Abstract
In image processing, the regularization term is always hard to choose. In this paper, we introduce a model based on the fractional derivative, which is derived from the classical Beltrami model. The proposed regularization term offers an ideal compromise between feature preservation, avoidance of staircasing and the loss of image contrasts. This model can outperform the high order regularization models and also the total fractional-order variation model. Also, we rigorously analyse the theoretical properties of the fractional derivative and show the existence and uniqueness of the proposed minimization problem in a suitable functional framework. In addition, to solve the variational problem, we consider the primal-dual projected gradient algorithm. Numerical experiments show that the proposed model produces competitive results compared to some classical regularizations, especially, it avoids the staircase effect and preserves edges and features of the image while reducing noise.
Similar content being viewed by others
References
Acar, R., Vogel, C.R.: Analysis of bounded variation penalty methods for ill-posed problems. Inverse Prob. 10(6), 1217 (1994)
Tikhonov, A.N., Arsenin, V.Y.: Methods for solving ill-posed problems. Wiley (1977)
Agrawal, O.P.: Fractional variational calculus in terms of Riesz fractional derivatives. J. Phys. A: Math. Theor. 40(24), 6287 (2007)
Alahyane, M., Hakim, A., Laghrib, A., Raghay, S.: Fluid image registration using a finite volume scheme of the incompressible Navier Stokes equation. Inverse Probl. Imaging 12(5), 1055–1081 (2018)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems, vol. 254. Clarendon Press, Oxford (2000)
Angenent, S., Haker, S., Tannenbaum, A., Kikinis, R.: On the Laplace–Beltrami operator and brain surface flattening. IEEE Trans. Med. Imaging 18(8), 700–711 (1999)
Attouch, H., Buttazzo, G., Michaille, G.: Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization, vol. 17. SIAM, Philadelphia (2014)
Aujol, J.-F.: Some first-order algorithms for total variation based image restoration. J. Math. Imaging Vis. 34(3), 307–327 (2009)
Bai, J., Feng, X.-C.: Fractional-order anisotropic diffusion for image denoising. IEEE Trans. Image Process. 16(10), 2492–2502 (2007)
Bai, J., Feng, X.-C.: Image denoising using generalized anisotropic diffusion. J. Math. Imaging Vis. 60, 1–14 (2018)
Bergmann, R., Weinmann, A.: A second-order tv-type approach for inpainting and denoising higher dimensional combined cyclic and vector space data. J. Math. Imaging Vis. 55(3), 401–427 (2016)
Bertalmio, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting. In: Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, pp. 417–424. ACM Press/Addison-Wesley Publishing Co. (2000)
Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3(3), 492–526 (2010)
Cai, J.-F., Osher, S., Shen, Z.: Split Bregman methods and frame based image restoration. Multiscale Model. Simul. 8(2), 337–369 (2009)
Carbone, L., De Arcangelis, R.: Unbounded Functionals in the Calculus of Variations: Representation, Relaxation, and Homogenization. Chapman and Hall, London (2001)
Chang, Q., Chern, I-Liang.: Acceleration methods for total variation-based image denoising. SIAM. J. Sci. Comput 25(3), 982–994 (2003)
Chambolle, A., Lions, P.-L.: Image recovery via total variation minimization and related problems. Numer. Math. 76(2), 167–188 (1997)
Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011)
Chan, T., Marquina, A., Mulet, P.: High-order total variation-based image restoration. SIAM J. Sci. Comput. 22(2), 503–516 (2000)
Chan, T. F., Esedoglu, S., Park, F.: A fourth order dual method for staircase reduction in texture extraction and image restoration problems. In: 2010 17th IEEE International Conference on Image Processing (ICIP), pp. 4137–4140. IEEE (2010)
Chen, D., Chen, Y., Xue, D.: Fractional-order total variation image restoration based on primal-dual algorithm. In: Abstract and Applied Analysis, vol. 2013. Hindawi (2013)
Chen, D., Sun, S., Zhang, C., Chen, Y., Xue, D.: Fractional-order tv-l 2 model for image denoising. Cent. Eur. J. Phys. 11(10), 1414–1422 (2013)
Demengel, F., Temam, R.: Convex-functions of a measure and applications. Indiana Univ. Math. J. 33(5), 673–709 (1984)
Esser, E., Zhang, X., Chan, T.F.: A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science. SIAM J. Imaging Sci. 3(4), 1015–1046 (2010)
Evans, L.: Measure Theory and Fine Properties of Functions. Routledge, London (2018)
Frohn-Schauf, C., Henn, S., Witsch, K.: Multigrid based total variation image registration. Comput. Vis. Sci. 11(2), 101–113 (2008)
