Construction and application of symmetric tight wavelet frames

Abstract

To ascertain the most desirable application properties, we extensively studied variously generalized wavelets and successfully used them in many applications. Without introducing other types of wavelets in this thesis, we are particularly focused on construction and applications of symmetric tight wavelet frames (STWFs), which are derived from a refinable function. The notion of tight wavelet frames could be considered as a natural generalization of orthonormal wavelets if redundancy is introduced into the wavelet system. By allowing redundancy, we gain the necessary flexibility to achieve such properties as symmetry and, more importantly, the short support and high vanishing moments of compactly supported wavelets. In addition, redundancy allows for the approximate shift invariance behavior caused by the dense time-scale plane. We also show some applications of STWFs for image processing, particularly image fusion and image denoising, and we demonstrate the possibility of using an STWF-based approach for such applications.