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Abstract
The use of supervised neural networks is a new approach to solving digital image correlation (DIC) problems, but the existing methods solely adopt the black-box neural network, i.e., the mapping from speckle image pair (reference image and deformed image) to multiple deformation fields (displacement fields and strain fields) is directly established without considering the physical constraints between the fields, causing a low level of accuracy that is even inferior to that of Subset-DIC. In this work, we proposed a deep learning model by introducing strain-displacement relations into a neural network, in which the effect of errors both in displacement and strain are considered in the network training. The back-propagation process of the proposed model is derived, and the solution scheme is implemented by Python. The performance of the proposed model is evaluated by simulation and real DIC experiments, and the results show that adding physical constraints to the neural network can significantly improve prediction accuracy.
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1. Introduction
Digital image correlation (DIC) is a full-field measurement method widely used in the field of experimental mechanics due to its easy experimental setup, low demand for measurement environment, full-field measurement and high precision [1,2]. DIC obtains the deformation (displacement and strain) fields by tracking the speckle patterns between the two images recorded before and after deformation, i.e., the reference image and the deformed image [3]. After the speckle image acquisition is completed, the traditional Subset-DIC obtains the displacement field through the following steps. Firstly, the calculation points are arranged on the reference image according to the measurement requirements (generally evenly distributed at equal intervals) [4], and the subset centred at each point is selected and a suitable subset shape function is chosen. Secondly, the displacement fields are calculated by repeating the correlation optimization algorithm for each calculation point. Thirdly, the strain field is obtained by using some special numerical differential algorithm [5]. As can be seen from the previous description, traditional DIC requires a selection of a series of key parameters when analyzing the deformation field from the speckle image, such as calculation point spacing, subset size and shape function form [6], etc. This selection process heavily impacts the calculation results and must be completed by qualified personnel with extensive experience. Recent works have offered some insights by introducing a deep learning approach to DIC, that is, the mapping from the speckle image pair (reference image and deformed image) to deformation fields represented by a black-box Convolutional Neural Network (CNN) [7,8]. Boukhtache et al. [9] used a typical supervised deep neural network, the U-Net-like model [10], for the DIC solution: a neural network model that can directly predict the full-field displacement was developed with the speckle image pair as the input and the displacement field as the output. Yang et al. [11] further used two U-Net-like models to predict the full-field displacement and full-field strain respectively. Their work proved that deep learning is feasible in solving the DIC problem, meaning the DIC solving process can be represented by a neural network, and the deformation field is directly inferred from the reference image and deformation image. These approaches have avoided setting parameters to some extent and skipped the optimization process for correlation. In addition, the data processing architecture of deep learning differs from traditional DIC [12] in that the method first trains the network model with a large number of data pairs, a step that is time-consuming but can be completed in advance. After the model training is completed, only one forward propagation process is needed to predict the deformation field from the image [13]. Therefore, in terms of experimental application, the deep learning method for solving the deformation field is much faster than the traditional method, and real-time measurement is predictably achievable. However, previous deep learning DIC methods fail to consider the physical constraints between deformation fields, resulting in poor spatial resolution.
Existing work using a neural network to solve the deformation fields from speckle image pair either use one network to get only the displacement field [9], or use two networks to map displacement and strain as two independent physical quantities [11]. In fact, as there is a definite relationship between displacement and strain in mechanics, the two are not independent physical quantities. Therefore, it is possible to introduce the strain-displacement relations into the neural network as a constraint to improve the training and prediction process. Introducing physical constraints such as strain-displacement relations into the neural network for solving DIC will potentially further improving the full-field measurement accuracy of the deformation fields [14]. In this paper, the strain-displacement relations are introduced into the network model for solving DIC, a new loss function considering both displacement and strain error is proposed, the back-propagation considering strain-displacement relations is derived, and it is demonstrated that the errors of both displacement and strain can be considered simultaneously with only one network. The training and solving process of the network is realized by a program. Finally, the effect of the improved neural network is examined with simulated images and real experimental images, and the results show that the proposed method improves the accuracy while maintaining the inference speed. For ease of reading, the U-Net-like model is referred to as DIC Neural Network (DIC-NN), and our network is referred to as DIC Physical Constrained Neural Network (DIC-PCNN).
