image. In reality, while implementing the DIC technique, a region of interest (ROI) must be specified or defined first in the reference image before correlation analysis. Normally, the regularly spaced pixels within the ROI, instead of all the pixels within the ROI, are considered as points to be computed and analyzed. To calculate the displacement components of each point of interest, a square subset of (2M +1) × (2M+1) pixels centered at the current point is selected and used to determine its corresponding location in the deformed image. Subsequently, the matching is completed through optimizing the predefined correlation criterion. Although the technique is simple in principle and implementation, existing DIC technique really has several evident deficiencies. For example, it is evident that the subsets surrounding the points located near or at the boundaries of the specified ROI may contain unwanted or foreign pixels from other regions. Figure 1 shows an example of image correlation computation of a dog-bone specimen acquired in a simple uniaxial tensile test, and it is clear that the subsets centered at boundary points (e.g., the two squares highlighted in red) contain unwanted pixels of background image. In such a case, the local deformation distributions within these subsets are discontinuous and thus cannot be approximated with the commonly used continuous displacement mapping function. As a result, erroneous displacement calculations with relatively very low cross-correlation coefficient occur at these locations. In the existing standard DIC method, in order to achieve reliable and accurate measurement, these boundary points are either intentionally excluded from calculation or automatically discarded after calculation by use of a global threshold applied to the computed cross-correlation coefficients. In this sense, the existing subset-based DIC technique cannot be taken as a genuine full-field deformation methodology, as the deformation information of the boundary points of ROI is absent. Nevertheless, the deformation information around a discontinuous boundary (e.g., a crack, notch or hole) is generally of great importance and interest in many applications such as residual stress/strain measurement, facture mechanics as well as finite element analysis. So far, to the author’s best knowledge, there still lacks a simple but effective technique to tackle the boundary points using the standard subset-based DIC method. A genuine full-field DIC algorithm seems highly demanded.

 

Fig. 1. An example of correlation calculation using conventional subset-based DIC algorithm: the two representative subsets surrounding the boundary points of the specified ROI contain unwanted points of background intensity.

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Apart from the subset-based DIC method, it is noted that some other techniques, for instance, the pointwise DIC method based on a genetic algorithm developed by Jin and Bruck [8], the continuum DIC method based on B-spline deformation function proposed by Chen et al., [9] and the DIC method based on finite element method presented and improved by Sun et al., [10] and Hild et al., [11–13] may be able to determine the displacement of each pixel within a selected ROI. However, these techniques have some inherent limitations and are less practical than the subset-based DIC technique. For example, the pointwise DIC algorithm optimizes the displacements of all pixels at a time using the genetic algorithm. Before the optimization, the initial guess of displacement parameters of each pixel must be provided using a subset-based DIC algorithm. Thus, the implementation of the pointwise DIC algorithm is complicated and time-consuming. Moreover, as for the continuum DIC method with B-spline function, as indicated in Ref. [9], it normally produces significant errors in the boundary sections due to a lack of neighboring sub-regions to impose positional and derivative constraints on the “free boundaries”.

In addition to the boundary point issue, irregular ROIs, rather than rectangular ROIs, are often defined to approximate the complex shapes of the specimens in many practical experiments. It is well-known that the DIC technique based on the most widely used Newton-Raphson (NR) algorithm requires an accurate initial guess to converge rapidly and reliably [14,15]. In existing DIC technique, the determined deformation vector of current point is used as the initial guess of the next point. This straightforward initial guess transfer scheme can be easily carried out for a simple rectangular ROI. However, for an irregular ROI defined for an object comprising geometric discontinuities, the initial guess transfer between neighboring points will be a challenging issue to ensure reliable and accurate measurements [16]. This paper aims to effectively cope with the boundary point issue as well as the irregular ROI issue with a simple yet universal approach.

In this work, to deal with the points near the boundaries of the specified ROI. A binary mask M_v for identifying the valid points is constructed after the ROI is specified. All the pixel points within the ROI are considered as valid points and labeled as 1 in the mask M_v. For other pixel points outside the ROI but within each square subset centered at boundary points, the proposed approach automatically identifies the valid and invalid points during calculation; subsequently, only the labeled valid points are used in correlation analysis and invalid points are excluded. For this reason, a subset centered at a point close to or located at the ROI boundaries may contain less than (2M +1) × (2M+1) valid pixels. Technically, the calculation for a square subset with fewer pixel points is equivalent to the one for a subset of irregular shape rather than square shape. When the number of valid points contained in a subset is less than (2M +1) × (2M+1) but much more than the number of unknown parameters, the deformation parameters can still be accurately computed, as will be shown later. Additionally, to cope with irregular ROI, a reliability-guided digital image correlation (RG-DIC) method [16] is employed. In the technique, the calculation begins with a selected seed point (or starting point) within the ROI, and is then guided by the cross-correlation coefficients of computed points. This scanning strategy can ensure a successful correlation and displacement calculation at the boundary points of a regular or irregular ROI.

