Inverse Compositional Gauss-Newton Algorithm Research Articles

Particle tracking-aided digital volume correlation for clay–sand soil mixtures

In this study, a novel, interdisciplinary method is introduced that merges fundamental geomechanics with computer vision to develop an advanced hybrid feature-aided digital volume correlation (DVC) technique. This technique is specifically engineered to measure and compute the full-field strain distribution in fine-grained soil mixtures. A clay–sand mixture specimen composed of quartz sand particles and kaolinite was created. Its mechanical properties and deformation behaviour were then tested using a mini-triaxial apparatus, combined with micro-focus X-ray computed tomography (μCT). The CT slices underwent image processing for denoising, segmentation of distinct phases, reconstruction of sand particles and feature extraction within the soil specimen. The proposed approach incorporated a two-step particle tracking method, which initially uses particle volume and surface area features to establish a preliminary matching list for a reference particle and then use the iterative closest point method for precise target particle matching. The soil specimen's initial displacement field was then mapped onto the DVC method's grid, and further refined through sub-voxel registration by way of a three-dimensional inverse compositional Gauss–Newton algorithm. The proposed method's effectiveness and efficiency were validated by accurately calculating the displacement and strain fields of the soil mixture sample, and comparing the results with those from a traditional DVC method. Given the soil's compositional and microstructural characteristics, these image-matching techniques can be integrated to create a versatile, efficient, and robust DVC system, suitable for a variety of soil mixture types.

Algorithm Analysis and Optimization of a Digital Image Correlation Method Using a Non-Probability Interval Multidimensional Parallelepiped Model

Digital image correlation (DIC), a widely used non-contact measurement technique, often requires empirical tuning of several algorithmic parameters to strike a balance between computational accuracy and efficiency. This paper introduces a novel uncertainty analysis approach aimed at optimizing the parameter intervals of a DIC algorithm. Specifically, the method leverages the inverse compositional Gauss–Newton algorithm combined with a prediction-correction scheme (IC-GN-PC), considering three critical parameters as interval variables. Uncertainty analysis is conducted using a non-probabilistic interval-based multidimensional parallelepiped model, where accuracy and efficiency serve as the reliability indexes. To achieve both high computational accuracy and efficiency, these two reliability indexes are simultaneously improved by optimizing the chosen parameter intervals. The optimized algorithm parameters are subsequently tested and validated through two case studies. The proposed method can be generalized to enhance multiple aspects of an algorithm’s performance by optimizing the relevant parameter intervals.

A prediction–correction method for fast and accurate initial displacement field estimation in digital image correlation

Initial displacement estimation is one of the most critical issues in digital image correlation. A better initial value can greatly improve the convergence rate and accuracy of the algorithms with subpixel accuracy. This paper developed an efficient estimation method to yield high-quality initial displacement fields. This method finds the initial displacement of each subset in a prediction–correction way, in which the displacement of the seed point is found by exhaustive search, but the other subsets are first predicted by an extrapolation scheme and then corrected by a monotonous search strategy. This method was tested by extensive experiments and validated by comparing with the well-known exhaustive search and adaptive rood pattern search methods, and then it was combined with the inverse compositional Gauss–Newton algorithm to perform subpixel-optimization experiments. The results demonstrated excellent features of accuracy, effectiveness, and convergence. Finally, we presented a three-dimensional surface reconstruction experiment using the proposed method, obtaining a geometric accuracy with a relative error of 0.016%.

