Self-Adaptive Moving Least Squares Measurement Based on Digital Image Correlation

Abstract

Digital image correlation (DIC) is a non-contact measurement technique used to evaluate surface deformation of objects. Typically, pointwise moving least squares (PMLS) fitting is applied to process the noisy data from DIC to obtain an accurate strain field. In this study, a self-adaptive pointwise moving least squares (SPMLS) method was developed to optimize the process of window size selection, thereby attaining superior accuracy in measurements. The premise of this method is that the noise in the displacement field follows white Gaussian noise. Under this assumption, it analyses the random errors and systematic errors of the PMLS method under different calculation window sizes. The optimal size of the calculation window is determined by minimizing the errors. Subsequently, the strain field is computed based on the optimized calculation window. The results were compared with a typical PMLS method. Whether calculating low-gradient strain fields or high-gradient strain fields, the computational accuracy of SPMLS is close to the optimal accuracy of PMLS. This study effectively addresses the inherent challenge of manually selecting window size in the PMLS method.

1. Introduction

Non-contact measurement is a technique for obtaining relevant information about the target object without physically touching its surface. It has been applied in various fields, such as infrared thermometry and thermal imaging, to collect human body data without affecting human health [1,2]. Deep-learning-based computer vision techniques leverage the capabilities of deep learning algorithms to analyze images captured by optical cameras, thereby improving the precision and efficiency of material assessment [3,4]. A laser scanner has the capability to generate point clouds, offering detailed information regarding deformation [5]. Fiberoptic sensors are used to monitor the structural integrity of infrastructure by measuring strain, temperature, and vibration [6]. Nonetheless, the cost and efficiencies associated with implementing these advanced methods remain significant issues.

Digital image correlation (DIC) is a non-contact optical measurement technique that can measure surface deformation of objects and has been applied in various fields including biology, geotechnical engineering, and bridge engineering [7,8,9,10]. Notably, the DIC system stands out as a cost-effective solution that facilitates real-time measurements of materials [11]. This technique captures the displacement field by analyzing changes in speckle patterns on the object’s surface before and after displacement, relying on a surface with sufficient texture or speckles, no obstructions, and appropriate illumination for clear image capture [12]. Subsequently, based on the displacement field data, the strain field of the object can be computed [13,14,15,16]. Therefore, DIC provides complete displacement and strain data for every point across the surface of the object, enabling the detection of localized deformations and stress concentrations [17]. In particular, DIC also generates visual representations of deformation and strain patterns, simplifying data interpretation for researchers and engineers [18,19,20]. However, the displacement field measured by DIC is affected by noise and system errors, leading to unreliable calculation results [21]. Therefore, data processing is a necessary step to obtain accurate strain fields [22].

In the literature, numerous data processing methods have been developed to improve the reliability of the DIC technique [23,24]. Pointwise least squares (PLS) is a typical data processing method developed based on the principle of least squares. PLS is specifically employed for fitting data within local regions to obtain precise measurements of displacement values [25]. PLS plays an important role in the DIC technique, enabling accurate calculation of displacements and deformations from image sequences [23,25]. However, the accuracy of PLS is significantly influenced by the choice of kernel function and window size, especially in scenarios involving high-gradient strain fields [26].

Another approach to mitigate the impact of noise is through the incorporation of Tikhonov regularization in both the fast Hermite element method (HEM) [27] and the regular polynomial smoothing method (RPSM) [28]. However, these methods necessitate extensive calculations and complex programming, making them time-consuming and impractical for real-time applications. Recently, moving least squares (MLS) has emerged as another crucial technique adopted in the DIC technique due to its remarkable smoothing and denoising capabilities [29,30]. The MLS creates shape functions in meshless methods, facilitating the creation of a smooth displacement field and consequently resulting in a reliable strain field. Building upon the principles of MLS, pointwise moving least squares (PMLS) was developed to enhances computational efficiency while maintaining high accuracy [31]. PMLS achieves this by treating the support domain as an independent window.

Table 1. Comparison of advantages and disadvantages of various calculation methods.

Method	Advantages	Disadvantages
HEM	1. High-order interpolation capabilities; 2. Better curve approximation.	1. Requires high data smoothness; 2. Noisy or fluctuating data reduce accuracy; 3. Needs sufficient data points for effective interpolation.
RPSM	1. Easy to implement with kernel functions; 2. High accuracy and stability with Tikhonov regularization and GCV function.	1. Difficult to choose optimal polynomial order and window size; 2. High computational complexity.
MLS	1. Handles irregularity and non-uniformity effectively; 2. Provides precise smoothing results.	1. High computationally complex; 2. Difficult to find optimal parameters
SPMLS	1. Adaptively selects computation window size, improving strain calculation accuracy; 2. Higher accuracy and stability in noisy conditions.	Systematic error formula is derived under specific window conditions and not applicable to incomplete windows.

lists the advantages and disadvantages of some methods.