The paper is structured as follows. After establishing the architecture of the proposed model and the loss function, the principle of this architecture is demonstrated and the derivation of the analytical solution of back-propagation with the addition of strain-displacement relations is completed. Then, the automatic differentiation of the model solution is derived, which is followed by comparing the results of the DIC-NN and the DIC-PCNN in simulation and real experiments to report their precision. In conclusion, this paper discusses the performance of proposed methods and gives prospects for applications.
2. Method
2.1. Problem set-up
As is shown in Fig. 1, DIC-PCNN is based on a U-Net-like structure with a modification in the output part, i.e., the introduction of strain-displacement relations from the elastic theory that establishes the physical connection between displacement field and strain field. Also, the new loss function is defined so that it takes the effects of both displacement and strain errors into consideration. The input, output and feature map of the hidden layer of DIC-PCNN are all three-dimensional arrays of n × n × m (n is the size, m is the number of channels), where the input of the model is a set of 256 × 256 × 2 grayscale data representing the reference image and the deformed image, and the output layer is divided into two parts, i.e., an array of 256 × 256 × 2 representing the displacement fields u and v, and an n × n × 3 array representing the three strain fields ${\varepsilon _x},{\varepsilon _y},{\gamma _{xy}}$.
The U-Net-like structure in Fig. 1 (the dash-dot line part) is similar to [9,15] in that it is in the form of a contracting path - expansive path architecture. The contracting path consists of 4 “block + max-pool” operations, where after each operation, the feature map of the layer is obtained, n is halved, and m is doubled. Correspondingly, the expansive path consists of 4 transpose convolution (upsampling) operations, where after each operation, the feature map has a doubled n and halved m, which is then merged with its counterpart from the corresponding contracting path [16], doubling the number of the channels. After the above operations, the n of the feature map is restored to 256, the displacement field is gained after a convolution operation, and the strain field can be obtained from strain-displacement relations.
In summary, the DIC-PCNN is a function from image pair (reference image, deformation image) to deformation field (displacement field, strain field), denoted as
The key to solving the neural network is to train the network with the existing data-set to obtain the optimal parameters $(\mathbf{w},\mathbf{b})$ [17]. The core step of the training, an iterative optimization process in its essence, is to compute the gradient of the loss function, i.e., $\frac{{\partial C}}{{\partial \mathbf{w}}}$ and $\frac{{\partial C}}{{\partial \mathbf{b}}}$. It is inefficient, however, to directly calculate the gradient of C to each weight, hence the back-propagation algorithm introduces an artificially defined parameter ${\delta ^l}$, i.e., the partial derivative of the C with respect to ${q^l}$ (elements in the feature map of lth layer), to avoid repeated calculations in the chain derivation [18] and also to calculate $\frac{{\partial C}}{{\partial \mathbf{w}}}$ and $\frac{{\partial C}}{{\partial \mathbf{b}}}$. Since DIC-PCNN is only modified in the output layer (Lth layer), this paper only derives the back-propagation for the output layer.