Experimental assessment of noise robustness of the forward-additive, symmetric-additive and the inverse-compositional Gauss-Newton algorithm in digital image correlation

Digital Image Correlation (DIC) is a well-known optical experimental method based on the assumption that pixel intensity does not change with motion. This assumption is never exactly satisfied because various sources of noise exist (read-, shot-, dark-noise of the Charge Coupled Device (CCD), thermal dilatation of lens, non-constant illumination,...); moreover, to accurately track displacements, pixel intensity has to be interpolated at non-integer locations, thus introducing both phase and intensity errors due to the use of theoretically incorrect interpolating kernels.DIC can be implemented using various formulations, the most used being the Forward-Additive Gauss-Newton (FA-GN) and the Inverse-Compositional Gauss-Newton formulation (IC-GN). Even though both formulations give the same results at the first order, their speed, converging characteristics, and noise robustness differ considerably in particular with regard to the noise bias. Indeed, Shao et al. [1], using a theoretical analysis of a mono-dimensional signal and a numerical simulation on FFT-shifted images, recently showed that the latter formulation is not affected by noise bias, providing the correct differential operator is used.In this work, a fully experimental assessment of the noise-bias sensitivity of both algorithms is performed, confirming the theoretical findings and the numerical simulations available in the literature.

Path independent stereo digital image correlation with high speed and analysis resolution

Path independent strategy is introduced into stereo digital image correlation, in which high accuracy is attained through the inverse compositional Gauss-Newton (IC-GN) algorithm with the second and the first order shape functions for stereo matching and temporal matching, respectively. To gain strong adaptibility to large and complex deformation or considerable parallax betweent the two views, two robust methods are developped to estimate the initial guess for the iterative IC-GN algorithm. One is aided by scale invariant feature transform (SIFT) features and the other is aided by epipolar constraint. Powered by parallel computing on GPU, the proposed stereo DIC demonstrates high computation efficiency. Real-time stereo DIC is realized and used to measure the large deformation at common video recording rate (∼24 fps) with high ananlysis resolution (∼6300 points of interest in region of interest).

3-D strain estimation in sandstone using improved digital volumetric speckle photography algorithm

Accurately and stably estimating the strain in rocks is important to understand their failure mechanism. Digital volumetric speckle photography (DVSP) combined with X-ray computed tomography (CT) has been used to investigate the strain in rocks. This study introduced the inverse compositional Gauss–Newton algorithm into DVSP to improve the accuracy and precision of displacement and strain measurements. Granules with different particle sizes were used to simulate the mesostructure of sandstone, and a series of baseline experiments were conducted to evaluate the accuracy of the improved algorithm, which had a performance that was approximately three times better than DVSP alone. Subsequently, the algorithm was applied to red sandstone subjected to the Brazilian test and an indentation experiment. The interior deformation characteristics of the specimen were clearly depicted, which clarified the failure mechanism of the rock material.

Real-time 3D Digital Image Correlation for Large Deformation and Rotation Measurements Based on a Deformation Transfer Scheme

With the recent increase in digital image correlation (DIC) applications, it is important to improve the computational efficiency of DIC and realize the closed loop control of systems. A real-time three-dimensional (3D) DIC method for large deformation and rotation measurements is proposed in this paper. To solve the problem of initial guess estimation under large deformation and rotation, the convergence radius of the inverse compositional Gauss-Newton (IC-GN) algorithm considering large rotation and deformation is analyzed. Then, a temporal-spatial deformation transfer scheme is proposed for efficient temporal and stereo matching in 3D-DIC. The highly efficient IC-GN algorithm with an improved interpolation look-up table is proposed to further improve the computational efficiency. The multipoint and full-field real-time 3D deformation measurements are performed by parallel CPU computation. The convergence radius of IC-GN are estimated to be below 3 pixels for rotation and below 5 pixels for tensile deformation. The accuracy of the initial guess estimation by deformation transfer is improved and can meet the requirements of large deformation and rotation measurements. Benefiting from the temporal-spatial deformation transfer scheme, the efficient IC-GN algorithm, the efficient temporal and stereo matching strategy and the parallel computation, real-time multipoint deformation measurements at 60 Hz with 50 points and full-field deformation measurements at 15 Hz with 5000 points are realized. Real-time multipoint and full-field 3D deformation measurements for large deformations and rotations are realized. High-accuracy, real-time and fully automatic 3D deformation measurement sensors are expected.

