1. Introduction

Digital image correlation (DIC) is a practical, flexible, and reliable optical metrology widely used for shape, motion, and deformation measurements [1–8]. This technique retrieves full-field displacements by matching subsets in different images. The matching process demands that the texture of the specimen contains abundant information content; therefore artificial speckle patterns are generally prepared. Since the qualities of the speckle patterns profoundly influence the metrological performance of DIC, both scholars and practitioners are concerned with the design and implementation of the optimal speckle patterns [9,10]. After long-term practice and research, several empirical rules have been summarized: the ideal speckle patterns should be isotropic, non-periodic, high information content, and good contrast [11]. However, for a precision metrology, the quantitative evaluation is more essential; quantitative quality assessment of speckle patterns is based on the evaluation of metrological performance of DIC, and requires exhaustive knowledge of DIC errors. Influenced by many factors [12,13], the displacement measurement errors of DIC are typically several hundredth pixels [14]; these errors can be classified in two categories: random errors (variability) and systematic errors (bias).

Systematic errors are mainly caused by under-matched shape function [15], interpolation error [16], and image noise [17]. Considering that the systematic errors due to under-matched shape function rely on the underlying displacement fields [15,18], this paper deals with the ultimate error regime exclusively [19], in which case the shape function is perfectly matched. In the absence of noise, imperfect interpolation will introduce sinusoidal-shaped systematic errors, which are referred to as interpolation bias [20]. In our previous work, a concept of interpolation bias kernel was introduced, the inherent nature of interpolation bias was revealed, and an interpolation bias prediction algorithm, which exploits both interpolation bias kernel and the information of power spectrum, was proposed [21]. Actual images are inevitably contaminated by image noise. Wang et al. indicated that, if image noise exists, the systematic errors of DIC comprise two terms: the first term is interpolation bias; the second term is noise-induced bias, which is due to the combination of non-ideal interpolation and image noise [17]. In the context of high noise level or low image contrast, the noise-induced bias is more significant than the interpolation bias. Unfortunately, the framework provided by Wang is merely available for traditional interpolation methods such as linear and cubic interpolation, but is not applicable to generalized interpolation methods such as commonly used BSpline [13,20] and OMOMS [22]. To overcome the limitations, Sutton et al. investigated the noise-induced bias through a concept called interpolation phase error. It is noted that the noise-induced bias is proportional to the noise energy and inverse proportional to the sum of squared intensity gradients [11]. Nevertheless, the dependence of noise-induced bias upon subpixel positions for generalized interpolation is still unknown.

The aim of the work is to present thorough analysis of noise-induced bias for convolution-based interpolation (both traditional interpolation and generalized interpolation belong to convolution-based interpolation). The paper is organized as follows. In part 2, theoretical analysis of noise-induced bias is presented. In part 3, the theoretical analysis is confirmed by numerical simulations. In part 4, actual subpixel translation experiment is performed to validate the correctness of the theoretical analysis. In part 5, the differences with previous finding are discussed, and a more intuitionistic explanation of the cause of noise-induced bias is proposed. In part 6, conclusions of this work are drawn.

2. Theoretical analysis of noise-induced bias

2.1 Convolution-based interpolation, traditional interpolation, and generalized interpolation

Considering that the readers of digital image correlation community may not familiar with the interpolation theory developed by the signal processing community, this section provides a brief introduction to convolution-based interpolation, traditional interpolation, and generalized interpolation. The readers are referred to review papers such as [23–25] for details.

According to the Nyquist-Shannon sampling theorem, if the band-width B of a band-limited function f(x) satisfies B<0.5 (the Nyquist frequency), f(x) can be completely recovered by its samples f(k)

In the early researches, it was common to choose φ_int(x) as a piecewise-polynomial with finite support for convenience. These interpolation methods are referred to as traditional interpolation. Two examples of traditional interpolation are linear and Keys interpolation [26]. Figure 1(a1) shows the interpolation bases corresponding to linear and Keys interpolation, superimposed on the desired sinc function. Due to the significant differences between the sinc function and these interpolation bases, the accuracies of these interpolation methods are frequently insufficient. In the frequency domain, interpolation transfer functions shown in Fig. 1(b) clearly demonstrate the poor accuracy of linear and Keys interpolation: it is evident that the differences between the ideal low-pass filter (the transfer function of sinc interpolation) and the transfers functions corresponding to linear and Keys interpolation are significant.

 

Fig. 1 Comparison of traditional interpolation and generalized interpolation. (a1) Interpolation bases of traditional interpolation, including linear interpolation and Keys interpolation, superimposed on the desired sinc function. (a2) Cardinal interpolation bases of generalized interpolation, including cubic BSpline, cubic OMOMS, quintic BSpline, and septic BSpline, superimposed on the desired sinc function. (b) Interpolation transfer functions of several interpolation methods, including linear interpolation, Keys interpolation, cubic BSpline, cubic OMOMS, quintic BSpline, septic BSpline, and the transfer function of sinc interpolation – the ideal low-pass filter.

