In this thesis, we have studied the problem of color image regularization, which is a low-level process and is often used as a key pre-processing step in many image processing applications. Most of these applications require that the regularization stage can preserve as much important image features (edges and corners etc.) as possible, while still being able to effectively remove noise and unwanted details. Although there are many existing regularization methods, few of them can produce both efficient noise removal and good edge preservation. To achieve better edge-preserving regularization performance, we have proposed a locally adaptive edge-preserving regularization framework for color images. The basic idea of our proposed framework is to treat edge regions and homogenous regions adaptively by applying different regularization process to them. We proposed a locally adaptive regularization term in Chapter 3, which is better adapted to local edge geometry. Besides that, an automatically calculated adaptive data fidelity term was also proposed to help better preserve edges. Experimental results are presented to show that our proposed framework achieved a good balance between noise removal and edge preservation comparing with other methods. In Chapter 4, we further extended our regularization framework to handle impulse noise by extending a grayscale impulse noise detection method to color images and used together with our proposed regularization framework. We also considered the case of mixed impulse and Gaussian noise by proposing an innovative two-phase framework inspired from color image inpainting. Finally, we proposed to use a semi-local Zernike moments in our regularization framework to get more robust performance for highly-noisy images. Possible extension of our proposed framework to the nonlocal version was also discussed and suggested as future research directions.