Diese Studie befasst sich mit fortschrittlichen Algorithmen, die darauf ausgelegt sind, systematische Fehler in der digitalen Bildkorrelation (DIC) abzumildern, einer Technik, die für die vollflächige Deformationsmessung weit verbreitet ist. Der Schwerpunkt liegt auf der Ausweitung der Recovery-Methode auf Formfunktionen zweiter Ordnung und der Einführung der Zero-Error-Point-Methode (ZEP) für Verschiebungen dritter Ordnung. Die Forschung untersucht die theoretischen Prinzipien hinter diesen Fehlern und bietet experimentelle Validierung durch simulierte und physikalische Experimente. Zu den wichtigsten Ergebnissen zählen die Wirksamkeit der Recovery-Methode für Formfunktionen zweiter Ordnung und die überlegene Leistung der ZEP-Methode bei der Verringerung systematischer Fehler im Vergleich zu herkömmlichen Ansätzen. Die Studie untersucht auch die Auswirkungen dieser Methoden auf zufällige Fehler, die durch Bildrauschen verursacht werden, um ihre Anwendbarkeit in praktischen Szenarien sicherzustellen. Indem sie die Beschränkungen bestehender Methoden aufgreifen und neue Lösungen vorschlagen, bieten diese Forschungen wertvolle Erkenntnisse für Fachleute, die sich mit komplexen Deformationsmessungen in verschiedenen Branchen befassen.
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Abstract
Background
In complex deformation measurements, systematic errors caused by undermatched shape functions are the primary source of errors in Digital Image Correlation (DIC). There are two important ones among the current undermatched systematic error mitigation methods, Recovery method and Improved Quasi-Gauss Point (IQGP) method, that have shown effectiveness in mitigating such errors, though each has its own inherent limitations. The Recovery method is derived based on first-order shape function, while the IQGP method is setup under the assumption of the second-order displacement field in subset.
Objective
This study aims to extend and improve both the Recovery method and IQGP method respectively to address these limitations and enhance their applicability while comparing the performance between themselves and with other current methods.
Methods
As for the Recovery method, the effectiveness in mitigating undermatched systematic errors for second-order shape functions is deduced and verified, which broadens its applicability. As for the IQGP method, a new method called Zero-Error Point (ZEP) method is proposed based on the similar principles while accepting the third-order displacement assumption which basically leads to better and wider adaptability compared to the IQGP method. Other classic undermatched systematic error mitigation method and deconvolution method are also involved into analysis and discussion here.
Results
The extended Recovery method can now mitigate the undermatched error of second-order shape functions compare to the original one just for the first-order shape function, and the improved IQGP method based on the third-order displacement field can achieve an accuracy improvement of nearly 0.4 pixels compared to the traditional IQGP according to experiment results.
Conclusion
These advancements enhance the performance of undermatched systematic errors algorithms of DIC, thus improving the ability of DIC in deformation characterization under inhomogeneous deformation.
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Introduction
Digital Image Correlation (DIC) has emerged as a widely utilized and powerful technique of full-field deformation measurement in experimental mechanics [1, 2]. It has wide applications across diverse industries, such as aerospace [3, 4], automotive [5, 6], and materials engineering [7, 8], etc. The key factor of its wide application is the ability to extract displacement and strain information just in a non-contact and low-cost way by using image registration algorithms [9]. Through minimizing the zero-mean normalized sum of squared difference (ZNSSD) criterion between reference and deformed subsets, it can effectively track the movement of target subset [10]. Shape function is another critical role of this process, enabling description of deformation fields inside subsets. The commonly used shape functions include zero-order (rigid translation), first-order (affine transformation), and second-order (quadratic transformation) functions [11‐13], each one suited to different levels of deformation complexity.
Despite the great versatility and wide applicability, the accuracy of DIC is still challenged by various factors, like random errors and systematic errors. The random errors refer to the error arise with image noise, which has already been well studied [14‐16]. The systematic errors, on the other side, can be caused by various different factors like gray-level interpolation [17‐20], lens distortion [21], camera self-heating [22, 23] and undermatched shape functions [24, 25]. Undermatched systematic errors refer to the errors caused by using low-order shape function to describe high-order displacement in subsets, which is one of the primary error sources of DIC when dealing with inhomogeneous or complex deformation measurement [26, 27].
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Traditionally, such undermatched systematic errors can be diminished by applying higher-order shape function or choosing smaller subset size. However, both approaches have certain limitations. As for using higher-order shape function, certain random noise can be amplified due to usage of higher-order shape function [15, 28], while computational costs and processing time increased sharply since higher-order shape functions involve more additional parameters that must be optimized [25]. Using smaller subset seems a good strategy in some cases. However, the smallest applicable subset size is affected by many factors including the speckle quality like speckle size and speckle density, image quality like image contrast and certain hardware parameters [29], which might lead to the circumstances that the size of the subset in measurement can not be set small enough. Besides, in practical measurements, sometimes, the pattern and extent of the real displacement remain unknown beforehand, which makes it challenging to choose an appropriate subset size beforehand. Therefore, setting up smaller subset size to diminish the undermatched systematic errors may encounter certain limitations in practical cases. Based on the limitations of both two traditional methods mentioned above, researchers have begin to focus on researching the underlying principles of undermatched systematic errors of DIC and proposing algorithms to diminish such undermatched systematic errors when using low-order shape function to describe complex deformation.
To reduce such undermatched systematic errors in DIC when using low-order shape function, researchers have developed various approaches including classic methods proposed by Xu [26] and Wang [30], Recovery method [31], AAR method [32] and Improved Quasi-Gauss Point (IQGP) method [33], which are well-performing approaches in undermatched systematic errors mitigation. The Recovery method uses the low-pass filtering characteristic of DIC calculation to get more accurate displacement value with undermatched error dimished. As for the IQGP method(which is based on previous Quasi-Gauss method [25]), it takes the theoretically analyzed zero-undermatched-error-points as the calculation points in subset to avoid the undermatched error introduced in displacement calculation.
The Recovery method and IQGP method discussed above, while demonstrating good performance, each has inherent limitations at the current status. The current Recovery method is derived based on the first-order shape function, leaving the scenario involving the second-order shape function unexplored. The IQGP method, on the other hand, operates under the assumption of second-order displacement within the subset, without considering the case involving third-order displacement. In fact, in practical complex deformation measurement scenario, especially when the subset size can not be set enough small and the displacement exhibits high gradient, second-order shape function might not be able to describe the deformation in subset appropriately. Likewise, the algorithm based on the second-order displacement assumption might also potentially fail. Thus, the mitigation methods for the undermatched systematic errors of the second-order shape function also need to be studied and investigated, while the algorithms based on higher-order displacement assumption need to be explored.
This study aims to address these limitations and challenges mentioned above and enhance and extend the applicability of both the Recovery method and IQGP method while discussing and comparing with some other current undermatched systematic error mitigation methods. Section "Extension of Recovery Method" focuses on the Recovery method, including a review of its fundamental principles and theoretical verification of its effectiveness when applied to second-order shape functions. Section "Enhanced Method Based on IQGP Method" introduces a novel method for mitigating undermatched systematic errors, building upon principles similar to those of the current IQGP method but taking third-order displacement into consideration. Section "Other Related Work" presents experimental results demonstrating the effectiveness of the Recovery method with second-order shape functions, as well as the newly proposed method. Section "Investigation of Potential Change of Random Error Caused by Image Noise" draws conclusions from this work.
