A Combined First and Second Order Variational Approach for Image Reconstruction

Acar, R., Vogel, C.R.: Analysis of bounded variation penalty methods for ill-posed problems. In: Inverse Problems, pp. 1217–1229 (1994)

Alvarez, L., Mazorra, L.: Signal and image restoration using shock filters and anisotropic diffusion. SIAM J. Numer. Anal. 31(2), 590–605 (1994)

Amar, M., Cicco, V.: Relaxation of quasi-convex integrals of arbitrary order. Proc. R. Soc. Edinb., Sect. A, Math. 124(05), 927–946 (1994)

Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, Oxford (2000)

Benning, M.: Singular Regularization of Inverse Problems (2011)

Benning, M., Brune, C., Burger, M., Müller, J.: Higher-order TV methods—Enhancement via Bregman iteration. J. Sci. Comput. 54(2–3), 269–310 (2013)

Benning, M., Burger, M.: Error estimates for general fidelities. Electron. Trans. Numer. Anal. 38(44–68), 77 (2011)

Bergounioux, M., Piffet, L.: A second-order model for image denoising. Set-Valued Var. Anal. 18(3–4), 277–306 (2010)

Bertozzi, A., Esedoglu, S., Gillette, A.: Analysis of a two-scale Cahn-Hilliard model for binary image inpainting. Multiscale Model. Simul. 6(3), 913–936 (2008)

Bertozzi, A.L., Greer, J.B.: Low-curvature image simplifiers: global regularity of smooth solutions and Laplacian limiting schemes. Commun. Pure Appl. Math. 57(6), 764–790 (2004)

Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3(1), 1–122 (2011)

Braides, A.: Γ-Convergence for Beginners. Oxford Lecture Series in Mathematics and Its Applications. (2002)

Bredies, K.: Recovering piecewise smooth multichannel images by minimization of convex functionals with total generalized variation penalty. Preprint (2012). http://math.uni-graz.at/mobis/publications/SFB-Report-2012-006.pdf

Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3, 1–42 (2009)

Bredies, K., Lorenz, D.: Mathematische Bildverarbeitung: Einführung in Grundlagen und moderne Theorie. Vieweg+Teubner, Wiesbaden (2011)

Bredies, K., Valkonen, T.: Inverse problems with second-order total generalized variation constraints. In: Proceedings of SampTA 2011—9th International Conference on Sampling Theory and Applications, Singapore (2011)

Burger, M., He, L., Schönlieb, C.: Cahn-Hilliard inpainting and a generalization for gray value images. SIAM J. Imaging Sci. 2(4), 1129–1167 (2009)

Burger, M., Osher, S.: Convergence rates of convex variational regularization. Inverse Probl. 20, 1411 (2004)

Burger, M., Resmerita, E., He, L.: Error estimation for Bregman iterations and inverse scale space methods in image restoration. Computing 81(2), 109–135 (2007)

Buttazzo, G., Freddi, L.: Functionals defined on measures and applications to non equi-uniformly elliptic problems. Ann. Mat. Pura Appl. 159(1), 133–149 (1991)

Cai, J., Chan, R., Shen, Z.: A framelet-based image inpainting algorithm. Appl. Comput. Harmon. Anal. 24(2), 131–149 (2008)

Chambolle, A., Lions, P.: Image recovery via total variation minimization and related problems. Numer. Math. 76, 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., Esedoglu, S.: Aspects of total variation regularized L 1 function approximation. SIAM J. Appl. Math. 65(5), 1817–1837 (2005)

Chan, T., Kang, S., Shen, J.: Euler’s elastica and curvature-based inpainting. SIAM J. Appl. Math. 63(2), 564–592 (2002)

Chan, T., Marquina, A., Mulet, P.: High-order total variation-based image restoration. SIAM J. Sci. Comput. 22(2), 503–516 (2001)

Chan, T.F., Esedoglu, S., Park, F.: Image decomposition combining staircase reduction and texture extraction. J. Vis. Commun. Image Represent. 18(6), 464–486 (2007)

