A New Class Of Orthogonal Function Systems And Their Applications In Image And Graphics Processing

With the development of modern digital technology, the study to systems of orthogonal functions has already become the international hot research subject. Based on the advanced achievements in this field, this thesis discusses mainly the systems of orthogonal functions and their applications in image and graphics processing by means of some techniques and methods from image processing, finite element theory and digital geometry etc. This thesis firstly explains the backgrounds, developments, current situations and applications of the system of orthogonal functions, then focus on two new systems of orthogonal functions: U systems and V systems. The new applications of these two classes of orthogonal functions systems are given in this thesis. Based on the classical high dimension systems of orthogonal functions and the construction methods of U system and V system, we also try to generalize the U systems and V systems to higher dimension cases and get some elementary results.The details of the thesis are as follows:1. The summarization of the systems of orthogonal functions: The systems of orthogonal functions can be divided into two types: standard and nonstandard. Nowadays, the standard systems of orthogonal functions include Fourier triangle system, polynomial orthogonal system, Walsh system, Haar system, U systems and V systems. By calssifying these systems of orthogonal functions and their characteristics, the thesis tries to offer relevant information for the new system of orthogonal functions.2. The new applications of U systems and V systems in image and graphics processing:Based on the unique characteristics of U systems and V systems, these two classes of orthogonal functions systems found lots of applications in practical fields. The thesis has used them to commercial bills and polygonal vector graphics. The experimental results show that the proposed methods can resist some common manipulations, such as translation, rotation, scaling and local modification.3. The system of orthogonal functions in the triangular domain: the research about high dimension function systems is mainly using the form of tensor production. How to construct high dimension system of orthogonal functions with non- tensor production is still an open problem. The thesis tries to extend U systems and V systems to two dimensions cases in the base of Walsh system and Haar system in triangular domain. Some elementary results are given.