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Research Article

A high order PDE-constrained optimization for the image denoising problem

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Pages 1821-1863 | Received 21 Jan 2020, Accepted 05 Dec 2020, Published online: 30 Dec 2020

References

  • Kirsch A. An introduction to the mathematical theory of inverse problems. Vol. 120. Germany: Springer Science & Business Media; 2011.
  • Chan TF, Esedoglu S. Aspects of total variation regularized l 1 function approximation. SIAM J Appl Math. 2005;65(5):1817–1837.
  • Rudin LI, Osher S, Fatemi E. Nonlinear total variation based noise removal algorithms. Phys D Nonlinear Phenom. 1992;60(1-4):259–268.
  • Hintermüller M, Holler M, Papafitsoros K. A function space framework for structural total variation regularization with applications in inverse problems. Inverse Probl. 2018;34(6):064002.
  • Liu J, Ni G, Yan S. Alternating method based on framelet l 0-norm and tv regularization for image restoration. Inverse Probl Sci Eng. 2019;27(6):790–807.
  • Laghrib A, Ezzaki M, El Rhabi M, et al. Simultaneous deconvolution and denoising using a second order variational approach applied to image super resolution. Comput Vis Image Underst. 2018;168:50–63.
  • Chantas G, Galatsanos NP, Molina R, et al. Variational bayesian image restoration with a product of spatially weighted total variation image priors. IEEE Trans Image Process. 2010;19(2):351–362.
  • Rodríguez P, Wohlberg B. Efficient minimization method for a generalized total variation functional. IEEE Trans Image Process. 2009;18(2):322–332.
  • Frick K, Marnitz P, Munk A. Statistical multiresolution estimation for variational imaging: with an application in poisson-biophotonics. J Math Imaging Vis. 2013;46(3):370–387.
  • Dong Y, Hintermüller M, Rincon-Camacho MM. Automated regularization parameter selection in multi-scale total variation models for image restoration. J Math Imaging Vis. 2011;40(1):82–104.
  • Bertalmío M, Caselles V, Rougé B, et al. Tv based image restoration with local constraints. J Sci Comput. 2003;19(1–3):95–122.
  • Almansa A, Ballester C, Caselles V, et al. A tv based restoration model with local constraints. J Sci Comput. 2008;34(3):209–236.
  • Strong DM, Aujol J-F, Chan TF. Scale recognition, regularization parameter selection, and Meyer's g norm in total variation regularization. Multiscale Model Simulat. 2006;5(1):273–303.
  • Gilboa G, Sochen NA, Zeevi YY. Estimation of optimal pde-based denoising in the snr sense. IEEE Trans Image Proces. 2006;15(8):2269–2280.
  • Vogel CR. Computational methods for inverse problems. Vol. 23. US: SIAM; 2002.
  • De los Reyes JC, Schönlieb C-B. Image denoising: learning the noise model via nonsmooth pde-constrained optimization. Inverse Problems Imaging. 2013;7(4):1183–1214.
  • Chan T, Marquina A, Mulet P. High-order total variation-based image restoration. SIAM J Sci Comput. 2000;22(2):503–516.
  • Osher S, Solé A, Vese L. Image decomposition and restoration using total variation minimization and the h. Multiscale Model Simul. 2003;1(3):349–370.
  • Papafitsoros K, Schönlieb C-B. A combined first and second order variational approach for image reconstruction. J Math Imaging Vis. 2014;48(2):308–338.
  • Chen Y, Ranftl R, Pock T. Insights into analysis operator learning: from patch-based sparse models to higher order mrfs. IEEE Trans Image Process. 2014;23(3):1060–1072.
  • Haber E, Tenorio L. Learning regularization functionals–a supervised training approach. Inverse Probl. 2003;19(3):611.
  • De los Reyes JC, Schönlieb C-B, Valkonen T. The structure of optimal parameters for image restoration problems. J Math Anal Appl. 2016;434(1):464–500.
  • De los Reyes JC, Schönlieb C-B, Valkonen T. Bilevel parameter learning for higher-order total variation regularisation models. J Math Imaging Vis. 2017;57(1):1–25.
  • Chung C, De los Reyes JC, Schönlieb C-B. Learning optimal spatially-dependent regularization parameters in total variation image restoration. arXiv preprint arXiv:1603.09155.
  • Kunisch K, Pock T. A bilevel optimization approach for parameter learning in variational models. SIAM J Imaging Sci. 2013;6(2):938–983.
  • Calatroni L, Cao C, De Los Reyes JC, et al. Bilevel approaches for learning of variational imaging models. Variat Methods Imaging Geom Control. 2017;18(252):2.
  • Van Chung C, De los Reyes J, Schönlieb C. Learning optimal spatially-dependent regularization parameters in total variation image denoising. Inverse Probl. 2017;33(7):074005.
  • Hintermüller M, Rautenberg CN. Optimal selection of the regularization function in a weighted total variation model. part i: modelling and theory. J Math Imaging Vis. 2017;59(3):498–514.
  • Hintermüller M, Rautenberg CN, Wu T, et al. Optimal selection of the regularization function in a weighted total variation model. part ii: algorithm, its analysis and numerical tests. J Math Imaging Vision. 2017;59(3):515–533.
  • Perona P, Shiota T, Malik J. Anisotropic diffusion. In: Geometry-driven diffusion in computer vision. Springer; 1994. p. 73–92.