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Extension of Recovery Method
This section incorporates introduction of Recovery method’s basic idea and verification of it in the case of second-order shape function.
Basic Principle
In local DIC, displacement results can be interpreted as the outcome of applying a Savitzky-Golay (S-G) filter to the actual displacement within the subset [24]. By using this property, the traditional Recovery method mitigates undermatched systematic errors by employing a linear combination of the results obtained through repeated applications of the S-G filter on DIC-derived displacements [31]. For simplicity, the derivation of the formula presented below is confined to one-dimensional cases. In this case, the S-G filter kernel corresponding to the first-order shape function is expressed as follow where the subscript “1” indicates that this filter kernel function corresponds to the case of first-order shape function:
$${g}_{1}(\overline{x })=\frac{1}{2M+1}$$
(1)
where \(\overline{x }\) denotes the coordinate of each position in the local coordinate system with origin lies at each subset center, i.e. the range of \(\overline{x }\) is confined to interval [-M, M]. It can be seen that filter kernel of first-order shape function is a is actually a mean filter with a window size of 2 M + 1(M represents half size of subset) from Eq. (1). Assume the real displacement within subset can be effectively approximated by a higher-order polynomial containing a linear combination of monomials. One common monomial with order n can be represented as:
$$f(x)={a}_{n}{x}^{n}$$
(2)
where \({a}_{n}\) is a constant coefficient, x represents the coordinate value in the global coordinate system, with its origin located at the upper-left corner of the reference image. Thus, the DIC results of first-order shape function can be estimated in discrete integration format as follow according to work of Schreier and Sutton [24]:
when the highest order of the actual displacement within the subset is assumed either in second- or third-order,\({f}_{1},{f}_{2}\) can be expressed as below:
In that case, the actual displacement f inside subset can be estimated as follow:
$$f=2{f}_{1}-{f}_{2}$$
(8)
Similarly, when the highest order of the actual displacement in subset is either fourth or fifth order, the actual displacement f can be approximated as:
$$f=3{f}_{1}-3{f}_{2}+{f}_{3}$$
(9)
When the highest order of actual displacement in subset is sixth or seventh order, the actual displacement f can be estimated as:
$$f=4{f}_{1}-6{f}_{2}+4{f}_{3}-{f}_{4}$$
(10)
Therefore, the undermatched systematic errors in the results obtained by using the first-order shape function can be mitigated by just employing Eqs. (8), (9), or (10), depending on the order of the actual displacement. A similar deduction can be extended to cases where the highest order of the actual displacement within the subset exceeds the seventh order for the case of first-order shape function.
Verification of Effectiveness for Second-Order Shape Function
The current Recovery method discussed earlier is derived under the assumption of first-order shape function, while the case of second-order shape function remains unexplored and unverified. The corresponding verification of second-order shape function is presented below.
Unlike the derivation presented Section "Basic Principle", the following analysis is based on two-dimensional displacement field, which is more suitable for normal actual scenarios.
The first step is to determine the filter kernel for second-order shape function under two-dimensional displacement field. According to the work of Schreier and Sutton [24], when the cross-correlation of DIC is maximized, indicating that the shape function \(W(\overline{x },\overline{y },\overrightarrow{p})\) closely approximates the actual displacement field(\(u(\overline{x },\overline{y })\) in x direction and \(v(\overline{x },\overline{y })\) in y direction) within the subset, the parameter vector \(\overrightarrow{p}\) can be further split into the part for u and v as:
where \({u}_{0}^{c},{u}_{x}^{c},{u}_{y}^{c},{u}_{xx}^{c},{u}_{xy}^{c},{u}_{yy}^{c},{v}_{0}^{c},{v}_{x}^{c},{v}_{y}^{c},{v}_{xx}^{c},{v}_{xy}^{c},{v}_{yy}^{c}\) are the parameters need to be determined in second-order shape function. \(\overrightarrow{\xi }\) and \(\overrightarrow{\eta }\) can be determined through the least square method by the optimization equations in the following Eq. (13):
In this way, the parameter vector \(\overrightarrow{p}\) of shape function can be estimated and determined. \(\overline{x },\overline{y }\) in Eq. (13) represent the local coordinates established within each subset, with the origin located at the subset center. M denotes half of the subset size. \({W}_{\xi }(\overline{x },\overline{y },\overrightarrow{\xi })\) and \({W}_{\eta }(\overline{x },\overline{y },\overrightarrow{\eta })\) represent the shape function in x and y direction, respectively. When it comes to second-order shape function, \({W}_{\xi }(\overline{x },\overline{y },\overrightarrow{\xi })\) and \({W}_{\eta }(\overline{x },\overline{y },\overrightarrow{\eta })\) can be expressed as:
For simplicity, the following derivation is just based on the consideration of x direction displacement(u), other more complex cases can be derived following the similar principle here. By substituting Eq. (14) into (13) and following the derivation procedures of previous work of Schreier and Sutton [24], the below Eq. (15) can be derived through simplification, whose full deducing procedure can be checked in Appendix:
In that case, by performing derivation operations similar to those of Sutton et al., the S-G filter kernel function of second-order shape function under two-dimensional displacement field can be denoted as:
where the subscript “2” indicates that this filter kernel function corresponds to the case of second-order shape function. However, it can be noticed that the S-G filter kernel function of second-order shape function in Eq. (17) is different from the one presented in Schreier and Sutton’s previous paper [24]. The reason lie in the assumption difference between them: In previous derivation made by Schreier and Sutton, the displacement u was assumed to depend solely on the x-coordinate, implying uniformity in the y-direction, while in this work, u is dependent on both x and y coordinates. This fundamental difference leads to the derived S-G filter kernel function of second-order shape function are not the same with each other. Another thing that needs to be noticed is that in Eq. (17), the range of \(\overline{x }\) and \(\overline{y }\) both are confined to interval [-M, M] since they are the values of local coordinate system, if Eq. (17) are changed into global coordinate system, it will be transformed into:
where \(f(x,y)\) denotes the actual two-dimensional displacement field within susbet. In the subsequent analysis, the similar notation of \({f}_{i}\) used in Eq. (6) is also adopted here:
Assuming that the actual displacement field within the subset can be accurately approximated by using a fourth-order Taylor expansion, i.e., the displacement field is of fourth order, \(f(x-m,y-n)\) in Eq. (19) can be expressed as:
In Eq. (19), the integration domain is a square region symmetric about the origin of the local coordinate system within the subset and the integration is discrete. Such discrete integration of the filter kernel function associated with second-order shape function within this domain can be derived as:
In the meanwhile, it can be observed that the filter kernel function \(g(x,y)\) is an even function meaning that the integration \(\sum_{m=-M}^{M}\sum_{n=-M}^{M}{g}_{2}(m,n){m}^{a}{n}^{b}\) results in a non-zero value only when both parameters a and b are even numbers. By using the characteristics analyzed above, when substituting Eq. (21) into Eq. (19), Eq. (19) can be reformulated as follows:
Since theoretically there is no undermatched systematic errors when using second-order shape function to calculated second-order displacement. Consequently, Eq. (23) can be further simplified as below:
It can be verified that Eqs. (25) and (26) are still satisfied when the actual displacement field within the subset is of the fifth order by following the same procedures above. Consequently, for the case of second-order shape function, when the actual displacement is of the fourth or fifth order—i.e., when the displacement can be effectively approximated by a fourth- or fifth-order Taylor expansion—the actual displacement f in subset can also be computed through Eq. (8) which corresponds to the case of using first-order shape function when the actual displacement field in subset is of the second or third order.
