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Enhancement of damagedimage prediction based on digital twin technology
Journal of Cloud Computing volumeÂ 12, ArticleÂ number:Â 177 (2023)
Abstract
Digital twins have revolutionized the field of image enhancement by applying their unique capabilities. A digital twin refers to a virtual replica of a physical object or system, which can be utilized to simulate and analyze realworld scenarios. In image enhancement, digital twins map entities to images, identify damaged areas, and restore them to their original state. This process involves utilizing the digital twin method to understand the underlying structure and characteristics of the image. The damaged areas can be accurately modeled and repaired using techniques like the CahnHilliard equation. Additionally, neural network models are leveraged to measure the effectiveness of the image restoration process. Compared with the firstorder numerical scheme, the secondorder method can improve the prediction accuracy by more than 40% in some cases. Through these advancements, digital twins have significantly enhanced imagesâ€™ quality, clarity, and visual appeal, contributing to various photography, healthcare, and remote sensing applications.
Introduction
Digital twins are virtual replicas of physical objects that can be used to simulate the behavior and performance of their realworld counterparts. Many experts [1,2,3] emphasize that digital twin modeling is the cornerstone for accurately representing physical entities. Image inpainting, a technique used to restore damaged areas in an image by leveraging information from surrounding regions, can be applied in digital twin technology. This method can be categorized into three types: nontexture repair, texture repair, and generative image restoration. Nontexture rehabilitation restores structural details such as boundaries, corners, and curvature, while texture repair corrects global information within the damaged region. Recently, generative image restoration based on deep learning has also made notable advancements.
One of the traditional approaches for image inpainting is the PDEbased model introduced by Bertalmio [4]. This method can be implemented in digital twins to emulate the expertise of museum experts who restore cultural artifacts using a nonlinear partial differential equation. Its fundamental principle involves diffusing sharp boundaries into the areas requiring inpainting. To achieve this, the method creates intensity contours around the repaired region, gradually spreading the gray level from the border of the damaged area to its interior. The isophetesâ€™ direction serves as the boundary condition for the inpainted area. As a result, this method can accurately model the behavior and performance of physical objects in digital twins.
In addition to the aforementioned PDEbased models, there are also alternative approaches to image inpainting. One such method is the curvaturedriven diffusion model that employs EulerLagrange equations to estimate missing regions within the image. Another popular approach is variational models, which formulate inpainting as an energy minimization problem. The Total Variation (TV) denoising and segmentation model, introduced by Chan and Shen [5], is a wellknown example of such models. This model proves effective for image restoration tasks by incorporating a penalty term that preserves the solutionâ€™s approximation outside the inpainted regions.
However, the TV model has limitations in dealing with significant gaps or intervals. To address this issue, Chan introduced the integral of the curvature along the boundary contour to the penalty term, enhancing the modelâ€™s ability to bridge larger intervals. This modified TV model for image inpainting can be utilized in digital twin technology to simulate the restoration of damaged objects accurately and efficiently.
Combining digital twin technology and image restoration based on the CahnHilliard equation brings a powerful approach to enhancing and repairing damaged images. Digital twin technology provides a virtual replica of the image, allowing for a deep understanding of its structure, features, and underlying physical phenomena. Applying the CahnHilliard equation, a mathematical model used to describe phase separation and interface dynamics, the damaged areas in the image can be effectively modeled and repaired.
The CahnHilliard equation considers factors such as diffusion and surface energies, enabling it to capture complex patterns and enhance restoration. The equation simulates the diffusion of information within the damaged regions, promoting the reconstruction of missing details and smoothing out irregularities. This modeling approach helps restore the image to its original state by minimizing the energy associated with the damaged areas.
Existing CH models only have firstorder accuracy in time, and simulating longterm evolution processes will increase the number of calculations. For this purpose this paper uses a secondorder accuracy numerical model.
Furthermore, the utilization of digital twin technology allows for a comprehensive assessment of the effectiveness of the image restoration process. Neural network models can measure the restored imageâ€™s quality, fidelity, and visual appeal. These models learn from a vast dataset of highquality images and their corresponding damaged versions, enabling them to evaluate the success of the restoration algorithm and provide quantitative metrics for assessment.
In this paper, we make the following contributions:

we give an energystable secondorder numerical format of the CH equation for image restoration.