The rest of this paper is organized as follows. In section 2, the basic principle of DIC is briefly described, a modified ZNSSD criterion is defined for the correlation analysis of boundary points and a scanning strategy guided by the correlation coefficient of computed points is employed to ensure successful initial guess transfer between consecutive points. In section 3, the performance of the proposed method is verified by processing the experimental images of two specimens with complex shape. In section 4, we conclude the paper.

2. Genuine full-field deformation measurement with the proposed DIC technique

2.1 Principles of standard subset-based digital image correlation

As mentioned previously, in practical implementation of DIC, a ROI in the reference image must be specified first and further divided into evenly spaced virtual grids. The displacements are computed at each point of the virtual grids to obtain the full-field deformation. The basic principle of the subset-based DIC is schematically illustrated in Fig. 2. To track the same point (or pixel), a square reference subset of (2M +1) × (2M+1) pixels centered at the current point P(x ₀, y ₀) from the reference image is chosen and used to find its corresponding location in the target image. Once the location of the target subset in the deformed image is found, the displacement components of the reference and target subset centers can be determined.

 

Fig. 2. Basic principle of subset-based DIC method: tracking the same pixel point in the reference and deformed image yields its displacement vector

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To obtain accurate estimation for the displacement components of the same point in the reference and target images, the following Zero-mean Normalized Sum of Squared Differences (ZNSSD) criterion [17], which is insensitive to the scale and offset of illumination lighting fluctuations, is utilized to evaluate the similarity of reference and target subsets:

where f(x, y) is the gray level intensity at coordinates (x, y) in the reference subset of the reference image and g(x’, y’) is the gray level intensity at coordinates (x’, y’) in the target subset of the deformed image, $f_m=\frac{1}{{\left(2M+1\right)}^2}\sum _{x=-M}^M\sum _{y=-M}^M\left[f\left(x,y\right)\right]$ and $g_m=\frac{1}{{\left(2M+1\right)}^2}\sum _{x=-M}^M\sum _{y=-M}^M\left[g(x\prime ,y\prime )\right]$ are the mean intensity values of reference and target subsets, respectively. p denotes the desired deformation vector.

It is worth noting that the ZNSSD correlation coefficient is actually related to the commonly used Zero-mean Normalized Cross-Correlation (ZNCC) coefficient according to the following equation [17]:

Although the ZNSSD and ZNCC coefficients have a direct linear relation, the optimization of Eq. (1) is much easier that Eq. (2). However, because the ZNCC coefficient falls into a range of [-1, 1] whereas the ZNSSD coefficient has a range of [0, 4], the ZNCC coefficient is more straightforward to show the similarity between the reference subset and target subset. In practice, the following measure is recommended: with Eq. (1), the ZNSSD correlation coefficient is optimized (or minimized) using Newton-Raphson (NR) method to determine the displacement components of each calculation point; then, the ZNSSD coefficient is converted to the ZNCC coefficient according to the rightmost side of Eq. (2). The ZNCC coefficient will be used as a reliability index to guide the correlation calculation in the RG-DIC technique described later.

In Fig. 1 the point Q(x, y) in the reference subset can be mapped to the point Q’(x’, y’) in the target subset according to the so-called “displacement mapping function”. If the subset is sufficient small, the subset deformation pattern can be well approximated with the commonly used first-order displacement mapping function [14]

In certain cases, the subset may undergo heterogeneous deformation. Then, a second-order displacement mapping function [17,6,7], which is capable of approximating more complicated deformation of the deformed subset, should be used and will provide more accurate measurement.

Minimization or optimization the ZNSSD criterion defined by Eq. (1) involves solving a non-linear function of six (or twelve) unknown parameters depending on the displacement mapping function adopted. Equation (1) can be optimized to get the desired in-plane displacement components in the x and y directions using the following classic NR iteration method [14].

where p ₀ is the initial guess of the solution, which can be achieved through a simple integer pixel displacement searching process or other approach [3]; p is the next iterative approximation solution; ∇C(p ₀) is the gradients of correlation coefficient and ∇∇C(p ₀) is the second-order derivative of correlation coefficient, also known as Hessian matrix, which can be further approximated to simplify the calculation without loss of accuracy [15].