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In order to enhance the interpolation accuracy, generalized interpolation was presented. The essential difference between the generalized interpolation and the traditional interpolation is the introduction of interpolation coefficients c(k). For generalized interpolation, the reconstruction function satisfies

For generalized interpolation [Eq. (3)], it can be represented in the form of Eq. (2), so that generalized interpolation belongs to convolution-based interpolation. At this time, φ_int(x) is referred to as cardinal interpolation basis. It should be emphasized that φ_int(x) and φ(x) are completely different for generalized interpolation; φ_int(x) does not have close-form and the support of φ_int(x) is infinite; thus the cardinal interpolation bases are mainly used for theoretical analysis. The relation between φ_int(x) and φ(x) is

The introduction of interpolation coefficients offers new possibilities and thus enhances the interpolation accuracy significantly. Figure 1(a2) depicts the cardinal interpolation bases of several generalized interpolation methods, including cubic BSpline (BSpline3), cubic OMOMS (OMOMS3), quintic BSpline (BSpline5), and septic BSpline (BSpline7). These cardinal interpolation bases have infinite support. Due to they are more analogous to sinc function, the interpolation accuracies of methods shown in Fig. 1(b) are higher than methods shown in Fig. 1(a). From the view of frequency domain, the transfer functions shown in Fig. 1(b) clearly present the superiority of generalized interpolation.

Generalized interpolation is greatly superior to traditional interpolation and thus is widely used. In digital image correlation community, because traditional interpolation such as cubic interpolation introduces large bias errors, it is suggested to employ generalized interpolation, for instance, BSpline and OMOMS interpolation, to enhance the interpolation accuracy [20,22].

2.2 Deficiencies of existing methods

In [17], the noise-induced bias was reported and the mathematical expressions of noise-induced bias for linear and cubic interpolation were derived. Nevertheless, mathematical framework provided by [17] needs to present a subpixel intensity using several nearby samples. This procedure can be performed for traditional interpolation whose supports are finite; however, it cannot be performed for generalized interpolation whose supports are infinite. Considering the widely use and high accuracy of generalized interpolation, it is necessary to extend the analysis to generalized interpolation, and this is the aim of this paper.

2.3 Systematic errors of digital image correlation

At first, the 1-dimensional case is considered. Suppose the original function is f(x); the domain of f(x) is (-∞,∞). The samples of f(x) are f(n); the sequence f(n) can be regarded as the reference image. f(x) is translated along the x axis by u₀ units, thereby the translated function f(x-u₀). The samples of f(x-u₀) are f(n-u₀); the sequence f(n-u₀) can be regarded as the deformed image. Interpolation is performed to reconstruct f(x-u₀) by its samples f(n-u₀), and the reconstructed function g(x) is given as [Eq. (2)]

DIC measures the real displacement u₀ by optimizing specific correlation criterion. If the sum of squared difference (SSD) criterion is employed, the measured displacement u is given as

Noise is inevitable in practice; the existence of noise will induce additional errors. Suppose the noise in the reference image is ε_f(n); then the observed intensity f_noise(n) is

The measured displacement u satisfies C_SSD’(u) = 0, and thus

2.4 Noise-induced bias for convolution-based interpolation

Equation (15) is complicated. Moreover, it contains terms about g(x), which are not convenient to calculate in practice. It is preferable that the estimation equation merely contains terms about f(x). Therefore, it is essential to simplify Eq. (15). If ${\displaystyle {\sum }_{n=-\infty }^{\infty }{g^{\prime }}^2\left(n+u_0\right)}\gg N{\sigma }^2$, the systematic errors can be approximated as

2.5 Sinusoidal approximation of noise-induced bias

Although Eq. (17) is simple, it still has some deficiencies in practice. As cardinal interpolation basis φ_int(x) does not have close-form for generalized interpolation, Φ(x) does not have close-form as well. Therefore, in practice, only numerical methods are available to calculate Φ(x), which is still quite inconvenient. In order to overcome this deficiency, in this paper, Φ(x) is approximated by its Fourier representation. Φ(x) is a periodic function with period 1, so that Φ(x) can be resolved into Fourier series. The Fourier coefficients of Φ(x) are expressed using the interpolation transfer function ${\widehat{\phi }}_{\mathrm{int}}\left(\nu \right)$.

Define $\Theta \left(x\right)=-{\phi }_{\mathrm{int}}\left(x\right){{\phi }^{\prime }}_{\mathrm{int}}\left(x\right)$, so that

In practice, the subset size is finite. Although the theoretical analysis in this section assumes that the domain of the original function is (-∞,∞), if the subset is extended by adding zeros, this condition will be satisfied. Since the intensities outside the subset are zeros, the infinite sum ${\Sigma }_{n=-\infty }^{\infty }$ in Eq. (15), Eq. (17), and Eq. (22) can be substituted by ${\Sigma }_{n=-\left(N-1\right)/2}^{\left(N-1\right)/2}$, here N denotes the subset size.

