Research On Variational Modelling And Algorithm For Image Restoration

Image is an important information media, which plays a critical role in all walks of life. During the processing of recording, storing and transferring, images may be de-graded, such as blurring, noise corruption, and losing contrast and details. These degra-dations are usually due to the limitations of the imaging device and condition, such as out-of-focus of the camera, noise inside the charge-coupled device, corruption in the transmission channel. To improve the quality of degraded images and recover clear im-ages is an important branch in the field of image processing. Image restoration refers to removal or minimization of known degradations in an image. It is an important prepro-cessing step for mid-level and high-level image processing.Image restoration belongs to the class of ill-posed problems. For the images de-graded by different types of noise, we establish the variational image restoration models from the regularization point of view. Then we propose the corresponding algorithms and analyze the convergence property. Numerical experiments demonstrate the effectiveness and superiority of the proposed methods. The main contributions of this thesis are shown as the following five parts:1. With the inspiration of the work[SIAM J. Sci. Comput., vol.31, pp.2322-2341, 2009], the l1-TV model is extended to the color image deblurring under impulsive noise. The numerical algorithm is developed under the framework of alternating minimization. The convergence of the algorithm is analyzed by using the five-point property. Numerical results illustrate the effectiveness of the proposed method.2. For coherent image acquisition systems, such as synthetic aperture radar, the observed images are often contaminated by multiplicative noise. Total variation (TV) regularization has been widely studied for multiplicative noise removal in the literature due to its edge-preserving feature. To overcome the staircase effects usually caused by TV regularization, a hybrid model is proposed. This model takes advantage of the good nature of the TV norm and the high-order TV norm to balance the edges and the smooth regions. Besides, a spatially regularization parameter updating scheme is used. Numerical results demonstrate the good performance of the proposed method.3. Images captured by image acquisition systems using photon-counting devices are often corrupted by Poisson noise. A model based on the TV regularization and the high-order TV regularization is proposed to deblur Poissonian images. The theorem on the existence and uniqueness of the proposed model is derived. This model can recover sharp edges and attenuate the staircase artifact. The alternating direction method of multipliers (ADMM) is employed to solve the minimization problem. Numerical experiments are implemented to illustrate the superiority of the proposed model.4. A new regularization function named total variation with group sparsity (OGS-TV) is proposed. For the image denoising and deblurring problems under Gaussian noise, a restoration model based on the OGS-TV regularization is proposed. This model can alleviate the staircase effect to a great extent. A key sub-problem is efficiently solved by the majorization-minimization method. The numerical examples are given to illustrate the superiority and efficiency of the proposed method.5. A multiplicative noise removal model is proposed by incorporating the regulariza-tion function OGS-TV. The corresponding algorithm is developed under the framework of the ADMM. Compared with the methods based on total variation and total generalized variation, the experimental results demonstrate that the proposed method behaves much better in terms of peak signal to noise ratio and structural similarity index.