Variational Image Colorization Models Using Higher-Order Mumford–Shah Regularizers

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.