3. Numerical results

3.1 Systematic errors

To validate the correctness of the proposed theoretical analysis, numerical simulations were performed. In the numerical tests, three 1-dimensional speckle patterns with different speckle sizes were utilized. These speckle patterns are composed of individual Gaussian speckles [29]; the intensity f(x) at x is given as

 

Fig. 2 1-dimensional speckle patterns utilized in the numerical tests. The speckle radii R of these Gaussian patterns [see Eq. (23)] are (a) 1.5, (b) 2.0, and (c) 3.0 respectively.

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In order to evaluate the systematic errors, for given speckle pattern, 20 translated speckle patterns were generated by translating the speckle positions x_k 0.05 units each time numerically, corresponding to displacements range from 0.05 pixels to 1 pixel. Exact sample values were utilized to remove the quantization errors. Additive Gaussian white noise with standard deviations range from 0.01 to 0.05 were added. In order to evaluate the systematic errors and random errors, the noise addition repeated for M times. The corresponding mean bias error and standard deviation error are given as [30]

The DIC algorithm was implemented by a self-written program: Gauss-Newton method and SSD criterion were utilized; as the underlying deformation is rigid motion, zero-order shape function was employed; the convergence criterion was that the iterative increment $\left|\Delta u\right|$ was less than 1 × 10⁻⁷; in order to explore the influence of interpolation, both cubic and quintic BSpline interpolation algorithms were employed. The subsets were chosen as the interval [-60, 60], so that there were 121 points in the subsets in total.

The theoretical estimations were obtained by Eq. (15). As the reasons pointed out in section 2, the infinite sum ${\Sigma }_{n=-\infty }^{\infty }$ in Eq. (15) was substituted by ${\Sigma }_{n=-\left(N-1\right)/2}^{\left(N-1\right)/2}$; in this simulation, as indicated above, the subset size N = 121. The cardinal interpolation basis φ_int(x) does not have close-form, so that φ_int(x) was obtained by numerical method; this method is based on the fact that φ_int(x) is the reconstruction function of δ(x) (Dirac delta function); the derivative φ’_int(x) and the second derivative φ”_int(x) were calculated in a similar way. g(n + u₀), g’(n + u₀), and g”(n + u₀) were calculated numerically from the reconstructed function g(x).

The systematic errors by DIC and by theory [Eq. (15)] are illustrated in Fig. 3. Figure 3(a1)-3(a2), 3(b1)-3(b2), and 3(c1)-3(c2) correspond to speckle patterns shown in Fig. 2(a), 2(b), and 2(c) respectively. Figure 3(a1)-3(c1) correspond to the cubic BSpline interpolation while Fig. 3(a2)-3(c2) correspond to the quintic BSpline.

 

Fig. 3 Systematic errors correspond to (Left) cubic BSpline and (Right) quintic BSpline interpolation at different noise level (noise standard deviations range from 0 to 0.05). From the top to the bottom: systematic errors correspond to speckle patterns shown in Fig. 2(a), 2(b), and 2(c) respectively.

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Figure 3 demonstrates that the systematic errors by DIC and by theory show excellent agreement. Figure 3 indicates that (1) the noise-induced bias can be much larger than the interpolation bias (interpolation bias corresponds to the systematic errors in the absence of noise); (2) a speckle pattern with large speckle size introduces less interpolation bias, but is more sensitive to noise; (3) if the intensity noise is significant, the bias of a pattern with large speckle size is probably much larger than that of a pattern with small speckle size; (4) for given speckle pattern, high-order BSpline interpolation method reduces both interpolation bias and noise-induced bias.

3.2 Noise-induced bias

This paper proceeds to validate the correctness of the theoretical results on noise-induced bias. The noise-induced bias by DIC was obtained by subtracting the interpolation bias (the systematics errors in the absence of noise) from the systematic errors in the presence of noise [see Eq. (16)]. The noise-induced bias by theoretical analysis was obtained by Eq. (17).

Figure 4 shows both noise-induced bias by DIC and by theoretical analysis. The DIC results are clearly consistent with our theoretical estimations. Figure 4 indicates that (1) the noise-induced bias increase as noise level increase; (2) the noise-induced bias of fine speckle patterns are less than that of coarse patterns; (3) for given speckle pattern and noise level, high order BSpline interpolation introduces less noise-induced bias.

 

Fig. 4 Noise-induced bias correspond to (Left) cubic BSpline and (Right) quintic BSpline interpolation at different noise level (noise standard deviations range from 0.01 to 0.05). From the top to the bottom: noise-induced bias for speckle patterns shown in Figs. 2(a), 2(b), and 2(c) respectively.

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Observation (1) and (2) have been reported and well explained [11]; observation (3) has been reported in [11], but its physical insight still remains unexplained. In the following section, this phenomenon will be explained.

