Data-driven tight frame construction and image denoising

Sparsity based regularization methods for image restoration assume that the underlying image has a good sparse approximation under a certain system. Such a system can be a basis, a frame, or a general over-complete dictionary. One widely used class of such systems in image restoration are wavelet tight frames. There have been enduring efforts on seeking wavelet tight frames under which a certain class of functions or images can have a good sparse approximation. However, the structure of images varies greatly in practice and a system working well for one type of images may not work for another. This paper presents a method that derives a discrete tight frame system from the input image itself to provide a better sparse approximation to the input image. Such an adaptive tight frame construction scheme is applied to image denoising by constructing a tight frame tailored to the given noisy data. The experiments showed that the proposed approach performs better in image denoising than those wavelet tight frames designed for a class of images. Moreover, by ensuring that the system derived from our approach is always a tight frame, our approach also runs much faster than other over-complete dictionary based approaches with comparable performance on denoising.