Research And Application Of Parseval Frame Wavelet In Multiresolution Analysis

Frame multiresolution analysis theory is an extension of orthogonal multiresolution analysis theory.This paper mainly studies the frame multiresolution analysis wavelet in L2(Rd).Firstly,we focus on the operator parameterization of multiresolution analysis of L2(Rd)frame wavelets.The results of operator parameterization of all orthogonal wavelet sets in L2(R)and operator parameterization theory of orthogonal multiresolution analysis wavelet sets show that all orthogonal multiresolution analysis wavelet sets can be parameterized by a set of unitary operators satisfying a certain commutative property.After deeply studying the theoretical results,we extend this idea to the frame multiresolution analysis wavelet set in L2(Rd)(d?1),and get the result that the frame multiresolution analysis wavelet set can be parameterized by a coisometric operator satisfying some commutative property.Secondly,a new high-dimensional frame multiresolution analysis wavelet is constructed by using the high-dimensional frame wavelet multiplier,and the image edge is processed by using the new wavelet.The specific research contents of this part are as follows:Let a be any 2 × 2 real dilation matrix,and let ? be its Fourier transform for any A-dilation wavelet ?.If the inverse Fourier transform of f(?)is still A-dilation wavelet for any A-dilation wavelet ?,then the measurable function f is called A-dilation wavelet multiplier.Under the condition that a is a 2 × 2 integer matrix and|det(A)|=2,we construct(?)and other six A-dilation matrix Haar type and Shannon type multiresolution analysis frame wavelets,and improve the image edge detection algorithm by using the constructed wavelets,and select the suitable ones for image edge detection through programmed experiments the type of high-dimensional wavelet measured.After the improvement of the traditional image edge detection method based on the wavelet transform algorithm,four methods of the original image edge detection are tested and selected,and finally the optimal wavelet type suitable for image edge detection is selected.What we get are:(1)Wavelet detection with phase:edge detection of a-expansion wavelet image with phase is the best.(2)The detection effect of different dilation matrix wavelets:(?)two expansion matrix wavelets have the best edge detection effect.(3)For the detection effect of two kinds of wavelets with the same dilation matrix:the edge detection effect of Shannon type frame wavelet image is the best.