The NR algorithm has been shown as one of the optimal algorithms for deformation parameters determination with superior stability and highest measurement accuracy [4]. Since deformation and rotation are included into the subset shape functions, one significant advantage of the DIC technique with NR algorithm is that it is insensitive to the deformation and rotation of the deformed image. However, as a nonlinear optimization algorithm, the NR algorithm requires an accurate initial guess to converge rapidly and accurately, where the convergence radius is estimated smaller than a few pixels as proved by Vendroux and Knauss [15]. Additionally, to speed up the calculation and also maintain its reliability, the computed deformation vector of the current point is normally used as the initial guess of the next point according to the continuous deformation assumption. Consequently, the transfer of initial guess between consecutive points is a key issue to achieve a successful computation for the NR algorithm.

2.2 Modified correlation criterion for boundary points

As mentioned earlier, the subsets centered at the points near the boundaries of ROI may contain less than (2M +1) × (2M+1) pixels as schematically illustrated in Fig. 3. Because the valid points within ROI have already been labeled as 1 in a binary mask M_v after specifying the ROI, they can be easily identified and therefore used in the following calculation. Accordingly, the following modified ZNSSD criterion, slightly different from Eq. (1), is defined for the boundary points to evaluate the similarity between the reference and target subsets:

where S denotes the set of valid points within the interrogated subset, $f_m=\frac{1}{N_v}\sum _{x,y\in S}\left[f\left(x,y\right)\right]$ and $g_m=\frac{1}{N_v}\sum _{x,y\in S}\left[g(x\prime ,y\prime )\right]$ are the mean intensity values of the reference and target subsets, respectively. N_v is the number of valid points within the subset centered at boundary points.

Using the same NR algorithm described above, the modified ZNSSD correlation coefficient can be optimized to minimum to determine the desired deformation parameter vector p provided that the number of valid points within the subset is larger than the number of desired unknowns (this can be always satisfied in real cases). Because the pixels used for correlation coefficient optimization is less than (2M +1) × (2M+1) pixels, the accuracy and precision of the determined displacements is normally a little lower than that computed with a full square subset. However, we should note that the measurement precision can be improved by automatically adjusting the subset size using the technique proposed in our recent work [19,20].

 

Fig. 3. Example of tracking a subset centered at a boundary point

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2.3 Reliability-guided digital image correlation

In the existing standard DIC algorithm, a ROI is defined and the computation generally starts from the upper left point of the ROI. Then, the correlation calculation is carried out point by point along each row or column as schematically shown in Fig. 4. According to the continuous deformation assumption, the determined deformation parameters of the former point are used as the initial guess of the current point. This simple scanning strategy is normally quite effective and very easy to implement when a simple rectangular ROI is defined. However, in many practical applications, the test specimen may have complex shape containing various geometric discontinuities (e.g., cracks, notches and complex cut-outs). In these cases, an irregular ROI, rather than a rectangular ROI, is normally specified in the reference image.

 

Fig. 4. Schematic of the calculation procedure with existing standard subset-based DIC: (a) row by row, (b) column by column.

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If a complex ROI is defined in the reference image to approximate the complex shape of the test object, the initial guess for some points will be a troublesome and challenging problem. In this case, if one point is wrongly computed, the incorrect result of deformation parameters will be passed on to the next point and lead to a propagation of error. In this work, a simple yet effective scanning strategy guided by the ZNCC coefficient of computed points is used for reliable initial guess transfer between consecutive points. Since the ZNCC coefficient value represents the reliability degree of the correlation analysis, the method is also called reliability-guided DIC method in our recent work [16].

To clearly indicate the implementation procedure of the RG-DIC algorithm, a flowchart is plotted as shown in Fig. 5. Before the implementation of the RG-DIC method, a queue Q and two binary masks M_v and M_c with the same size as the digital images are built. After a point has been analyzed, it is subsequently inserted into the queue Q according to the magnitude of its correlation coefficient. The binary mask M_v is to identify the valid points (set to 1) to be included in the analysis and the invalid points (set to 0) to be excluded. It is necessary to note that the binary mask M_v is constructed once the ROI is specified in the reference image, and all the pixels within the defined ROI are considered as valid points. The other binary mask M_c denotes the valid points that have been computed. The initial value of each pixel in M_c is set to 0; if one point has been computed, its corresponding position in M_c is set to 1. The implementation procedure of the proposed RG-DIC method comprises the following three steps:

Step 1: Specify a seed point (or starting point) in the reference image, which can be accurately and reliably searched in the deformed image. The deformation parameters and the ZNCC coefficient of the single seed point can be computed using the NR method. Afterwards, the correlated seed point is marked as 1 in the binary mask M_c and inserted into queue Q.

