Constructions And Properties For Two-direction Vector-valued Wavelets With Mulit-scale

Wavelet analysis has been developing into a branch of mathematics on the basis ofFourier analysis and functional analysis. Vector-valued wavelets belong to wavelettheory, and is widely used in digital films, doppler image, vector signal processing,multi-channel image transmission and so on. Two-direction vector-valued wavelet hasmore flexibility for the selection of filters than multiwavelets. As we known, two-scalerefinable equation plays a important role in construction and applications of wavelets.The scaling vectors functions with nonnegative masks play a very bases role inengineering and technology. Beginning with two-direction vector-valued refinablefunction with two-scale, the construction of orthogonal two-direction vector-valuedwavelets with two-scale is researched by means of two-direction vector-valuedmuilresolution analysis and matrix theory. The properties for biorthogonal two-directionvector-valued bivariate wavelet packets are characterized.Firstly, the research background of the thesis is summarized. The developmentprocess of wavelet analysis and some basic theories are outlined.Secondly, orthogonal two-direction vector-valued wavelets with multi-scale isresearched. The concept of two-direction vector-valued multiresolution analysis isintroduced. The two-direction vector-valued wavelets and wavelet packets with multi-scale are presented. We present a construction algorithm of compactly supportedorthogonal two-direction vector-valued wavelets by means of the matrix theory and thetime-frequency analysis method. Properties of a class of orthogonal two-directionvector-valued wavelet packets are characterized. Lastly, we research the construction for biorthogonal bivariate two-directionvector-valued wavelets. We present a procedure for constructing the biorthogonalbivariate two-direction vector-valued wavelets by virtue of matrix theory. We alsoinvestigate property of biorthogonal bivariate two-direction vector-valued waveletpackets by means of operator theory and integral transform theory. Biorthogonalrelationship formule concerning the two-direction vector-valued wavelet packets arepresented. Moreover, Riesz bases ofL2(R2, C n)is constructed from the biorthogonaltwo-direction vector-valued bivariate wavelet packets.