we introduce the multigrid solution method of this numerical format, which can solve the problem quickly. Compared with the Newton method, it does not need to calculate the Jacobian matrix and saves memory.
The full text is organized as follows. Related work section introduces related work, Methodology section introduces the numerical format and multigrid algorithm, and Results and discussion section gives examples to prove the superiority of the secondorder format. Finally, we provide some discussions and conclusions in Conclusion section.
Related work
The combination of digital twin technology and the CahnHilliard equationbased image restoration presents a promising solution for the enhancement of damaged images. By leveraging the virtual representation provided by digital twins and the mathematical modeling capabilities of the CahnHilliard equation, it becomes possible to restore images with improved accuracy, clarity, and visual quality. This application holds significant potential in fields like medical imaging, forensics, and historical document restoration, where damaged images need to be recovered and analyzed.
Gillette and Bertozzi [6] were the first to apply the CahnHilliard equation model to image restoration work. They added a fidelity item based on the CH equation, which has a good repair effect on binary images.
\(\lambda (x)\) is the mark of the damaged area, W(u) is the double well potential function, and one of the authorâ€™s suggestions is \(u^{2}(u  1)^{2}\)
Neumann zeroflux boundary value condition, \(\Omega\) is the image area, D is the area to be repaired, if
The model can fill in a wide range of information gaps very well. In solving the model, the author uses a numerical format of convex decomposition.
Burger [7] obtained the \(TVH^{1}\) model after seeking the limit of the energy functional of Bertozziâ€™s model and removed the free energy item. This model can be applied to grayscale images. \(TVH^{1}\) has the same minimum point as the CH equation, but the model slowly converges to a steady state.
Kim [8] pointed out that when repairing an image with broken stripes, the nonsmooth break at the boundary and the smooth repair function can cause stepped repair at the edge. For this reason, the author introduces a preprocessing method, which first solves an anisotropic diffusion equation and then solves the improved CH equation.
To address the above problems, Zou [9] proposes to combine the PeronaMalik equation and the CH equation. By adding the nonlinear diffusion coefficient before the diffusion term, the CH equation of anisotropic diffusion is obtained, which ultimately achieves a smooth repair effect at the boundary.
Since then, different scholars have attempted to expand the application of the CH equation in image restoration. Since the free energy term in the CH model specifies a stable point, it is only suitable for binary images. For grayscale pictures and color images, it is necessary to remove the free energy term or consider the complexvalued CH model [7, 10] in vector form, or use multivalued potential functions instead of doublewell potentials [11]. Other considerations include the inpainting of color images using the multiphase flow CH model [12], and the consideration of fractional CH equations [13].
The CH equation essentially represents the average curvature, but the steadystate calculation time in the CH equation is greatly affected by \(\epsilon\). In [6], the author suggested using a twostage algorithm but did not provide specific guidance. This paper proposes selecting the value of \(\epsilon\) based on the size of the damaged area. For larger damaged areas, a multistage algorithm can be employed to expedite the convergence speed. Additionally, the calculation of the next stage begins after the previous step enters a steady state to prevent oscillation at the boundary.
Carrillo et al. [14] applied the CH model for image preprocessing in the neural network image recognition task. They used the MINIST dataset to verify the CH model as a preprocessor, which can improve the recognition accuracy of damaged data. The above image inpainting models based on the CH equation are all based on the convex decomposition method and the numerical format with firstorder precision on time. Since the spatial step is a fixed value of 1 for the image, although the spatial accuracy is secondorder in the standard discrete format, the time step will still affect the overall error. If one use a small time step, it will increase a lot of calculations.
This paper presents a timebased secondorder numerical scheme based on a convex decomposition method, which has the good properties of unconditional energy stability and unique solvability and is solved using a multigrid method [15]. Based on the work of Carrillo et al. [14], the experimental process of handwritten data is reexamined, proving that the secondorder numerical format can achieve a better restoration effect and improve the prediction accuracy of the neural network model (TableÂ 1).
