M-band Linear-phase Orthogonal Filter-banks And Wavelets

Although multi-band wavelet has been widely studied,the design of multi-band wavelet with linear phase is still a great challenge,especially when the number of channels of multi-band wavelet is relatively large.In this paper,we mainly focus on a study the design of two kinds of multi-band orthogonal wavelets,and the first is real linear phase multi-band orthogonal wavelet,the other is multi-band orthogonal complex wavelet,in which multi-band orthogonal complex wavelet includes symmetric orthogonal wavelet and linear phase orthogonal complex wavelet.This paper is organized as follows: In the first chapter,we review the basic concepts of multi-band orthogonal wavelets,the development history and the background of applications.In the second chapter,we mainly discuss the semi-rank factorization theory of real linear phase multi-band orthogonal wavelets.Firstly,we study a particular decomposition form of vector space,obtained two individual orthogonal projection matrices,and examine the fundamental properties of order-1 paraunitary blocks constructed by these two kinds of orthogonal projection matrices.Secondly,based on these two kinds of orthogonal projection matrices,no matter the number of channels of filter banks is even or odd,we proposed a complete and minimized semi-rank factorization theory of linear phase para-unitary filter banks,and a direct connection between linear phase and orthogonal projection matrix is established.Especially,for odd band linear phase filter banks,the factorization is brand new.Finally,we discuss the sufficient and necessary conditions for the para-unitary filter banks have first or second order regularity under the semi-rank factorization.In the third chapter,we mainly discuss the design and applications of multi-band wavelet filters.In the semi-rank factorization,there are two kinds of fundamental matrices: the starting matrix and the orthogonal projection matrix .The orthogonal matrix plays a critical role in the parameterizations of these two kinds of matrices.So,first of all,we discuss the problem of parameterizations of the orthogonal matrices by the Givens rotation matrix.Secondly,according to the application,we design 3,4,5,6,7,8-band wavelet filter banks with linear phase.Finally,we test the performance of 8-band wavelet in image compression coding.The experimental results show that the optimal 8-band wavelet designed in this paper has excellent performance.In Chapter 4,we extend the real linear phase multi-band orthogonal wavelet to the complex form.In the complex domain,according to the symmetry of the filter,two types of filter banks can be obtained.One is called symmetric multiband orthogonal complex wavelet.In the filter banks of this kind of wavelet,the real and imaginary parts of each filter are symmetric or antisymmetric at the same time and have the same symmetry centre.Another type we called linear phase multi-band orthogonal complex wavelet.In this type of filter banks,each filter is either conjugate symmetric or conjugate antisymmetric,so it has a linear phase.In this chapter,we also discussed the semi-rank factorization of these two types of multi-band orthogonal complex wavelets and the structures of the corresponding matrices.We find that there are no other types of orthogonal complex wavelets with linear phase in the complex domain except the Haar-type wavelets.Finally,we designed 2,3,4-band orthogonal complex wavelets with symmetry and 3,4-band orthogonal complex wavelets with linear phase.Chapter 5 is the last chapter of this paper.In this chapter,we summarized the main work done in this paper.Secondly,we briefly discuss the problems to be solved and the challenges to be faced in the future.