Several challenges have been identified in the application of data processing methods for the DIC technique, according to the literature review:

(1)

It remains unknown as to how to optimize the calculation parameters in data processing methods. The calculation parameters for the aforementioned methods require manual selection [31,32], and the choice of these parameters significantly affects the accuracy of the calculation results. Among these parameters, the window size plays a crucial role influencing both random errors and systematic errors, ultimately determining the results of these processing methods [21,31,32,33,34]. Random errors usually arise from data quality, while systematic errors stem from the mismatch between the basis function and strain. In regions with low strain gradients, random errors generally have a greater effect than systematic errors [33]. In this case, a large window is required to reduce the impact of random errors. For regions subjected to high strain gradients, systematic errors have a greater impact [33], and utilizing a smaller calculation window can effectively mitigate system errors.

(2)

It is challenging to obtain accurate strain fields from current methods. Currently, the primary methods for acquiring strain fields involve local smoothing methods [23,25] and global smoothing methods [35]. Local smoothing methods offer advantages such as simplicity and efficient computation, but their accuracy is significantly influenced by the choice of fitting window [36]. Global smoothing methods are more complex, leading to higher computational expenses. Despite efforts by some researchers to enhance these methods, their computational efficiency remains limited [27]. In addition, there are certain differences in the optimal window selection based on high-gradient deformation fields and low-gradient deformation fields [37,38].

These gaps in knowledge have hindered broader applications of the DIC technique, as it remains unclear as to how to properly interpret sensing data in the presence of complex deformations. To address these challenges, the overarching goal of this study was to develop a self-adaptive pointwise moving least squares (SPMLS) method. This approach allows for the automatic adjustment of the optimal window in the PMLS method, thereby significantly mitigating the impact of variations in window size on measurement accuracy. Specifically, this research features two main contributions:

(1)

This study develops the SPMLS method, designed to autonomously determine the optimal window size for strain field calculations. Specifically, it derives the formulas for calculating both systematic and random errors associated with the PMLS method, particularly in the context of employing quadratic basis functions. By combining these two types of errors, the total error is obtained. Subsequently, the total error is calculated for different window sizes, and the window size corresponding to the minimum total error is selected for strain calculations. This approach not only effectively improves calculation accuracy but also resolves the issue of manually selecting the calculation window size.

(2)

This paper utilizes two simulation experiments to evaluate the performance of the SPMLS method, testing both high-gradient and low-gradient strain fields with various intensities of Gaussian noise added. By comparing the root mean square error (RMSE) of the PMLS and SPMLS methods under these conditions, the results confirm that the SPMLS method selects a calculation window size close to the optimal value, significantly enhancing calculation accuracy. This underscores the superiority of the SPMLS method over manual window size selection.

The remainder of the paper is structured as follows: Section 2 introduces the principle of the proposed methods; Section 3 presents the experimental program and discusses the results; Section 4 performs a case study; and Section 5 summarizes the conclusions.

2. Methodology

2.1. Digital Image Correlation

Digital-image-related methods calculate the displacement and strain of the material surface by comparing images before and after deformation. The surface of the material under investigation was prepared with speckle patterns, which provide a random, high-contrast texture. Images of the speckled surface were captured using a camera. Initially, images were taken of the undeformed state. The material was then subjected to some form of stress or load, causing it to deform. During deformation, additional images were captured to record the changes in the speckle pattern. The images were analyzed using specialized DIC method to calculate the displacement and strain of the materials. As shown in Figure 1, a reference subset was established based on the point $P(x_0,y_0)$ to be measured in the reference image before the object was deformed. The deformed image was searched through the predefined correlation function to obtain the most matching target subset, whose center point was $P^{\prime }(x_0^{\prime },y_0^{\prime })$ [39].

The point $Q(x_i,y_j)$ in the reference subset can be mapped to the point $Q^{\prime }(x_i^{\prime },y_j^{\prime })$ in the target subset through functional expressions, as shown in Equation (1) [23].

$$\left\{\begin{array}{l}{x^{\prime }}_i=x_i+u+{\displaystyle \frac{\partial u}{\partial x}}\Delta x+{\displaystyle \frac{\partial u}{\partial y}}\Delta y\\ {y^{\prime }}_j=y_j+v+{\displaystyle \frac{\partial v}{\partial x}}\Delta x+{\displaystyle \frac{\partial v}{\partial y}}\Delta y\end{array}\right.$$

(1)

where u and v are the displacements of P in the x and y directions; $\Delta x$ and $\Delta y$ are the distances between P and Q in the x and y directions, respectively; and ${\displaystyle \frac{\partial u}{\partial x}},{\displaystyle \frac{\partial u}{\partial y}},{\displaystyle \frac{\partial v}{\partial x}}$, and ${\displaystyle \frac{\partial v}{\partial y}}$ are the parameters describing the displacement gradients of the subset.

