Systematic errors in digital image correlation due to undermatched subset shape functions

Digital image correlation techniques are commonly used to measure specimen displacements by finding correspondences between an image of the specimen in an undeformed or reference configuration and a second image under load. To establish correspondences between the two images, numerical techniques are used to locate an initially square image subset in a reference image within an image taken under load. During this process, shape functions of varying order can be applied to the initially square subset. Zero order shape functions permit the subset to translate rigidly, while first-order shape functions represent an affine transform of the subset that permits a combination of translation, rotation, shear and normal strains. In this article, the systematic errors that arise from the use of undermatched shape function, i.e., shape functions of lower order than the actual displacement field, are analyzed. It is shown that, under certain conditions, the shape functions used can be approximated by a Savitzky-Golay low-pass filter applied to the displacement functions, permitting a convenient error analysis. Furthermore, this analysis is not limited to the displacements, but naturally extends to the higher-order terms included in the shape functions. This permits a direct analysis of the systematic strain errors associated with an undermatched shape function. Detailed numerical studies are presented for the case of a second-order displacement field and first- and second-order shape functions. Finally, the relation of this work to previously published studies is discussed.     

Assessment of Digital Image Correlation Measurement Errors: Methodology and Results

Optical full-field measurement methods such as Digital Image Correlation (DIC) are increasingly used in the field of experimental mechanics, but they still suffer from a lack of information about their metrological performances. To assess the performance of DIC techniques and give some practical rules for users, a collaborative work has been carried out by the Workgroup “Metrology” of the French CNRS research network 2519 “MCIMS (Mesures de Champs et Identification en Mécanique des Solides / Full-field measurement and identification in solid mechanics, http://www.ifma.fr/lami/gdr2519)”. A methodology is proposed to assess the metrological performances of the image processing algorithms that constitute their main component, the knowledge of which being required for a global assessment of the whole measurement system. The study is based on displacement error assessment from synthetic speckle images. Series of synthetic reference and deformed images with random patterns have been generated, assuming a sinusoidal displacement field with various frequencies and amplitudes. Displacements are evaluated by several DIC packages based on various formulations and used in the French community. Evaluated displacements are compared with the exact imposed values and errors are statistically analyzed. Results show general trends rather independent of the implementations but strongly correlated with the assumptions of the underlying algorithms. Various error regimes are identified, for which the dependence of the uncertainty with the parameters of the algorithms, such as subset size, gray level interpolation or shape functions, is discussed.     

Assessment of Digital Image Correlation Measurement Accuracy in the Ultimate Error Regime: Improved Models of Systematic and Random Errors

The literature contains many studies on assessment of DIC uncertainties, particularly in the ultimate error regime, when the shape function used to describe the material transformation perfectly matches the actual transformation. For pure sub-pixel translations, bias and random errors obtained for experimental or synthetic images show more complex evolution versus the fractional part of displacement than those predicted by the existing theoretical models. Indeed, small deviations arise, mainly around integer values of imposed displacements for noisy images, and they are interpreted as the unrepresentativeness of the underlying hypotheses of the theoretical models. In a first step, differences between imposed and measured displacements are analysed: random error is independent of fractional displacement, and systematic error does not decrease for values close to integer displacements whatever the noise level. Therefore, new prediction models are proposed based on the analysis of identified phenomena from synthetic speckle-pattern 8-bit images. The statistical approach used in this paper generalizes the methods proposed in the literature and mimics the experimental methodology usually used for displacement measurements performed in different subsets in the same image. Two closed-form expressions for the systematic and random errors and a linear interpolation scheme are developed. These models, depending only on image properties and the imposed displacement, are built with a very limited number of parameters. It is then possible to predict the evolution of bias and random errors from one to four images.     