3.3 Sinusoidal approximation

Last section demonstrates the correctness of our theoretical results on noise-induced bias. Therefore, for the sinusoidal approximation, the issue is how many terms is enough to well represent the interpolation noise coupling function Φ(x). To tackle this problem, the coupling function Φ(x) corresponding to Keys, cubic BSpline, quintic BSpline, and septic BSpline were calculated numerically. The corresponding Fourier coefficients a_m were calculated as well. Because the Fourier coefficients merely depend on the interpolation transfer function, for given interpolation method, a_m are constants. The interpolation transfer functions of aforementioned interpolation methods are [23]

Table 1. Fourier series coefficients of interpolation-noise coupling function.

View Table

 

Fig. 5 Interpolation-noise coupling function and its corresponding first order Fourier series.

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4. Experimental verification

Actual subpixel translation experiment was performed to verify the correctness of the mathematical derivations of noise-induced bias.

The experiment method has been published in [21]. To clarify the situation, the principle is explained briefly. The experiment set-up is shown in Fig. 6(a). The idea of this method is to display a speckle pattern on the computer screen and then translate the speckle pattern by an image processing software [Fig. 6(b)]; the integer pixel translation on the computer screen will introduce a subpixel translation on the sensor plane [Fig. 6(c)]. The experiment was carried out as follows. (1) A camera (OK_IM1161, Beijing JoinHope Image Technology Ltd.) with 12mm lens was rigidly fixed to a variation-isolation table; then a tablet computer (Microsoft Surface Pro 3) was placed on the table. The distance was adjusted so that 10 pixel on the computer screen roughly corresponded to a pixel of the camera sensor. (2) A laser pen was placed on the camera and its direction was justified until the laser point was at the middle of the image. A CD disk was placed on the computer screen to reflect the laser beam; meanwhile, the position of the computer was adjusted until the reflected laser beam coincided with the incident laser beam, which ensured the perpendicular of the computer screen and the optical axis. Then the CD disk and the laser pen were carefully taken away. (3) A speckle pattern was displayed on the computer screen. The speckle pattern was translated using an image processing software. During the translation, a wireless keyboard was used. For each translation step, 100 frames were captured to analyze the noise distribution.

 

Fig. 6 Subpixel translation experiment. (a) Experiment set-up. (b) A speckle pattern was displayed on a computer screen; then the pattern was translated using an image processing software. (c) Schematic of the principle: integer pixel translation on the computer screen will introduce a subpixel position in the camera sensor plane.

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As before, the noise-induced bias was obtained by subtracting the interpolation bias (bias without noise) from the systematic errors (bias with noise). Therefore, it is necessary to obtain the bias both with and without noise. The data processing flows corresponding to bias with and without noise are different; they are shown in Fig. 7 schematically. The computation of bias without noise is shown in Fig. 7(a): the deformed images were averaged first to obtain a virtual noiseless deform image, and then the average deform image correlated with the reference image to obtain the measured displacement. The computation of bias with noise is shown in Fig. 7(b): the reference image correlated with 100 noisy images, thereby 100 measured displacements corrupted by image noise; these 100 measured displacements were averaged to obtain an average displacement. In both cases, the measured displacements were fitted by a linear model and the differences with the linear fit were the bias. The reference image was the average of 100 images and thus was virtually noiseless.

 

Fig. 7 Computational flow chart. (a) Bias without noise: the deformed images are averaged first to obtain a virtual noiseless deform image; then the average deform image correlates with the reference image. (b) Bias with noise: the noisy deformed images correlate with the reference image; then the measured displacements are averaged.

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The DIC algorithm was implemented by a self-written program. Gauss-Newton method was utilized. In order to be consistent with the theoretical analysis, SSD criterion was employed. Zero order shape function was used. The subset size was chosen as 137 × 78. In order to investigate the influence of speckle size, three speckle patterns were utilized in this work as shown in Figs. 8(a1)-8(c1). The densities of these speckle patterns are 65%. The speckle sizes are roughly 4, 6, and 8 pixels respectively.

 

Fig. 8 Subpixel translation experimental. From left to right, results for speckle patterns with different speckle size. From top to bottom, (a1)-(c1) Subsets for correlation: the speckle size are roughly 4, 6, and 8 pixels respectively. (a2)-(c2) Standard deviations of the image noise: it can be seen that the noise level depend on the image intensity. (a3)-(c3) Systematic errors with and without noise (denoted as noisy and noiseless in the figure): the systematic errors without noise are the interpolation bias; the systematic errors with noise include both interpolation bias and noise-induced bias [Eq. (16)]. The DIC results were obtained following the computational flow chart shown in Fig. 7. The theory estimations were based on the interpolation bias prediction formula in [21] and the noise-induced bias estimation formula Eq. (26). (a4)-(c4) Noise-induced bias: the DIC results were obtained by subtracting interpolation bias from the systematic errors, the theoretical estimations were based on Eq. (26).

