Good signal representation and the corresponding signal processing algorithms lie at the heart of the signal processing research effort. Since the 1980's wavelet analysis has become more and more a mature tool in many applications such as image compression due to some key advantages over the traditional Fourier analysis. In this thesis we first develop a wavelet-based statistical framework and an efficient algorithm for solving the linear inverse problems with application to image restoration. The result is an efficient method that produces state-of-the-art results for such problems and has potential further applications in other areas. To overcome the issues such as the blocking artifacts in using orthogonal wavelets, we next investigate the design issue of more flexible basis representations based on frames. In particular, we develop a quasi image rotation method that is based on pixel reassignment and hence retains the original image statistics. When combined with translation operators, this method provides very efficient and desirable frames for image processing. Given a frame, due to the large number of redundant basis functions in it, how to efficiently implement a frame-based algorithm is the key issue. We show this through the example of optimal signal denoising in the presence of added zero-mean white noise. We show that the optimal solution exists yet the computation toward the solution is very heavy. We develop a framework that allows for fast approximations to the optimal solution and has clear physical interpretation. This method is in essence different from the other various approximate approaches such the basis pursuit and has applications in other areas such as image segmentation. We also develop a complexity regularized iterative algorithm for getting sparse solutions to the frame-based signal denoising problem.