The displacement accuracy obtained by the DIC method was sufficient, but the strain accuracy was poorer due to noise influence. Therefore, it was necessary to process the data obtained by the DIC method. The PMLS method is one approach to fitting the initial displacements obtained by DIC to derive the strain [27,31].

2.2. Pointwise Moving Least Squares Method

Employing the PMLS method to fit noisy displacement data derived from the DIC method can result in a more precise strain field, enhancing the accuracy of the analysis. The PMLS method introduces a weighting function based on the PLS method, which allows for a more accurate representation of the strain field compared to the PLS method [31]. The PLS method assigns equal weight to each point within the calculation window, making its calculation accuracy vulnerable to changes in the window size and outliers. In contrast, the PMLS method, which extends from PLS, integrates a weighting function to enhance stability, making it more suitable at calculating high-gradient strain fields. Through the pointwise moving least squares (PMLS) method, the horizontal strain (${\epsilon }_{xx}$), vertical strain (${\epsilon }_{yy}$), and shear strain (${\gamma }_{xy}$) can be obtained [31]:

$$\left\{\begin{array}{l}Aa=Bu\\ Ab=Bv\end{array}\right.$$

(2)

$$\left\{\begin{array}{l}{\epsilon }_{xx}={\displaystyle \frac{\partial u}{\partial x}}\\ {\epsilon }_{yy}={\displaystyle \frac{\partial v}{\partial y}}\\ {\gamma }_{xy}={\displaystyle \frac{\partial u}{\partial y}}+{\displaystyle \frac{\partial v}{\partial x}}\end{array}\right.$$

(3)

$$A\left(x,y\right)=\left[p\left(-M,-M\right),\dots ,p\left(i,j\right),\dots ,p\left(M,M\right)\right]W{\left[p\left(-M,-M\right),\dots ,p\left(i,j\right),\dots ,p\left(M,M\right)\right]}^T$$

(4)

$$B\left(x,y\right)=\left[p\left(-M,-M\right),\dots ,p\left(i,j\right),\dots ,p\left(M,M\right)\right]W$$

(5)

where $u\left(x,y\right)$ and $v\left(x,y\right)$ are the horizontal displacement field and vertical displacement field obtained by the DIC method, respectively; $a(x,y)$ and $b(x,y)$ are coefficients of basis function; M is the half-width of the computation window size and is a positive integer; $w\left(x+i,y+j\right)$ is the weighted function; W is a diagonal matrix with $w\left(x+i,y+j\right)$ on its main diagonal, serving as a weight function matrix; and $p\left(x,y\right)$ is the basis function, expressed as

$$p\left(x,y\right)={\left[\begin{array}{ccc}\begin{array}{ccc}\begin{array}{ccc}\begin{array}{ccc}1& x& y\end{array}& x^2& xy\end{array}& y^2& \cdots \end{array}& x{y}^{k-1}& y^k\end{array}\right]}^T$$

(6)

where $k$ represents the degree of the basis function, and the degree of calculation accuracy is evaluated. A higher degree indicates higher calculation accuracy.

This article used the quadratic basis function. $a(x,y)$ is the coefficient of the basis function. Taking the horizontal direction as an example, it can be expressed as follows:

$$a\left(x,y\right)=A^{-1}Bu$$

(7)

$w\left(x+i,y+j\right)$ is a weighted function with compact support. Normally, Gaussian functions or spline functions are used as weighting functions in the DIC technique. However, the results of Gaussian functions are usually influenced by shape parameters [40], which is why they were not employed in this article. Therefore, a fifth-degree spline function is used as weighted function, as shown below:

$$w\left(s\right)=1-10s^3+15s^4-6s^5$$

(8)

where $s=\sqrt{{\left({\displaystyle \frac{\left|i\right|}{M}}\right)}^2+{\left({\displaystyle \frac{\left|j\right|}{M}}\right)}^2}$.

2.3. Self-Adapting Pointwise Moving Least Squares

The SPMLS method, based on the PMLS method, can adaptively select the computation window size, further improving the accuracy of strain calculation. PMLS can be expressed as $Aa=Bu$, where the DIC displacement field u can be expressed as $u=u_{theory}+u_{error}$, where $u_{theory}$ is the actual displacement value, and $u_{error}$ is the displacement error; $a$ can be expressed as $a=a_{theory}+a_{error}$, where $a_{theory}$ is the real solution of deformation, and $a_{error}$ is the deformation error. The errors in the calculation process of the PMLS method can be divided into systematic errors and random errors. During calculation, the accurate strain field cannot be measured if the DIC displacement field itself deviates greatly. Therefore, this method only considers the DIC displacement field noise and the error caused by the mismatch between the basis function and the real deformation. This assumes that it is an independent Gaussian white noise with a mean of zero and a standard deviation of $\sigma$ in each window. This assumption requires high accuracy of the DIC displacement field. In the actual measurement, the displacement field deviation can be reduced by using low-pass filtering to pre-process the image and use appropriate shape functions for approximation.

