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.     

数字图像相关的噪声导致系统误差及散斑质量评价标准

数字图像相关中散斑质量评价标准应该综合考虑系统误差和随机误差的作用。之前的工作考虑了无图像噪声情况下插值引起的系统误差,本文则进一步研究了有图像噪声情况的系统误差,并与随机误差综合考虑提出了完善的散斑质量评价参数。本文推导了有噪声情况下系统误差的解析形式,揭示了噪声引入系统误差产生的内在本质在于插值引起噪声不确定性对亚像素位置的依赖。依据理论分析,插值噪声耦合函数的概念被引入,它由插值基函数平移平方和的斜率决定,表征了噪声引入系统误差随亚像素位置的变化。插值耦合函数将之前的研究成果纳入统一的理论体系,并从本质上解释了高阶B-样条插值对应的噪声引起系统误差较小的现象。数值模拟与本文的理论分析显示一致,在真实的亚像素平移实验验证中,本文将公式推广到非均匀噪声情况,并与实验结果获得了较好的吻合。基于对系统误差的理论分析,综合考虑系统误差和随机误差影响,提出了两种计算误差评估参数:总误差的最大值和平方平均值,并提出了快速估计算法且通过数值模拟进行了验证。计算误差评估参数实际也是一种散斑质量评价参数,提出的评估参数弥补了现存散斑质量评价参数未足够考虑插值影响的缺陷,是更完善的散斑质量评价标准。本文应用新提出的散斑评价参数对一些常见散斑图进行了评价,并将其用于对模拟散斑图的优化。     

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.     

数字图像相关中的亚像素位移定位算法进展

数字图像相关方法 (DIC) 已经作为一种常用的光学计量有效手段应用于实验力学及其它科学研究和工程应用领域中. 经过 20 多年的发展, 该方法日渐成熟和完善. 作为提高测量精度的亚像素位移定位算法被认为是该方法的关键技术之一. 本文对二维数字图像相关的基本原理及其中为提高测量精度而提出的各种亚像素位移定位算法做了概括性的介绍. 在总结已有研究成果的基础上, 分析了各种方法优缺点. 并对数字图像相关方法的最新进展作了简要介绍, 随着算法精度、效率以及硬件设备性能的提高, 该方法必将会获得更广泛的应用.      

Statistical Analysis of the Effect of Intensity Pattern Noise on the Displacement Measurement Precision of Digital Image Correlation Using Self-correlated Images

This paper discusses the effect of intensity pattern noise on the displacement measurement precision of digital image correlation (DIC). Through mathematical deduction, a formula is presented to estimate the displacement measurement error caused by intensity pattern noise. The resulting formula synthetically reflects the effects of the variance of noise, the intensity variance and the subset size on the displacement measurement precision. To verify the correctness of the resulting formula, two experiments are done. The first one is a real self-correlation experiment, and aims to analyze the effect of the subset size, while the second is a numerical self-correlation experiment, and is to analyze to the effect of the different noise levels. The experimental results are in good agreement with the theoretical predictions. The project is supported by the National Science Foundation of China (10572103).     

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.

     

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.     

Robust particle image velocimetry using gradient method with upstream difference and downstream difference

In the first part of this paper, sub-pixel displacement detection methods in particle image velocimetry (PIV) measurements were discussed. Some schemes for calculating the spatial gradient in a gradient method, which is one of sub-pixel displacement detection methods, were evaluated. It was found that the second order central finite difference scheme tends to underestimate the gradient of the tracer-particle’s intensity. Consequently it leads to an overestimation of sub-pixel displacements. A first-frame upstream and second-frame downstream finite difference scheme was then proposed and evaluated for one-dimensional flow measurements. Furthermore, the proposed difference scheme was extended to apply to two-dimensional flows. It was shown that taking the direction of the sub-pixel correction into account and using fewer adjacent pixels are important for PIV images consisting of small tracer particle images. Finally, a robust procedure for the execution of the gradient method was proposed, which can eliminate the effect of noises. It was found that the proposed scheme is not only highly accurate but is highly robust to noise.     

基于数字图像体相关的物体内部三维位移场分析

在二维数字图像相关算法的基础上,推导了三维数字图像体相关算法,并应用于物体内部的三维位移场分析。用计算机模拟方法对SR-CT重建的物体内部三维图像施加已知变形,对变形前后三维数字图像进行体相关运算,获得三维位移场。在计算亚像素位移时,本文提出了一种三维的基于灰度梯度的算法,给出了梯度函数,阐述了数字模拟三维体相关的计算过程,通过数字模拟实验证明了算法的正确性和可靠性。     

A Space-Time PGD-DIC Algorithm:

The aim of this study is to develop a new regularized Digital Image Correlation (DIC) method for time dependent measurements. The correlation problem is written as a minimization problem over the space-time domain in a general formulation including 2D-DIC and Stereo DIC (SDIC). The unknown time-resolved displacement field is found as a sum of products of space and time functions, similarly to the Proper Generalized Decomposition in computational mechanics. It is shown that the space fields are less sensitive to noise as time regularity acts as a physical regularization of the space fields. The proposed method is illustrated by vibration measurement under harmonic excitation in 2D-DIC and SDIC.     