数字图像相关法测量局域变形场中形函数和模板尺寸的影响

数字图像相关法测量均匀变形场已被普遍接受, 其测量结果可与应变片测量结果比较. 然而, 在工程测量中, 针对局域变形场(应变高度集中, 如波特文-勒夏特利埃带、试件缺口附近和裂纹尖端等), 应变片受限于其尺寸, 其测量结果是接触面内的平均应变值. 此时, 采用数字图像相关法能够测量这些局域变形场. 但形函数和模板尺寸等计算参数对计算结果影响很大, 这也导致使用者很难判断计算结果的可靠性. 论文通过对合金拉伸实验获得的不同应变梯度的波特文-勒夏特利埃带和模拟生成的带的计算分析, 发掘了形函数和模板尺寸作用于计算结果的深层机制, 证明了二阶形函数比一阶形函数更适用于高度非均匀的局域变形场. 提出了在局域应变场测量中, 当一阶和二阶形函数计算结果的相对误差小于10% 时, 二阶形函数的结果是可靠的判据.      

Mesh-Based Digital Image Correlation Method Using Non-Uniform Elements for Measuring Displacement Fields with High Gradient

A new mesh-based digital image correlation (mesh-DIC) scheme using non-uniform 8-node quadrilateral (Q8) elements is proposed to process displacement fields with regions of both high and low gradient. The effect of element size on Q8-DIC is extensively analyzed, and the results show that when the shape function of an element matches or over-matches the gradient of the displacement field, a larger element size can achieve better results. Conversely, when the shape function under-matches the gradient of the displacement field, a suitable element size should be chosen. Furthermore, a practical and effective method based on the residual error of the DIC calculation is proposed to automatically refine mesh with non-uniform elements. The effectiveness of this method is verified using simulated speckle images that contain one- and two-dimensional non-uniform displacements. The proposed method is also used to improve the accuracy of two experiments that measure displacement fields with high gradient.     

Distortion of Digital Image Correlation (DIC) Displacements and Strains from Heat Waves

“Heat waves” is a colloquial term used to describe convective currents in air formed when different objects in an area are at different temperatures. In the context of Digital Image Correlation (DIC) and other optical-based image processing techniques, imaging an object of interest through heat waves can significantly distort the apparent location and shape of the object. There are many potential heat sources in DIC experiments, including but not limited to lights, cameras, hot ovens, and sunlight, yet error caused by heat waves is often overlooked. This paper first briefly presents three practical situations in which heat waves contributed significant error to DIC measurements to motivate the investigation of heat waves in more detail. Then the theoretical background of how light is refracted through heat waves is presented, and the effects of heat waves on displacements and strains computed from DIC are characterized in detail. Finally, different filtering methods are investigated to reduce the displacement and strain errors caused by imaging through heat waves. The overarching conclusions from this work are that errors caused by heat waves are significantly higher than typical noise floors for DIC measurements, and that the errors are difficult to filter because the temporal and spatial frequencies of the errors are in the same range as those of typical signals of interest. Therefore, eliminating or mitigating the effects of heat sources in a DIC experiment is the best solution to minimizing errors caused by heat waves.     

On the use of digital image correlation for slip measurement during coupon scale fretting fatigue experiments

Digital image correlation (DIC) is a full field three dimensional measurement technique that can quantify displacements and strains of a surface. In this paper, digital image correlation is used as a slip measurement technique during coupon scale fretting fatigue experiments. Slip measured with the novel DIC technique is compared to conventional slip measurement techniques as clip gauges and modified clip gauge measurements proposed by Wittkowsky et al. Slip measurements with the DIC system show lower slip values and higher tangential contact stiffness’s compared to (modified) clip gauge measurements. Slip measured with DIC is obtained closer to the contact compared to clip gauges, eliminating the influence of elastic deformations or fitting parameters. During the fretting fatigue experiments are two equal contacts simultaneously tested. However, the slip of both contacts is not identical with outliers of more than 10% difference in slip amplitude.     