In a similar way, when the actual displacement field within the subset is of the sixth or seventh order,\({f}_{1}\),\({f}_{2}\) and \({f}_{3}\) can be represented as follows:
It is straightforward to verify that for the second-order shape function, when the actual displacement within the subset is of the sixth or seventh order, the actual displacement f can be computed through Eq. (9). Likewise, when the actual displacement within the subset is of the eighth or ninth order, the actual displacement f can be calculated through Eq. (10) by following the similar procedures mentioned above.
In summary, the Recovery method is equally effective in mitigating undermatched systematic errors for the second-order shape function. However, since the Recovery method involves performing S-G filtering on the displacement field, its effectiveness in mitigating undermatched systematic errors may be impaired in regions near the edge of the region of interest (ROI), where displacement data may be insufficient.
The above derivation demonstrate the Recovery method’s effectiveness and validness for second-order shape function. However, by following the similar procedures above, the Recovery method for shape functions of any arbitrary order beyond second-order can be deduced and constructed to handle with its undermatched systematic errors.
Enhanced Method Based on IQGP Method
This section proposes an enhanced method based on the principles of Improved Quasi-Gauss Point (IQGP) method followed by the brief review of the former IQGP method.
Basic Principle of IQGP Method
Apart from the Recovery method, the IQGP method is another important approach for mitigating systematic errors caused by undermatched shape function. Based on the former Quasi-Gauss Point (QGP) method [25], this method reduces such errors by selecting the points where the undermatched systematic errors are theoretically zero, rather than the subset center point, as the calculation points within the subset. However, the QGP method is based on the assumption of one-dimensional displacement. To address this limitation and extend its application to two-dimensional displacement scenarios, the IQGP method is proposed as below.
Assume that the actual displacement field within the subset is of second order and can be expressed as:
As previously analyzed in Section "Verification of Effectiveness for Second-Order Shape Function", the shape function and the actual displacement field within the subset should satisfy the relationship described in Eq. (13). When the first-order shape function is applied, its parameters can be expressed in terms of the actual displacement coefficients, as shown in Eq. (30). Under these circumstances, the systematic errors introduced by the undermatched shape function can be denoted by subtracting the actual displacement from the shape function which is represented through the coefficients. The corresponding undermatched systematic error can be expressed:
where \({u}_{{e}_{2nd}}^{1st},{v}_{{e}_{2nd}}^{1st}\) denote the undermatched systematic error of first-order shape function under second-order displacement field in x and y direction, respectively. Take the x direction as an example, when the undermatched systematic error is equal to zero, there exists the following equation:
A rotation is then applied to the local coordinate system \(O{\overline{x} }\overline{y }\) in subset, such that it is rotated clockwise by an angle θ to form a new local coordinate system \(O{\overline{x} }_{0}{\overline{y} }_{0}\). The transformation relationship between corresponding points in these two coordinate systems is expressed as follows:
From Eq. (37), it can be observed that when the actual deformation within the subset corresponds to a two-dimensional second-order displacement, the distribution of zero-undermatched-systematic-error points within the subset forms a conic section.
Assume that in Eq. (35), there exist solutions unrelated to uxx, uyy and uxy. In this case, the following equations hold:
By comparing Eq. (39) with the solution of the traditional QGP method, it is evident that the locations of zero-undermatched-systematic-error points under a two-dimensional second-order displacement within the subset correspond to the positions of the traditional Quasi-Gauss points rotated clockwise by an angle θ, as determined by Eq. (36). This approach offers enhanced robustness and applicability, as it accounts for two-dimensional displacement scenarios. Given its enhancements, this method is referred to as the Improved Quasi-Gauss Point (IQGP) method.
Enhanced Method Aimed for Third-Order Displacement
The IQGP method introduced in the Section "Basic Principle of IQGP Method" is based on the assumption of second-order displacement. Therefore, a more advanced method aimed for third-order deformation is needed.
When the actual displacement within the subset is of third order, which is denoted as:
where \({u}_{0}\),\({v}_{0}\) denote the actual displacement value at the center of the subset, ux, uy, uxx, uxy, uyy, uxxx, uxxy, uxyy, uyyy, vx, vy, vxx, vxy, vyy, vxxx, vxxy, vxyy, vyyy are coefficients of various displacement derivatives. Assume that the first-order shape function is used, which can be expressed as:
where \({W}_{\xi }(\overline{x },\overline{y },\overrightarrow{p})\) and \({W}_{\eta }(\overline{x },\overline{y },\overrightarrow{p})\) denote first-order shape function in x and y direction. Since the shape function and the actual displacement field within the subset should satisfy the relationship described in Eq. (13), by substituting Eqs. (41) and (40) into Eq. (13), the parameters in shape function can be expressed in terms of the actual three-order displacement coefficients as:
It can be observed that Eq. (44) is a cubic equation with two unknowns, making it difficult to obtain a direct theoretical solution. To address this, certain practical approach is employed to determine the positions of the zero-undermatched-systematic-error points within the subset. This method involves making one variable fixed and solving for the other, effectively transforming the equation to a cubic form with just single unknown, which is easier to solve. The detailed description and steps of this method are outlined as follows:
As for the y-direction, fix and take \(\overline{y }\) value as:\(\overline{y }\) = − M, − M + 1…0…M − 1, M and then solve the equation that is only dependent on \(\overline{x }\). If the solution of \(\overline{x }\) lies within the interval [− M, M], the corresponding coordinates (\(\overline{x }\),\(\overline{y }\)) are output as one of the solutions in the subset.
As for the x-direction, fix and take \(\overline{x }\) value as:\(\overline{x }\) = − M, − M + 1…0…M − 1, M and then solve the equation that is only dependent on \(\overline{y }\). If the solution of \(\overline{y }\) lies within the interval [− M, M], the corresponding coordinates (\(\overline{x }\),\(\overline{y }\)) are output as one of the solutions in the subset.
By following the steps above, a list of zero-undermatched-systematic-error points in the subset can be obtained, which can then be used as the calculation points to mitigate systematic errors caused by undermatched shape functions. This method is named as the Zero-Error-Point method, or ZEP method.