Significance of Parallel Computing on the Performance of Digital Image Correlation Algorithms in MATLAB

Digital Image Correlation (DIC) is a powerful tool used to evaluate displacements and deformations in a non-intrusive manner. By comparing two images, one from the undeformed reference states of the sample and the other from the deformed target state, the relative displacement between the two states is determined. DIC is well-known and often used for post-processing analysis of in-plane displacements and deformation of the specimen. Increasing the analysis speed to enable real-time DIC analysis will be beneficial and expand the scope of this method. Here we tested several combinations of the most common DIC methods in combination with different parallelization approaches in MATLAB and evaluated their performance to determine whether the real-time analysis is possible with these methods. The effects of computing with different hardware settings were also analyzed and discussed. We found that implementation problems can reduce the efficiency of a theoretically superior algorithm, such that it becomes practically slower than a sub-optimal algorithm. The Newton–Raphson algorithm in combination with a modified particle swarm algorithm in parallel image computation was found to be most effective. This is contrary to theory, suggesting that the inverse-compositional Gauss–Newton algorithm is superior. As expected, the brute force search algorithm is the least efficient method. We also found that the correct choice of parallelization tasks is critical in attaining improvements in computing speed. A poorly chosen parallelization approach with high parallel overhead leads to inferior performance. Finally, irrespective of the computing mode, the correct choice of combinations of integer-pixel and sub-pixel search algorithms is critical for efficient analysis. The real-time analysis using DIC will be difficult on computers with standard computing capabilities, even if parallelization is implemented, so the suggested solution would be to use graphics processing unit (GPU) acceleration.

3D SIFT aided path independent digital volume correlation and its GPU acceleration

A novel path independent digital volume correlation (DVC) method is proposed, in which the internal deformation field is determined with the aid of the features extracted from the volume images. The deformation vector at each point of interest is first estimated by tracking the motion of the nearby keypoints obtained through 3D scale-invariant feature transform (SIFT), and then fed as the initial guess into the 3D inverse compositional Gauss-Newton algorithm to achieve high accuracy result. Benefiting from the robustness of SIFT to various deformation and distortion of image, the proposed DVC method demonstrates outstanding performance to automatically process the volume images containing large and complex deformation, which keep challenging to current DVC methods for decades. The proposed DVC method is further powered by the parallel computing on GPU. An ultrafast computation speed (about 10,000 POI/s) can be reached on a personal computer when dealing with the volume images of normal sizes. The 3D SIFT aided DVC method, which achieves unprecedented balance between accuracy, adaptability, and efficiency, shows great potential in the quantitative analysis of internal deformation.

Accurate and simple digital volume correlation using pre-interpolation

Existing incremental digital volume correlation methods can reduce the number of errors introduced by interpolation calculations in the inverse-compositional Gauss–Newton (IC-GN) algorithm iteration. However, the accuracy of these existing methods is insufficient for some conditions as the curve-fitting method has high computational efficiency but lacks accuracy. A simple pre-interpolation method is proposed to improve the accuracy and computational efficiency of digital volume correlation. First, the pretreatment of a deformed volume image is calculated by the cubic spline interpolation method with the most often chosen interpolation step of 1/2 sub-voxel. Next, the pre-interpolation is calculated only once and the block calculation techniques solve the memory problem. Then, the reference sub-volume in the updated reference volume image is translated into the nearest half-integer voxel position instead of the integer voxel position or other sub-voxel positions. The pre-interpolation method is applied to both the IC-GN algorithm and the curve-fitting method. Experimental results show that the maximum mean bias error and the maximum standard deviation of the improved IC-GN algorithm are reduced by 34% and 75%, respectively. The improved curve-fitting has better accuracy and computational efficiency than the IC-GN algorithm under small strain and the curve-fitting method can achieve about 3.2 times speedup than the IC-GN algorithm.