Spatial DIC Errors due to Pattern-Induced Bias and Grey Level Discretization

Digital image correlation (DIC) is an optical metrology method widely used in experimental mechanics for full-field shape, displacement and strain measurements. The required strain resolution for engineering applications of interest mandates DIC to have a high image displacement matching accuracy, on the order of 1/100th of a pixel, which necessitates an understanding of DIC errors. In this paper, we examine two spatial bias terms that have been almost completely overlooked. They cause a persistent offset in the matching of image intensities and thus corrupt DIC results. We name them pattern-induced bias (PIB), and intensity discretization bias (IDB). We show that the PIB error occurs in the presence of an undermatched shape function and is primarily dictated by the underlying intensity pattern for a fixed displacement field and DIC settings. The IDB error is due to the quantization of the gray level intensity values in the digital camera. In this paper we demonstrate these errors and quantify their magnitudes both experimentally and with synthetic images.

     

Adaptive Rotated Gaussian Weighted Digital Image Correlation (RGW-DIC) for Heterogeneous Deformation Measurement

Background

Digital image correlation (DIC) has advanced to become a flexible, reliable and fast optical method for the measurement of non-contact and full-field surface deformation. However, the accuracy of existing methods in measuring heterogeneous deformation fields—especially for the high gradient strain field – can be improved.

Objective

In state-of-art local DIC applications, several methods have been put forward to adapt a subset to unknown deformation. Although improvements in performance using these methods are obtained, results are still ungratified for severely heterogeneous deformation such as the Star 2 and Star 5 images from DIC Challenge 2.0.

Methods

In this paper, a rotated Gaussian weighted zero-mean normalized sum of squared difference (RGW-ZNSSD) criterion function is proposed as the basis for RGW-DIC subset size adaptation. RGW-DIC can automatically determine the optimum weight distribution, hence self-adaptivity in subset size and orientation are achieved simultaneously.

Results

The effectiveness of the proposed RGW-DIC is verified using DIC-challenge 2.0 images and simulated sinusoidal deformation images. Results reveal that the adaptively determined subset weight distribution can significantly improve the accuracy of heterogeneous deformation measurement compared with traditional DIC and DIC with isotropic Gaussian weight functions.

Conclusions

The proposed RGW-DIC can be applied to unknown severely heterogeneous deformation measurement.

     

Cross Polarization for Improved Digital Image Correlation

Digital image correlation (DIC) is a surface deformation measurement technique for which accuracy and precision are sensitive to image quality. This work presents cross polarization, the use of orthogonal linear polarizers on light source(s) and camera(s), as an effective method for improving optical DIC measurements. The benefits of cross polarization are characterized through quantitative and statistical comparisons from two experiments: rigid body translation of a flat sample and uniaxial tension of a superelastic shape-memory alloy (SMA). In both experiments, cross polarization eliminated saturated pixels that degrade DIC measurements, and increased image contrast, which enabled higher spatial precision by using smaller subsets. Subset sizes are usually optimized for correlation confidence interval (typically with subsets of 21×21 px or larger), but can be decreased to achieve the highest possible spatial precision at the expense of increased correlation confidence intervals. Smaller subset sizes (such as 9×9 px) require better images to maintain correlation within error thresholds. By comparing DIC results from a uniaxial SMA tension test with unpolarized and cross-polarized images, we show that for 9×9 px subsets, the loss of valid DIC data points was reduced almost ten-fold with cross polarization. The only disadvantage we see to cross polarization is the decrease in specimen illumination due to transmission losses through the polarizers, which can easily be accommodated with sufficiently intense light sources. With the installation of relatively inexpensive linear polarizing filters, an optimum optical DIC setup can provide even better DIC measurements by delivering images without saturated pixels and with higher contrast for increased DIC spatial precision.     