The strain system error is mainly caused by a mismatch between the basis function and the order of deformation to be fitted. In this study, the quadratic basis function in the PMLS method was taken as an example to deduce the systematic error in the x direction. For the PMLS method, $Aa\left(x,y\right)=Bu$, which is expressed as

$$\left(\begin{array}{cccccc}{\displaystyle {\sum }_{i,j}{w}_{ij}}& {\displaystyle {\sum }_{i,j}{w}_{ij}}i& {\displaystyle {\sum }_{i,j}{w}_{ij}}j& {\displaystyle {\sum }_{i,j}{w}_{ij}{i}^2}& {\displaystyle {\sum }_{i,j}{w}_{ij}}ij& {\displaystyle {\sum }_{i,j}{w}_{ij}{j}^2}\\ {\displaystyle {\sum }_{i,j}{w}_{ij}}i& {\displaystyle {\sum }_{i,j}{w}_{ij}{i}^2}& {\displaystyle {\sum }_{i,j}{w}_{ij}}ij& {\displaystyle {\sum }_{i,j}{w}_{ij}}{i}^3& {\displaystyle {\sum }_{i,j}{w}_{ij}}{i}^{2}j& {\displaystyle {\sum }_{i,j}{w}_{ij}}i{j}^2\\ {\displaystyle {\sum }_{i,j}{w}_{ij}}j& {\displaystyle {\sum }_{i,j}{w}_{ij}}ij& {\displaystyle {\sum }_{i,j}{w}_{ij}{j}^2}& {\displaystyle {\sum }_{i,j}{w}_{ij}}{i}^{2}j& {\displaystyle {\sum }_{i,j}{w}_{ij}}i{j}^2& {\displaystyle {\sum }_{i,j}{w}_{ij}}{j}^3\\ {\displaystyle {\sum }_{i,j}{w}_{ij}{i}^2}& {\displaystyle {\sum }_{i,j}{w}_{ij}}{i}^3& {\displaystyle {\sum }_{i,j}{w}_{ij}{i}^2}j& {\displaystyle {\sum }_{i,j}{w}_{ij}{i}^4}& {\displaystyle {\sum }_{i,j}{w}_{ij}{i}^{3}j}& {\displaystyle {\sum }_{i,j}{w}_{ij}{i}^2}{j}^2\\ {\displaystyle {\sum }_{i,j}{w}_{ij}}ij& {\displaystyle {\sum }_{i,j}{w}_{ij}}{i}^{2}j& {\displaystyle {\sum }_{i,j}{w}_{ij}}i{j}^2& {\displaystyle {\sum }_{i,j}{w}_{ij}}{i}^{3}j& {\displaystyle {\sum }_{i,j}{w}_{ij}{i}^2}{j}^2& {\displaystyle {\sum }_{i,j}{w}_{ij}}i{j}^3\\ {\displaystyle {\sum }_{i,j}{w}_{ij}{j}^2}& {\displaystyle {\sum }_{i,j}{w}_{ij}i{j}^2}& {\displaystyle {\sum }_{i,j}{w}_{ij}{j}^3}& {\displaystyle {\sum }_{i,j}{w}_{ij}{i}^2}{j}^2& {\displaystyle {\sum }_{i,j}{w}_{ij}i{j}^3}& {\displaystyle {\sum }_{i,j}{w}_{ij}{j}^4}\end{array}\right)\left(\begin{array}{c}a\left(x,y\right)\\ a_x\left(x,y\right)\\ a_y\left(x,y\right)\\ a_{xx}\left(x,y\right)\\ a_{xy}\left(x,y\right)\\ a_{yy}\left(x,y\right)\end{array}\right)=\left(\begin{array}{c}{\displaystyle {\sum }_{i,j}{w}_{ij}u\left(x+i,y+j\right)}\\ {\displaystyle {\sum }_{i,j}i{w}_{ij}u\left(x+i,y+j\right)}\\ {\displaystyle {\sum }_{i,j}j{w}_{ij}u\left(x+i,y+j\right)}\\ {\displaystyle {\sum }_{i,j}{i}^2{w}_{ij}u\left(x+i,y+j\right)}\\ {\displaystyle {\sum }_{i,j}ij{w}_{ij}u\left(x+i,y+j\right)}\\ {\displaystyle {\sum }_{i,j}{j}^2{w}_{ij}u\left(x+i,y+j\right)}\end{array}\right)$$