The ZEP method demonstrated above is mainly focused on the case of third-order displacement in local subsets since in practical DIC measurements, the selected subset size won’t be fixed at a too large value in order to maintain the accuracy and computational speed especially in complex deformation measurement, which makes that the adoption of higher-order displacement assumption(like fifth- or sixth-order displacement assumption) in local subsets seems not so worthwhile and necessary. However, the ZEP method does have the ability to be used in higher-order(even any arbitrary order) displacement assumption in local subsets, which is going to be described below:
Assume the displacement field within the subset can be represented by an nth-order polynomial, i.e., the subset displacement field is in nth-order. The actual displacement within the subset can be denoted as:
where \({a}_{ij}\) and \({b}_{ij}\) denote the scalar weights of each monomial \({\overline{x} }^{i}{\overline{y} }^{j}\) in the polynomial expansion above. Then, by applying the relationship shown in Eq. (13), the parameters in first-order shape function in Eq. (41) can be expressed in terms of \({a}_{ij}\) and \({b}_{ij}\) in the nth-order polynomial equations above. In this way, we can describe the undermatched systematic errors distribution functions inside the subset by using the \({a}_{ij}\) and \({b}_{ij}\) terms under the assumption that the shape function we used is first-order while the real displacement in subset is nth-order. Then by solving such undermatched systematic errors distribution functions, the zero-undermatched-systematic-error points in the subset can be obtained. The high-order equation with two unknowns can be solved following the similar procedures described above in Section "Enhanced Method Aimed for Third-Order Displacement", which is to fix one variable and solve another and store the solution if it lies within the subset range. That is still the case of arbitrary order actual displacement with first-order shape function. Following the similar principle, the case of nth-order actual displacement with mth-order shape function can be deduced and derived as long as it is still the undermatched shape function cases(n > m). Thus, through ZEP method, for any arbitrary mth-order shape function, it is capable to formulate the corresponding undermatched systematic error functions under the case of assuming actual displacement is in nth-order in local subset(n > m).
Other Related Work
The above two sections (Sections "Extension of Recovery Method" and "Enhanced Method Based on IQGP Method") are mainly focused on the Recovery method and ZEP method. In this section, some other current mitigation methods of systematic errors caused by undermatched shape function are mentioned and analyzed here for subsequent comparison in later section.
Classic Method
Classic method refers to the undermatched error mitigation method through directly computing the error estimation value and subsequently subtracting this error from the DIC results [32]. The classic method for first-order shape function is deduced by Xu et al. [26] while the subsequent derivation for second-order shape function is accomplished by Wang et al. [30].
For the case of first-order shape function, the systematic errors caused by the undermatched shape function at the center of the subset can be theoretically denoted as:
Therefore, the classic method for first-order shape function can be presented by subtracting such undermatched systematic errors from the first-order DIC results as:
Similar case of second-order shape function is also presented here where the undermatched systematic errors at the center of the subset is shown in Eq. (48) and the corresponding classic method of it is shown in Eq. (49):
Another well-recognized related work is the deconvolution method proposed by Grédiac et al. [34]. By applying an iterative deconvolution procedure, the undermatched systematic error in DIC results can be diminished. However, through derivation, the core formula is the same with the classic method above, i.e. it can be seen as a iterative operation of classic method. The corresponding derivation is listed below:
For simplicity, assuming only the displacement in x-direction: u is considered while other more complex scenarios can be derived in the similar principle. In this case, the Eq. (51) can be transformed into:
where the integration in Eq. (52) is discrete by definition. when it comes to the case of first-order shape function, the S-G kernel filter function w can also be denoted with the format shown in Eq. (53) as below:
Therefore, it can be clearly found that the formula used in deconvolution method is just the same with the classic method combining with the iterative operation.
Extension of Deconvolution Method and Classic Method into Higher-Order Shape Function Cases
It can be noticed that in the derivation of the Recovery method’s effectiveness for second-order shape function case, the Taylor expansion used in Eq. (21) is very similar to the Taylor expansion used in deconvolution method:
However, when it comes to the case of second-order shape function. Such Taylor expansion in Eq. (58) may cause issue, since according to the previous Section "Verification of Effectiveness for Second-Order Shape Function", the corresponding S-G kernel filter function w of second-order shape function can be represented in Eq. (18). In that case, it can be easily deduced that: \({\iint }_{({\eta }_{1},{\eta }_{2})\in {\mathfrak{R}}^{2}}w({\eta }_{1},{\eta }_{2})\eta d{\eta }_{1}d{\eta }_{2}=0\) and \({\iint }_{({\eta }_{1},{\eta }_{2})\in {\mathfrak{R}}^{2}}w({\eta }_{1},{\eta }_{2}){\eta }_{k}{\eta }_{l}d{\eta }_{1}d{\eta }_{2}=0\) following the previous Eq. (24) and corresponding analysis in Section "Verification of Effectiveness for Second-Order Shape Function". Thus, as analysed here, the original deconvolution method based on Eq. (58) is not able to mitigate the undermatched systematic errors of second-order DIC results. Such phenomena can also be attributed to the fact that the second-order shape function can describe the second-order displacement in subset. Therefore, if the Taylor expansion is just assumed in second-order, the undermatched systematic errors for second-order shape function is theoretically zero.
Nevertheless, if the fourth-order Taylor expansion used in Eq. (21) is taken into account, that problem can be solved. In fact, the Eq. (21) can be seen as the extended version of Eq. (58) from second-order displacement assumption to fourth-order displacement assumption. Following the same analytical procedure and reasoning as presented in Section "Verification of Effectiveness for Second-Order Shape Function", the second-order DIC result \(\widetilde{q}\) can also be represent by Eq. (25), in which case, the corresponding correction term \(\delta \widetilde{q}\) of deconvolution method can be denoted as:
By comparing with the classic method, it is clear that the Eq. (63) of the deconvolution method for the second-order shape function is essentially the same with the Eq. (49) of the classic method for the second-order shape function, while the only difference is deconvolution method incorporates the iterative operation. This confirms the previous conclusion regarding the fundamental equivalence between deconvolution method and classic method.
Based on the preceding analysis, the expressions of higher-order (more than or equals to third-order) shape functions for both deconvolution method and classic method can be derived following these steps:
1.
Follow the principles in Eq. (13) and use the least square method to help determine the \({u}_{0}^{c}\) equation.
2.
Through \({u}_{0}^{c}\) equation derive the S-G kernel filter function of the desired order shape function(assume the desired order is i-th)
3.
Represent DIC results in convolution format and assume the real displacement in subset is in j-th order, i.e. it can be approximated by j-th order Taylor expansion.( j > i)
4.
Simplify the equation to get the concise version of the undermatched systematic error formula.
5.
Direct form will be the classic method while the iterative operation turns it into deconvolution method.
In summary, the above procedure enables the derivation and representation of both deconvolution method and classic method for shape functions of any arbitrary order.
Investigation of Potential Change of Random Error Caused by Image Noise
The proposed methods described in the above Section "Extension of Recovery Method" and "Enhanced Method Based on IQGP Method" can reduce the undermatched systematic errors caused by undermatched shape function in DIC theoretically. However, when using these methods to mitigate such errors, whether the random errors caused by image noise will change is a potential issue needs to be investigated. The investigation here is followed with the setup and operation in classic research work of random errors by Yu and Pan [35].
