DISCRETE FRAMES AND TIGHT FRAMES FOR SPARSE IMAGE REPRESENTATION

作     者：YUFEI ZHAO 

作者单位：NATIONAL UNIVERSITY OF SINGAPORE 

学位级别：博士

导师姓名：Hui Ji

授予年度：2016年

摘      要：In recent years, sparse approximation has played a fundamental role in many signal processing areas. The sparsity-induced regularization methods for image recovery are implemented based on the assumption that the underlying images can be sparsely approximated under the given system. Herein, over-complete systems, especially tight frames, possess advantages in sparse image representation and have been widely used in applications. In the first part of this dissertation, we focus on constructing discrete (tight) frames using Gabor atoms to meet the needs for sparse image modeling. Gabor systems have many advantages in sparse representation, for example accurate local time-frequency analysis and strong orientation selectivity. However, the discretiza- tion of continuous Gabor frames is non-trival in the sense that the resulted discrete system may lose the frame property, as well as fast implementation algorithms. Mo- tivated by these, we study the general theory of discrete Gabor frames by developing Gramian and dual Gramian analysis in C N. Consequently, we derive a necessary and sufficient condition for discrete tight Gabor frames and construct two classes of discrete tight Gabor frames as examples. Further, to remove the non-zero DC (di- rect current) offset, we revise the tight Gabor frame to Gabor induced frames with closed-form dual frames and the decomposition and recontruction processes can be implemented via filter bank based fast algorithms. The orientation selectivity of the resulted Gabor induced frame is optimal, i. e. the associated filters provide all the possible directions defined on discrete uniform grid. A weakness of the Gabor system is that it lacks the multi-scale property since all its atoms are of fixed size. One way to solve this problem is to consider multi-scale Gabor frames composed of several Gabor frames with windows of various lengths. The other way is to construct tight frame with both Gabor and MRA structures. Specifically, we take a set