Step 2: If the queue Q is not empty, remove the first point, which has the maximum ZNCC coefficient, from the top of the queue. Next, analyze each of its four (or eight) neighboring points: if it is a valid point (i.e., M_v =1) and has not been computed (i.e., M_c =0), it will then be computed using the NR method to obtain its deformation parameters and ZNCC correlation coefficient. Note that the computed deformation parameters of the removed point are used as initial guess for its neighbors. Then, each of the just calculated points is labeled as 1 in the binary mask M_c and inserted into the queue Q according to its ZNCC coefficient value.

Step 3: Repeat step 2 until the queue Q is empty, which means that all the valid points in the ROI have been computed and the image correlation computation is completed.

 

Fig. 5. Flowchart of the scanning strategy used in RG-DIC method

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From the above description, it can be seen that the correlation analysis is always performed from the points with highest ZNCC coefficient to the points with lowest ZNCC coefficient. If a point has been incorrectly computed with a low ZNCC coefficient due to decorrelation or other reasons, the computation of its neighbors will be postponed to near the end of the processing. As a result of this approach, the unreliable results of these points will not affect the correlation of its neighbors, and thus, the possible error propagation of existing algorithm can be avoided.

3. Experimental results

To verify the effectiveness and practicality of the proposed method, two representative image pairs of specimens with relatively complex shapes will be processed to reproduce the genuine full-field displacement fields within the specified ROI. In the following calculation, the NR algorithm with first-order displacement mapping function is employed to optimize the ZNSSD criterion to get the desired deformation parameters for each point; the subset used is 33×33 pixels and the grid step (i.e., the distance between neighboring calculation points) is set to 2 pixels. In the examples, only displacement fields are presented. Despite this, we should mention that the strain fields can be readily determined from the computed displacement fields with a pointwise least-squares strain estimator, about which more details can be found in Refs. [3, 5].

3.1 Deformation measurement of a dog-bone specimen with two semicircular cut-outs

In the first validation test, a dog-bone shape specimen made of alumina with a size 2mm×8mm×0.13mm is used. Prior to the experiment, an artificial speckle pattern was made on the specimen by spraying white and black paints on its surface. The specimen was clamped tightly at both ends and the corresponding image was recorded as reference image. Afterwards, the horizontal uniaxial tensile loading was exerted on the right side and a deformed image was recorded. The image pair shown in Fig. 6 will be processed with the proposed DIC technique to extract the full-field displacement fields.

 

Fig. 6. Experimental images of a dog-bone specimen subjected to uniaxial tensile loading: (a) reference image, (b) deformed image, (c) a binary image shows all the valid points within the specified ROI.

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Since the specimen is of relatively complex shape with two semicircular cut-outs, the ROI is approximated with a polygon as shown in Fig. 6(a). Figure 6(c) intuitively shows all the valid points (highlighted in white) defined in the reference image, which are labeled as 1 in the corresponding binary mask M_v. For the points within an irregular ROI, the conventional DIC method is prone to yield incorrect results at some locations. Using the proposed approach, the calculation starts from the seed point (the seed point and its surrounding subset are plotted in red in Fig. 6(a)) and is then guided by the ZNCC coefficients of computed points. Figure 7 gives two intermediate stages of the process along with the final results of the computed u-and v-displacement fields. Note that the void areas within the contour plots shown in Figs. 7 denote the points that will be computed later. In contrast to the ZNCC coefficient distributions shown later in Fig. 8(e), it can be clearly seen that the displacements were determined in an order from the points with higher ZNCC coefficients to the points with lower ZNCC coefficients.

 

Fig. 7. Two intermediate results and the finial results of the computed u-displacement (top), v-displacement (bottom) using the proposed method. The void areas within the contour plots denote the points with relatively lower ZNCC coefficients and will be processed later.