Using DIC to Identify Microscale Strain Fields from In-situ SEM Images of a Pearlitic Steel

Digital Image Correlation (DIC) is used to analyze in-situ obtained SEM images of a pearlitic steel. Rather than using a synthetic speckle the microstructure of the material (cementite lamellae embedded in a ferrite matrix) is used as a natural speckle. The impact of the DIC method parameters on the identified motion (displacements and strains) is studied and it is shown that the method is robust, in the sense of being insensitive to the subset size, when it comes to determining the local subset displacements. However, a sufficiently large subset size is required in order for the local subset strains to converge.     

Fast, Robust and Accurate Digital Image Correlation Calculation Without Redundant Computations

High-efficiency and high-accuracy deformation analysis using digital image correlation (DIC) has become increasingly important in recent years, considering the ongoing trend of using higher resolution digital cameras and common requirement of processing a large sequence of images recorded in a dynamic testing. In this work, to eliminate the redundant computations involved in conventional DIC method using forward additive matching strategy and classic Newton–Raphson (FA-NR) algorithm without sacrificing its sub-pixel registration accuracy, we proposed an equivalent but more efficient DIC method by combining inverse compositional matching strategy and Gauss-Newton (IC-GN) algorithm for fast, robust and accurate full-field displacement measurement. To this purpose, first, an efficient IC-GN algorithm, without the need of re-evaluating and inverting Hessian matrix in each iteration, is introduced to optimize the robust zero-mean normalized sum of squared difference (ZNSSD) criterion to determine the desired deformation parameters of each interrogated subset. Then, an improved reliability-guided displacement tracking strategy is employed to achieve further speed advantage by automatically providing accurate and complete initial guess of deformation for the IC-GN algorithm implemented on each calculation point. Finally, an easy-to-implement interpolation coefficient look-up table approach is employed to avoid the repeated calculation of bicubic interpolation at sub-pixel locations. With the above improvements, redundant calculations involved in various procedures (i.e. initial guess of deformation, sub-pixel displacement registration and sub-pixel intensity interpolation) of conventional DIC method are entirely eliminated. The registration accuracy and computational efficiency of the proposed DIC method are carefully tested using numerical experiments and real experimental images. Experimental results verify that the proposed DIC method using IC-GN algorithm and the existing DIC method using classic FA-NR algorithm generate similar results, but the former is about three to five times faster. The proposed reliability-guided IC-GN algorithm is expected to be a new standard full-field displacement tracking algorithm in DIC.     

非连续数字图像相关方法在裂纹重构中的应用

数字图像相关方法作为一种新的非接触式位移测量方法,在力学工程中有广泛的应用前景,然而受限于标准方法对图像变形的连续性要求,这种高效的测量方法在断裂力学领域的推广受到了限制. 为解决这一问题,提出采用引入子区分离数学模型,代替标准方法的连续模型,来对非连续区域进行精确识别和匹配的非连续数字图像相关方法. 研究子区被裂纹等非连续分割后原始像素点的位移情况,并引入裂纹张开向量用以表征被分割子区的主区和副区的位移关系;从而建立子区分离模型的数学表达式,并且为所提出的模型设计相应的图像相关算法;然后将所提出的非连续数字图像相关方法应用于重构平板拉伸试验开裂过程中图像的位移. 研究结果表明,相比于标准的数字图像相关方法,所提出的非连续数字图像相关方法解决了图像相关法在非连续区域失效的问题,提高了数字图像相关方法对位移测量的正确率,特别是能够准确重构裂纹面及附近的位移场,其测量精度能够达到亚像素级别.      

Error Assessment in Stereo-based Deformation Measurements

Increasing interest in the use of digital image correlation (DIC) for full-field surface shape and deformation measurements has led to an on-going need for both the development of theoretical formulae capable of providing quantitative confidence margins and controlled experiments for validation of the theoretical predictions. In the enclosed work, a series of stereo vision experiments are performed in a manner that provides sufficient information for direct comparison with theoretical predictions using formulae developed in Part I. Specifically, experiments are performed to obtain appropriate optimal estimates and the uncertainty margins for the image locations/displacements, 3-D locations/displacements and strains when using the method of subset-based digital image correlation for image matching. The uncertainty of locating the 3-D space points using subset-based pattern matching is estimated by using theoretical formulae developed in Part I and the experimentally defined confidence margins for image locations. Finally, the uncertainty in strains is predicted using formulae that involves both the variance and covariance of intermediate variables during the strain calculation process. Results from both theoretical predictions and the experimental work show the feasibility and accuracy of the predictive formulae for estimating the uncertainty in the stereo-based deformation measurements.     