The reference speckle pattern image used here, as shown in Fig. 1, was captured beforehand in a practical experiment. The image has a resolution of 250 × 250 pixels, with an average speckle size of approximately 4.5 pixels and a spatial occupancy of around 46.7%. Other properties, such as speckle randomness, also conform to the relevant standards [36]. A 0.5 pixel in-plane translation along x-direction is applied to the reference image using cubic spline interpolation implemented through MATLAB to generate the deformation image for random error analysis. The Region of Interest(ROI) is set to a region of 150 × 150 pixels, corresponding to the coordinate range 50 ≤ x ≤ 200 and 50 ≤ y ≤ 200 in the global coordinate system, which is shown in Fig. 1 as the square region. To simulate the effect of image noise, zero-mean white Gaussian noise with standard deviation of 4% is applied to the deformation image. The ROI here incorporating 22801 pixel points is calculated and the results are evaluated by standard deviation(SD) errors, which is the same process as described in Yu and Pan’s work. The subset size here is chosen in the range from 11× 11 pixels to 71 × 71 pixels(11 × 11, 21 × 21, 31 × 31, 41 × 41, 51 × 51, 61 × 61, 71 × 71).
Fig. 1
Reference speckle image and ROI region with the global coordinate system Oxy originating at the top left
Figure 2 shows the SD errors by using first- and second-order shape functions in the measured displacement described above. The phenomenon is consistent with experimental results and conclusions in the previous papers focusing on random errors [15, 35]: when the subset size is fixed, the SD errors of second-order shape function nearly twice the errors of first-order shape function. Besides, the SD errors of both first- and second-order will diminish with the increase of the subset size.
Fig. 2
SD errors of first- and second-order shape function under different subset size with 0.5 pixel in-plane translation in x-direction
Here, the Recovery method is applied to both first- and second-order shape functions’ scenarios to see what kind of changes will show up in the scale of SD errors of the revised displacements by using Recovery method onto results of first- and second-order shape function. The results are shown in Fig. 3(a) and (b). It can be seen that, the revised displacement by using the Recovery method contains more random errors compared to the original displacement from Fig. 3(a) and (b). Besides, it is clear that, after using Recovery method considering higher-order displacement in subset, the SD errors of revised displacements of both first- and second-order shape function become larger. This can be explained by the features of Recovery method: Recovery method mitigates the undermatched systematic errors through combining the results of different times of DIC calculation(DIC can be seen as a process of S-G filtering as described in Sect. "Basic Principle"), which means that random noise will be accumulated causing SD errors increase. Moreover, the Recovery method that takes higher-order displacement assumption into account uses the corresponding results of low-order displacement assumption, which leads to the random errors existing in low-order displacement based Recovery method will be passed through and accumulated in the results of higher-order displacement based Recovery method leading to the phenomenon that higher-order displacement based Recovery method’s results contain more SD errors.
Fig. 3
SD errors of original results of first- and second-order shape function and revised results by using Recovery method with different displacement assumption under different subset size: a first-order shape function case. b second-order shape function case
According to Fig. 3(a), it can be notice that the SD errors of original first-order shape function DIC and its corresponding Recovery methods’ results diminish quickly and sharply with the subset size increasing from 11 × 11 pixels to 21 × 21 pixels. When the subset size reaches 31 × 31 pixels, even the result by 6th/7th displacement assumption based Recovery method has SD error scale smaller than 0.01 pixel. Similar status also shown up in the cases of second-order shape function in Fig. 3(b), where the SD error scale of the result by 8th/9th displacement assumption based Recovery method is around 0.03 pixels with subset size set up to 21 × 21 pixels. It can be observed that when using the Recovery method in DIC with small subset size, the effects of random errors caused by image noise is not tiny in first-order shape function case and even pretty obvious in the case of second-order shape function. However, when setting subset size up and larger than 21 × 21 pixels, such effects in both first- and second-order shape functions are minimal and negligible.
Thus, using Recovery method in the scenario of subset size up to or larger than 21 × 21 pixels is more suitable and reliable compared to using it in for the cases with smaller subset size. Besides, when the subset size is fixed at small value, the local displacement within the subset is more likely to be homogenous deformation rather than high-order deformation, which makes it less meaningful to do undermatched systematic errors mitigation operation in this case [32]. An important thing needs to be mentioned is the requirement on size of speckles and subset. Speckle size should be large enough [29], and the selected subsets must include enough amount of speckles to ensure the robustness and image contrast [37], especially when doing the complex displacement measurement. If the speckle is small, it is more likely to disappear or lose correlation features. Similarly when subset doesn’t contain enough speckles, the correlation is more likely to fail in complex deformation cases. Based on all of these described above, although Recovery method is more applicable to the scenario of subset size up to or larger than 21 × 21 pixels, that circumstance is acceptable and valid in physical measurement scenarios when taking random errors caused by image noise into account.
Similar procedures are done for the case of ZEP method compared with IQGP and QGP method. Not like the Recovery method, the ZEP method and other two methods here(IQGP and QGP method) are just applicable for first-order shape function. Therefore, all these three methods are used to improve the original first-order DIC results and see how random error scale will change with that. The SD error results are shown in Fig. 4. Similar to the phenomenon of Recovery method shown in Fig. 3(a) and (b), the SD error of results from QGP, IQGP and ZEP methods are larger than the ones of original first-order DIC results. It might be attributed to the influence of image noise on the zero-undermatched-error points positions’ estimation in these three methods, leading to the increase of SD error scale. The SD error scale of QGP, IQGP and ZEP methods are very similar to each other. Besides, even under the case of 11 × 11 pixels subset size where the SD error of QGP, IQGP and ZEP methods are in largest value shown in Fig. 4, their value are around 0.025 pixels which are not so obvious. Not to mention when the subset size fixed at larger value, the SD errors of these three methods diminish quickly to a very small scale. Like the analysis done above for the case of Recovery method. It can be concluded that ZEP method has similar SD error scale compared to previous QGP and IQGP method, although their SD error scale is nearly twice the one of original first-order DIC, the overall SD errors are small and insignificant. Thus, ZEP method is still valuable and applicable under a wide range of subset size when dealing with undermatched systematic errors mitigation even taking the random errors caused by image noise into consideration.
Fig. 4
SD errors of original results of first-order shape function and revised results by using QGP, IQGP and ZEP method under different subset size
In summary, the random error induced by image noise change do change when using Recovery method and ZEP method. Nevertheless, considering the size requirement of speckles, subsets, such random error change is limited and nearly negligible in physical measurement scenarios.
Results of Experiment
In this section, both simulated and physical experiments are conducted for verification. All algorithms are implemented using MATLAB software for computational analysis.
Simulated Experiment
Simulated experiments are designed to evaluate the effectiveness of the Recovery method for the second-order shape function and the proposed ZEP method. These experiments are conducted by applying specific types of complex deformations to a reference speckle pattern image to generate corresponding deformed images. The performance of these two methods in mitigating undermatched systematic errors is then assessed, in which way their effectiveness can be evaluated.