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Figure 8 illustrates the computed full-field u-displacement, v-displacement and ZNCC coefficient distributions with the proposed technique and the existing standard technique. The computed contour maps have been imposed on the ROI of the reference image for an intuitive comparison. By use of the modified ZNSSD correlation coefficient defined in Eq. (5) and the ZNCC coefficient determined by the rightmost side of Eq. (2), the algorithm can automatically discard the invalid points within each subset surrounding the boundary points. As a result, the displacements of the points located at or near the boundaries of ROI can be reliably and accurately determined, as clearly seen in Figs. 8(a) and 8(c). Also, from the ZNCC coefficient distributions shown in Fig. 8(e), it can be seen that the ZNCC coefficients of the boundary points are almost the same as those of the central points computed with a regular square subset. The relatively high ZNCC coefficients of the boundary points help to clearly prove the validity and practicality of the proposed approach. In contrast, the computational results obtained by using the existing standard DIC technique with the common ZNCC criterion are shown in the right side of Figs. 8. It is evident from Figs. 8(b) and 8(d) that the computed u- and v-field displacements for the points located near the ROI boundaries are unreliable. The ZNCC coefficients of these boundary points shown in Fig. 8(f) are much lower than those in Fig. 8(e), which clearly proves the advantage of the proposed technique.

 

Fig. 8. Comparison of the results obtained with the proposed technique and existing technique: (a)(b) u-displacement field, (c)(d) v-displacement field, and (e)(f) ZNCC coefficient distributions

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3.2 Deformation measurement of a specimen with a single-edge-crack

In the second validation test, the reference and deformed images of a PMMA specimen with a premade single-edge-crack on its left side are used, as shown in Fig. 9. The specimen geometry and experimental details can be found in Ref. [21]. Using this type of specimen, both mode-I and mode-□ stress intensity factors of the material can be determined using a non-linear least-squares algorithm [21] from the measured full-field displacement fields. In this experiment, the specimen is subjected to uniaxial tensile loading exerted along the vertical direction. In Fig. 9(b), considerable deformation can be observed around the opening crack.

 

Fig. 9. Experimental images of a specimen with a crack subjected to uniaxial tensile loading: (a) reference image (the specified ROI is highlighted with yellow color), (b) deformed image.

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Since the specimen contains a discontinuous area (i.e., a single-edge-crack), a polygonal ROI is specified as illustrated in Fig. 9(a). All the points within the ROI are considered as valid points to be analyzed. With the proposed DIC technique, the full-field displacement fields of the ROI can be determined. Figure 10 shows the contour maps of the computed u-displacement, v-displacement and ZNCC coefficient distributions, which are intentionally imposed on the ROI of the reference image for an intuitive look. It is clear that genuine full field displacement fields including the displacements of the points located at the boundaries of ROI have been reliably and accurately calculated. As expected, the u- and v-displacement distributions are almost symmetric. Moreover, from the ZNCC coefficient distributions plotted in Fig.10(c), it is seen that the ZNCC coefficients of the boundary points are almost the same as those of the central points computed with a regular square subset, which verify the validity and practicality of the proposed approach once again.

 

Fig. 10. From left to right: computed u-displacement field, v-displacement field and ZNCC coefficient distributions

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It should be noted that the subsets centered at the boundary points may cross over the crack and be divided into two separate parts by the crack. Very recently, a subset splitting algorithm [22] has been proposed to cope with this problem by dividing the subset into a master subset and a slave subset, and the displacements of both the master and slave subsets are determined. Slightly different from that algorithm, in this work, the proposed approach yields a region growing way to automatically identify the subset containing the subset center, and only this simply-connected subset (i.e., master subset in the subset splitting algorithm) is used for computation.

4. Conclusion

Existing standard subset-based DIC technique is typically not able to determine the displacements of subsets containing invalid and unwanted pixel points, and the deformation information of the points near and at the boundaries of the specified ROI is generally absent. For this reason, the conventional subset-based DIC method does not really provide genuine full-field deformation measurements. In this work, an automatic, robust and universally applicable DIC approach is proposed to overcome the limitation of the conventional standard DIC method. To reliably and accurately determine the genuine full-field deformation of an arbitrarily specified ROI in the reference image, a modified ZNSSD criterion is defined to evaluate the similarity of the subsets with a smaller number of valid pixels than usual. In addition, an automatic scanning strategy guided by the ZNCC coefficients of correlated points is employed to ensure successful initial guess transfer between consecutive points. The proposed improved DIC technique is very robust and effective, and it is universally applicable to the genuine full-field deformation measurements of objects with complex or arbitrary shapes. The validity and practicality of the proposed method has been successfully demonstrated by processing two pairs of experimental images.