Fast and Efficient Algorithms for TV Image Restoration

In this thesis, we study two aspects in image processing. Part I is on the fast and efficient algorithms for the TV-L1 image restoration. Part II is on the fast and efficient algorithms for the positively constraint maximum penalized TV image restoration.;In Part I of the thesis, we focus on the fast and efficient algorithms for the TV-L1 minimization problem which can be applied to recover the blurred images corrupted by impulse noise. We construct the half-quadratic algorithm (HQA) for TV-L1 image restoration based on the half-quadratic technique. By introducing the proximal point algorithm into the HQA, we then obtain a modified HQA. We call it the proximal point half-quadratice algorithm (PHA). We introduce the PHA aiming to decrease the condition number of the coefficient matrix as updating the iterator in HQA. Until recently, there have been many efficient methods to solve the TV-L1 minimization problem. Examples are the primal-dual method, the fast total variational deconvolution method (FTVDM), and the augmented Lagrangian method (ALM). By numerical results of the FTVDM and ALM, we see that the images restored by these methods may sometimes appear to be blocky. Come back to our methods. The HQA and the PHA are both fast and efficient algorithms to solve the TV-L1 minimization problem. We prove that our algorithms are both majorize-minimize algorithms for solving a regularized TV-L1 problem. Given the assumption ker(∇)∩ker(BT B) = {0}, the convergence and linear convergence of the HQA is then easily obtained. Without such an assumption, a convergence result of PHA is also obtained. We apply our algorithms to deblur images corrupted with impulse noise. The results show that the HQA is faster and more accurate than the ALM and FTVDM for salt-and-pepper noise and comparable to the two methods for random-valued impulse noise. The PHA is comparable to the HQA in both recovered effect and computing consuming. Comparing with ALM and FTVDM, the PHA is faster and more accurate than ALM and FTVDM for salt-and-pepper noise and comparable to the two methods for random-valued impulse noise. Furthermore, the recovered images by the HQA and the PHA are less blocky.;Part II of the thesis focuses on the positively constraint maximum penalized total variation image restoration. We develop and implement a multiplicative iteration approach for the positively constrained total variation image restoration. We call our algorithm MITV. The MITV algorithm is based on the multiplicative iterative algorithm originally developed for tomographic image reconstruction. The advantages of the MITV are that it is very easy to derive and implement under different image noise models and it respects the positivity constraint. Our method can be applied to kinds of noise models, the Gaussian noise model, Poisson noise model and the impulse noise model. In numerical test, we apply our algorithm to deblur images corrupted with Gaussian noise. The results show that our method give better restored images than the forward-backward splitting algorithm.