Two complex deformation fields with high displacement gradients are selected for experimentation: (1) a one-dimensional sinusoidal displacement field. A typical function of it can be denoted as:
$$u(x,y)=asin(\frac{2\pi x}{b})$$
(64)
In this study, the parameters in Eq. (64) are fixed as a = 1.5 and b = 90. The actual displacement field corresponding to Eq. (64) is shown in Figs. 5(a) and (b). (2) a two-dimensional Gaussian displacement field. A typical function of it can be expressed as:
Fig. 5
Actual displacement field of the sinusoidal displacement: a three-dimensional figure; b curve graph
In this experiment, the parameters in Eq. (65) are set to c = 5 and d = 25. The corresponding actual displacement field described by Eq. (65) is depicted in Figs. 6(a) and (b). The performance evaluation in the Gaussian displacement experiment is mainly focusing on the cross-section at y = 122(as shown in Fig. 6(b)), where the deformation intensity is near its maximum. Both deformation fields described above are used for verification. In order to highlight the undermatched effects of the second-order shape function, the half-subset size, denoted as M, is set to 30 and 40 pixels for the second-order shape function in the experiments. On the other hand, when it comes to the first-order shape function, the half-subset size is set to 20 and 30 pixels. The grid step is fixed at 1 pixel throughout the experiments. As for a comparison, the results of deconvolution method(which is the same with classic method) of first-order and second-order shape function are also incorporated in the experiment. The iteration operation is not included here, since the main topic of this work is not focused on the iteration analysis and influence.
Fig. 6
Actual displacement field of the Gaussian displacement: a three-dimensional figure; b two-dimensional figure
In this section, the effectiveness of the Recovery method in mitigating undermatched systematic errors for the second-order shape function is evaluated using the two displacement fields described above. The corresponding results are presented in Figs. 7, 8, 9 and 10.
Fig. 7
The result of undermatched systematic errors mitigation for second-order shape function by using Recovery method in sinusoidal displacement with M set to 30: a estimated displacement result of Recovery method under different order displacement assumption; b corresponding errors between the various estimation values shown in Fig. 7a and the real displacement
The result of undermatched systematic errors mitigation for second-order shape function by using Recovery method in sinusoidal displacement with M set to 40: a estimated displacement result of Recovery method under different order displacement assumption; b corresponding errors between the various estimation values shown in Fig. 8a and the real displacement
The result of undermatched systematic errors mitigation for second-order shape function by using Recovery method in Gaussian displacement y = 122 position with M set to 30: a estimated displacement result of Recovery method under different order displacement assumption; b corresponding errors between the various estimation values shown in Fig. 9a and the real displacement
The result of undermatched systematic errors mitigation for second-order shape function by using Recovery method in Gaussian displacement y = 122 position with M set to 40: a estimated displacement result of Recovery method under different order displacement assumption; b corresponding errors between the various estimation values shown in Fig. 10a and the real displacement
In this experiment, the Recovery methods with different highest order of the actual displacement assumption, corresponding to Eqs. (8), (9) and (10), are used. The Eqs. (8), (9) and (10) independently correspond to the case of assumed actual displacement in fourth/fifth order, sixth/seventh order and eighth/nineth order. These Recovery methods and the deconvolution method for the second-order shape function are used to correct the original second-order DIC results. The displacement values of corrected ones and the original second-order DIC(labeled as “Disp of 2nd-shape function”) along with the ground truth displacement(labeled as “Real Displacement”) are compared in U figures, with “Recovery method of n-th disp” represents the corrected displacement value by using Recovery method with actual displacement assumed in n-th order. The difference between these displacement value and the ground truth displacement are calculated and labeled as Ue. In other words, Ue reflects the extent of undermatched systematic errors of the corrected displacement by different Recovery method, deconvolution method for the second-order shape function and the original second-order DIC results.
According to the results shown in Figs. 7, 8, 9 and 10, the Recovery method, as analyzed in Section "Extension of Recovery Method", effectively reduces systematic errors caused by undermatched shape functions for the second-order shape function. The effectiveness of the Recovery method based on fourth/fifth-order displacement assumption is very close to the one of deconvolution method corresponding to the second-order shape function, while demonstrates the good performances of the Recovery method for second-order shape function. Furthermore, as the assumed displacement order increases, the estimated results generally converge more closely to the actual displacement values, which can be seen in Figs. 7(a), 8(a), 9(a) and 10(a). However, it is also observed that in some cases, higher-order displacement assumptions may result in more errors(higher peak values in error curves) especially near the edge region shown in Figs. 7(b) and 8(b). This phenomenon can be attributed to the combination operation of prior lower-order-displacement-assumption-based results in the Recovery method, which leads to the errors from prior results still propagate to subsequent higher-order-displacement-assumption-based results, leading to error accumulation. Especially when the region near the edge of the ROI where no sufficient data can be used for Recovery method is under obvisouly complex deformation, such error accumulation phenomenon can be obvious. As seen in the case of Figs. 7(b) and 8(b), the region near the edge is under the complex deformation, but in such region, it doesn’t have enough data for the S-G filtering calculation(requires a full window of neighboring pixels) to generate the data needed in the Recovery method, which leads to the case that near the ROI boundary only a partial window is available, leading under complex deformation to error accumulation in each calculation. That can interpret the phenomenon in Figs. 7(b) and 8(b) that at the edge region of the right side of ROI, Recovery method based on higher-order displacement leads to larger errors at that region. Another phenomenon that can be noticed is that, with the increase of the size of the subset size, the undermatched systematic error of original second-order DIC becomes bigger, in which case using higher-order-assumption-based Recovery method can lead to better result compare to the one based on the lower order assumption, especially for the case of Gaussian displacement shown in Figs. 9(b) and 10(b). In conclusion, the Recovery method can effectively reduce the undermatched systematic errors in second-order shape function. However, caution should be exercised when employing higher-order-displacement-assumption-based Recovery method, especially when it comes to the edge area of the ROI where the edge region is also under complex deformation.
ZEP Method
This section evaluates the performance of the ZEP method in mitigating systematic errors caused by undermatched shape functions. Since both IQGP method and ZEP method are based on first-order shape function. Only first-order shape function is used in this experiment. The two displacement fields mentioned earlier are also both used here. For a more comprehensive assessment and comparison, the IQGP method is also included. The corresponding results are shown in Figs. 11, 12, 13 and 14. Similar to the previous process described in Section "Recovery Method of Second-Order Shape Function". IQGP method, ZEP method and deconvolution method of first-order shape function are used to correct the original displacement result calculated by first-order DIC. The displacement values of corrected displacement by IQGP method, ZEP method and deconvolution method of first-order shape function, the original first-order DIC along with the ground truth displacement are compared in U figures, while Ue denotes the difference between these displacement value and the ground truth displacement. In other words, Ue figures can shows the magnitude of the undermatched systematic errors.
