Automated Finite Element Solution of Diffusion Models for Image Denoising

[1] ALK ÄMPER, M.—LANGER, A.: Using DUNE-ACFem for non-smooth minimization of bounded variation functions, Archive of Numerical Software 5 (2017), 3–19.Search in Google Scholar

[2] ALNÆS, M.—BLECHTA, J.—HAKE, J.—JOHANSSON, A.— KEHLET, B.—LOGG, A.— RICHARDSON, C.—RING, J.—ROGNES, M. E. —WELLS, G. N.: The FEniCS project version 1.5, Archive of Numerical Software 3 (2015).Search in Google Scholar

[3] BARTELS, S.: Total variation minimization with finite elements: convergence and iterative solution, SIAM Journal on Numerical Analysis 50 (2012), 1162–1180.10.1137/11083277XSearch in Google Scholar

[4] CHAMBOLLE, A.—POCK, T.: Approximating the total variation with finite differences or finite elements,In: Handbook of Numerical Analysis, Vol. 22, Elsevier, 2021. pp. 383–417.10.1016/bs.hna.2020.10.005Search in Google Scholar

[5] CHARBONNIER, P.—BLANC-FERAUD, L.—AUBERT, G.—BARLAUD, M.: Two deterministic half-quadratic regularization algorithms for computed imaging,In: Proceedings of 1st International Conference on Image Processing, IEEE, Vol. 2, 1994, pp. 168–172.Search in Google Scholar

[6] FENICS PROJECT: . FEniCS project 2019.1.0. 2019 [Online; accessed on 05-Mai-2021], https://fenicsproject.org/Search in Google Scholar

[7] HANDLOVI ČOV Á, A.—MIKULA, K.—SGALLARI, F.: Variational numerical methods for solving nonlinear diffusion equations arising in image processing, Journal of Visual Communication and Image Representation 13 (2002), 217–237.10.1006/jvci.2001.0479Search in Google Scholar

[8] HINTERM ÜLLER, M.—RINCON-CAMACHO, M.: An adaptive finite element method in L²-TV-based image denoising, Inverse Problems & Imaging 8 (2014), no. 3, 685—711.Search in Google Scholar

[9] HJOUJI, A.—EL-MEKKAOUI, J.—JOURHMANE, M.: Mixed finite element method for nonlinear diffusion equation in image processing, Pattern Recognition and Image Analysis 29 (2019), 296–308.10.1134/S1054661819020020Search in Google Scholar

[10] LANGTANGEN, H. P.—LOGG, A.: Solving PDEs in Python: the FEniCS Tutorial I. Springer Nature, 2017.10.1007/978-3-319-52462-7Search in Google Scholar

[11] LANGTANGEN, H. P.—MARDAL, K.-A.: Introduction to Numerical Methods for Variational Problems. Springer International Publishing, Cham, 2019.10.1007/978-3-030-23788-2Search in Google Scholar

[12] LOGG, A.—MARDAL, K.-A.—WELLS, G.: Automated Solution of ifferential Equations by the Finite Element Method: The FEniCS Book Vol. 84. Springer Science & Business Media, 2012.10.1007/978-3-642-23099-8Search in Google Scholar

[13] PERONA, P.—MALIK, J.: Scale-space and edge detection using anisotropic diffusion, IEEE Transactions on Pattern Analysis and Machine Intelligence 12 (1990), 629–639.10.1109/34.56205Search in Google Scholar

[14] RUDIN, L. I.—OSHER, S.—FATEMI, E.: Nonlinear total variation based noise removal algorithms, Experimental mathematics: Computational issues in nonlinear science (Los Alamos, NM, 1991). Phys. D: 60 (1992), no. 1, 259–268.Search in Google Scholar

[15] SCHERZER, O.—WEICKERT, J.: Relations between regularization and diffusion filtering,J.Math. Imaging and Vision 12 (2000), no. 1, 43–63.Search in Google Scholar

[16] THE DEFELEMENT CONTRIBUTORS: . DefElement: an encyclopedia of finite element definitions. [Online; accessed 09-March-2021]. https://defelement.com, 2021.Search in Google Scholar

[17] VILLA, U.—PETRA, N.—GHATTAS, O.: hIPPYlib: An Extensible Software Framework for Large-Scale Inverse Problems Governed by PDEs: Part I: Deterministic Inversion and Linearized Bayesian Inference, ACM Transactions on Mathematical Software (TOMS) 47 (2021), 1–34.10.1145/3428447Search in Google Scholar