According to Eq. (2), the loss function consists of a displacement part and a strain part, and the displacement value is output by the feature map in Lth layer. Therefore, when solving the ${\delta ^L}$ of Lth layer in DIC-PCNN, both partial derivatives of $\textrm{MS}{\textrm{E}_u}$ and $\textrm{MS}{\textrm{E}_\varepsilon }$ with respect to ${q^l}$ need to be considered separately. The $\frac{{\partial \textrm{MS}{\textrm{E}_u}}}{{\partial {q^L}}}$ can be expressed as
Expand Eq. (6), we have
For the strain part, the elastic theory states that strain can be expressed as a function of displacement, and strain-displacement relation is
Since the output strain layer has no weight connection, the strain error affects the weight of the Lth layer. The partial derivatives of $\textrm{MS}{\textrm{E}_\varepsilon }$ with respect to ${q^l}$ are
The shear strain ${\gamma _{xy}}$ in the above equation is calculated jointly by u and v. We expand the left part of the Eq. (11a)
Similarly, we have $\frac{{\partial \textrm{MS}{\textrm{E}_\varepsilon }}}{{\partial {\varepsilon _{y(i,j)}}}},\frac{{\partial {\varepsilon _{y(i,j)}}}}{{\partial q_{x,y,v}^L}},\frac{{\partial {\gamma _{xy(i,j)}}}}{{\partial q_{i,j,v}^L}}$. In summary, ${\delta ^L}(\mathbf{\varepsilon })$ is
Combining Eqs. (9) and (16), we have
It can be seen from Eq. (17) that the ${\delta ^L}$ of the Lth layer includes the contribution of both errors in the displacement field and the strain field. After obtaining the ${\delta ^L}$, the ${\delta ^l}$ of hidden layer can be solved recursively [7]. Therefore, the displacement and strain errors can be transmitted layer by layer with an impact on training of the entire network.
2.2. Problem-solving process
The previous section derived the back-propagation algorithm with the constraint of strain-displacement relations in our model. But for digital images, the data is discrete, and the derivative of the discrete displacement field needs to be calculated. The first-order derivative of the displacement can be approximated by finite differences using the displacement values at the sample points [19]. In this section, the central difference is used to calculate the strain. For the strain at point $P(x,y)$, we have
The partial derivatives of C with respect to ${\varepsilon _x}$ is equal to that of C to ${u_{i,j + h}},{u_{i,j - h}}$. Calculating the partial derivatives of the loss function with respect to $(\mathbf{w},\mathbf{b})$ is generally realized by reverse mode automatic differentiation (AD) [20,21]. Table 1 and Fig. 2 give the flowchart of the reverse mode AD to compute the gradient of Eq. (19). The AD follows the chain rule and breaks an overall expression down to sub-expressions, where the latter in the forward sweep are calculated and the output saved as intermediate variables, and then the derivatives are computed recursively in the reverse sweep for each sub-expression using the intermediate variables [22]. The derivation of $C({\varepsilon _y})$ and $C({\gamma _{xy}})$ is similar to that of $C({\varepsilon _x})$, and thus extended to all points in the deformation field.
Table 1. Reverse mode automatic differentiation flowchart of Eq. (19)
The $\textrm{MS}{\textrm{E}_\varepsilon }$ in Eq. (4) can be written as
Combine Eqs. (2) and (20) and substitute them into the deep learning framework to call the built-in function [23] to calculate the gradients, and then call an optimization algorithm [24] to update the network parameters.
Theoretically, both displacement and strain components could be obtained by the model above. However, it is worth noting that we introduce the strain-displacement relationship into the loss function to promote the training of the U-Net-like model and to improve the accuracy of the displacement field, not directly obtain the strain results for experimental analysis. In fact, the commonly used Subset-DIC is also optimized by both displacement and strain during the iterative optimization process (when first-order deformation is considered, the parameter space is $\left[ {u,\frac{{\partial u}}{{\partial x}},\frac{{\partial u}}{{\partial y}},v,\frac{{\partial v}}{{\partial x}},\frac{{\partial v}}{{\partial y}}} \right]$[4]). From a more in-depth level, the effect of strain components in the iterative optimization process is to assist the solution of displacement. That is, improve the accuracy of the solution of displacement components by introducing the deformation, not only the rigid body translation, of the subset. However, practical analysis and test results show that the strain obtained directly from the optimization procedure is not very accurate, so the strain fields are often calculated by the displacement fields using the numerical derivation method. Like Subset-DIC, the strain field directly obtained by DIC-PCNN does not have high accuracy, and it will require us to obtain strain from the displacement field later by using numerical methods [4].