Digital image correlation based on variable circle template in dual camera matching

In the measurement of the 3-D shape of a large curved object using traditional 3-D-digital image correlation (DIC), the left and right camera images are relatively rotated or deformed too much, which will cause the matching to fail. We propose a 3D-DIC matching method. Compared with traditional methods having fixed-size templates, this method is robust to relative rotation and deformation between dual cameras and is suitable for 3D-DIC measurement of nonplanar objects. In the proposed method, we use polar coordinates to define the matching template as a variable circle, which is convenient for interpolation calculation and changing the template size. The zero-mean normalized cross-correlation method is used to determine the position of an entire pixel on the distorted image. This step can provide an initial value close to the true value for subsequent iterations of the inverse compositional Gauss–Newton algorithm. The performance of the proposed method is verified by experiments. Compared with the traditional matching method, the proposed method is suitable for matching large curved surfaces between 3D-DIC dual cameras and extends the application of DIC technology.

Elimination of systematic error in digital image correlation caused by intensity interpolation by introducing position randomness to subset points

A strategy called random subset is presented to reduce systematic error in digital image correlation caused by intensity interpolation. In classical digital image correlation techniques, the subset points fall at integer locations; by contrast, the proposed random subset strategy imposes random offsets on the conventional locations, and consequently the subset points are located at non-integer positions. The introduction of position randomness to subset points makes the random subset strategy distinctly different from the classical digital image correlation technique. The random offsets can follow various probability distributions. In this work, two types of random offset were studied as examples, and the maximum of the random offsets was adjusted to modify performance. Using the random subset strategy, random-subset-based subpixel registration algorithms were developed in this study, including the random-subset-based forward additive Gaussian–Newton algorithm and the random-subset-based inverse compositional Gauss–Newton algorithm. The measurement accuracy of the random subset-based subpixel registration algorithms was investigated. Numerical simulations and real-world experiments demonstrated that as the maximum of the random offsets increases, the amplitude of the sinusoidal-shaped systematic error changes from positive to negative. Hence, if an appropriate random offset is selected, systematic error can be considerably reduced. In fact, the accuracy enhancement is greater than one order of magnitude, and systematic error is virtually eliminated.

High-efficiency and high-accuracy digital speckle correlation for full-field image deformation measurement

Many works around the digital speckle correlation (DSC) are to improve the computational efficiency and measurement accuracy in recent years. This work aims to improve the efficiency and accuracy of DSC for both single-point and full-field points used in mechanical properties test of materials. For this purpose, first, a subpixel initial guess for the inverse compositional Gauss–Newton algorithm (IC-GN) with the first-order shape function is introduced for single-point image registration. Then, with the aid of strategy of subset image edge extend interpolation (SIEEI), the efficiency and accuracy of full-field displacement and deformation measurement are improved simultaneously. The SIEEI is employed to reduce mean squared difference errors caused by traditional bicubic interpolation algorithm. Comparative studies between the traditional IC-GN algorithm and the proposed algorithms are presented using simulated speckle images and CCD images. The proposed method achieves more executive efficiency and more accuracy for single-point and full-field points image registration. The computational efficiency of the proposed algorithms increases 7.5% for full-field registration using CCD images. The mean of squared difference errors of the SIEEI method is less than the traditional bicubic interpolation algorithm. The presented approach shows great potentials for challenging application, such as mechanical properties test of materials.

Comparative Analysis of Warp Function for Digital Image Correlation-Based Accurate Single-Shot 3D Shape Measurement.

