Wavelet transforms and PDE techniques in image compression.

摘要

In this thesis, we propose using Partial Differential Equation (PDE) techniques in wavelet image processing to reduce edge artifacts generated by wavelet approximations. We have been exploring in two directions: the first is to apply the main idea of the well developed ENO schemes for numerical shock capturing to modify standard wavelet transforms so that a fewer number of large high frequency coefficients are generated. Therefore less distortion near edges is created when the coefficients are thresholded. Another direction is to use minimization techniques, in particular the minimization of total variation (TV), to select and modify the retained standard wavelet coefficients so that the reconstructed images have less oscillations near edges. Numerical experiments show that both approaches improve the reconstructed image quality in wavelet compression and in denoising.