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To estimate the noise-induced bias, the image noise should be known. The standard deviations of the image noise were evaluated by 100 frames and the results are depicted in Figs. 8(a2)-8(c2). In accord with [31], the image noise depends on corresponding image intensity. Hence, it is essential to extend previous theoretical analysis to cases of non-uniform noise. Besides, images are 2-dimensional signals. After some mathematical derivations, the formula of noise-induced bias for non-uniform noise is given as

As indicated in [21], experimental results of BSpline are sensitive to environmental noise; thus Keys interpolation was employed. The interpolation basis of Keys interpolation is [26]

To estimate noise-induced bias by Eq. (26), three terms are required: the interpolation noise coupling function, the mean of noise variance, and the mean of squared intensity gradients. In this experiment, Φ(u₀) is shown in Eq. (28), the $\overline{{\sigma }^2}$ are 3.4053, 3.4175, 3.3956 respectively, and $\overline{{f^{\prime }}_x^2}$ are 1152.9, 983.37, and 809.38 respectively.

Figures 8(a3)-8(b3) illustrate the systematic errors with noise and without noise. The DIC results were obtained following the computational flow chart shown in Fig. 7. The theoretical estimation of the interpolation bias was based on the formula in [21]; the estimation of the noise-induced bias was based on Eq. (26); the estimation of the systematic error was the sum of estimations of interpolation bias and noise-induced bias. Figures 8(a4)-8(c4) show the noise-induced bias by experiment and theory, which show good agreement. In this experiment, the noise-induced bias is serval thousandth pixels, about one-tenths of the interpolation bias. As the speckle density is the same, the intensity gradient of a fine pattern is larger than that of a coarse pattern, resulting in a smaller noise-induced bias.

5. Discussion

5.1 Comparison with findings in literatures

Findings in [17] are special cases of theoretical analysis presented in this paper. For example, if linear interpolation is used, the linear interpolation basis is

5.2 Cause of noise-induced bias

In order to obtain insight into noise-induced bias, it is crucial to know its cause. In [11], Sutton et al. noted that the noise-induced bias is due to the position-dependence of noise caused by the low-pass effect of interpolation. This position-dependence means that, even though the variance of noise is${\sigma }_0^2$ at every integer position, the variance of interpolated intensity σ²(u) is still a function of subpixel position u

Suppose function f(x); its samples f(k) are corrupted by noise ε(k). If convolution-based interpolation is used, the interpolated intensity at a subpixel position u is

To validate the correctness of Eq. (33), numerical simulations were performed for the 1-dimentional speckle pattern shown in Fig. 2(a). Additive Gaussian white noise with standard deviation σ₀ = 0.01 were added at integer pixel positions. In order to simulate the randomness of the noise, the addition repeated for 1000000 times. Keys, BSpline, and OMOMS interpolation were employed to interpolate intensities at subpixel positions in the interval [-1,1]. Figure 9 illustrates the standard deviations of the interpolated intensity σ(u) by numerical experiments and by Eq. (33); the simulation and theoretical results show excellent agreement. Remarkably, it can be found that (1) the fluctuation of high-order BSpline is more unobvious; (2) as 0 is the non-differentiable point of the interpolation basis of OMOMS [23], σ(u) of OMOMS is C⁰ continuity; (3) for Keys and BSpline interpolation methods, because of continuity and symmetry, σ(u) achieve their extreme values at integer and half-pixel positions.

 

Fig. 9 The position-dependence of noise. The standard deviation of added noise at integer positions is 0.01. The theoretical estimations are based on Eq. (33).

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On the basis of Eq. (33), the form of the interpolation-noise coupling function Φ(u) can be explained qualitatively (recalling that$\Phi \left(u\right)=-{\displaystyle \sum _{k=-\infty }^{\infty }{\phi }_{\mathrm{int}}\left(u-k\right){{\phi }^{\prime }}_{\mathrm{int}}\left(u-k\right)}$). If SSD criterion is employed [see Eq. (8)], the expectation of C_SSD(u) is

6. Conclusion

The purpose of this paper is to provide theoretical formulae to estimate noise-induce bias for both traditional interpolation and generalized interpolation. The conclusions are drawn as follows:

The primary motivation of this work is to provide a valid quality assessment of speckle patterns and thus facilitating the standardization of speckle patterns for DIC.

Appendix

A. Detailed derivation of the systematic errors

Since

(35)$$\Gamma \left(u\right)={\displaystyle \sum _{n=-\infty }^{\infty }{\left[f\left(n\right)-g\left(n+u\right)\right]}^2}.$$

Therefore

(36)$$\begin{array}{l}\frac{1}{2}{\Gamma }^{\prime }\left(u\right)=-{\displaystyle \sum _{n=-\infty }^{\infty }\left[f\left(n\right)-g\left(n+u\right)\right]g^{\prime }\left(n+u\right)},\\ \frac{1}{2}{{\Gamma }^{\prime }}^{\prime }\left(u\right)={\displaystyle \sum _{n=-\infty }^{\infty }{g^{\prime }}^2\left(n+u\right)}-{\displaystyle \sum _{n=-\infty }^{\infty }\left[f\left(n\right)-g\left(n+u\right)\right]{g^{\prime }}^{\prime }\left(n+u\right)}.\end{array}$$