3. Results and discussion
3.1 Training and test of the neural network
To train the model, this paper refers to the data-set production method in [9,11] to generate synthetic speckle images and their corresponding standard displacement field, and the strain field corresponding to the latter is obtained by Eq. (18). The reference images in the speckle image pair are selected from real experiments, as is shown in Fig. 3, and the simulated deformation image is generated following the method in Ref. [9]. A total of 2,460 sets of data were generated, and the upper and lower bounds of the displacement in the data-set were set to 2 pixels and -2 pixels, respectively. Figure 3 shows two examples from our data-set. Finally, the above data-set was used as the train set, and the “star displacement” field shown in Fig. 5 was cropped to 4 sets of data as the test set.
Fig. 3. Two samples in the training set, including reference image, (the first speckle image in each row), deformation fields (the middle 5 graphs in each row) $u,v,{\varepsilon _x},{\varepsilon _y},{\gamma _{xy}}$ used for simulation and one of the deformed images (the last speckle image in each row).
In the train set and test set, we introduce random noise of different magnitudes to the reference image and the deformed image to enhance the noise resistance of the model. DIC-NN and DIC-PCNN have the same number of weights and biases and use the same training procedure. The hardware is NVIDIA Tesla P100 GPU.
For this work, the code is written via PyTorch, the mini-batch size is set to 32, Adam [24] with a weight decay coefficient at 0.03 is used for the optimization algorithm, and the learning rate l starts from 0.0005 subject to dynamic adjustment along the training process, and the method refers to [25].
Figure 4 shows the loss of displacement and strain in the training processes of DIC-NN and DIC-PCNN in a total of 100 epochs. The strain output by the neural network is calculated by the displacement through Eq. (18). It can be seen that the addition of the strain-displacement relations improves the stability and the eventual accuracy of the network, indicating that the error of the strain part is conducive to the training of the network.
Fig. 4. Displacement loss and strain loss comparison between DIC-NN and DIC-PCNN, where the x coordinate represents the number of epochs (an epoch indicates that all data has been sent to the network and a process of forward propagation & reverse propagation has been completed). (a) displacement loss of 1-50 epochs; (b) displacement loss of 50-100 epochs; (c) strain loss of 1-50 epochs; (d) strain loss of 50-100 epochs.
3.2 Assessment of the results with simulation experiment
The “star displacement” poses challenges to DIC [26,27]. It is formed by synthetic sine waves and used to evaluate the performance of various full-field measurements in experimental mechanics [28], suitable for quantifying the effect of the spatial frequency on the measured displacement field. As is shown in Fig. 5, the sine wave frequency decreases linearly from left to right.
Fig. 5. Star displacement and speckle image. (a) Star displacement field used to generate the deformation image, (b) reference image (obtained from a real experiment).
This section compares the Subset-DIC, DIC-NN and DIC-PCNN to analyze the overall performance of the three methods. The step size of Subset-DIC is set to 1 pixel and the subset size is set to 2M + 1 = 15 pixels. For computational speed and the ease of merging the images, the length and width of the input image are generally multiples of 2. The neural network computational results shown in this section are four 256 × 256 matrix arrays merged into a 256 × 1024 matrix array.
It should be noted that the results calculated by the DIC-NN and DIC-PCNN are inaccurate at the edge of the image because of the border deviations [29], and the affected area is approximately 4 pixels wide. One way to reduce the border deviation for the continuity of displacement is to obtain the data at the splicing area by interpolating the data on both sides, and the other is to move the detection box in step size of a certain length (for example, when the step size is 1/2 the size of the detection box, the data in the middle of a detection box can be used to replace the data at the edge of the adjacent detection box). Figure 6 shows the predicted displacement fields and the global error maps between the predicted value and the true value, and the border results obtained by interpolating the data on both sides of the border. Figure 7 shows the mean absolute error (MAE) of v-displacement per column (the data is selected from column 12 to column 1012 and from row 12 to row 244), and the MAE is defined as follows
where ${v_p}$ and ${v_t}$ are the predicted and the true displacements respectively, and N is the height of the error map, and j is column index. The MAE per column shows the calculation accuracy of three methods at differential spatial frequencies.Fig. 6. Comparison of calculation “star field” results among three methods. (a) V-displacement calculated by Subset-DIC; (b) the error map (difference between the result and the true value) of v-displacement calculated by Subset-DIC; (c) v-displacement calculated by DIC-NN; (d) the error map of v-displacement calculated by DIC-NN; (e) v-displacement calculated by DIC-PCNN; (f) the error map of v-displacement calculated by DIC-PCNN.