Digital image correlation (DIC)-based stereo 3D shape measurement is a kind of single-shot method, which can achieve high precision and is robust to vibration as well as environment noise. The efficiency of DIC has been greatly improved with the proposal of inverse compositional Gauss-Newton (IC-GN) operators for both first-order and second-order warp functions. Without the algorithm itself, both the registration accuracy and efficiency of DIC-based stereo matching for shapes with different complexities are closely related to the selection of warp function, subset size, and convergence criteria. Understanding the similarity and difference of the impacts of prescribed subset size and convergence criteria on first-order and second-order warp functions, and how to choose a proper warp function and set optimal subset size as well as convergence criteria for different shapes are fundamental problems in realizing efficient and accurate 3D shape measurement. In this work, we present a comparative analysis of first-order and second-order warp functions for DIC-based 3D shape measurement using IC-GN algorithm. The effects of subset size and convergence criteria of first-order and second-order warp functions on the accuracy and efficiency of DIC are comparatively examined with both simulation tests and real experiments. Reference standards for the selection of warp function for different kinds of 3D shape measurement and the setting of proper convergence criteria are recommended. The effects of subset size on the measuring precision using different warp functions are also concluded.

Optimized digital speckle patterns for digital image correlation by consideration of both accuracy and efficiency.

The technique of digital image correlation (DIC), which has been widely used for noncontact deformation measurements in both the scientific and engineering fields, is greatly affected by the quality of speckle patterns in terms of its performance. This study was concerned with the optimization of the digital speckle pattern (DSP) for DIC in consideration of both the accuracy and efficiency. The root-mean-square error of the inverse compositional Gauss-Newton algorithm and the average number of iterations were used as quality metrics. Moreover, the influence of subset sizes and the noise level of images, which are the basic parameters in the quality assessment formulations, were also considered. The simulated binary speckle patterns were first compared with the Gaussian speckle patterns and captured DSPs. Both the single-radius and multi-radius DSPs were optimized. Experimental tests and analyses were conducted to obtain the optimized and recommended DSP. The vector diagram of the optimized speckle pattern was also uploaded as reference.

Interpolation bias for the inverse compositional Gauss–Newton algorithm in digital image correlation

It is believed that the classic forward additive Newton–Raphson (FA-NR) algorithm and the recently introduced inverse compositional Gauss–Newton (IC-GN) algorithm give rise to roughly equal interpolation bias. Questioning the correctness of this statement, this paper presents a thorough analysis of interpolation bias for the IC-GN algorithm. A theoretical model is built to analytically characterize the dependence of interpolation bias upon speckle image, target image interpolation, and reference image gradient estimation. The interpolation biases of the FA-NR algorithm and the IC-GN algorithm can be significantly different, whose relative difference can exceed 80%. For the IC-GN algorithm, the gradient estimator can strongly affect the interpolation bias; the relative difference can reach 178%. Since the mean bias errors are insensitive to image noise, the theoretical model proposed remains valid in the presence of noise. To provide more implementation details, source codes are uploaded as a supplement.

Digital image correlation based on a fast convolution strategy

In recent years, the efficiency of digital image correlation (DIC) methods has attracted increasing attention because of its increasing importance for many engineering applications. Based on the classical affine optical flow (AOF) algorithm and the well-established inverse compositional Gauss–Newton algorithm, which is essentially a natural extension of the AOF algorithm under a nonlinear iterative framework, this paper develops a set of fast convolution-based DIC algorithms for high-efficiency subpixel image registration. Using a well-developed fast convolution technique, the set of algorithms establishes a series of global data tables (GDTs) over the digital images, which allows the reduction of the computational complexity of DIC significantly. Using the pre-calculated GDTs, the subpixel registration calculations can be implemented efficiently in a look-up-table fashion. Both numerical simulation and experimental verification indicate that the set of algorithms significantly enhances the computational efficiency of DIC, especially in the case of a dense data sampling for the digital images. Because the GDTs need to be computed only once, the algorithms are also suitable for efficiently coping with image sequences that record the time-varying dynamics of specimen deformations.