Influence of individual variations of particle image intensities on high-resolution PIV

Individual variations of intensity of tracer particles, e.g., due to out-of-plane displacements between exposures, strongly limit the achievable accuracy of correlation-based PIV processing. The RMS error originated by this effect correlates with the spatial resolution that can be achieved with the processing algorithm making especially high-resolution algorithms like iterative image deformation affected by this error. Both aspects are shown, the gain of resolution by iterative image deformation and the loss of accuracy due to individual variations of particle intensities.     

数字图像相关方法中匹配及过匹配形函数的误差分析

基于图像子区的数字图像相关方法需采用合适的形函数来近似目标图像子区的真实变形。由于实际测量时目标子区的局部变形往往是未知的,实际采用的不同阶次(零阶、一阶和二阶)的形函数不可避免地产生误匹配(欠匹配和过匹配)问题,从而引入位移测量的系统或随机误差。尽管由欠匹配形函数引起的系统误差已被充分认识,由过匹配形函数引起的位移误差仍缺少理论解释。本文首先推导出采用一阶和二阶形函数的数字图像相关方法的随机误差理论公式,随后采用一系列数值实验验证了理论公式的准确性。结果显示:过匹配形函数不会引入额外的系统误差,但会增加随机误差,且二阶形函数的随机误差是一阶形函数的二倍。考虑到由欠匹配一阶形函数引入的系统误差往往远大于过匹配二阶形函数的随机误差,因此在未能确知变形的情况下,推荐使用二阶形函数。     

Recent Progress in Digital Image Correlation

In this paper, we report the following important progress recently made in the basic theory and practical implementation of digital image correlation (DIC) for deformation measurement. First, we answer a basic but confusing question to the users of DIC: what is a good speckle pattern for DIC? We present a simple, easy-to-compute yet effective global parameter, called mean intensity gradient, for quality assessment of the entire speckle pattern. Second, we provide an overview of various correlation criteria used in DIC for evaluating the similarity of the reference and deformed subsets, and demonstrate the equivalence of three robust and most widely used correlation criteria, i.e., a zero-mean normalized cross-correlation (ZNCC) criterion, a zero-mean normalized sum of squared difference (ZNSSD) criterion and a parametric zero-mean normalized sum of squared difference (PSSD_ab) criterion with two additional unknown parameters, which elegantly unifies these correlation criteria for subset-based pattern matching. Third, we describe an iterative least squares (ILS) algorithm for accurate subpixel motion detection, which is proved to be equivalent to the existing Newton–Raphson algorithm, but the principle and implementation of ILS algorithm is more straightforward and easier. Finally, to overcome the two limitations of existing subset-based DIC technique, we introduce a robust and generally applicable reliability-guided DIC technique, in which the calculation path is guided by the ZNCC coefficients of computed points, to determine the genuine full-field deformation of an object with complex shape.     

Sensitivity of in-Plane Strain Measurement to Calibration Parameter for out-of-Plane Specimen Rotations

