Multiscale Sharpening and Smoothing in Besov Spaces with Applications to Image Enhancement

In this paper we use multiscale characteristics of wavelet decompositions and their relationship to smoothness spaces such as Besov spaces to derive a framework for smoothing and sharpening of signals and images. As a result, we derive a multiscale generalization of traditional techniques, such as unsharp masking, while using the smoothness parameter α in the Besov space B α q ( L p ) to provide a unifying framework for the two operations sharpening and smoothing. As a result multiscale smoothing or sharpening is defined as a switching between different smoothness spaces. The degree of sharpening or smoothing is linked to the Besov space parameter α. Combined with wavelet denoising the nonlinear image enhancement in Besov spaces via wavelets provides a tool for high-quality low-cost image processing. For the example of a document, that has been blurred by a scanning process, we demonstrate how information on the smoothing properties of an input device combined with an image model provide enough information to determine the right amount of multiscale sharpening, i.e., for inverting the smoothing process, that is suitable to obtain a deblurred image. Multiscale sharpening then leads to a switching from a Besov space with large degree of smoothness to the one with a lower degree of smoothness. This technique combined with wavelet denoising provides visually pleasant images with crisp text.