A semi-local paradigm for wavelet denoising

Wavelet denoising methods have been proven useful for many one- and two-dimensional problems. Most existing methods can in principle be carried over to three-dimensional problems, such as the denoising of volumetric positron emission tomography (PET) images, but they may not be sufficiently flexible in allowing some regions of an image to be denoised more aggressively than others. In this paper, we propose a semi-local paradigm for wavelet denoising. The semi-local paradigm involves the division of an image into suitable blocks, which are then individually denoised. To denoise the blocks, we use our modification of the generalized cross validation (GCV) technique of Jansen and Bultheel to choose thresholding parameters; we also present risk estimators to guide some of the other choices involved in the implementation. Experiments with phantom PET images show that the semi-local paradigm provides superior denoising compared to standard application of the GCV technique. An asymptotic analysis demonstrates that, under some regularity conditions, semi-local denoising is asymptotically consistent on the logarithmic scale. The paper concludes with a discussion on the nature of semi-local denoising and some topics for future research.