A PDE-based Mathematical Method in Image Processing: Digital-Discrete Method for Perona-Malik Equation


  • Ahmet Yıldırım Ege University, Faculty of Science, Department of Mathematics, 35100, Bornova, İzmir, Turkey
  • İsmet Karaca Ege University, Faculty of Science, Department of Mathematics, 35100, Bornova, İzmir, Turkey


Perona-Malik equation, digital-discrete method, digital topology, finite difference method, image processing


In this study, we propose a new and effective algorithm for image processing. The method based on the combination of digital topology, partial differential equations and finite difference scheme is called the digital-discrete method. We try to solve the Perona-Malik equation using the digital-discrete method. We use the MATLAB package program when analyzing images. The analyzes we make on the images show how the algorithm is useful, effective and open to development.


. K.Mikula, N.Ramarosy, Semi-implicit finite volume scheme for solving nonlinear diffusion equations in image processing, Numerische Mathematik 89 (2001) 561–590

. A.Sarti, K.Mikula, F.Sgallari, C. Lamberti, Evolutionary partial differential equations for biomedical image processing, Journal of Biomedical Informatics 35 (2002) 77–91

. J.Weickert, Efficient image segmentation using partial differential equations and morphology, Pattern Recognition 34 (2001) 1813-1824

. K.Mikula, Image processing with partial differential equations, Modern Methods in Scientific Computing and Applications 75 (2002) 283-321

. F.Gibou, Partial differential equations-based segmentation for radiotherapy treatment planning, Mathematical Biosciences and Engineering 2(2) (2005) 209-226

. S.Angenent, E.Pichon, A.Tannenbaum, Mathematical Methods in Medical Image Processing, Bulletin (New Series) of the American Mathematical Society 43(3) (2006) 365–396

. A.Kuijper, Image Processing with Geometrical and Variational PDEs, 31st annual workshop of the Austrian Association for Pattern Recognition (OAGM/AAPR), OEAGM07 (Schloss Krumbach, Austria, May 3-4, (2007) 89-96

. E.Nadernejad , H.Koohi , H.Hassanpour, PDEs-Based Method for Image Enhancement, Applied Mathematical Sciences, 2(20) (2008) 981-993

. S.Kim. H.Lim, Fourth-order partial differential equations for effective image denoising, Seventh Mississippi State - UAB Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conf. 17 (2009) 107–121

. Z.Lin, W.Zhang, X.Tang, Designing Partial Differential Equations for Image Processing by Combining Differential Invariants, Microsoft Research Newsletter, MSR-TR-2009-192

. L,J.Belaid, An Overview of the Topological Gradient Approach in Image Processing: Advantages and Inconveniences, Journal of Applied Mathematics Volume 2010, Article ID 761783, 19 pages

. O.Niang, A.Thioune, M.C.E.Gueirea, E.Delechelle, J.Lemoine, Partial Differential Equation-Based Approach for Empirical Mode Decomposition: Application on Image Analysis, IEEE Transactions on Image Processing 21(9) (2012) 3991-4001

. C.Shen, C. Li, P.Wang, An Adaptive Partial Differential Equation for Noise Removal, Proceedings of the 2nd International Symposium on Computer, Communication, Control and Automation 4 (2013) 55-58

. B.Ghanbari, L.Rada, K.Chen, A restarted iterative homotopy analysis method for two nonlinear models from image processing, International Journal of Computer Mathematics 91(3) (2014) 661–687

. X.Y. Liu, C.H. Lai, K.A. Pericleous, A fourth-order partial differential equation denoising model with an adaptive relaxation method, International Journal of Computer Mathematics, 92(3) (2015) 608-622

. U.A.Nnolim, Analysis of proposed PDE-based underwater image enhancement algorithms, arXiv:1612.04447v1 [cs.CV]

. R.Yu, C.Zhu, X.Hou, L.Yin, Quasi-Interpolation Operators for Bivariate Quintic Spline Spaces and Their Applications, Mathematical and Computational Applications 22(1) (2017) 10: 15 pages

. Y.Huang, N.Guo, M.Seok, Y.Tsividis, K.Mandli, S.Sethumadhavan, Hybrid Analog-Digital Solution of Nonlinear Partial Differential Equations, In Proceedings of MICRO-50, Cambridge, MA, USA, October 14–18 (2017) 14 pages

. M.Benseghir, F.Z.Nouri, P.C.Tauber, A New Partial Differential Equation for Image Inpainting, Boletim da Sociedade Paranaense de Matemática 39(3) (2021) 137-155

. L.M. Chen, Digital Functions and Data Reconstruction, Digital-Discrete Methods, Springer-Verlag New York (2013)

. A.Rosenfeld, ‘Continuous’ functions on digital pictures, Pattern Recognition Letters 4(3) (1986) 177-184

. L. Chen, The necessary and sufficient condition and the efficient algorithms for gradually varied fill, Chinese Science Bulletin 35 (10) (1990) 870–873

. E.Khalimsky, Motion, deformation, and homotopy in finite spaces, In Proceedings IEEE international conference on systems, man, and cybernetics, Chicago (1987) 227–234

. T.Y.Kong, A digital fundamental group, Computers and Graphics 13 (1989) 159–166

. L.Boxer, Digitally continuous functions, Pattern Recognition Letters 15(8) (1994) 833–839

. A.Rosenfeld, , Contraction of digital curves, University of Maryland’s Technical Report (1996)

. G.Agnarsson, L.Chen, On the extension of vertex maps to graph homomorphisms, Discrete Mathematics 306(17) (2006) 2021–2030

. L.Chen, A digital-discrete method for smooth-continuous data reconstruction, Journal of the Washington Academy of Sciences 96(2) (2010) 47–65

. L.Chen, F.Luo, Harmonic Functions for Data Reconstruction on 3D Manifolds, arXiv:1102.0200v3 [math.NA]

. L.Chen, The properties and the algorithms for gradually varied fill, Chinese Journal of Computers 14(3) (1991)

. T.H.Cormen, C.E.Leiserson, R.L.Rivest, Introduction to algorithms, MIT, New York (1993)

. E.Erdem, Nonlinear Diffusion, Hacettepe University, Department of Computer Engineering Lecture Notes (2013) 1-15

. O.Demirkaya, M.H. Asyali, P.K. Sahoo, Image Processing with MATLAB: Applications in Medicine and Biology, CRC Press: Taylor& Francis Group, Boca Raton, USA (2009)

. P.Perona, J.Malik, Scale-Space and Edge Detection Using Anisotropic Diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence 12(7) (1990) 629-639

. G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, Oxford Applied Mathematics and Computing Science Series (1985)




How to Cite

Ahmet Yıldırım, & İsmet Karaca. (2021). A PDE-based Mathematical Method in Image Processing: Digital-Discrete Method for Perona-Malik Equation. American Scientific Research Journal for Engineering, Technology, and Sciences, 84(1), 118–129. Retrieved from https://asrjetsjournal.org/index.php/American_Scientific_Journal/article/view/7274