(9)

where $i,j\in [-M,M]$, M is the size of half window; $w_{ij}=w\left(x+i,y+j\right)$ is the weight of each point. Given that $w_{ij}$ is solely dependent on distance, it exhibits symmetry. Since $i,j\in [-M,M]$, it follows that ${\displaystyle {\sum }_{i,j}{w}_{ij}i=0}$, ${\displaystyle {\sum }_{i,j}{w}_{ij}j=0}$, ${\displaystyle {\sum }_{i,j}{w}_{ij}ij=0}$, ${\displaystyle {\sum }_{i,j}{w}_{ij}{i}^{2}j=0}$, ${\displaystyle {\sum }_{i,j}{w}_{ij}i{j}^2=0}$, ${\displaystyle {\sum }_{i,j}{w}_{ij}{i}^{3}j=0}$, and ${\displaystyle {\sum }_{i,j}{w}_{ij}i{j}^3=0}$. Upon substituting the aforementioned values of zero for the respective parameters into Equation (9), it is derived that

$$\left(\begin{array}{cccccc}{\displaystyle {\sum }_{i,j}{w}_{ij}}& 0& 0& {\displaystyle {\sum }_{i,j}{w}_{ij}{i}^2}& 0& {\displaystyle {\sum }_{i,j}{w}_{ij}{j}^2}\\ 0& {\displaystyle {\sum }_{i,j}{w}_{ij}{i}^2}& 0& 0& 0& 0\\ 0& 0& {\displaystyle {\sum }_{i,j}{w}_{ij}{j}^2}& 0& 0& 0\\ {\displaystyle {\sum }_{i,j}{w}_{ij}{i}^2}& 0& 0& {\displaystyle {\sum }_{i,j}{w}_{ij}{i}^4}& 0& {\displaystyle {\sum }_{i,j}{w}_{ij}{i}^2}{j}^2\\ 0& 0& 0& 0& {\displaystyle {\sum }_{i,j}{w}_{ij}{i}^2}{j}^2& 0\\ {\displaystyle {\sum }_{i,j}{w}_{ij}{j}^2}& 0& 0& {\displaystyle {\sum }_{i,j}{w}_{ij}{i}^2}{j}^2& 0& {\displaystyle {\sum }_{i,j}{w}_{ij}{j}^4}\end{array}\right)\left(\begin{array}{c}a\left(x,y\right)\\ a_x\left(x,y\right)\\ a_y\left(x,y\right)\\ a_{xx}\left(x,y\right)\\ a_{xy}\left(x,y\right)\\ a_{yy}\left(x,y\right)\end{array}\right)=\left(\begin{array}{c}{\displaystyle {\sum }_{i,j}{w}_{ij}u\left(x+i,y+j\right)}\\ {\displaystyle {\sum }_{i,j}i{w}_{ij}u\left(x+i,y+j\right)}\\ {\displaystyle {\sum }_{i,j}j{w}_{ij}u\left(x+i,y+j\right)}\\ {\displaystyle {\sum }_{i,j}{i}^2{w}_{ij}u\left(x+i,y+j\right)}\\ {\displaystyle {\sum }_{i,j}ij{w}_{ij}u\left(x+i,y+j\right)}\\ {\displaystyle {\sum }_{i,j}{j}^2{w}_{ij}u\left(x+i,y+j\right)}\end{array}\right)=\left(\begin{array}{c}{P}_1\\ P_2\\ P_3\\ P_4\\ P_5\\ P_6\end{array}\right)$$

(10)

For simplicity, let $Bu=P$, which yields $a_x\left(x,y\right){\displaystyle {\sum }_{i,j}{w}_{ij}}{i}^2={\displaystyle {\sum }_{i,j}i{w}_{ij}u\left(x+i,y+j\right)=P_2}$. Perform Taylor expansion on $u\left(x+i,y+j\right)$ and obtain the strain of second-order PMLS as

$$a_x\left(x,y\right)={\displaystyle \frac{\partial u\left(x,y\right)}{\partial x}}+{\displaystyle \frac{{\displaystyle {\sum }_{i,j}\left(\frac{1}{6}{w}_{ij}{i}^4\frac{{\partial }^{3}u\left(x,y\right)}{\partial x^3}+\frac{1}{2}{w}_{ij}{i}^2{j}^2\frac{{\partial }^{3}u\left(x,y\right)}{\partial x\partial y^2}\right)}}{{\displaystyle {\sum }_{i,j}{w}_{ij}{i}^2}}}$$

(11)

The strain system error of second-order PMLS is

$$e_s=a_x\left(x,y\right)-{\displaystyle \frac{\partial u\left(x,y\right)}{\partial x}}={\displaystyle \frac{{\displaystyle {\sum }_{i,j}\left(\frac{1}{6}{w}_{ij}{i}^4\frac{{\partial }^{3}u\left(x,y\right)}{\partial x^3}+\frac{1}{2}{w}_{ij}{i}^2{j}^2\frac{{\partial }^{3}u\left(x,y\right)}{\partial x\partial y^2}\right)}}{{\displaystyle {\sum }_{i,j}{w}_{ij}{i}^2}}}$$

(12)

Each displacement gradient was obtained by the central difference method. However, the random error of the DIC displacement field affects the accuracy of the displacement gradient. For cases where the random error has a large impact, the displacement gradient is not reliable, and the strain system error obtained thereby is also inaccurate. The solution to the strain random error must first be based on the $u_{error}$ assumption, and the covariance matrix of $u_{error}$ is obtained as ${\sigma }^{2}I$. Here, $I$ is the identity matrix, and $\sigma$ is the random error of the DIC displacement field [41].