Since

(37)$$\begin{array}{l}\Lambda \left(u\right)=2{\displaystyle \sum _{n=-\infty }^{\infty }\left[f\left(n\right)-g\left(n+u\right)\right]\left[{\epsilon }_f\left(n\right)-{\epsilon }_g\left(n+u\right)\right]}\\ +{\displaystyle \sum _{n=-\infty }^{\infty }\left[{\epsilon }_f^2\left(n\right)-2{\epsilon }_f\left(n\right){\epsilon }_g\left(n+u\right)+{\epsilon }_g^2\left(n+u\right)\right]}.\end{array}$$

Therefore

(38)$$\begin{array}{l}\frac{1}{2}{\Lambda }^{\prime }\left(u\right)=-{\displaystyle \sum _{n=-\infty }^{\infty }{g}^{\prime }\left(n+u\right)\left[{\epsilon }_f\left(n\right)-{\epsilon }_g\left(n+u\right)\right]}-{\displaystyle \sum _{n=-\infty }^{\infty }\left[f\left(n\right)-g\left(n+u\right)\right]{\epsilon }_g{}^{\prime }\left(n+u\right)}\\ -{\displaystyle \sum _{n=-\infty }^{\infty }{\epsilon }_f\left(n\right){\epsilon }_g{}^{\prime }\left(n+u\right)}+{\displaystyle \sum _{n=-\infty }^{\infty }{\epsilon }_g\left(n+u\right){\epsilon }_g{}^{\prime }\left(n+u\right)}.\end{array}$$

The corresponding expectation

(39)$$\begin{array}{l}\mathbb{E}\left(\frac{1}{2}{\Lambda }^{\prime }\left(u\right)\right)=\mathbb{E}\left({\displaystyle \sum _{n=-\infty }^{\infty }{\epsilon }_g\left(n+u\right){{\epsilon }^{\prime }}_g\left(n+u\right)}\right)=\mathbb{E}\left({\displaystyle \sum _{n=-\infty }^{\infty }{\displaystyle \sum _{k_1=-\infty }^{\infty }{\epsilon }_g\left(k_1\right){\phi }_{\mathrm{int}}\left(n+u-k_1\right)}{\displaystyle \sum _{k_2=-\infty }^{\infty }{\epsilon }_g\left(k_2\right){{\phi }^{\prime }}_{\mathrm{int}}\left(n+u-k_2\right)}}\right)\\ =\mathbb{E}\left({\displaystyle \sum _{n=-\infty }^{\infty }{\displaystyle \sum _{k_1=-\infty }^{\infty }{\displaystyle \sum _{k_2=-\infty }^{\infty }{\epsilon }_g\left(k_1\right){\epsilon }_g\left(k_2\right){\phi }_{\mathrm{int}}\left(n+u-k_1\right){{\phi }^{\prime }}_{\mathrm{int}}\left(n+u-k_2\right)}}}\right)\\ ={\displaystyle \sum _{k=-\infty }^{\infty }{\sigma }^2\left(k\right){\displaystyle \sum _{n=-\infty }^{\infty }{\phi }_{\mathrm{int}}\left(n+u-k\right){{\phi }^{\prime }}_{\mathrm{int}}\left(n+u-k\right)}}\\ \text{=}{\displaystyle \sum _{k=-\infty }^{\infty }{\sigma }^2\left(k\right)}\cdot {\displaystyle \sum _{n=-\infty }^{\infty }{\phi }_{\mathrm{int}}\left(n+u\right){{\phi }^{\prime }}_{\mathrm{int}}\left(n+u\right)}.\end{array}$$

It is assumed that N points are contaminated by noise with standard deviation σ, and hence

(40)$$\mathbb{E}\left(\frac{1}{2}{\Lambda }^{\prime }\left(u\right)\right)=N{\sigma }^2{\displaystyle \sum _{k=-\infty }^{\infty }{\phi }_{\mathrm{int}}\left(u-k\right){{\phi }^{\prime }}_{\mathrm{int}}\left(u-k\right)}.$$

The form of Λ”(u) is

(41)$$\begin{array}{l}\frac{1}{2}{{\Lambda }^{\prime }}^{\prime }\left(u\right)=-{\displaystyle \sum _{n=-\infty }^{\infty }{g^{\prime }}^{\prime }\left(n+u\right)\left[{\epsilon }_f\left(n\right)-{\epsilon }_g\left(n+u\right)\right]}+2{\displaystyle \sum _{n=-\infty }^{\infty }{g}^{\prime }\left(n+u\right){{\epsilon }^{\prime }}_g\left(n+u\right)}-{\displaystyle \sum _{n=-\infty }^{\infty }\left[f\left(n\right)-g\left(n+u\right)\right]{{{\epsilon }^{\prime }}^{\prime }}_g\left(n+u\right)}\\ -{\displaystyle \sum _{n=-\infty }^{\infty }{\epsilon }_f\left(n\right){{{\epsilon }^{\prime }}^{\prime }}_g\left(n+u\right)}+{\displaystyle \sum _{n=-\infty }^{\infty }{{\epsilon }^{\prime }}_g^2\left(n+u\right)}+{\displaystyle \sum _{n=-\infty }^{\infty }{\epsilon }_g\left(n+u\right){{{\epsilon }^{\prime }}^{\prime }}_g}\left(n+u\right).\end{array}$$