基于光学和扫描显微平台的微尺度实验力学检测技术和装置研究

本文基于光学和扫描显微平台,介绍了本研究组在微尺度实验力学检测技术和设备方面的最新研究成果。在检测技术方面涉及显微散斑干涉技术、微标记阵列检测技术、晶粒变形分析技术、光学探针动静态变形分析技术;在检测系统和装置方面介绍了新近开发的双视场薄膜检测系统、散斑微干涉系统、微标记检测平台、AFM和SEM单轴拉伸装置、三维微定位与加载系统、微力传感器及其标定装置、微动平台驱动装置等。探讨了微尺度实验力学检测中的问题和新的检测技术,给出了一些典型的应用和相关装置。     

Mechanical model of organ of Corti

According to the vibration characteristics of the organ of Corti(OC), seven hypotheses are made to simplify the structure of the model, and a mechanical OC model is established. Using the variational principle, a displacement analytical expression is solved under a certain pressure. The results are in good agreement with experimental data, showing the validity of the formula. Combined with the damage caused by noise in clinic, it is found that the hardening of outer hair cells and outer stereocilia can lead to loss of hearing and generation of threshold shift. In addition, the results show that high frequency resonance occurs at the bottom of the basilar membrane(BM), and low frequency resonance occurs at the top of the BM. This confirms the frequency selective characteristics of the BM. Further, using this formula can avoid interference of the environment and the technical level of the test personnel, and can evaluate performance of the OC objectively.     

Use of 3-D Digital Image Correlation to Characterize the Mechanical Behavior of a Fiber Reinforced Refractory Castable

Refractory castables exhibit very low fracture strain levels when subjected to tension or bending. The main objective of this work is to show that 3-D digital image correlation (3-D DIC) allows such low strain levels to be measured. Compared to mechanical extensometer measurements, 3-D DIC makes it possible to reach similar strain resolution levels and to avoid the problem of position dependance related to the heterogeneous nature of the strain and to strain localization phenomena. First, the 3-D DIC method and the experimental set-up are presented. Secondly, an analysis of the 3-D DIC method is performed in order to evaluate the resolution, the standard uncertainty and the spatial resolution for both displacement and strain measurements. An optimized compromise between strain spatial resolution and standard uncertainty is reached for the configuration of the experimental bending test. Finally, the macroscopic mechanical behavior of a fiber reinforced refractory castable (FRRC) is studied using mechanical extensometry and 3-D DIC in the case of tensile and four-point bending tests. It is shown that similar results are obtained with both methods. Furthermore, in the case of bending tests on damaged castable, 3-D DIC results demonstrate the ability to determine Young’s modulus from heterogeneous strain fields better than by using classical beam deflection measurements.     

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.     

Surface profiling using sequential sampling and inverse methods. Part II: Implementation and experimental results

Sequential sampling with inverse calculation is demonstrated to provide a practical and reliable method for profiling surfaces, independent of rigid-body motions. Examples are given where various arrangements of displacement sensors can provide single-sided, double-sided, and parallel profilling. Double-sided measurements also give specimen thickness, and parallel measurements give surface twist. In common with other inverse calculation methods, there is a tradeoff between achievable spatial resolution and sensitivity to measurement noise. Such sensitivity, which typically doubles the measurement noise in the computed profiles, can be minimized by appropriate choice of sensor configuration, measurement method, and use of regularization. Practical guidelines for all these features are given, together with example experimental results for four typical sensor configurations.     

Template matching for improved accuracy in molecular tagging velocimetry

In 2D molecular tagging velocimetry (MTV), tags are written into a fluid flow with a laser grid and imaged at discrete times. These images are analyzed to calculate Lagrangian displacement vectors, often by direct cross correlation. The cross correlation method is inherited from particle imaging velocimetry, where the correlated images contain a random pattern of particles. A template matching method is presented here which takes advantage of the known geometry of laser written tag grids in MTV to achieve better accuracy. Grid intersections are explicitly located in each image by correlation with a template with several linear and rotational degrees of freedom. The template is a continuous mathematical function, so the correlation may be optimized at arbitrary sub-pixel resolution. The template is smooth at the spatial scale of the image noise, so random error is substantially suppressed. Under typical experimental conditions at low imaging resolution, displacement uncertainty is reduced by a factor of 5 compared to the direct cross correlation method. Due to the rotational degrees of freedom, displacement uncertainty is insensitive to highly deformed grids, thus permitting longer delay times and increasing the relative accuracy and dynamic range of the measurement. In addition, measured rotational displacements yield velocity gradients which improve the fidelity of interpolated velocity maps.     