$$\sigma ={\displaystyle \frac{\sqrt{2}{\sigma }_0}{\sqrt{{\displaystyle \sum {\displaystyle \sum f_x^2}}}}}$$

(13)

where ${\sigma }_0$ is the standard deviation of noise, and $f_x$ is the grey level gradient in the horizontal direction, representing the difference in greyscale values between two adjacent points along that direction. Then, the variance matrix of the deformation error $a_{error}$ in the PMLS method is obtained [33]:

$$\mathrm{Var}\left(a_{error}\right)={\sigma }^2\mathrm{diag}\left[A^{-1}B{\left(A^{-1}B\right)}^{\mathrm{T}}\right]$$

(14)

The strain random error $e_r$ is the arithmetic square root of $\mathrm{Var}{\left(a_{error}\right)}_{\left(2,2\right)}$. From this, the calculation formula of the total strain error is obtained:

$$e_a=\sqrt{c{\left(e_s\right)}^2+{\left(e_r\right)}^2}$$

(15)

It is noted that the systematic error in displacement gradients is sensitive to noise interference during the calculation process [33]. Gaussian low-pass filtering (GLPF) can attenuate the influence of noise interference on calculation results to some extent. However, determining suitable calculation parameters in GLPF is still challenging [42]. Therefore, the weight coefficient $c$ is introduced. As shown in Equation (16), when the displacement gradient of the point to be fitted is much larger than the random error (not less than 10), then the impact of the random error on the displacement gradient can be ignored and the $c$ value is 1; otherwise, the $c$ value is 0.

$$c=\left\{\begin{array}{cc}0& u_x<10\sigma \\ 1& u_x\ge 10\sigma \end{array}\right.$$

(16)

where u_x is the displacement gradient, u_x = [u(x + 1, y) − u(x − 1, y)]/2.

In images, each point to be fitted comes with a set of error values that vary depending on the selected window size. This paper compares the total strain error under different windows and uses the window with the smallest total strain error as the optimal window to calculate the strain.

The calculation process of SPMLS method is shown in Figure 2. The general process can be summarized as follows: (1) The images used for DIC processing have been prepared with speckle patterns. Each speckle point is used as a point of interest. (2) Calculate random errors and system errors. The random errors in the displacement field corresponding to different half-window sizes (M) are calculated using Equation (13). Then, the total strain error (e_r) and weight (c) can be determined based on the calculated values $\sigma$ and by applying Equations (15) and (16). Next, substitute each displacement gradient into Equation (12) to obtain the system error for each M. (3) Select the optimal window size. Using the results from Step 2, calculate the total strain error for each M. The value of M that corresponds to the minimum total error indicates the optimal window size. (4) Calculate strain. Use the optimal window size to calculate the strain. After repeating the above steps for all points of interest, the final output is the calculated strain field of the specified object.

Note: Points of interest are the points within a designated region of interest that are selected for calculation, with the interval between each point measured in pixels.

3. Experimental Program

3.1. Simulation Study

This section may be divided by subheadings. It should provide a concise and precise description of the experimental results, their interpretation, and the experimental conclusions that can be drawn.

In this study, an open-source software called Glare (20211121) [43] was utilized to generate speckle patterns and corresponding deformation images. The selected speckle parameters in the software are as follows: diameter of 4 pixels, compactness of 60%, and a degree of 40%. This article does not consider the impact of speckle parameter settings on the calculation results, which will be analyzed in subsequent research. The size of the image was 260 pixels × 260 pixels, and the region of interest was 220 pixels × 220 pixels, with 48,400 points of interest. The strain $u_1=2\sin \left(2\pi x/260\right)$ and $u_2=2\sin \left(4\pi x/260\right)$, with Gaussian noise with a mean of 0 and different standard deviations of 0.01 (2.55 bit) and 0.02 (5.10 bit). The DIC method adopts a first-order shape function, calculates a subset size of 21 pixels × 21 pixels, and uses the normalized least square distance (ZNSSD) function for matching.

3.2. Simulation Results

Figure 3 shows the reference image and deformed image after adding noise with a standard deviation of 0.01. The PMLS method, SPMLS method, PLS method, and the existing SPLS method [33] (can be regarded as a special case of the SPMLS method with $w_{ij}$ being constant) were used to fit the displacement field under different noise conditions. Due to the need for sufficient valid points in second-order basis function calculations, a minimum window size of 5 was used. To balance efficiency with processing time, the maximum window size was set at 31. Window sizes can be adjusted as needed for specific computations. The root mean square error was used as the criterion for evaluating the accuracy of strain calculation results.

