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Research on deep learning image processing technology of second-order partial differential equations

  • S.I. : Machine Learning and Big Data Analytics for IoT Security and Privacy (SPIOT 2021)
  • Published:
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Abstract

Image classification can effectively manage and organize images, laying a good foundation for the work in multiple fields of image processing. With the rise of Internet technology and social networks, the number of digital images has increased dramatically and there are more and more applications. People also use intuitive pictures instead of words when expressing emotions and information. A large number of digital images need to be managed, analyzed, and retrieved. This urgently requires more efficient and accurate image classification technology. Deep learning is a learning method that extends traditional neural networks, simulating the process of gradual abstraction of human brain cognition. The number of hidden layers is deepened, and features can be learned automatically. This paper has completed the traditional image noise detection and segmentation experiment based on convolutional neural network. We introduced the various parts of the convolutional neural network, including the convolutional layer, activation function, fully connected layer, etc. Noise detection is completed based on Fast RCNN on the MSTAR data set containing the background. The model fully combines the advantages of the isotropic model, the Perona-Malik model, and the second-order directional derivative. In order to better maintain the edge structure information of the image, the structure information is extracted based on the original noise image, and the improved ID model and the improved PM model are directionally diffused along the edge tangent direction of the original noise image. Considering the local structure information of the image, we use the slice similarity modulus value as the edge detector of the improved PM model, and use the slice similarity modulus value to construct a new weighting function to adaptively balance the relative weights of the improved ID model and the improved PM model. Simulation and measured data verify the effectiveness of this network in removing image coherent speckle noise. We compare and analyze it with existing denoising methods. The use of visual evaluation and objective evaluation indicators to evaluate the denoising effect and calculation efficiency shows the advantages of the network in this paper in terms of denoising effect, calculation time and space complexity.

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References

  1. Beck C, Weinan E, Jentzen A (2019) Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations. J Nonlinear Sci 29(4):1563–1619

    Article  MathSciNet  MATH  Google Scholar 

  2. Berg J, Nyström K (2018) A unified deep artificial neural network approach to partial differential equations in complex geometries. Neurocomputing 317:28–41

    Article  Google Scholar 

  3. Nabian MA, Meidani H (2019) A deep learning solution approach for high-dimensional random differential equations. Probab Eng Mech 57:14–25

    Article  Google Scholar 

  4. Weinan E, Yu B (2018) The deep ritz method: a deep learning-based numerical algorithm for solving variational problems. Commun Math Stat 6(1):1–12

    Article  MathSciNet  MATH  Google Scholar 

  5. Liu Z, Yang Y, Cai Q (2019) Neural network as a function approximator and its application in solving differential equations. Appl Math Mech 40(2):237–248

    Article  MathSciNet  MATH  Google Scholar 

  6. Fujii M, Takahashi A, Takahashi M (2019) Asymptotic expansion as prior knowledge in deep learning method for high dimensional BSDEs. Asia-Pacific Finan Mark 26(3):391–408

    Article  MATH  Google Scholar 

  7. Kumar A, Ahmad MO, Swamy MNS (2019) A framework for image denoising using first and second order fractional overlapping group sparsity (HF-OLGS) regularizer. IEEE Access 7:26200–26217

    Article  Google Scholar 

  8. Berner J, Grohs P, Jentzen A (2020) Analysis of the generalization error: empirical risk minimization over deep artificial neural networks overcomes the curse of dimensionality in the numerical approximation of black-scholes partial differential equations. SIAM J Math Data Sci 2(3):631–657

    Article  MathSciNet  MATH  Google Scholar 

  9. Lu L, Meng X, Mao Z et al (2021) DeepXDE: a deep learning library for solving differential equations. SIAM Rev 63(1):208–228

    Article  MathSciNet  MATH  Google Scholar 

  10. Sun H, Hou M, Yang Y et al (2019) Solving partial differential equation based on Bernstein neural network and extreme learning machine algorithm. Neural Process Lett 50(2):1153–1172

    Article  Google Scholar 

  11. Fang C, Zhao Z, Zhou P et al (2017) Feature learning via partial differential equation with applications to face recognition. Pattern Recogn 69:14–25

    Article  Google Scholar 

  12. Monga V, Li Y, Eldar YC (2021) Algorithm unrolling: Interpretable, efficient deep learning for signal and image processing. IEEE Signal Process Mag 38(2):18–44

    Article  Google Scholar 

  13. Zidan MA, Jeong YJ, Lee J et al (2018) A general memristor-based partial differential equation solver. Nat Electron 1(7):411–420