A q-Factor-Based Digital Image Correlation Algorithm (qDIC) for Resolving Finite Deformations with Degenerate Speckle Patterns

Digital image correlation (DIC) has become a widely utilized non-contact, full-field displacement measurement technique for obtaining accurate material kinematics. Despite the significant advances made to date, high resolution reconstruction of finite deformations for images with intrinsically low quality speckle patterns or poor signal-to-noise content has not been fully addressed. In particular, large image distortions imposed by materials undergoing finite deformations create significant challenges for most classical DIC approaches. To address this issue, this paper describes a new open source DIC algorithm (qDIC) that incorporates cross-correlation quality factors (q-factors), which are specifically designed to assess the quality of the reconstructed displacement estimate during the motion reconstruction process. A q-factor provides a robust assessment of the uniqueness and sharpness of the cross-correlation peak, and thus a quantitative estimate of the subset-based displacement measure per given image subset and level of applied deformation. We show that the incorporation of energy- and entropy-based q-factor metrics leads to substantially improved displacement predictions, lower noise floor, and reduced decorrelation even at significant levels of image distortion or poor speckle quality. Furthermore, we show that q-factors can be utilized as a quantitative metric for constructing a hybrid incremental-cumulative displacement correlation scheme for accurately resolving very large homogeneous and inhomogeneous deformations, even in the presence of significant image data loss.     

Displacement fields denoising and strains extraction by finite element method

Optical full-field measurement methods are now widely applied in various domains. In general, the displacement fields can be directly obtained from the measurement, however in mechanical analysis strain fields are preferred. To extract strain fields from noisy displacement fields is always a challenging topic. In this study, a finite element method for smoothing displacement fields and calculating strain fields is proposed. An experimental test case on a holed aluminum specimen under tension is applied to validate this method. The heterogeneous displacement fields are measured by digital image correlation (DIC). By this proposed method, the result shows that the measuring noise on experimental displacement fields can be successfully removed, and strain fields can be reconstructed in the arbitrary area.     

A Robust Finite Element-based Filter for Digital Image and Volume Correlation Displacement Data

Background

Digital Image and Volume Correlation (DIC and DVC) are non-contact measurement techniques that are used during mechanical testing for quantitative mapping of full-field displacements. The relatively high noise floor of DIC and DVC, which is exasperated when differentiated to obtain strain fields, often requires some form of filtering. Techniques such as median filters or least-squares fitting perform poorly over high displacement gradients, such as the strain localisation near a crack tip, discontinuities across crack flanks or large pores. As such, filtering does not always effectively remove outliers in the displacement field.

Objective

This work proposes a robust finite element-based filter that detects and replaces outliers in the displacement data using a finite element method-based approximation.

Methods

A method is formulated for surface (2D and Stereo DIC) and volumetric (DVC) measurements. Its validity is demonstrated using analytical and experimental displacement data around cracks, obtained from surface and full volume measurements.

Results

It is shown that the displacement data can be filtered in such a way that outliers are identified and replaced. Moreover, data can be smoothed whilst maintaining the nature of the underlying displacement field such as steep displacement gradients or discontinuities.

Conclusions

The method can be used as a post-processing tool for DIC and DVC data and will support the use of the finite element method as an experimental–numerical technique.

     

On the Use of NURBS Functions for Displacement Derivatives Measurement by Digital Image Correlation

In this paper, we propose to investigate the potential improvement of using Non-Uniform Rational B-Spline (NURBS) functions for displacement measurements by digital image correlation (DIC). The aim is at improving the performance of DIC to capture with low uncertainty and low noise levels not only the displacement field but also its derivatives. Indeed, when the displacement field is used to feed constitutive law identification procedures, displacement derivatives are required and thus may be measured with robustness. Two examples illustrate the potential of NURBS for DIC: a compressive test on a wood sample and a bending test on a steel beam. For the latter, beam kinematics are adopted and NURBS are used in order to capture the variation of the curvature (second derivative of the displacement) along the beam axis. For these two examples, an error study based on a decomposition of the error into the correlation error and the interpolation error, is carried out and shows the great potential of NURBS functions for DIC.