Methodology
Here, we use the secondorder numerical scheme of [15] and the geometric multigrid solver [16, 17] for the resulting nonlinear problem. Consider energy first
The original CH equation is the gradient flow in the \(H^{1}\) space, and it is no longer in the form of gradient flow after adding the fidelity item. After the two terms of the energy functional are varied in the space of \(H^{1}\) and \(L^2\) respectively, the equation of the continuous situation is obtained:
The boundary condition is the zeroflux Neumann boundary (let \(u = \epsilon \Delta \phi  \frac{1}{\epsilon }(\phi ^{3}  \phi )\))
Initial value:
For a detailed definition of discrete function space, please refer to [15]. Here is the format after the discretization of time and space directly:
\(\epsilon\) is the interface width parameter, \(\phi _{ij}\) is the unknown pixel value of the image at position (i, j), \(\phi _{ij}^n\) is the known pixel value of the image at position (i, j) at the nth time step, and \(\Delta _h\) is a discrete secondorder difference operator.
For CH equations that do not contain fidelity terms, this scheme has unconditional energy stability, unique solvability [15], and the dependence of the convergence constant on the polynomial [18]. After rewriting using the auxiliary variable u, we get:
We use the nonlinear geometric multigrid algorithm for solving the numerical format above. For the original CH equation, which is the gradient flow in \(H^{1}\) space, the numerical scheme processed by convex decomposition corresponds to the minimization of a convex functional. Then, the convergence of the multigrid solver can be obtained. For the geometric multigrid algorithm, in addition to the conventional Interpolation and Ristriction operators, it is mainly necessary to define the residual function and the Smoothing operator in the code implementation.
After arranging the known and unknown items in the equation, the following form is obtained:
Express the residual function r as the difference between the Operator and Source term \(r=NS.\)
For the sake of simplicity, the definition is as follows:
The Operator term is defined as a nonlinear operator \(N=(N^1,N^2)^T\) of shape \(2\times N\times N\), and the component form is:
The Source term is defined as the source \(S=(S^{(1,n)},S^{(2,n)})^T\) of shape \(2\times N\times N\), and the component form is
Solving the residual equation is equivalent to solving \(N(\phi )=S^n\) The Smooth operator is the local linearization of the Operator term, defined as:
Zeroflux Neumann boundary condition:
The initial value is the input data \(\phi (0)=\phi ^0\). A multigrid algorithm in the format of the Vcycle full approximation scheme (FAS) is used here. Unlike the multigrid iteration using Newtonâ€™s method for global linearization, a nonlinear disturbance equation is solved on the coarse grid in the FAS format:
In the FAS format, only local linearization is done in the smoothing operator.
Multigrid works on grids with hierarchies. It does not solve the equations precisely on the coarsest mesh but still uses the smoothing step to obtain an approximate solution. Then, use the standard interpolation and restriction operators to transfer the information between the two layers of grids. The calculation process of the multigrid cycle can refer to [17]. with the secondorder extrapolation formula to establish initial value estimates at different time points, we could obtain the complete multigrid process with time step iterations with an outer cycle. The stop condition is defined by the norm of the residual function using normalization.
Parameters and Neural Network Models
In the original CH equation, \(\epsilon\) describes the width \(\delta\) of the boundary layer, and there is an approximate relationship \(\delta = 4.164 \sqrt{\epsilon ^2}\). Meanwhile, in the energy, \(\epsilon\) is Used to weight the gradient term, and the gradient term \(\nabla \phi \) is used to describe the boundary. The corresponding energy minimization process requires a reduction in the number of edges. The original physical change is in two stages. The first stage is phase separation, and the stable state of interfaces with multiple \(\delta\) is rapidly formed in the region; the second stage is coarsening, and the boundary in the steady state is merging, showing that small areas merge into larger sizes.
Since \(\epsilon\) determines the width of the boundary layer, smaller epsilon values correspond to sharper boundaries. A twostage algorithm is used in the image restoration process: the first stage sets a considerable \(\epsilon\) value so that the boundaries connect by diffusion. The second stage sets a small \(\epsilon\) value to make the edges sharper after reconnection.
Since the smallest unit is a pixel point in the image, the grid space step size h in the corresponding difference method is a fixed value of 1. In the CH equation, the more points on the boundary, the smaller the error. According to the relationship between the boundary layer thickness and the parameter \(\epsilon\), it is necessary to set a more significant \(\epsilon\) value when repairing a large damaged area.