Charbonnier, P., Blanc-Féraud, L., Aubert, G., Barlaud, M.: Two deterministic half-quadratic regularization algorithms for computed imaging. In: ICIP (2)’94, pp. 168–172 (1994)

Combettes, P., Wajs, V., et al.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4(4), 1168–1200 (2006)

Dal Maso, G.: Introduction to Γ-Convergence. Birkhäuser, Basel (1993)

Dal Maso, G., Fonseca, I., Leoni, G., Morini, M.: A higher order model for image restoration: the one dimensional case. SIAM J. Math. Anal. 40(6), 2351–2391 (2009)

Demengel, F.: Fonctions à Hessien borné. Ann. Inst. Fourier 34, 155–190 (1985)

Demengel, F., Temam, R.: Convex functions of a measure and applications. Indiana Univ. Math. J. 33, 673–709 (1984)

Didas, S., Weickert, J., Burgeth, B.: Properties of higher order nonlinear diffusion filtering. J. Math. Imaging Vis. 35, 208–226 (2009)

Dong, Y., Hintermüller, M., Knoll, F., Stollberger, R.: Total variation denoising with spatially dependent regularization. In: ISMRM 18th Annual Scientific Meeting and Exhibition Proceedings, p. 5088 (2010)

Duval, V., Aujol, J., Gousseau, Y.: The TVL1 model: a geometric point of view. Multiscale Model. Simul. 8(1), 154–189 (2009)

Eckstein, J.: Splitting methods for monotone operators with applications to parallel optimisation. Ph.D. Thesis (1989)

Ekeland, I., Témam, R.: Convex Analysis and Variational Problems. Classics in Applied Mathematics. SIAM, SIAM (1999)

Esser, E., Zhang, X., Chan, T.: A general framework for a class of first order primal-dual algorithms for TV minimization. UCLA CAM report 09-67 (2009)

Evans, L.: Partial Differential Equations, 2nd edn. Graduate Studies in Mathematics, vol. 19. AMS, Providence (2010)

Feng, X., Prohl, A.: Analysis of total variation flow and its finite element approximations. Modél. Math. Anal. Numér. 37(3), 533 (2003)

Frick, K., Marnitz, P., Munk, A.: Statistical multiresolution estimation in imaging: fundamental concepts and algorithmic framework. Electron. J. Stat. 6, 231–268 (2012)

Frohn-Schauf, C., Henn, S., Witsch, K.: Nonlinear multigrid methods for total variation image denoising. Comput. Vis. Sci. 7(3), 199–206 (2004)

Gabay, D.: Chapter IX applications of the method of multipliers to variational inequalities. In: Fortin, M., Glowinski, R. (eds.) Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems. Studies in Mathematics and Its Applications, vol. 15, pp. 299–331. Elsevier, Amsterdam (1983)

Gilboa, G., Sochen, N., Zeevi, Y.: Image enhancement and denoising by complex diffusion processes. IEEE Trans. Pattern Anal. Mach. Intell. 26(8), 1020–1036 (2004)

Goldstein, T., Osher, S.: The split Bregman method for L1 regularized problems. SIAM J. Imaging Sci. 2, 323 (2009)

Green, P.J.: Bayesian reconstructions from emission tomography data using a modified EM algorithm. IEEE Trans. Med. Imaging 9(1), 84–93 (1990)

Hinterberger, W., Scherzer, O.: Variational methods on the space of functions of bounded Hessian for convexification and denoising. Computing 76, 109–133 (2006)

Kanizsa, G.: La Grammaire du Voir. Diderot (1996)

Lai, R., Tai, X., Chan, T.: A ridge and corner preserving model for surface restoration. UCLA CAM report 11-55 (2011)

Lefkimmiatis, S., Bourquard, A., Unser, M.: Hessian-based norm regularization for image restoration with biomedical applications. IEEE Trans. Image Process. 21(3), 983–995 (2012)

