Efficient alternating minimization methods for variational edge-weighted colorization models

Abstract

Alternating minimization algorithms are developed to solve two variational models, for image colorization based on chromaticity and brightness color system. Image colorization is a task of inpainting color from a small region of given color information. While the brightness is defined on the entire image domain, the chromaticity components are only given on a small subset of image domain. The first model is the edge-weighted total variation (TV) and the second one is the edge-weighted harmonic model that proposed by Kang and March (IEEE Trans. Image Proc. 16(9):2251–2261, 2007). Both models minimize a functional with the unit sphere constraints. The proposed methods are based on operator splitting, augmented Lagrangian, and alternating direction method of multipliers, where the computations can take advantage of multi-dimensional shrinkage and fast Fourier transform under periodic boundary conditions. Convergence analysis of the sequence generated by the proposed methods to a Karush-Kahn-Tucker point and a minimizer of the edge-weighted TV model are established. In several examples, we show the effectiveness of the new methods to colorize gray-level images, where only small patches of colors are given. Moreover, numerical comparisons with quadratic penalty method, augmented Lagrangian method, time marching, and/or accelerated time marching algorithms demonstrate the efficiency of the proposed methods.