An Improved Digital Image Correlation Calculation in the Case of Substantial Lighting Variation

In the last decade, digital image correlation (DIC) has become the most widely used non-contact technique for measuring full-field motion and deformation in image sequences. As permanent hardware improvements make it possible to readily obtain digital images with higher and higher resolutions at ever-increasing frame rates, the efficiency of image registration methods is becoming increasingly important. Another aspect of this problem is the desire to improve the accuracy of these methods, which in general implies additional computational time. This problem is addressed by applying the inverse compositional Gauss-Newton (IC-GN) algorithm, which is modified to include the parametric sum of squared differences (PSSD) correlation criterion. However, the classical IC-GN algorithm with the sum of squared differences (SSD) correlation criterion cannot be used in cases in which lighting variations are of considerable importance. In this paper, a solution to this disadvantage is presented. The lighting variation is directly included in the IC-GN minimization procedure, which reduces the additional calculation time. Moreover, this IC-GN algorithm does not need to re-evaluate and invert the Hessian matrix. This results in a more computationally efficient algorithm. Experimental results show that the proposed algorithm is insensitive to the linear intensity variation in the target image; consequently, it has a faster convergence rate than the IC-GN algorithm with the zero-mean normalized sum of squared differences (ZNSSD) correlation criterion. As a result, computational time can be reduced by up to 30% in the case of convergence condition 𝜖=0.001. Owing to the above mentioned advantages of the proposed IC-GN algorithm, it could become a new standard for efficient and robust full-field displacement tracking in image sequences with varying lighting intensity.

Automated initial guess in digital image correlation aided by Fourier–Mellin transform

The state-of-the-art digital image correlation (DIC) method using iterative spatial-domain cross correlation, e.g., the inverse-compositional Gauss–Newton algorithm, for full-field displacement mapping requires an initial guess of deformation, which should be sufficiently close to the true value to ensure a rapid and accurate convergence. Although various initial guess approaches have been proposed, automated, robust, and fast initial guess remains to be a challenging task, especially when large rotation occurs to the deformed images. An integrated scheme, which combines the Fourier–Mellin transform-based cross correlation (FMT-CC) for seed point initiation with a reliability-guided displacement tracking (RGDT) strategy for the remaining points, is proposed to provide accurate initial guess for DIC calculation, even in the presence of large rotations. By using FMT-CC algorithm, the initial guess of the seed point can be automatically and accurately determined between pairs of interrogation subsets with up to ±180 deg of rotation even in the presence of large translation. Then the initial guess of the rest of the calculation points can be accurately predicted by the robust RGDT scheme. The robustness and effectiveness of the present initial guess approach are verified by numerical simulation tests and real experiment.

Self-calibration single-lens 3D video extensometer for high-accuracy and real-time strain measurement.

The accuracy of strain measurement using a common optical extensometer with two-dimensional (2D) digital image correlation (DIC) is not sufficient for experimental applications due to the effect of out-of-plane motion. Although three-dimensional (3D) DIC can measure all three components of displacement without introducing in-plane displacement errors, 3D-DIC requires the stringent synchronization between two digital cameras and requires complicated system calibration of binocular stereovision, which makes the measurement rather inconvenient. To solve the problems described above, this paper proposes a self-calibration single-lens 3D video extensometer for non-contact, non-destructive and high-accuracy strain measurement. In the established video extensometer, a single-lens 3D imaging system with a prism and two mirrors is constructed to acquire stereo images of the test sample surface, so the problems of synchronization and out-of-plane displacement can be solved easily. Moreover, a speckle-based self-calibration method which calibrates the single-lens stereo system using the reference speckle image of the specimen instead of the calibration targets is proposed, which will make the system more convenient to be used without complicated calibration. Furthermore, an efficient and robust inverse compositional Gauss-Newton algorithm combined with a robust stereo matching stage is employed to achieve high-accuracy and real-time subset-based stereo matching. Tensile tests of an Al-alloy specimen were performed to demonstrate the feasibility and effectiveness of the proposed self-calibration single-lens 3D video extensometer.