Fig. 7. Comparison of the mean absolute error of v-displacement per column among three methods (Subset-DIC, DIC-NN, DIC-PCNN).
The results in Fig. 6 and Fig. 7 show that the error of Subset-DIC decreases sharply first and then smoothly from left to right according to the horizontal axis, with the decreasing of the heterogeneity of the displacement field. Correspondingly, the error of DIC-PCNN and DIC-NN decreases gradually from left to right and almost converges in the uniform deformation area. In the non-uniform displacement area, the error of Subset-DIC is far greater than that of DIC-NN and DIC-PCNN, while in the uniform displacement area, the error of Subset-DIC is slightly lower than that of DIC-NN and DIC-PCNN. In almost all regions, the error of DIC-PCNN is less than that of DIC-NN.
Table 2 makes quantitative statistics on the results, and it can be seen that for the star displacement field, the MAE of Subset-DIC is 2.63 times larger than that of DIC-PCNN in columns 12-256. In the medium spatial frequency region (columns 257-512), the MAE of DIC-PCNN is 24.84% lower than that of Subset-DIC and 41.21% lower than that of DIC-NN. In the part of the low spatial frequency region (columns 513-1012), the Subset-DIC has the highest accuracy, with an MAE of approximately 0.02 pixels, while the MAE of DIC-PCNN is maintained at around 0.05 pixels. DIC-NN has the lowest accuracy with an MAE of 0.0747 pixels. Generally speaking, the results of DIC-NN and DIC-PCNN in the non-uniform displacement area are far better than those of Subset DIC, and the results in the uniform displacement area are slightly worse than those of Subset DIC. In addition, the magnitude of the error map and MAE per column of DIC-PCNN are smaller than those of DIC-NN in whole areas. This is because, in the uniform displacement area, the subset in Subset-DIC can suppress image noise and smooth the displacement results. The DIC-NN has the lowest accuracy as its displacement field output has no constraints, while the DIC-PCNN accuracy is higher because it uses strain-displacement relations to constrain the displacement of one point and its surrounding points. Although DIC-NN and DIC-PCNN also have implicit “subsets” due to their convolutional layers, perhaps this subset is smaller than that of Subset-DIC, and its smoothing effect is less pronounced.
Table 2. Absolute error of the three methods (Subset-DIC, DIC-NN, DIC-PCNN)
In terms of computation duration, the time consumed for DIC-PCNN and DIC-NN is the same when inferring displacements as they have the same number of parameters at the inference stage. We have simply tested the calculation speed of the DIC-PCNN and found that the average inference speed is 9.28 + E7 points of interest per second processed with Nvidia Tesla P100 GPU, which is similar to that described in Ref. [9].
3.3 Assessment of the results with real experiment
In this section, we will show the application of DIC-PCNN, DIC-NN and Subset-DIC in calculating the deformation field in the three-point bending test of aluminum specimens with prefabricated cracks evaluate the generalization ability of neural network method in actual experiments. (See [30] for the experimental method and detailed steps of data acquisition.) Figure 8 is the experimental image and region of interest (ROI). Here, 1 mm of the specimen size is converted to 21.2 pixels in the image. In 300 pixels × 500 pixels ROI, four rectangular detection boxes are arranged to obtain four groups of speckle image pairs. Splicing the adjacent detection boxes can avoid border deviations of CNN. The step size of Subset-DIC is 2 pixels and the subset size is 13 pixels. The images taken under the load of 2 kN are selected as the reference images and the images taken under the load of 2.25 kN are selected as deformation images.
Fig. 8. Diagram of ROI and two rows for comparison of the displacement field in the reference image of the three-point bending experiment.
