Summary: The {\it L. I. Rudin, S. Osher} and {\it E. Fatemi} [Physica D 60, No. 1--4, 259--268 (1992;

Zbl 0780.49028)] model (ROF) for image restoration has been extensively studied due to its edge preserving capability, but for images without edges (jumps), the solution to this model has the undesirable staircasing effect. To improve the model, {\it M. Lysaker, A. Lundervold} and {\it X.-C. Tai} [“Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time”, IEEE Transactions on Image Processing 12, No. 12, 1579--1590 (2003)] (LLT) proposed a better second-order functional suitable for restoring smooth images but it is difficult to preserve discontinuities for non-smooth images. It turns out that results from convex combinations of ROF model and LLT model can preserve the main advantages of both models (see [{\it M. Lysaker, S. Osher} and {\it X.-C. Tai}, “Noise removal using smoothed normals and surface fitting”, IEEE Transactions on Image Processing 13, No. 10, 1345--1357 (2004); Qi. Chang, X.-C. Tai and L. Xing, “A compound algorithm of denoising using second-order and fourth-order partial differential equations”, Numer. Math. Theor. Meth. Appl. 2, No. 4, 353--376 (2009)]). In this paper, we first propose an applicable homotopy algorithm based fixed point method for the LLT model. We then propose two new variants of convex combination models. Numerical experiments are shown to demonstrate the advantages of these combination models and the robustness of our homotopy algorithm.