Image denoising via optimal control of the diffusivity function in a PDE-constrained problem

Kawtar  Lehriz; Omar  Gouasnouane; Noureddine  Moussaid; Anouar  Ben-Loghfyry

doi:10.5269/bspm.77629

Kawtar Lehriz
Omar Gouasnouane
Noureddine Moussaid
Anouar Ben-Loghfyry

Abstract

In this paper, we suggest a new picture denoising model for PDE-constrained problem that is based on the popular Perona-Malik model. We present a novel method based on the optimal control of the diffusivity function to overcome the staircasing effect and contrast loss, two intrinsic drawbacks of the Perona-Malik equation. Through adaptively controlling the diffusion process, our model successfully improves noise reduction while maintaining important image characteristics. Comparisons with some models show how much better our method is at preserving structural features and improving denoising performance.

Downloads

Download data is not yet available.

References

G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, 2nd ed., vol. 147, Springer, New York, 2006.

R. C. Gonzalez and R. E. Woods, Digital Image Processing, 4th ed., Pearson, 2018.

K. Zhang, W. Zuo, Y. Chen, D. Meng, and L. Zhang, “Beyond a Gaussian denoiser: Residual learning of deep CNN for image denoising,” IEEE Trans. Image Process., vol. 26, no. 7, pp. 3142–3155, 2017.

S. W. Zamir et al., “Restormer: Efficient Transformer for High-Resolution Image Restoration,” in Proc. IEEE/CVF Conf. Computer Vision and Pattern Recognition (CVPR), 2022, pp. 5728–5739.

A. Charkaoui, A. Ben-Loghfyry, A class of nonlinear parabolic PDEs with variable growth structure applied to multiframe MRI super-resolution, Nonlinear Analysis: Real World Applications, 83(2025), p.104259.

A. Charkaoui, A. Ben-Loghfyry, A novel multi-frame image super-resolution model based on regularized nonlinear diffusion with Caputo time fractional derivative, Communications in Nonlinear Science and Numerical Simulation, 139(2024), p.108280.

A. Chambolle, “An algorithm for total variation minimization and applications,” Journal of Mathematical Imaging and Vision, vol. 20, no. 1–2, pp. 89–97, 2004.

A. Ben-Loghfyry, A. Hakim, A novel robust fractional-time anisotropic diffusion for multi-frame image super-resolution, Advances in Computational Mathematics, 49(6), p.79.

R. Addouch, N. Moussaid, O. Gouasnouane and A. Ben-Loghfyry, Total fractional-order variation and bilateral filter for image denoising, Mathematical modeling and computing, (11, Num. 3) (2024), pp.642-653.

A. Ben-Loghfyry, A fractional-time PDE-constrained parameter identification for inverse image noise removal problem, Journal of the Franklin Institute, 362(2), 2025, p.107443.

A. Charkaoui, A. Ben-Loghfyry and S. Zeng, A novel parabolic model driven by double phase flux operator with variable exponents: Application to image decomposition and denoising, Computers & Mathematics with Applications, 174(2024), pp.97-141.

A. Charkaoui, A. Ben-Loghfyry, Anisotropic equation based on fractional diffusion tensor for image noise removal, Mathematical Methods in the Applied Sciences, 47(12), 2024, pp.9600-9620.

O. Gouasnouane, N. Moussaid, S. Boujena and K. Kabli, A nonlinear fractional partial differential equation for image inpainting, Math. Model. Comput, 9(2022), pp.536-546.

A. Ben-loghfyry and A. Hakim, Caputo fractional-time of a modified Cahn-Hilliard equation for the inpainting of binary images, Journal of Mathematical Modeling, 11(2), 2023, pp.357-373.

F.Y. Shih, Image processing and pattern recognition: fundamentals and techniques, 2010, John Wiley & Sons.

M. Kaur and V. Kumar,A comprehensive review on image encryption techniques, Archives of Computational Methods in Engineering, 27(1), 2020, pp.15-43.

A. Krull, T.-O. Buchholz, and F. Jug, “Noise2Void – Learning denoising from single noisy images,” in Proc. IEEE/CVF Conf. Computer Vision and Pattern Recognition (CVPR), 2019, pp. 2129–2137.

A. Buades, B. Coll, and J.-M. Morel, “A non-local algorithm for image denoising,” in Proc. IEEE Conf. Computer Vision and Pattern Recognition (CVPR), 2005, pp. 60–65.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D: Nonlinear Phenomena, vol. 60, no. 1–4, pp. 259–268, 1992.

C. Tian, L. Fei, W. Zheng, Y. Xu, and Z. Luo, “Deep learning on image denoising: An overview,” Neural Networks, vol. 131, pp. 251–275, 2020.

P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 12, no. 7, pp. 629–639, 1990.

J. Weickert, Anisotropic Diffusion in Image Processing, Teubner, Stuttgart, 1998.

G. Gilboa and S. Osher, “Nonlocal operators with applications to image processing,” Multiscale Modeling & Simulation, vol. 7, no. 3, pp. 1005–1028, 2008.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-posed Problems, Winston and Sons, Washington, DC, 1977.

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, 1971.

S. Bonettini, M. Prato, and S. Rebegoldi, “A scaled gradient projection method for constrained image deblurring,” Inverse Problems, vol. 34, no. 1, 2018.

J.-L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires, Dunod, Paris, 1969.

O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, Springer, New York, 1985.

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, SIAM, Philadelphia, 1999.

E. Casas, “Control of an elliptic problem with pointwise state constraints,” SIAM Journal on Control and Optimization, vol. 24, no. 6, pp. 1309–1318, 1986.

M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich, Optimization with PDE Constraints, Springer, Heidelberg, 2009.

J. Weickert, Anisotropic Diffusion in Image Processing, Teubner, Stuttgart, 2002.

G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3rd ed., Oxford University Press, Oxford, 1985.

L. N. Trefethen, Spectral Methods in MATLAB, Software, Environments, and Tools, Vol. 10, SIAM, Philadelphia, 2000.

K. Zhang, W. Zuo, and L. Zhang, “FFDNet: Toward a fast and flexible solution for CNN-based image denoising,” IEEE Trans. Image Process., vol. 27, no. 9, pp. 4608–4622, 2018.