The authors introduce a family of variational models for color image denoising which minimize linear growth energy functionals of the maps into the unit sphere in \(\mathbb{R}^{3}\). They develop some fully discrete finite element methods for computing solutions of the proposed models. They introduce the \(\mathbb{L}^{2}\) gradient flow, which is described by a system of nonlinear parabolic partial differential equations, for the energy functionals as a way to solve the variation problem and present the mathematical framework and setting for the gradient flow. Numerical results are presented to show the good performance of the proposed models and numerical methods.