Goldstein, T., Osher, S.: The split Bregman method for l1-regularized problems. SIAM J. Imaging Sci. 2(2), 323–343 (2009)
Govan, A.: Introduction to optimization. In: North Carolina State University, SAMSI NDHS, Undergraduate Workshop (2006)
Guidotti, P.: A new nonlocal nonlinear diffusion of image processing. J. Differ. Equ. 246(12), 4731–4742 (2009)
Guidotti, P., Lambers, J.V.: Two new nonlinear nonlocal diffusions for noise reduction. J. Math. Imaging Vis. 33(1), 25–37 (2009)
Hadri, A., Khalfi, H., Laghrib, A., Nachaoui, M.: An improved spatially controlled reaction-diffusion equation with a non-linear second order operator for image super-resolution. Nonlinear Anal. Real World Appl. 62, 103352 (2021)
Hakim, M., Ghazdali, A., Laghrib, A.: A multi-frame super-resolution based on new variational data fidelity term. Appl. Math. Model. 87, 446–467 (2020)
He, B., Yuan, X.: Convergence analysis of primal-dual algorithms for a saddle-point problem: from contraction perspective. SIAM J. Imaging Sci. 5(1), 119–149 (2012)
Janev, M., Pilipović, S., Atanacković, T., Obradović, R., Ralević, N.: Fully fractional anisotropic diffusion for image denoising. Math. Comput. Model. 54(1–2), 729–741 (2011)
Laghrib, A., Afraites, L., Hadri, A., Nachaoui, M.: A non-convex PDE-constrained denoising model for impulse and gaussian noise mixture reduction. Inverse Probl. Imaging (2022)
Laghrib, A., Alahyane, M., Ghazdali, A., Hakim, A., Raghay, S.: Multiframe super-resolution based on a high-order spatially weighted regularisation. IET Image Proc. 12(6), 928–940 (2018)
Laghrib, A., Chakib, A., Hadri, A., Hakim, A.: A nonlinear fourth-order PDE for multi-frame image super-resolution enhancement. Discrete Contin. Dyn. Syst.-B 25(1), 415 (2020)
Laghrib, A., Ezzaki, M., El Rhabi, M., Hakim, A., Monasse, P., Raghay, S.: Simultaneous deconvolution and denoising using a second order variational approach applied to image super resolution. Comput. Vis. Image Underst. 168, 50–63 (2018)
Laghrib, A., Ghazdali, A., Hakim, A., Raghay, S.: A multi-frame super-resolution using diffusion registration and a nonlocal variational image restoration. Comput. Math. Appl. 72(9), 2535–2548 (2016)
Lysaker, M., Lundervold, A., Tai, X.-C.: Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time. IEEE Trans. Image Process. 12(12), 1579–1590 (2003)
Machado, J.T., Kiryakova, V., Mainardi, F.: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1140–1153 (2011)
Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
Papafitsoros, K., Schönlieb, C.-B.: A combined first and second order variational approach for image reconstruction. J. Math. Imaging Vis. 48(2), 308–338 (2014)
Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198. Elsevier, Amsterdam (1998)
Poschl, C., Scherzer, O.: Characterization of minimizers of convex regularization functionals. Contemp. Math. 451, 219–248 (2008)
Rosman, G., Dascal, L., Tai, X.-C., Kimmel, R.: On semi-implicit splitting schemes for the Beltrami color image filtering. J. Math. Imaging Vis. 40(2), 199–213 (2011)
Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992)
Shen, J., Kang, S.H., Chan, T.F.: Euler’s elastica and curvature-based inpainting. SIAM J. Appl. Math. 63(2), 564–592 (2003)
Shor, N.Z.: Minimization Methods for Non-differentiable Functions, vol. 3. Springer, Berlin (2012)
Valkonen, T., Bredies, K., Knoll, F.: Total generalized variation in diffusion tensor imaging. SIAM J. Imaging Sci. 6(1), 487–525 (2013)
Vese, L.A., Osher, S.J.: Image denoising and decomposition with total variation minimization and oscillatory functions. J. Math. Imaging Vis. 20(1–2), 7–18 (2004)
Zhang, D., Chen, K.: A novel diffeomorphic model for image registration and its algorithm. J. Math. Imaging Vis. 60, 1261–1283 (2018)
Zhang, J., Chen, K.: A total fractional-order variation model for image restoration with nonhomogeneous boundary conditions and its numerical solution. SIAM J. Imaging Sci. 8(4), 2487–2518 (2015)
Zhang, J., Wei, Z., Xiao, L.: Adaptive fractional-order multi-scale method for image denoising. J. Math. Imaging Vis. 43(1), 39–49 (2012)
Zosso, D., Bustin, A.: A primal-dual projected gradient algorithm for efficient Beltrami regularization. Comput. Vis. Image Underst. pp. 14–52 (2014)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ben-Loghfyry, A., Hakim, A. & Laghrib, A. A denoising model based on the fractional Beltrami regularization and its numerical solution. J. Appl. Math. Comput. 69, 1431–1463 (2023). https://doi.org/10.1007/s12190-022-01798-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-022-01798-9