The image processing steps are as follows: Step 1, input the image to be repaired and the mask image. Step 2, transform the pixel value of the image to be repaired into the range of [1,1]. Step 3, uses the transformed result as the initial value and the mask image pixel value to set the indication coefficient. Step 4, establish the numerical format of the CahnHilliard equation and use the FAS format multigrid solution. Step 5, transform the calculation result into the original image value range.
In the following experiments, the inpainting effect of the secondorder CH scheme is verified by using the prediction effect changes before and after preprocessing on the neural network model. To this end, a neural network model is trained on the handwritten digit dataset MINIST. Then, make different damage patterns to the test data, use the neural network model to predict, and obtain the prediction accuracy of the damaged data; then use the CH model to perform image restoration and get the prediction accuracy of the repaired data. Finally, compare the improvement in prediction accuracy before and after restoration. The neural network model building and calculation process following Carrillo et al. [14].
We use two random damage modes: 1) Destroy rows randomly select several rows for shading. 2) Destroy pixels and randomly select several pixels for masking.
The MNIST dataset is handwritten digit data. Each image is a 28*28 grayscale image.
In the numerical solution of the CH equation, the time step \(dt=0.1\), the space step \(h=1\), and the definition domain is a square area of [0,28]*[0,28]. Using a twostage approach, \(\epsilon _1=1.5,\epsilon _2=0.5\), values changed at time point \(t=2\). Termination time \(T=6\). Penalty coefficient
We use the prediction accuracy of the neural network model to evaluate the effect of image restoration. The evaluation index used here is the improvement rate:
postAcc is the prediction accuracy rate after processing, and preAcc is the prediction accuracy rate before processing.
The signaltonoise ratio of an image is defined as:
where z is the original image and u is the inpainted image.
Results and discussion
Comparison of effect with and without CH filter
The evaluation index used here is the lift rate(Lift). It is defined as the relative change value of the prediction accuracy of the neural network model before and after preprocessing.
Because each time a new random number is used for the damaged position of each picture, the prediction accuracy rate obtained by each execution of the program is different. The mean of five experiments is used here as the final result. From TablesÂ 2 and 3, the results show that the secondorder formatâ€™s improvement rate is higher. Moreover, in the firstorder form, the promotion rate obtained is negative when the number of damaged lines is 24 or 26 in the broken line mode. In the pixelbroken way, the pixel ratio is \(30\%, 40\%\), the promotion rate is zero. Both are positive values here. Combined with the previous signaltonoise ratio analysis, we found that the firstorder format may not have selected the optimal time point for prediction.
Note: Compared with the original paper [14] , the initial prediction accuracy is different from that given here due to the use of different initial values.
The tabular data verifies that using the CH model to paint the image can improve the prediction effect of the image, and the improvement rate is higher than that of the firstorder scheme. In addition, in the firstorder format [14], the promotion rate obtained is negative when the number of lines breaks into 24 and 26 lines. When the pixel destruction ratio is 30%, 40%, the promotion rate is zero. In the secondorder format, both are positive values. It shows the superiority of the secondorder scheme in terms of accuracy improvement. In terms of calculation speed, use the fsolve function in the scipy package to solve nonlinear equations. After Carrillo et al. [14] changed the direct solution to the alternate direction row solution, they reduced the processing time of an image from 25s to 8s. In this paper, the nonlinear multigrid algorithm is used, and the calculation time of each print is 1s.
Case Analysis
A single image is compared before and after using the CH filter, and a larger value is used before the first time point \(t=2\), and the change process is dominated by diffusion. With smaller values after the first time point, the region edges sharpen quickly, forming sharp boundaries.
There will be marks on thicker banded edges. We can see that the image has a clearer boundary after 60 steps of evolution.
Comparison of results in pixel destruction mode. It can be seen from Figs.Â 1 and 2 that: a). A large \(\epsilon\) at the first time point makes the image blurred, and a small \(\epsilon\) value at the following few time points makes the picture gradually more evident, and the boundary sharper. b). Successive row corruption can make repairs difficult. Leave traces at intermediate time points.
SNR analysis
SNR is the signaltonoise ratio initially used to measure the percentage of signaltonoise. The larger the signaltonoise ratio, the better the image quality. The SNR is used here to measure how close the repaired image is to the original image. The signaltonoise ratio was used to compare the difference in image processing effects between the firstorder and secondorder schemes of the CH equation.