3.3. Discussion

As shown in Figure 4, the PLS method performed better than the PMLS method for calculating strain fields that were relatively gentle or exhibited gradual variations. This phenomenon arose due to the interplay of systematic and random errors in the context of calculating a uniform and smoothly varying strain field. When dealing with such scenarios, where systematic errors are minimal, the pre-eminence of random errors shapes the overall error profile. Notably, the PLS method, characterized by uniform weight distribution, yielded superior smoothing effects compared to the PMLS method, resulting in a strain field of heightened smoothness.

However, from a precision standpoint, the distinction between the two methods was marginal. In scenarios involving the computation of high-gradient strain fields, PMLS exhibited diminished accuracy when the calculation window size was small, in contrast to the more uniform weight allocation of the PLS method. As the calculation window size increased, systematic errors progressively assumed dominance, thereby accentuating the advantages of the PMLS method. This was reflected in its superior precision and stability over the PLS method. Regardless of the method employed, be it PLS or PMLS, an escalation in noise intensity universally leads to an attenuation of overall calculation accuracy.

The PLS method excelled in fitting strain fields governed predominantly by random errors, while the PMLS method found its niche in the computation of strain fields where systematic errors predominate. Both methodologies, however, exhibited variability in their results, with alterations in the calculation window size. Practical implementation introduces challenges due to the difficulty in precisely measuring strain gradients and noise, compounded by the unreliability of manually selected calculation window sizes, underscoring the inherent limitations of these approaches.

As shown in Figure 4a, when calculating displacement field $u_1$, the optimal accuracies of PMLS (1%), PLS (1%), SPMLS (1%), SPLS (1%), PMLS (2%), PLS (2%), SPMLS (2%), and SPLS (2%) were 0.0010, 0.0010, 0.0011, 0.0011, 0.0011, 0.0011, 0.0012, and 0.0012, respectively. The optimal window sizes for PMLS and PLS were relatively large. This is because the computation errors for low-gradient strain fields are dominated by random errors, and increasing the computation window size reduces random errors. The accuracy obtained by the SPMLS and SPLS was close to the optimal accuracy of the PMLS and PLS. It can be seen that both SPMLS and SPLS selected computation window sizes close to optimal when computing low-gradient strain fields. Both methods are suitable for computing low-gradient strain fields. As shown in Figure 4b, when calculating displacement field $u_2$, the optimal accuracies of PMLS (1%), PLS (1%), SPMLS (1%), SPLS (1%), PMLS (2%), PLS (2%), SPMLS (2%), and SPLS (2%) were 0.0040, 0.0040, 0.0041, 0.0041, 0.0042, 0.0042, 0.0043, and 0.0046 respectively. The optimal computation window sizes for PMLS and PLS were relatively small. This is because the computation errors for high-gradient strain fields are dominated by systematic errors, which increase as the computation window size increases. The calculation accuracy of the SPMLS method was nearly identical to the optimal value of the PMLS method, with a variance of less than 2.5%. The computational performance of SPLS (1%) was excellent, but SPLS (2%) performed poorly. This is because larger noise levels affect the determination of the optimal computation window size for both SPMLS and SPLS, with PLS being more sensitive to changes in computation window size compared to SPMLS.

Overall, the accuracy of computations for PMLS, SPMLS, PLS, and SPLS decreased with increasing strain field gradients and noise intensity. When computing low-gradient strain fields, both SPMLS and SPLS performed well. However, when computing high-gradient strain fields, especially under high-noise conditions, SPMLS outperformed SPLS.

4. Case Study

4.1. Specimens and Investigated Cases

The material came from the experiments in “sample13” and “sample12” in the public data of the Society for Experimental Mechanics (SEM). The data of “sample13” were gathered from a 2D experiment that utilized a ring light and bead-blasted surface preparation. The detailed data can be found in reference [44]. The size of the image was 1040 pixels × 1392 pixels, and the region of interest was 700 pixels × 700 pixels, with 490,000 points of interest.

“Sample12” constitutes a purely experimental dataset featuring a painted speckle pattern applied to a tensile specimen with a central hole. This dataset proves valuable for conducting studies on virtual strain gauge sizes, exploring the spatial resolution and noise robustness of DIC codes, as well as facilitating round-robin tests to compare different DIC formulations. The size of the image was 400 pixels × 1040 pixels, and the region of interest was 300 pixels × 800 pixels, with 240,000 points of interest.

4.2. Results and Discussion

The reference image and the deformed image are shown in Figure 5a,b and Figure 6a,b. The subset size was 31 pixels × 31 pixels, and the other calculation parameters were the same as in Section 3. The displacement field obtained by DIC analysis is shown in Figure 5c and Figure 6c. The displacement field was fitted using the quadratic basis function PMLS method and SPMLS method. Since the true strain field in actual experiments is unknown, the optimal parameters of the PMLS method cannot be accurately determined. Therefore, the PMLS method selection windows were 7, 13, 19, 25, and 31, which are represented as PMLS-7, PMLS-13, PMLS-19, PMLS-25, and PMLS-31, respectively. The calculation results of strain at the black line in Figure 5c and Figure 6c were selected to be compared with SPMLS, as shown in Figure 7.

