Abstract
This article introduces variational models for restoring a color image from a grayscale image with color given in only small regions. The models involve the chromaticity color component as in Kang and March (IEEE Trans Image Proc 16(9):2251–2261, 2007), but we make use of higher-order regularization to effectively recover color values of piecewise-smooth images. The first model involves a convex weighted higher-order regularization term, where the weight assists to inhibit the diffusion of chromaticity across the edges. To realize this proposed model, we solve its approximated version obtained by introducing a new variable. We prove the existence of minimizers for both the original and approximated problems, and determine the convergence of their respective solutions. Moreover, we introduce higher-order versions of a Mumford–Shah-like regularizing functional and utilize them for image colorization. The nonconvexity of the proposed functionals enables us to automatically restrain the dispersion of chromaticity across the edges. We also present fast and efficient iterative algorithms for solving the proposed models. Numerical results validate that our models perform more effectively than first-order regularization-based models.
Similar content being viewed by others
References
Ambrosio, L., Tortorelli, V.M.: Approximation of functionals depending on jumps by elliptic functionals via \(\gamma \)-convergence. Commun. Pure Appl. Math. 43, 999–1036 (1990)
Ambrosio, L., Tortorelli, V.M.: On the approximation of free discontinuity problems. Bollettino dellUnione Matematica Italiana B(7)(6), 105–123 (1992)
Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, vol. 147, 2nd edn. Applied Mathematical Sciences. Springer (2006)
Bergounioux, M., Piffet, L.: A second-order model for image denoising. Set Valued Anal. Var. Anal. 18(3–4), 277–306 (2010)
Besag, J.E.: Digital image processing: towards bayesian image analysis. J. Appl. Stat. 16(3), 395–407 (1989)
Blake, A., Zisserman, A.: Visual Reconstruction. MIT Press, Cambridge (1987)
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 (2010)
Bugeau, A., Ta, V.T., Papadakis, N.: Variational exemplar-based image colorization. IEEE Trans. Image Proc. 23(1), 298–307 (2014)
Candès, E.J., Wakin, M.B., Boyd, S.P.: Enhancing sparsity by reweighted \(l_1\) minimization. J. Fourier Anal. Appl. 14(5), 877–905 (2008)
Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20, 89–97 (2004)
Chan, T.F., Marquina, A., Mulet, P.: High-order total variation-based image restoration. SIAM J. Sci. Comput. 22(2), 503–516 (2000)
Charbonnier, P., Féraud, L., Aubert, G., Barlaud, M.: Deterministic edge-preserving regularization in computed imaging. IEEE Trans. Image Proc. 6(2), 298–311 (1997)
Daubechies, I., Devore, R., Fornasier, M., Gntrk, C.S.: Iteratively reweighted least squares minimization for sparse recovery. Commun. Pure Appl. Math. 63(1), 1–38 (2010)
Demengel, F.: Fonctions à hessien borné. Ann. Inst. Fourier 34(2), 155–190 (1985)
Douglas, J., Rachford, H.H.: On the numerical solution of heat conduction problems in two and three space variables. Trans. Am. Math. Soc. 82(3), 421–439 (1956)
Esser, E.: Applications of lagrangian-based alternating direction methods and connections to split-bregman. In: UCLA CAM Report 09–31 (2009)
Fang, L., Shen, C., Fan, J., Shen, C.: Image restoration combining a total variational filter and a fourth-order filter. J. Vis. Commun. Image Represent. 18(4), 322–330 (2007)
Fornasier, M.: Nonlinear projection digital image inpainting and restoration methods. J. Math. Imaging Vis. 24(3), 359–373 (2006)
Geman, D., Reynolds, G.: Constrained restoration and recovery of discontinuities. IEEE Trans. Pattern Anal. Mach. Intell. 14(3), 367–383 (1992)
Geman, D., Yang, C.: Nonlinear image recovery with half-quadratic regularization. IEEE Trans. Image Proc. 4(7), 932–946 (1995)
Geman, S., Geman, D.: Stochastic relaxation, gibbs distributions, and the bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. PAMI–6(6), 721–741 (1984)
Goldstein, T., Osher, S.: The split bregman method for l1-regularized problems. SIAM J. Imaging Sci. 2(2), 323–343 (2009)
Irony, R., Cohen-Or, D., Lischinski, D.: Colorization by example. In: Proceedings of Eurographics Symposium on Rendering, pp. 201–210 (2005)
Jung, M., Kang, M.: Efficient nonsmooth nonconvex optimization for image restoration and segmentation. J. Sci. Comput. 62(2), 336–370 (2015)
Kang, S.H., March, R.: Variational models for image colorization via chromaticity and brightness decomposition. IEEE Trans. Image Proc. 16(9), 2251–2261 (2007)
Lefkimmiatis, S., Bourquard, A., Unser, M.: Hessian-based norm regularization for image restoration with biomedical applications. IEEE Trans. Image Proc. 21(3), 983–995 (2010)
Levin, A., Lischinski, D., Weiss, Y.: Colorization using optimization. Proc. SIGGRAPH Conf. 23(3), 689–694 (2004)
Li, S.: Markov Random Field Modeling in Computer Vision, 1st edn. Springer, New York (1995)