In practice, out-of-plane motions usually are not avoidable during experiments. Since 2D–DIC measurements are vulnerable to parasitic deformations due to out-of-plane specimen motions, three-dimensional digital image correlation (StereoDIC or 3D–DIC) oftentimes is employed. The StereoDIC method is known to be capable of accurate deformation measurements for specimens subjected to general three-dimensional motions, including out-of-plane rotations and displacements. As a result, there has been limited study of the deformation measurements obtained when using StereoDIC to measure the displacement and strain fields for a specimen subjected only to out-of-plane rotation. To assess the accuracy of strain measurements obtained using stereovision systems and StereoDIC when a specimen undergoes appreciable out of plane rotation, rigid body out-of-plane rotation experiments are performed in the range ?40⁰?≤?θ?≤?40⁰ using a two-camera stereovision system. Results indicate that (a) for what would normally be considered “small angle” calibration processes, the measured normal strain in the foreshortened specimen direction due to specimen rotation increases in a non-linear manner with rotation angle, with measurement errors exceeding ±1400με and (b) for what would normally be considered “large angle” calibration processes, the magnitude of the errors in the strain are reduced to ±300με. To theoretically assess the effect of calibration parameters on the measurements, two separate analyses are performed. First, theoretical strains due to out-of-plane rigid body rotation are determined using a pinhole camera model to project a series of three-dimensional object points into the image plane using large angle calibration parameters and then re-project the corresponding sensor plane coordinates back into the plane using small angle calibration parameters. Secondly, the entire imaging process is also simulated in order to remove experimental error sources and to further validate the theory. Results from both approaches confirmed the same strain error trends as the experimental strain measurements, providing confidence that the source of the errors is the calibration process. Finally, variance based sensitivity analyses show that inaccuracy in the calibrated stereo angle parameter is the most significant factor affecting the accuracy of the measured strain.     

A Projected Finite Element Update Method for Inverse Identification of Material Constitutive Parameters in transversely Isotropic Laminates

In this paper, a novel application of Finite Element Update Method (FEUM) is proposed for the inverse identification of material constitutive parameters in transversely isotropic laminates. Two-dimensional Digital Image Correlation (2D–DIC) is used for full-field measurements which is required for the identification process. Instead of measuring the in-plane displacements, which is a well-known application of 2D–DIC, we seek to measure the pseudo-displacements resulting from out-of-plane (towards camera) deflection of plate under a point load. These pseudo-displacements are basically the perspective projection of the three dimensional displacement fields on the image-plane of the image acquisition system. The cost function in this method is defined in terms of these projections instead of the true displacements – and hence the name Projected Finite Element Update Method (PFEUM). In this article, identification of in-plane elastic moduli of Carbon Fiber Reinforced Plastic (CFRP) plate has been performed using plate bending experiments which show pre-dominantly out-of-plane deflection with little contribution from the in-plane displacements. Identification results are validated by direct experimental measurements of the unknown elastic constants as well as theoretical estimates based on volume ratio of constituents. The results show good conformance between estimated and target values for at least three material parameters namely E₁, E₂ and G₁₂. Effects of experimental noise on parameter estimates has also been evaluated to explain the observed deviation in estimated parameters with current test configuration.     

A Novel “Subset Splitting” Procedure for Digital Image Correlation on Discontinuous Displacement Fields

Digital Image Correlation (DIC) is an easy to use yet powerful approach to measure displacement and strain fields. While the method is robust and accurate for a variety of applications, standard DIC returns large error and poor correlation quality near displacement discontinuities such as cracks or shear bands. This occurs because the subsets used for correlation can only capture continuous deformations from the reference to the deformed image. As a result the regions around discontinuities are typically removed from the area of interest, before or after analysis. Here, a novel approach is proposed which enables the subset to split in two sections when a discontinuity is detected. This method enables the measurement of “displacement jumps”, and also of displacements and strains right by the discontinuity (for example a crack profile or residual strains in the wake). The method is validated on digitally created images based on mode I and mode II asymptotic displacement fields, for both sub-pixel and super-pixel crack opening displacements. Finally, an actual fracture experiment on a high density polyethylene (HDPE) specimen demonstrates the robustness of the method on actual images. Compared to other methods capable of handling discontinuities, this novel “subset-splitting” procedure offers the advantage of being a direct extension of the now popular standard DIC, and can therefore be implemented as an “upgrade” to that method.