The mean value of SNR at different time points for 2000 samples in row corruption mode was calculated. The abscissa is the number of time steps, and the number of damaged lines is 22, 24, and 26, respectively. It can be seen from the Fig.Â 3: a. At the first three time points, the SNR of the secondorder scheme is higher than that of the firstorder form [14] and reaches the maximum at the second time point. It shows that the 2nd order scheme has a better repair effect in the first stage of the calculation. b. The entire SNR curve rises first and then falls. The best repair effect is at the middle time point. c. The more destroyed rows, the smaller the maximum SNR that can be achieved. The more damaged points, the worse the repair effect.
The above analysis suggests that using the repair value of the last time point is unnecessary when using neural network prediction, and a better prediction effect can be obtained by using the repair value of the second time point.
Compared with the restoration effect of other CH models
A comparison of the repair effects of different variants of the CH model is shown in Fig.Â 4. The time step is 0.1, the end time is 110, and the image size is 128*128. All models are solved using the method based on time evolution. TableÂ 4 show that the repair effect of the fractional CH equation [13] is the worst. The existence of the fractional term increases the difficulty of solving and introduces a larger parameter search space. The same problem exists in the PeronaMalik method [9] and the preprocessing CH [8] method. The introduction of nonlinear terms increases the difficulty of solving the original equation and increases the parameter search space. In contrast, the problem of the CH model exists only in the choice of twostage parameters. Regarding calculation speed, the solution of the CH equation using the geometric multigrid algorithm is faster.
Discussion
In the previous section, we have highlighted the effectiveness of the CH highprecision numerical model in binary image restoration. This model offers a more accurate and efficient solution than traditional firstorder formats. Using a secondorder format, we can achieve higher signaltonoise ratios and greater prediction accuracy for image restoration tasks, resulting in superior output quality.
We can further enhance its capabilities by integrating the CH highprecision numerical model with a nonlinear multigrid solver that accelerates data preprocessing tasks. The multigrid solver enables faster calculations for large datasets by approximating solutions on successively coarser grids. This approach reduces the number of measures required, significantly improving computational efficiency.
In digital twin technology, integrating the CH highprecision numerical model and the nonlinear multigrid solver can be applied to various areas, such as medical imaging and engineering. For instance, in medical imaging, this approach can be used to restore damaged or degraded images, diagnose anomalies, and improve the accuracy of medical models. Similarly, in engineering, digital twins can be created to simulate system behavior and predict potential issues, such as equipment failure or performance degradation, allowing for proactive maintenance and minimizing the risk of downtime.
Conclusion
This paper introduces a digital twinbased approach with a modeling method by CH equation in image restoration tasks with high temporal precision. The proposed method utilizes a nonlinear multigrid solver, which enhances the signaltonoise ratio and prediction accuracy compared to the firstorder format. Additionally, the multigrid solver enables faster data preprocessing, making it particularly beneficial for handling large datasets. This study applies advanced techniques to image restoration for the first time by adopting a highorder numerical scheme for the CH equation.
The FAS format offers advantages by eliminating the need to calculate the Jacobian matrix and reducing memory requirements for extensive computations. However, its implementation for application problems faces challenges due to the demand for custom smoothing operators for different dimensions and equations, which may affect code implementation.
Furthermore, this article primarily focuses on a limited number of examples, mainly simple textures in handwritten datasets, and concentrates on addressing local texture restoration. Further investigations are necessary to showcase the full potential of highprecision solutions of the CH equation. Future work will apply the proposed approach to more complex datasets to comprehensively demonstrate its advantages.
Availability of data and materials
Data and materials are available upon request.
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Codes used in the study are available upon request.
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Funding
This work was supported by the National Natural Science Foundation of China(No.61070015), Guangdong Frontier and Key Technological Innovation Special Funds(Grant Scientific and Technological Project)(No.2014B010110004), Funds of The University Engineering and Technology Research Center Of Guangdong(No.2022GCZX013).
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Guo, J., Qi, D. Enhancement of damagedimage prediction based on digital twin technology. J Cloud Comp 12, 177 (2023). https://doi.org/10.1186/s1367702300563y
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DOI: https://doi.org/10.1186/s1367702300563y