||Video compression, being the hottest topic of the last twenty years, has recently become more and more mature. Coding standards like MPEG-2, H.264 and AVS are widely studied and used. Since a lossy quantization is employed in the entire coding scheme for a deeper compression, video details may be lost after it is compressed. More serious problem is the visual artifact brought in by lossy compression. A typical and well-known type of artifact is the blocking artifact which is caused by the block based compression scheme. Filtering along the block edges which is to suppress this artifact is already part of standards like H.264 and AVS. Such filters are basically stopping the cross edge high frequency based on the strength of the quantization and some other local information and their filter coefficients are fixed. However, as it is observed, in standards like AVS, ringing artifact becomes more serious a problem due to a differently designed quantization scheme. Meanwhile, edges may be blurred because they are high frequency information, and hard to preserve well after quantization. Recovering images from a batch of degraded ones is known as image restoration. Two interesting topics here could attract our attention. The first one the single image coding artifact removal. In this case, only the information inside one image is available. By restricting the variation of the coding distortion, we start from the minimum sum of square error(SSE) problem. Since such problem is ill-posed, we seek help from regularization and design different penalties by generalizing an existing one. These penalties are designed for removing both blocking and ringing artifacts while preserving edges. By applying a gradient descent method, the artifact removal process could be viewed as doing filtering in every iteration. A more detailed algorithm design is presented including both this updating and the stopping criteria of our method. We try to explain why some penalties perform well while some do not. Both PSNR comparison and subjective quality comparison are presented since we are concerned for both of them. Moreover, we have studied how these parameters affect our result. The other topic is how to combine different frames with almost the same contents together to generate this new image. We worked on this for a period of time and Chapter 3 briefly describes the algorithm. It shows that, for the ideal case, combining multiple frames could help in reducing these artifacts. However, in the real situation, because of the complexity of the true motion, such method does not work at all. However, during our study of the motion registration, we found a low-complexity way to do motion registration. The theoretical analysis is given in Chapter 3.3. It is found that such method is useful in some applications.