    Article  Google Scholar 

  14. Winovich N, Ramani K, Lin G (2019) ConvPDE-UQ: Convolutional neural networks with quantified uncertainty for heterogeneous elliptic partial differential equations on varied domains. J Comput Phys 394:263–279

    Article  MathSciNet  MATH  Google Scholar 

  15. Dwivedi V, Srinivasan B (2020) Physics informed extreme learning machine (PIELM)–a rapid method for the numerical solution of partial differential equations. Neurocomputing 391:96–118

    Article  Google Scholar 

  16. Avrutskiy VI (2020) Neural networks catching up with finite differences in solving partial differential equations in higher dimensions. Neural Comput Appl 32(17):13425–13440

    Article  Google Scholar 

  17. Zhang L, Cheng L, Li H et al (2021) Hierarchical deep-learning neural networks: finite elements and beyond. Comput Mech 67(1):207–230

    Article  MathSciNet  MATH  Google Scholar 

  18. Yang Y, Hou M, Sun H et al (2020) Neural network algorithm based on legendre improved extreme learning machine for solving elliptic partial differential equations. Soft Comput 24(2):1083–1096

    Article  MATH  Google Scholar 

  19. Pham H, Warin X, Germain M (2021) Neural networks-based backward scheme for fully nonlinear PDEs. SN Part Differ Equ Appl 2(1):1–24

    MathSciNet  MATH  Google Scholar 

  20. Wei X, Jiang S, Li Y et al (2019) Defect detection of pantograph slide based on deep learning and image processing technology. IEEE Trans Intell Transp Syst 21(3):947–958

    Article  Google Scholar 

  21. Eigel M, Schneider R, Trunschke P et al (2019) Variational Monte Carlo—bridging concepts of machine learning and high-dimensional partial differential equations. Adv Comput Math 45(5):2503–2532

    Article  MathSciNet  MATH  Google Scholar 

  22. Rai PK, Tripathi S (2019) Gaussian process for estimating parameters of partial differential equations and its application to the Richards equation. Stoch Environ Res Risk Assess 33(8):1629–1649

    Article  Google Scholar 

  23. Zhu Y, Zabaras N, Koutsourelakis PS et al (2019) Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data. J Comput Phys 394:56–81

    Article  MathSciNet  MATH  Google Scholar 

  24. Indolia S, Goswami AK, Mishra SP et al (2018) Conceptual understanding of convolutional neural network-a deep learning approach. Procedia Comput Sci 132:679–688

    Article  Google Scholar 

  25. Hiremath PS, Bhusnurmath RA (2017) Texture classification using partial differential equation approach and wavelet transform. Pattern Recognit Image Anal 27(3):473–479

    Article  Google Scholar 

  26. Qiao L, Zhu Y, Zhou H (2020) Diabetic retinopathy detection using prognosis of microaneurysm and early diagnosis system for non-proliferative diabetic retinopathy based on deep learning algorithms. IEEE Access 8:104292–104302

    Article  Google Scholar 

  27. Dai W, Li D, Tang D et al (2021) Deep learning assisted vision inspection of resistance spot welds. J Manuf Process 62:262–274

    Article  Google Scholar 

  28. Kim MW, Jeng GS, Pelivanov I et al (2020) Deep-learning image reconstruction for real-time photoacoustic system. IEEE Trans Med Imaging 39(11):3379–3390

    Article  Google Scholar 

  29. Buongiorno D, Cascarano GD, De Feudis I et al (2021) Deep learning for processing electromyographic signals: a taxonomy-based survey. Neurocomputing 452:549–565

    Article  Google Scholar 

  30. Vasconcelos CN, Vasconcelos BN (2020) Experiments using deep learning for dermoscopy image analysis. Pattern Recognit Lett 139:95–103

    Article  Google Scholar 

  31. Lei J, Li G, Zhang J et al (2016) Continuous action segmentation and recognition using hybrid convolutional neural network-hidden markov model model. IET Comput Vis 10(6):537–544

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Normal School, Shenyang University, Shenyang, 110044, Liaoning, China

    Qingzhe Wu

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  1. Qingzhe Wu
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Correspondence to Qingzhe Wu.

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Wu, Q. Research on deep learning image processing technology of second-order partial differential equations. Neural Comput & Applic 35, 2183–2195 (2023). https://doi.org/10.1007/s00521-022-07017-7

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  • DOI: https://doi.org/10.1007/s00521-022-07017-7

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