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)

Lysaker, M., Tai, X.C.: Iterative image restoration combining total variation minimization and a second-order functional. Int. J. Comput. Vis. 66(1), 5–18 (2006)

Nikolova, M.: Minimizers of cost-functions involving nonsmooth data-fidelity terms. Application to the processing of outliers. SIAM J. Numer. Anal. 40(3), 965–994 (2003)

Nikolova, M.: A variational approach to remove outliers and impulse noise. J. Math. Imaging Vis. 20(1), 99–120 (2004)

Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An iterative regularization method for total variation-based image restoration. Multiscale Model. Simul. 4, 460–489 (2005)

Papafitsoros, K., Sengul, B., Schönlieb, C.: Combined first and second order total variation inpainting using split Bregman (2013, to appear). http://www.maths.cam.ac.uk/postgrad/cca/files/ipol.pdf

Pock, T., Cremers, D., Bischof, H., Chambolle, A.: Global solutions of variational models with convex regularization. SIAM J. Imaging Sci. 3(4), 1122–1145 (2010)

Pöschl, C.: Tikhonov regularization with general residual term. Ph.D. Thesis, Leopold-Franzens-Universität Innsbruck, Austria (2008)

Pöschl, C., Scherzer, O.: Characterization of minimisers of convex regularization functionals. Contemp. Math. 451, 219–248 (2008)

Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992)

Scherzer, O.: Denoising with higher order derivatives of bounded variation and an application to parameter estimation. Computing 60(1), 1–27 (1998)

Schönlieb, C., Bertozzi, A., Burger, M., He, L.: Image inpainting using a fourth-order total variation flow. In: Proc. Int. Conf. SampTA09, Marseilles (2009)

Setzer, S.: Split Bregman algorithm, Douglas-Rachford splitting and frame shrinkage. In: Scale Space and Variational Methods in Computer Vision, vol. 5567, pp. 464–476 (2009)

Setzer, S., Steidl, G.: Variational methods with higher order derivatives in image processing. In: Approximation XII, pp. 360–386 (2008)

Setzer, S., Steidl, G., Teuber, T.: Infimal convolution regularizations with discrete ℓ 1-type functionals. Commun. Math. Sci. 9, 797–872 (2011)

Tai, X., Hahn, J., Chung, G.: A fast algorithm for Euler’s elastica model using augmented Lagrangian method. SIAM J. Imaging Sci. 4(1), 313–344 (2011)

Tikhonov, A.: Solution of incorrectly formulated problems and the regularization method. Sov. Math. Dokl. 5, 1035 (1963)

Vassilevski, P., Wade, J.G.: A comparison of multilevel methods for total variation regularization. Electron. Trans. Numer. Anal. 6, 255–270 (1997)

Vese, L.: A study in the BV space of a denoising-deblurring variational problem. Appl. Math. Optim. 44(2), 131–161 (2001)

Vogel, C.: A multigrid method for total variation-based image denoising. Prog. Syst. Control Theory 20, 323 (1995)

Wang, Y., Yin, W., Zhang, Y.: A fast algorithm for image deblurring with total variation regularization. CAAM technical report TR07-10, Rice University, Houston, Tex, USA (2007)

Wang, Z., Bovik, A.: Mean squared error: love it or leave it? A new look at signal fidelity measures. IEEE Signal Process. Mag. 26(1), 98–117 (2009)

Wang, Z., Bovik, A., Sheikh, H., Simoncelli, E.: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600–612 (2004)

Whittaker, E.T.: On a new method of graduation. Proc. Edinb. Math. Soc. 41, 63–75 (1923)

Wu, C., Tai, X.: Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models. SIAM J. Imaging Sci. 3(3), 300–339 (2010)

Yin, W., Osher, S., Goldfarb, D., Darbon, J.: Bregman iterative algorithms for ℓ 1-minimisation with applications to compressed sensing. SIAM J. Imaging Sci. 1, 142–168 (2008)