Note: True strain refers to the actual strain in a specific direction, which is not obtainable in real experiments. However, in simulation experiments, since the deformation function is known, the true strain at each point can be obtained and used to analyze the error of the strain obtained by various algorithms.

Figure 7 proves that the selection of window size in the PMLS method significantly affects its computational results. As shown in Figure 7a, when calculating “sample13”, due to significant strain variations, each region was affected differently by noise. In high-strain regions, the impact of noise was relatively minor, resulting in less variation in the strain curve. Conversely, in low-strain regions, the influence of noise was more pronounced. The difference in smoothness between PMLS-7 and PMLS-13 is attributed to smaller computation windows leading to larger random errors caused by noise. In contrast, PMLS-19, PMLS-25, and PMLS-31 demonstrated better smoothness. The strain curves of SPMLS closely resemble those of the latter three methods in low-strain regions, showcasing excellent noise reduction and smoothing capabilities. However, in high-strain regions, the strain curves of SPMLS are closer to those of PMLS-7 and PMLS-13, indicating poorer smoothness. This is because SPMLS tends to revert to DIC displacement field strain in high-gradient areas, thereby opting for smaller computation windows.

As shown in Figure 7b, when calculating “sample12”, the strain curves of PMLS-7, PMLS-13, PMLS-19, and PMLS-25 were all insufficiently smooth, and the smoothness decreased as the calculation window size became smaller. This is attributed to the relatively gentle strain gradients in the “sample12”, where the order of the quadratic kernel function is adequate for expressing the required deformation order. Therefore, during calculation, only random errors introduced by noise must be considered. In contrast, the strain curve of PMLS-31 was the smoothest, and its calculation accuracy was not inferior to the strain curves obtained using other calculation window sizes in the PMLS method. The strain curve obtained by the SPMLS method almost completely overlapped with that of PMLS-31. Through error analysis, the calculation window size obtained for over 95% of the errors was 31 pixels × 31 pixels. Notably, when comparing PMLS and SPMLS in calculating the strain curves for “sample12” and “sample13”, the maximum differences at the same position were 0.00279 and 0.00462, respectively. Figure 7 shows that, in practical calculations, the SPMLS method also determines the appropriate calculation window size by analyzing both systematic errors and random errors. It suggests that the SPMLS method possesses high adaptability and is well-suited for handling complex strain fields.

4.3. Discussion on Limitation

Although SPMLS exhibits good computational performance under different noise and deformation conditions, there are still some limitations:

(1)

The formula for systematic error is derived under the condition of a complete square calculation window of (2M + 1) × (2M + 1) and is not applicable to situations where the calculation window is incomplete, such as edge points of components or points near larger cracks.

(2)

The random error is based on the assumption that the noise within each calculation window is independent Gaussian noise, which may not align with actual computational scenarios. Therefore, the random error obtained from this formula still differs from the true random error, affecting the final calculation results. This can also be observed from the simulation results in Section 3.3, where the strain calculation accuracy obtained by the SPMLS method is only close to the best calculation accuracy of PMLS, but not better.

(3)

The SPMLS method employs quadratic basis functions, which result in significant errors when calculating high-gradient strains. Higher-order basis functions are needed to address this issue. However, the formula derivation for the SPMLS method with higher-order basis functions is excessively complex and requires further research.

5. Conclusions

This research has designed a novel approach that enables precise variable selection, automated properties prediction, and comprehensive interpretation of DIC sensing results. Based on the above investigations, the following conclusions were drawn:

This paper introduces a SPMLS method for fitting strain fields, addressing the issue of manually selecting window sizes in the conventional PMLS method. This approach determines the optimal window size by analyzing the overall error associated with the PMLS method and utilizes this window size to fit high-precision strain fields.

Through simulation tests and case studies, the computational accuracy of the SPMLS method was evaluated and compared with the PMLS method. The findings indicate that the accuracy of the SPMLS method in fitting high-gradient strain fields is comparable to the optimal performance of the PMLS method, while maintaining smoothness during the fitting process. These results validate the practical applicability of the SPMLS method.

The performance of SPLS and SPMLS is comparable when calculating low-gradient strain fields, but SPMLS performs better when computing high-gradient strain fields.

The ideal computational accuracy should be higher than the optimal accuracy of PMLS. However, based on the results from Section 3.3, it is evident that the computational accuracy of SPMLS is still lower than that of PMLS. This suggests that there is room for improvement in the method of selecting the computation window size.

SPMLS is a method based on PMLS for selecting the optimal window size. The computational accuracy is limited by PMLS. To improve computational accuracy, methods with accuracy higher than PMLS can be selected for error analysis.

Section 2.3 introduces Equation (16) as a theoretical approach, and further research is required to ascertain the optimal parameters.