Lia, F., Baob, Z., Liuc, R., Zhang, G.: Fast image inpainting and colorization by chambolle’s dual method. J. Vis. Commun. Image Represent. 22(6), 529–542 (2011)
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 Proc. 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)
Mota, J., Xavier, J., Aguiar, P., Puschel, M.: A proof of convergence for the alternating direction method of multipliers applied to polyhedral-constrained functions. arXiv:1112.2295 (2011)
Mumford, D., Shah, J.: Boundary detection by minimizing functionals. In: Proceedings of IEEE International Conference Acoustics, Speech Signal Process. pp. 22–26 (1985)
Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. XLVII, 577–685 (1989)
Nikolova, M.: Markovian reconstruction using a GNC approach. IEEE Trans. Image Proc. 8(9), 1204–1220 (1999)
Nikolova, M., Chan, R.H.: The equivalence of half-quadratic minimization and the gradient linearization iteration. IEEE Trans. Image Proc. 16(6), 1623–1627 (2007)
Nikolova, M., Ng, M.K., Tam, C.P.: Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction. IEEE Trans. Image Proc. 19(12), 3073–3088 (2010)
Nikolova, M., Ng, M.K., Zhang, S., Ching, W.: Efficient reconstruction of piecewise constant images using nonsmooth nonconvex minimization. SIAM J. Imaging Sci. 1(1), 2–25 (2008)
Ochs, P., Dosovitskiy, A., Brox, T., Pock, T.: An iterated \(\ell _1\) algorithm for non-smooth non-convex optimization in computer vision. IEEE Conference on Computer Vision and Pattern Recognition, 1759–1766 (2013)
Ochs, P., Dosovitskiy, A., Brox, T., Pock, T.: On iteratively reweighted algorithms for nonsmooth nonconvex optimization in computer vision. SIAM J. Imaging Sci. 8(1), 331–372 (2013)
Oh, S., Woo, H., Yun, S., Kang, M.: Non-convex hybrid total variation for image denoising. J. Vis. Commun. Image Represent. 24(3), 332–344 (2013)
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)
Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12(7), 629–639 (1990)
Pierre, F., Aujol, J.F., Bugeau, A., Papadakis, N., Ta, V.T.: Exemplar-based colorization in rgb color space. In: Proceedings of IEEE International Conference Image Processing, pp. 625–629 (2014)
Pierre, F., Aujol, J.F., Bugeau, A., Papadakis, N., Ta, V.T.: Luminance-chrominance model for image colorization. SIAM J. Imaging Sci. 8(1), 536–563 (2015)
Quang, M., Kang, S., Le, T.: Image and video colorization using vector-valued reproducing kernel hilbert spaces. J. Math. Imaging Vis. 37(1), 49–65 (2010)
Robini, M., Lachal, A., Magnin, I.: A stochastic continuation approach to piecewise constant reconstruction. IEEE Trans. Image Proc. 16(10), 2576–2589 (2007)
Sapiro, G.: Inpainting the colors. In: Proceedings of IEEE International Conference Image Processing, vol. 2, pp. 698–701 (2005)
Scherzer, O.: Denoising with higher order derivatives of bounded variation and an application to parameter estimation. Computing 60(1), 1–27 (1998)
Setzer, S.: Operator splittings, bregman methods and frame shrinkage in image processing. Int. J. Comput. Vis. 92, 265–280 (2011)
Setzer, S., Steidl, G.: Variational methods with higher order derivatives in image processing. Approximation 12, 360–386 (2008)
Shah, J.: A common framework for curve evolution segmentation and anisotropic diffusion. In: Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, San Francisco, June pp. 136–142 (1996)
Teboul, S., Féraud, L.B., Aubert, G., Barlaud, M.: Variational approach for edge-preserving regularization using coupled PDE’s. IEEE Trans. Image Proc. 7(3), 387–397 (1998)
Vese, L.A., Chan, T.F.: Reduced non-convex functional approximations for image restoration & segmentation. In: UCLA CAM report, pp. 97–56 (1997)
Vogel, C.R., Oman, M.E.: Fast robust total variation based reconstruction of noisy, blurred images. IEEE Trans. Image Proc. 7, 813–824 (1998)
Yatziv, L., Sapiro, G.: Fast image and video colorization using chrominance blending. IEEE Trans. Image Proc. 15(5), 1120–1129 (2006)
Yin, W., Osher, S., Goldfarb, D., Darbon, J.: Bregman iterative algorithms for l1-minimization with applications to compressed sensing. SIAM J. Imaging Sci. 1(1), 143–168 (2008)
Zhang, T.: Analysis of multi-stage convex relaxation for sparse regularization. J. Mach. Learn. Res. 11, 1081–1107 (2010)
Acknowledgments
Miyoun Jung was supported in part by the Hankuk University of Foreign Studies Research Fund and by the Basic Science Research Program through the NRF of Korea funded by the Ministry of Science, ICT, and Future Planning (2013R1A1A3010416). Myungjoo Kang was supported by the Basic Sciences Research Program through the NRF of Korea funded by the Ministry of Science, ICT, and Future Planning (2014R1A2A1A10050531 and 2015R1A5A1009350).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jung, M., Kang, M. Variational Image Colorization Models Using Higher-Order Mumford–Shah Regularizers. J Sci Comput 68, 864–888 (2016). https://doi.org/10.1007/s10915-015-0162-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-015-0162-9