Fig. 11
The result of undermatched systematic errors mitigation for first-order shape function by using ZEP and IQGP method in sinusoidal displacement with M set to 20: a estimated displacement result of ZEP and IQGP method; b corresponding errors between the estimation values shown in Fig. 11a and the real displacement
The result of undermatched systematic errors mitigation for first-order shape function by using ZEP and IQGP method in sinusoidal displacement with M set to 30: a estimated displacement result of ZEP and IQGP method; b corresponding errors between the estimation values shown in Fig. 12a and the real displacement
The result of undermatched systematic errors mitigation for first-order shape function by using ZEP and IQGP method in Gaussian displacement y = 122 position with M set to 20: a estimated displacement result of ZEP and IQGP method; b corresponding errors between the estimation values shown in Fig. 13a and the real displacement
The result of undermatched systematic errors mitigation for first-order shape function by using ZEP and IQGP method in Gaussian displacement y = 122 position with M set to 30: a estimated displacement result of ZEP and IQGP method; b corresponding errors between the estimation values shown in Fig. 14a and the real displacement
As illustrated in Figs. 11, 12, 13 and 14, the ZEP method proposed in Section "Enhanced Method Based on IQGP Method" demonstrates superior effectiveness in reducing undermatched systematic errors for the first-order shape function compared to the original IQGP method. Furthermore, as the subset size increases, the difference in performance between the two methods becomes more obvious, which can be noticed from comparing Figs. 11(b) with 12(b) and Figs. 13(b) with 14(b). In Fig. 14(b), when M is set to 30 pixels, the peak value of Ue curve of IQGP method is 0.96 compared to the same position the value of Ue curve of ZEP method is just 0.52. That phenonmen can be explained by the fact that the undermatched systematic errors are huge for the case of that subset size and the second-order displacement assumption, compared to the third-order displacement assumption, is less appropriate to describe the deformation inside subset especially at the region near the peak. It can be seen that when the undermatched effect is huge, the ZEP method can reach an accuracy improvement of nearly 0.4 pixels compared to the IQGP method. Moreover, the comparison with the deconvolution method reveals that the IQGP method achieves similar accuracy while the ZEP method demonstrates better performance, particularly as the subset size increases.Thus, through the Figs. 11, 12, 13 and 14, the superior performance of ZEP method has been validated.
Another thing needs to be examined is the computational cost of the ZEP method. As discussed in Section "Enhanced Method Aimed for Third-Order Displacement", a series of cubic equations need to be solved in ZEP method compared to the IQGP method. The computational costs of the ZEP method and IQGP method for sinusoidal and Gaussian displacements are presented in Table 1. As shown in the table, the ZEP method requires more time than the IQGP method due to its need for solving multiple cubic equations. However, the computational time of the ZEP method remains pretty small compared to that of the DIC calculation. Therefore, despite its higher cost, this difference is not significant enough to be a major concern.
Table 1
computational cost comparison
Displacement type
Subset Size (pixel)
Time consumption (s)
Sinusoidal displacement
M = 20
DIC calculation
179.3803
IQGP method
0.0596
ZEP method
4.0061
M = 30
DIC calculation
261.5186
IQGP method
0.0575
ZEP method
6.6968
Gaussian displacement
M = 20
DIC calculation
195.4708
IQGP method
0.0522
ZEP method
3.7865
M = 30
DIC calculation
297.8388
IQGP method
0.0561
ZEP method
5.9821
Physical Experiment
In this section, a physical experiment is conducted to validate the effectiveness of the Recovery method for the second-order shape function and the ZEP method in the practical measurement. A tensile test is performed on a centrally perforated Q235 low-carbon steel specimen to assess these methods. The specimen dimensions are 200 mm in length, 30 mm in width, and 4 mm in thickness, with a centrally prefabricated circular hole of 6 mm in diameter, as illustrated in Fig. 15. This test is a classic approach for generating inhomogeneous deformation, particularly in the region surrounding the hole, where complex displacement occurs.
Fig. 15
The centrally perforated Q235 low-carbon steel specimen (unit: mm)
The MTS 809 Axial/Torsional Test System is used to apply the load, while a Basler a2A4508-20umBAS camera, equipped with a Basler Lens C11-5020-12 M-P, captures images at a resolution of 4508 × 4096 pixels. During the experiment, the lower fixture moves downward at a rate of 2 mm/min. Prior to testing, an image of the specimen surface with speckles is captured as the reference image. The corresponding deformation image is taken at the moment when the displacement of the lower fixture reaches 2.8 mm. The setup of the system is shown in Fig. 16.
In DIC calculations, IC-GN sub-pixel registration algorithm [38] along reliability-guided method [39] are used. In this experiment, DIC calculations are performed by using first-order, second-order, and fourth-order shape functions, with the half-subset size set to 20 pixels. Although the discussion of fourth-order shape function in DIC is not so common, it is incorporated here to provide a approximated real displacement reference since the result of fourth-order shape function contains much less undermatched systematic errors especially compared to the first and second order DIC results. The reference displacement fields in the x- and y-directions obtained by using fourth-order DIC are shown in Figs. 17(a) and 17(b). To facilitate analysis and comparison, the displacement in the x-direction, u, at the cross-section x = 2166 is selected as the target for evaluation. The corresponding results are depicted in Figs. 18 and 19.
Fig. 17
The results of the tensile experiment of the Q235 low-carbon steel centrally perforated specimen by using the fourth-order shape function: a displacement field in the x-direction; b displacement field in the y-direction
a The results comparison of corrected results by applying Recovery method under various order displacement assumption to second-order DIC displacement with the original second-order DIC result and fourth-order DIC result; b Partial magnification of the square region marked in Fig. 18a
a The results comparison of corrected results by applying IQGP method and ZEP method to first-order DIC displacement with the original first-order DIC result and fourth-order DIC result; b Partial magnification of the square region marked in Fig. 19a
As shown in Fig. 18, the results obtained using the Recovery method under different displacement order assumptions are close to those of the original second- and fourth-order shape functions. This similarity can be attributed to the relatively minor and less pronounced undermatched systematic errors associated with the second-order shape function in this example, resulting in minimal variation across different results. On the other hand, as shown in Fig. 19, the result of original first-order DIC has obvious undermatched systematic errors where the ZEP method demonstrates strong effectiveness in mitigating undermatched systematic errors in the first-order shape function, further highlighting its robustness in improving displacement field accuracy. Compared to the IQGP method, result of ZEP method contains smaller and less oscillation.
Discussion: Comparison Between Recovery Method and ZEP Method
In the above Section "Results of Experiment", both simulated and physical experiments are been used to verify the effectiveness of both Recovery method(for the case of second-order shape function) and ZEP method(for the case of first-order shape function). In this section, an experimental comparison between the performance of Recovery method and ZEP method has been setup to get a deeper evaluation of both two methods.
The simulated experiment used here still follows the same setup introduced in Section "Simulated Experiment" while sinusoidal displacement and Gaussian displacement(as shown in Figs. 5 and 6) are still used in effectiveness evaluation in this section. Considering the simplicity and convenience, the comparison here is conducted in the case of first-order shape function. In other words, both the Recovery method and ZEP method are used to revise and mitigate the undermatched systematic errors in results of first-order DIC. Then the revised results by using the Recovery method and ZEP method are evaluated and compared. The experiment of both sinusoidal displacement and Gaussian displacement are conducted in three kinds of subset size: half subset size M fixed at 10, 20 and 30 pixels. Since the ZEP method is based on the third-order displacement assumption in local subset, the selected Recovery method is also based on second- or third-order displacement assumption for better comparison.