The corresponding expectation

(42)$$\begin{array}{l}\mathbb{E}\left(\frac{1}{2}{{\Lambda }^{\prime }}^{\prime }\left(u\right)\right)=\mathbb{E}\left({\displaystyle \sum _{n=-\infty }^{\infty }{{\epsilon }^{\prime }}_g^2\left(n+u\right)}+{\displaystyle \sum _{n=\infty }^{\infty }{\epsilon }_g\left(n+u\right){{{\epsilon }^{\prime }}^{\prime }}_g\left(n+u\right)}\right)\\ =\mathbb{E}\left(\begin{array}{l}{\displaystyle \sum _{n=-\infty }^{\infty }{\displaystyle \sum _{k_1=-\infty }^{\infty }{\epsilon }_g\left(k_1\right){{\phi }^{\prime }}_{\mathrm{int}}\left(n+u-k_1\right)}{\displaystyle \sum _{k_2=-\infty }^{\infty }{\epsilon }_g\left(k_2\right){{\phi }^{\prime }}_{\mathrm{int}}\left(n+u-k_2\right)}}\\ +{\displaystyle \sum _{n=-\infty }^{\infty }{\displaystyle \sum _{k_1=-\infty }^{\infty }{\epsilon }_g\left(k_1\right){\phi }_{\mathrm{int}}\left(n+u-k_1\right)}{\displaystyle \sum _{k_2=-\infty }^{\infty }{\epsilon }_g\left(k_2\right){{{\phi }^{\prime }}^{\prime }}_{\mathrm{int}}\left(n+u-k_2\right)}}\end{array}\right)\\ =\mathbb{E}\left({\displaystyle \sum _{k_1=-\infty }^{\infty }{\displaystyle \sum _{k_2=-\infty }^{\infty }{\epsilon }_g\left(k_1\right){\epsilon }_g\left(k_2\right){\displaystyle \sum _{n=-\infty }^{\infty }\left[{{\phi }^{\prime }}_{\mathrm{int}}\left(n+u-k_1\right){{\phi }^{\prime }}_{\mathrm{int}}\left(n+u-k_2\right)+{\phi }_{\mathrm{int}}\left(n+u-k_1\right){{{\phi }^{\prime }}^{\prime }}_{\mathrm{int}}\left(n+u-k_2\right)\right]}}}\right)\\ ={\displaystyle \sum _{k=-\infty }^{\infty }{\sigma }^2\left(k\right){\displaystyle \sum _{n=-\infty }^{\infty }\left[{{\phi }^{\prime }}_{\mathrm{int}}^2\left(n+u-k\right)+{\phi }_{\mathrm{int}}\left(n+u-k\right){{{\phi }^{\prime }}^{\prime }}_{\mathrm{int}}\left(n+u-k\right)\right]}}\\ ={\displaystyle \sum _{k=-\infty }^{\infty }{\sigma }^2\left(k\right)}\cdot {\displaystyle \sum _{n=-\infty }^{\infty }\left[{{\phi }^{\prime }}_{\mathrm{int}}^2\left(n+u\right)+{\phi }_{\mathrm{int}}\left(n+u\right){{{\phi }^{\prime }}^{\prime }}_{\mathrm{int}}\left(n+u\right)\right]}\\ =N{\sigma }^2{\displaystyle \sum _{k=-\infty }^{\infty }\left[{{\phi }^{\prime }}_{\mathrm{int}}^2\left(u-k\right)+{\phi }_{\mathrm{int}}\left(u-k\right){{{\phi }^{\prime }}^{\prime }}_{\mathrm{int}}\left(u-k\right)\right]}.\end{array}$$

Substitute Eq. (36), Eq. (40), and Eq. (42) into Eq. (14); we can get Eq. (15).

