Wavelet based image decomposition by variational functionals

摘要

We discuss a wavelet based treatment of variational problems arising in the context of image processing, inspired by papers of Vese-Osher and Osher -Sole-Vese, in particular, we introduce a special class of variational functionals, that induce a decomposition of images in oscillating and cartoon components. Cartoons are often modeled by BV functions. In the setting of Vese et.el. and Osher et.al. the incorporation of BV penalty terms leads to PDE schemes that are numerically intensive. We propose to embed the problem in a wavelet framework. This provides us with elegant and numerically efficient schemes even though a basic requirement, the involvement of the space BV. has to be softened slightly. We show results on test images of our wavelet algorithm with a B_1~1(L_1) penalty term, and we compare them with the BV restorations of Osher-Sole-Vese.