The results of comparison in case of sinusoidal displacement are shown in Fig. 20 while the results of Gaussian displacement are presented in Fig. 21. It can be noticed that when the subset size is relatively small(M = 10), the undermatched systematic errors are not so huge. In that case, both the Recovery method and ZEP method can successfully diminish the undermatched systematic errors and their accuracy and effectiveness are similar according to Figs. 20(b) and 21(b). However, when the subset size increases which means the undermatched systematic errors increase, the revised results by Recovery method seems to have more residual error compared to the revised results by ZEP method, which can be seen from Figs. 20(d), (f) and 21(d), (f). Thus, according to the experimental results here, ZEP method seems to have better performance on undermatched systematic error mitigation compared to the Recovery method. In contrast to the ZEP method, the Recovery method achieves higher-order undermatched systematic error correction through iterative combinations of prior S-G filtered data. It seems to be an advantage by eliminating the need to rederive the formula of zero-undermatched-error point coordinates—a requirement inherent to ZEP method when advancing to higher-order corrections. However, as mentioned in Section "Recovery Method of Second-Order Shape Function", the error accumulation and propagation in higher-order-displacement based Recovery method might be a issue requiring attention when using it.
Fig. 20
Displacement and corresponding error comparison of original first-order DIC result with the revised results by Recovery method based on 2nd/3rd-order displacement assumption and ZEP method in Sinusoidal displacement measurement: a, b: when M = 10; c, d: when M = 20; e, f: when M = 30
Displacement and corresponding error comparison of original first-order DIC result with the revised results by Recovery method based on 2nd/3rd-order displacement assumption and ZEP method in Gaussian displacement measurement: a, b: when M = 10; c, d: when M = 20; e, f: when M = 30
This study primarily focuses on improving algorithms of mitigating undermatched systematic errors. To enhance the applicability of the Recovery method, a corresponding theoretical derivation under the case of the second-order shape function is conducted, demonstrating its effectiveness. Besides, based on the case that the current IQGP method only accounts for second-order displacement, a novel approach(ZEP method) considering third-order displacement is proposed, offering greater adaptability for complex deformation scenarios. Finally, experimental results validate the effectiveness of both the Recovery method for second-order shape function and the newly developed approach, confirming their robustness in mitigating systematic errors. Theoretically, following the similar principles presented in this work, both two methods(Recovery method and ZEP method) can be used for the mitigation of undermatched systematic errors of any arbitrary order shape function.
Declarations
Competing Interests
The authors have no relevant financial or non-financial interests to disclose.
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To determine parameter \({{\varvec{u}}}_{0}^{{\varvec{c}}}\), the relevant equations can be extracted and rearranged from Eq. (69) to yield the following Eq. (70), which is the formula seen in previous Eq. (15):
Sutton MA, Matta F, Rizos D, Ghorbani R, Rajan S, Mollenhauer DH et al (2017) Recent progress in digital image correlation: background and developments since the 2013 W m murray lecture. Exp Mech 57:1–30. https://doi.org/10.1007/s11340-016-0233-3CrossRef
Li B, Yang GB, Zhu QR, Ni F (2011) Study on shear test of new style automotive structural adhesive using digital image correlation method. CMC-Comput Mater Continua 21:107–117
Cardoso D, Pereira R, Moura M, Correia N (2025) Assessment of fracture process zone length in carbon fiber-reinforced epoxy composites through digital image correlation. Theor Appl Fract Mech 136:104816. https://doi.org/10.1016/j.tafmec.2024.104816CrossRef
8.
Li D, Wei X, Gao Y, Jiang J, Xia W, Wang B (2024) Investigations on tensile mechanical properties of rigid insulation tile materials at elevated temperatures based on digital image correlation algorithm. Constr Build Mater 413:134925. https://doi.org/10.1016/j.conbuildmat.2024.134925CrossRef
Lu H, Cary PD (2000) Deformation measurements by digital image correlation: implementation of a second-order displacement gradient. Exp Mech 40:393–400. https://doi.org/10.1007/bf02326485CrossRef
Pan B, Wang B (2016) Digital image correlation with enhanced accuracy and efficiency: a comparison of two subpixel registration algorithms. Exp Mech 56:1395–1409. https://doi.org/10.1007/s11340-016-0180-zCrossRef
17.
Schreier HW, Braasch JR, Sutton MA (2000) Systematic errors in digital image correlation caused by intensity interpolation. Opt Eng 39:2915–2921. https://doi.org/10.1117/1.1314593CrossRef
18.
Su Y, Zhang QC, Fang Z, Wang YR, Liu Y, Wu SQ (2019) Elimination of systematic error in digital image correlation caused by intensity interpolation by introducing position randomness to subset points. Opt Lasers Eng 114:60–75. https://doi.org/10.1016/j.optlaseng.2018.10.012CrossRef
19.
Baldi A, Bertolino F (2015) Experimental analysis of the errors due to polynomial interpolation in digital image correlation. Strain 51:248–263. https://doi.org/10.1111/str.12137CrossRef
Schreier HW, Sutton MA (2002) Systematic errors in digital image correlation due to undermatched subset shape functions. Exp Mech 42:303–310. https://doi.org/10.1007/bf02410987CrossRef
25.
Pan B, Zou X (2020) Quasi-Gauss point digital image/volume correlation: a simple approach for reducing systematic errors due to undermatched shape functions. Exp Mech 60:627–638. https://doi.org/10.1007/s11340-020-00588-3CrossRef
Jones EMC, Iadicola MA, Bigger R, Blaysat B, Boo C, Grewer M et al (2018) A Good Practices Guide for Digital Image Correlation, International Digital Image Correlation Society https://doi.org/10.32720/idics/gpg.ed1
Hu JC, Miao H, Xu JC, Zhang RY, Lai LZ, Deng YM (2025) Adaptive undermatched systematic error mitigation algorithm considering spatial continuity in digital image correlation. Surface Topography: Metrology and Properties. https://doi.org/10.1088/2051-672X/ade7a4CrossRef
33.
Hu JC, Xu JC, Zhang RY, Lai LZ, Miao H (2025) An Improved Undermatched Systematic Error Compensation Method in Digital Image Correlation. J Exp Mech (In Chinese). https://doi.org/10.7520/1001-4888-24-063CrossRef
34.
Grédiac M, Blaysat B, Sur F (2019) A robust-to-noise deconvolution algorithm to enhance displacement and strain maps obtained with local DIC and LSA. Exp Mech 59:219–243. https://doi.org/10.1007/s11340-018-00461-4CrossRef
Pan B, Xie HM, Wang ZY, Qian KM, Wang ZY (2008) Study on subset size selection in digital image correlation for speckle patterns. Opt Express 16:7037–7048. https://doi.org/10.1364/oe.16.007037CrossRef
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