Multiscale sharpening and smoothing in Besov spaces with applications to image enhancement

In this paper we use multiscale characteristics of wavelet decompositions a nd 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 u nsharp masking, while using the smoothness parameter of in the Besov space B-alpha(q)(L-p) to provide a unifying framework for the two operations shar pening and smoothing. As a result multiscale smoothing or sharpening is def ined as a switching between different smoothness spaces. The degree of shar pening or smoothing is linked to the Besov space parameter cu. Combined wit h wavelet denoising the nonlinear image enhancement in Besov spaces via wav elets provides a tool for high-quality low-cost image processing. For the e xample of a document, that has been blurred by a scanning process, we demon strate how information on the smoothing properties of an input device combi ned with an image model provide enough information to determine the right a mount of multiscale sharpening, i.e., for inverting the smoothing process, that is suitable to obtain a deblurred image. Multiscale sharpening then le ads to a switching from a Besov space with large degree of smoothness to th e one with a lower degree of smoothness. This technique combined with wavel et denoising provides visually pleasant images with crisp text. (C) 2001 Ac ademic Press