| In this thesis, we develop some new models and efficient algorithms for image denoising. The total variation model of Rudin, Osher, and Fatemi(ROF) for image denoising is considered to be one of the most successful deterministic denoising models. It exploits the non-smooth total variation (TV) semi-norm to preserve discontinuities and to keep the edges of smooth regions sharp. Despite its simple form, the TV semi-norm results in a strongly nonlinear Euler-Lagrange equation and poses computational challenge in solving the model efficiently. Moreover, this model produces so-called staircase effect. In this thesis, we propose several new algorithms and models to solve these problems. We study the discretized ROF model and propose a new algorithm which does not involve partial differential equations. Convergence of the algorithm is analyzed. Numerical results show that this algorithm is efficient and stable. We then introduce a denoising model which utilizes high-order difference to approximate piece-wise smooth functions. This model eliminates undesirable staircases, and improves both visual quality and signal-to-noise ratio. Our algorithm is generalized to solve the high-order models. A relaxation technique is proposed for the iteration scheme, aiming to accelerate our solution process. Finally, we propose a method combining total variation and wavelet packets to improve performance on texture-rich images. The ROF model is utilized to eliminate noise, and a wavelet packet transform is used to enhance textures. The numerical results show that the combinational method exploits the advantages of both total variation and wavelet packets. |
