A Study On The Two-direction Wavelets And The Two-dimensional Of Four-direction Wavelets

Wavelet analysis develops from harmonic analysis，it gained wide attention assoon as being propounded.The construction of wavelet is the essential issue of wave-let analysis. It is widely known that two scaling refinly equations plays a key-role inthe construction and application of wavelet. Scaling function,which with a non-nega-tive mask,is often applied in engineering, and a lot of people have made researcheson it.In addition, he gavethe constructing algorithm of high approximation order andregularity of refinable functions. Two-direction wavelets is a much more common si-tuation compared to wavelet, and it makes a great contribution to the development ofwavelet theory. There are more and more people begin to research on it. Inspired byTwo-direction wavelets and in the hope of popularizing it.I introduced the of two-di-mensional four-direction scaling function and corresponding the two-dimensional fo-ur-direction wavelets, and made an in-depth analysis on the construction of two-dir-ection wavelets.The innovation of this paper includes:In Chapter Two,we mainly studied the two-dimension of four-direction withadilation.Based on two-direction wavelets theory, a four-direction wavelets with tec-tonic scale a is constructed by tensor product.Moreover, MRA, orthogonal and bio-rthogonal of this two-dimension of four-direction wavelets are given,relevant proper-ties of this two-dimensional wavelet is showed in the conclusion (This part has beenpublished in Jilin University Journal).In Chapter There,we mainly constructed the two-direction with arbitrary matrixdilation orthogonal nonseparable wavelets packet, gave the definition of the two-dir-ection with arbitrary matrix dilation orthogonal nonseparablewavelet packet and themulti-resollution of matrix dilation, acquired with arbitrary matrix dilation orthogon-al condition and We obtain related properties and conclusions of two-direction witharbitrary matrix dilation orthogonal nonseparable wavelet packets.(This past hasbeen published inMATHEMATICS IN PRACTICE AND THEORYV). In Chapter Four,we mainly constructed the two dimensional of four-directionwavelets packets,gave multi-resolution definition of the two dimensional four-direction wavelets and orthogonal condition and biorthogonal condition of the two di-mensional of four-direction bi-orthogonal wavelets package,and acquired the relati-ve qualities and conclusion of he two dimensional four-direction wavelets packets.(This past has been published in Hubei University Journal).In Chapter Five,we mainly studied the two-dimension of four-direction subsp-aces Shanon type sampling theorem. Based on multi-resolution analysis of the two-dimension of four-direction.After,constructing algorithm for to the uniform andnon-uniform of Shanon sampling theorem in the two-dimension of four-direction.In Chapter Six,we mainly studied two-direction combined vector-valued waveletpackets and wavelets base of dilation matrix, combined wavelet is a mixture of diffe-rent wavelet function which are on the same level.The results suggests that multidim-ensional of wavelet is much more flexible.On the basis of the combining waveletbasies and gave the constructing algorithm for two-direction combined dilationmatrix vector-valued wavelet packets and obtained someproperty of two-direction c-ombined dilation matrix vector-valued wavelet packets.