Wavelet Approximation and Image Restoration

Abstract

In this thesis mainly focused on the wavelet approximation theory and some applications of wavelet frame regularized method to image processing problems. The wavelet tight frame decomposition can provide sparse representation of piece-wise smooth images. Moreover, the coefficients of wavelet decomposition can provide good approximation to underlying solutions and their derivatives in smooth pieces partitioned by sharp edges. Therefore, using linear approximation of low-pass coefficients and sparse non-linear approximation of wavelet coefficients by thresholding operator, the general wavelet frame based image processing methods can recover the noise-free images with smooth regions and sharp edges.

First part of this thesis applied the wavelet tight frame was to two types of highly ill-posed image restoration problems: blind image inpainting and computed tomography (CT) image reconstruction. The second part of this thesis proved that the coefficients of wavelet decomposition can form quasi-projection operators to approximate the smooth functions and their derivatives with arbitrarily high approximation order, which demonstrates that most wavelet coefficients in smooth regions are usually close to zero in approximation sense.