  • Weickert J. Coherence-enhancing diffusion filtering. Journal of Computer Vision¡/DIFdel¿Int J Comput Vis. 1999;31(2-3):111–127.
  • Weickert J, Scharr H. A scheme for coherence-enhancing diffusion filtering with optimized rotation invariance. J Vis Commun Image Represent. 2002;13(1-2):103–118.
  • Burgeth B, Didas S, Weickert J. A general structure tensor concept and coherence-enhancing diffusion filtering for matrix fields. In: Visualization and processing of tensor fields. Springer; 2009. p. 305–323.
  • Elad M. On the origin of the bilateral filter and ways to improve it. Image Processing¡/DIFdel¿IEEE Trans Image Process. 2002;11(10):1141–1151.
  • El Mourabit I, El Rhabi M, Hakim A, et al. A new denoising model for multi-frame super-resolution image reconstruction. Signal Processing. 2017;132:51–65.
  • Bredies K, Kunisch K, Pock T. Total generalized variation. SIAM J Imaging Sci. 2010;3(3):492–526.
  • Valkonen T, Bredies K, Knoll F. Total generalized variation in diffusion tensor imaging. SIAM J Imaging Sci. 2013;6(1):487–525.
  • Chambolle A, Pock T. A first-order primal-dual algorithm for convex problems with applications to imaging. J Math Imaging Vis. 2011;40(1):120–145.
  • Zhang X, Burger M, Osher S. A unified primal-dual algorithm framework based on bregman iteration. J Sci Comput. 2011;46(1):20–46.
  • Laghrib A, Hakim A, Raghay S. An iterative image super-resolution approach based on bregman distance. Signal Proces Image Commun. 2017;58:24–34.
  • Afraites L, Hadri A, Laghrib A. A denoising model adapted for impulse and gaussian noises using a constrained-pde. Inverse Probl. 2020;36(2):025006.
  • Gilbarg D, Trudinger NS. Elliptic partial differential equations of second order. Springer; 2015.
  • Catté F, Lions P-L, Morel J-M, et al. Image selective smoothing and edge detection by nonlinear diffusion. SIAM J Numer Anal. 1992;29(1):182–193.
  • Brezis H. Analyse fonctionnelle. Masson: Paris; 1983.
  • Dautray R, Lions J. Mathematical analysis and numerical methods for science and technology: evolution problems I. US: Springer; 1992.
  • Zeidler E. Nonlinear functional analysis and its applications: III: variational methods and optimization. US: Springer Science & Business Media; 2013.
  • Aubin JP. Un théorème de compacité. Acad Sci Paris. 1963;256:5042–5044.
  • Simon J. Compact sets in the space lp(0,t;b). Ann. Mat. Pura Appl. 1987;146:65–96.
  • Clason C, Valkonen T. Primal-dual extragradient methods for nonlinear nonsmooth pde-constrained optimization. SIAM J Optim. 2017;27(3):1314–1339.
  • Wu C, Tai X-C. Augmented lagrangian method, dual methods, and split Bregman iteration for rof, vectorial tv, and high order models. SIAM J Imaging Sci. 2010;3(3):300–339.
  • Buades A, Coll B, Morel J-M. Non-local means denoising. Image Process On Line. 2011;1:208–212.
  • Maleki A, Narayan M, Baraniuk RG. Anisotropic nonlocal means denoising. Appl Comput Harmon Anal. 2013;35(3):452–482.
  • Papafitsoros K, Schoenlieb CB, Sengul B. Combined first and second order total variation inpainting using split Bregman. Image Process On Line. 2013;3:112–136.
  • Duval V, Aujol J-F, Gousseau Y. The tvl1 model: a geometric point of view. Multiscale Model Simul. 2009;8(1):154–189.
  • Nikolova M. A variational approach to remove outliers and impulse noise. J Math Imaging Vis. 2004;20(1-2):99–120.
  • De Boer JF, Cense B, Park BH, et al. Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography. Opt Lett. 2003;28(21):2067–2069.
  • Wang Z, Simoncelli EP, Bovik AC. Multiscale structural similarity for image quality assessment. In: The thirty-seventh asilomar conference on signals, systems & computers. Vol. 2. IEEE; 2003, p. 1398–1402.
  • Bredies K, Dong Y, Hintermüller M. Spatially dependent regularization parameter selection in total generalized variation models for image restoration. Int J Comput Math. 2013;90(1):109–123.
  • Hintermüller M, Papafitsoros K. Generating structured nonsmooth priors and associated primal-dual methods. In: Handbook of numerical analysis. Vol. 20. Elsevier; 2019. p. 437–502.
  • Hintermüller M, Papafitsoros K, Rautenberg CN, et al. Dualization and automatic distributed parameter selection of total generalized variation via bilevel optimization. arXiv preprint arXiv:2002.05614.
  • Yi D, Lee S. Fourth-order partial differential equations for image enhancement. Appl Math Comput. 2006;175(1):430–440.
  • Liu X, Huang L, Guo Z. Adaptive fourth-order partial differential equation filter for image denoising. Appl Math Lett. 2011;24(8):1282–1288.
  • Zhang X, Ye W. An adaptive fourth-order partial differential equation for image denoising. Comput Math Appl. 2017;74(10):2529–2545.
  • Casas E, Fernandez L. Distributed control of systems governed by a general class of quasilinear elliptic equations. J Diff Equ.
  • Ciarlet PG. The finite element method for elliptic problems. Vol. 40. US: Siam; 2002.
  • Eymard R, Herbin R, Linke A, et al. Convergence of a finite volume scheme for the biharmonic problem.

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