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**Efficient homotopy solution and a convex combination of ROF and LLT models for image restoration.**
*(English)*
Zbl 1264.35288

Summary: The L. I. Rudin, S. Osher and 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, M. Lysaker, A. Lundervold and 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 [M. Lysaker, S. Osher and 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.

### MSC:

35R35 | Free boundary problems for PDEs |

49J40 | Variational inequalities |

60G40 | Stopping times; optimal stopping problems; gambling theory |

65D18 | Numerical aspects of computer graphics, image analysis, and computational geometry |