Figure 9(a) is the displacement field $u(x,y)$ calculated by FEM, and here we use it as a standard value. Figures 9(b),(c),(e) show the results of Subset-DIC, DIC-NN and DIC-PCNN respectively. It can be seen that the results of Subset-DIC, DIC-NN, and DIC-PCNN are similar in terms of distribution, but DIC-PCNN and DIC-NN are noisier. Besides, DIC-PCNN has better results than DIC-NN. Considering that subset comes with a filter smoothing effect, we also use a simple moving average filter to smooth the results of Fig. 9(c) and Fig. 9(e) to obtain Fig. 9(d) and Fig. 9(f). The results show that the noise is significantly suppressed after filtering, and the results in Fig. 9(f) and Fig. 9(b) are very similar. The above conclusion can also be supported by Fig. 10(b) and Fig. 10(d).
Fig. 9. Comparison of u-displacement results with several methods. (a) FEM result of ROI in Fig. 8; (b) Subset-DIC result of ROI in Fig. 8; (c) DIC-NN result of ROI in Fig. 8; (d) DIC-NN results processed by moving average with 13 × 13 sliding window. (e) DIC-PCNN result of ROI in Fig. 8; (f) DIC-PCNN results processed by moving average with 13 × 13 sliding window.
Fig. 10. Comparison of u-displacement results in two rows. (a) u results among FEM, DIC-NN, DIC-NN (smoothed), DIC-PCNN, DIC-PCNN (smoothed) and Subset-DIC on Row 1 in Fig. 8; (b) deviations between DIC-NN and FEM, between DIC-NN (smoothed) and FEM, between DIC-PCNN and FEM, between DIC-PCNN (smoothed) and FEM, between Subset-DIC and FEM on Row 1; (c) u results among FEM, DIC-NN, DIC-NN (smoothed), DIC-PCNN, DIC-PCNN (smoothed) and Subset-DIC on Row 2; (d) deviations between DIC-NN and FEM, between DIC-NN (smoothed) and FEM, between DIC-PCNN and FEM, between DIC-PCNN (smoothed) and FEM, between Subset-DIC and FEM on Row 2.
Combining the theoretical derivation results in method section, it can be proved that adding mechanical laws to neural network can further deliver higher-precision results.
4. Conclusion
We introduce the strain-displacement relations into the neural network with the addition of physical constraints in solving DIC problems, which is called DIC-PCNN. Then derive the back-propagation algorithm and solution by reverse mode automatic differential with physical constraints, and the whole process is implemented with PyTorch. Then, a simulated experiment and a three-point bending experiment are implemented to compare the performance of several methods, i.e., Subset-DIC, DIC-NN and DIC-PCNN, in handling uniform and non-uniform deformation fields.
The results of simulated experiments and real experiments show that the deep learning method can be used for DIC deformation analysis, and the accuracy of the proposed DIC-PCNN is better than DIC-NN. The accuracy of DIC-PCNN is far better than that of Subset-DIC in non-uniform displacement fields and slightly poorer in uniform displacement fields, which is because the subset comes with smoothing and good noise suppression ability. DIC-PCNN can also achieve similar results to Subset-DIC in uniform displacement field if a good filtering method is chosen.
Introducing strain-displacement relations into black-box neural networks to solve digital image correlation problems can improve prediction accuracy. In addition, this method utilizes the high-speed computing power of the neural network to address the scenes with a real-time computing demand and does not need to select excessive parameters. This work can provide some ideas for other related optical metrology studies. However, supervised neural networks also have disadvantages, because a large amount of data is required to train the model. In the future, unsupervised neural networks will be developed to solve this problem.
Funding
National Natural Science Foundation of China (11727801, 12132009).
Acknowledgments
The authors would like to thank the National Natural Science Foundation of China for help identifying collaborators for this work. The authors would like to thank Professor Qibing Zhu from Jiangnan University and Associate Professor Qinwei Ma from Beijing Institute of Technology for their help in revised manuscript.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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