A Fully Conservative Parallel Numerical Algorithm with Adaptive Spatial Grid for Solving Nonlinear Diffusion Equations in Image Processing


  • Andrey D. Bulygin Tomsk State University V.E. Zuev Institute of Atmospheric Optics SB RAS (IAO SB RAS)
  • Denis A. Vrazhnov National Research Tomsk State University, Laboratory of Biophotonics




In this paper we present simple yet efficient parallel program implementation of grid-difference method for solving nonlinear parabolic equations, which satisfies both fully conservative property and second order of approximation on non-uniform spatial grid according to geometrical sanity of a task. The proposed algorithm was tested on Perona–Malik method for image noise ltering task based on differential equations. Also in this work we propose generalization of the Perona–Malik equation, which is a one of diffusion in complex-valued region type. This corresponds to the conversion to such types of nonlinear equations like Leontovich–Fock equation with a dependent on the gradient field according to the nonlinear law coefficient of diffraction. This is a special case of generalization of the Perona–Malik equation to the multicomponent case. This approach makes noise removal process more flexible by increasing its capabilities, which allows achieving better results for the task of image denoising.


Perona P., Malik J.: Scale-space and edge detection using anisotropic diffusion. IEEE Transactions on pattern analysis and machine intelligence 12(7), 629–639 (1990)

Wei G.W. : Generalized Perona–Malik equation for image restoration. IEEE Signal Processing Letters 6(7), 165–167 (1999)

Wang N., et al.: A hybrid model for image denoising combining modified isotropic diffusion model and modified Perona–Malik model. IEEE Access 6, 33568–33582 (2018), DOI: 10.1109/ACCESS.2018.2844163

Maiseli B., et al.: Perona–Malik model with self-adjusting shape-defining constant. Information Processing Letters. 137, 26–32 (2018), DOI: 10.1016/j.ipl.2018.04.016

Vrazhnov D.A., Shapovalov A.V., Nikolaev V.V.: Symmetries of differential equations in computer vision applications. Computer Research and Modeling 2(4), 369–376 (2010)

Bertalmio M., et al.: Image inpainting. In: Proceedings of the 27th annual conference on Computer graphics and interactive techniques. ACM Press, pp. 417–424. Addison-Wesley Publishing Co. (2000)

Tschumperl D., Deriche R.: Anisotropic diffusion partial differential equations in multichannel image processing: framework and applications. Advances in Imaging and Electron Physics (AIEP), pp. 145–209, AcademicPress (2007)

Tadmor E.: A review of numerical methods for nonlinear partial differential equations. Bulletin of the American Mathematical Society 49(4), 507–554 (2012), DOI: 10.1090/S0273-0979-2012-01379-4

Terekhov A.V.: A fast parallel algorithm for solving block-tridiagonal systems of linear equations including the domain decomposition method. Parallel Computing 39(6-7), 245–258 (2013), DOI: 10.1016/j.parco.2013.03.003




How to Cite

Bulygin, A. D., & Vrazhnov, D. A. (2019). A Fully Conservative Parallel Numerical Algorithm with Adaptive Spatial Grid for Solving Nonlinear Diffusion Equations in Image Processing. Supercomputing Frontiers and Innovations, 6(1), 14–18. https://doi.org/10.14529/jsfi190103