B. The dependence of sum of squared intensity gradients upon subpixel positions

The dependence of the sum of squared intensity gradients upon subpixel positions is characterized in this appendix. The Fourier transform of function f(x) is

(43)$$\widehat{f}\left(\nu \right)={\displaystyle {\int }_{-\infty }^{\infty }f\left(x\right)e^{-j2\pi \nu x}\text{d}x}.$$

The Fourier transform of g’(x + u₀) is

(44)$$ℱ\left\{g^{\prime }\left(x+u_0\right)\right\}=j2\pi \nu {\widehat{\phi }}_{\mathrm{int}}\left(\nu \right){\displaystyle \sum _{k=-\infty }^{\infty }{e}^{j2\pi k{u}_0}\widehat{f}\left(\nu -k\right)},$$

where ${\widehat{\phi }}_{\mathrm{int}}\left(\nu \right)$ is the transfer function of the interpolation method. Using the Possion summation formulae, the discrete time Fourier transform of the sequence g’(x + u₀) is

(45)$${\displaystyle \sum _{n=-\infty }^{\infty }{g}^{\prime }\left(n+u_0\right)e^{-j2\pi \nu n}}={\displaystyle \sum _{m=-\infty }^{\infty }j2\pi \left(\nu -m\right){\widehat{\phi }}_{\mathrm{int}}\left(\nu -m\right){\displaystyle \sum _{k=-\infty }^{\infty }{e}^{j2\pi k{u}_0}\widehat{f}\left(\nu -k-m\right)}}.$$

It’s rational to assume that f(x) satisfies sampling theorem, and thus k + m = 0. In this case, using the Parseval’s theorem, the sum of squared intensity gradients can be expressed in the frequency domain:

(46)$${\displaystyle \sum _{n=-\infty }^{\infty }{g^{\prime }}^2\left(n+u_0\right)}=4{\pi }^2{\displaystyle {\int }_{-1/2}^{1/2}{\left|{\displaystyle \sum _{m=-\infty }^{\infty }{e}^{-j2\pi m{u}_0}\left(\nu -m\right){\widehat{\phi }}_{\mathrm{int}}\left(\nu -m\right)}\right|}^2\cdot {\left|\widehat{f}\left(\nu \right)\right|}^2\text{d}\nu }.$$

Interpolation transfer functions have low-pass effect; therefore terms other than $\widehat{\phi }\left(\nu \right)$ and $\widehat{\phi }\left(\nu \pm 1\right)$ are negligible. In addition, if only fundamental frequency is involved, then

(47)$$\begin{array}{c}{\displaystyle \sum _{n=-\infty }^{\infty }{g^{\prime }}^2\left(n+u_0\right)}\approx 4{\pi }^2{\displaystyle {\int }_{-1/2}^{1/2}\left[E_{\text{DC}}\left(\nu \right)+E_{\mathrm{COS}}\left(\nu \right)\cos 2\pi u_0\right]{\left|\widehat{f}\left(\nu \right)\right|}^2\text{d}\nu };\\ E_{\text{DC}}\left(\nu \right)={\left(\nu -1\right)}^2{\widehat{\phi }}_{\mathrm{int}}^2\left(\nu -1\right)+{\nu }^2{\widehat{\phi }}_{\mathrm{int}}^2\left(\nu \right)+{\left(\nu +1\right)}^2{\widehat{\phi }}_{\mathrm{int}}^2\left(\nu +1\right),\\ E_{\mathrm{COS}}\left(\nu \right)=2\nu {\widehat{\phi }}_{\mathrm{int}}\left(\nu \right)\left[\left(\nu -1\right){\widehat{\phi }}_{\mathrm{int}}\left(\nu -1\right)+\left(\nu +1\right){\widehat{\phi }}_{\mathrm{int}}\left(\nu +1\right)\right].\end{array}$$

In order to verify the validity of Eq. (47), numerical experiments were performed. Cubic (4 tap), quintic (6 tap), and septic (8 tap) BSpline interpolation methods were utilized to obtain the sum of squared intensity gradients for speckle patterns shown in Fig. 2. Figures 10(a1)-10(a3) illustrate the sum of squared intensity gradients obtained by DIC and by Eq. (47) for speckle patterns shown in Figs. 2(a)-2(c). The theoretical estimations show good agreement with the results by DIC.

 

Fig. 10 (a1)-(a3) Sum of squared intensity gradients correspond to speckle patterns shown in Fig. 2; (b) E_COS(ν).

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Equation (47) implies that SSSIG can be divided into two parts: a direct component, and a cosine-shaped component. The amplitude of the cosine-component is determined by the integral of the product of the power spectrum and the kernel E_COS(ν). E_COS(ν) related to cubic, quintic, and septic BSpline interpolation methods are illustrated in Fig. 10(b). A high order BSpline interpolation method has small frequency response E_COS(ν), and therefore the fluctuation is less obvious. Because small speckle size gives rise to significant high frequency component and the absolute value of E_COS(ν) is large at high frequency, the fluctuations of fine patterns are larger than that of coarse patterns. However, the cosine component is general negligible compared to the direct component, and therefore it’s rational to consider ${\displaystyle \sum _{n=-\infty }^{\infty }{g^{\prime }}^2\left(n+u_0\right)}\approx {\displaystyle \sum _{n=-\infty }^{\infty }{{g^{\prime }}^2\left(n+u_0\right)|}_{u_0=0}}={\displaystyle \sum _{n=-\infty }^{\infty }{f^{\prime }}^2\left(n\right)}$.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11332010, 51271174, 11372300, 11127201, 11472266 and 11428206).

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