Properties And Application For Vector-valued Gabor Frames And Frame Wavelets

The notion for the frame is the generalization of Riesz basis,and it is of some prop-perties of Riesz basis.The frame theory is an important part of the wavelet analysis.It not only promotes the intensive study of functional analysis and operator theory,but also promotes the development of non-linear approximation theory.By means of operator algebra theory,matrix theory,frame multiresolution analysis and time frequency analysis,we mainly study construction algorithms for vector-valued Gabor frames and wavelet frames and their characteristics in this paper.A zero-watermarking algori-thm associated with M-band orthogonal wavelets is proposed.Several novel results are obtained.Firstly,the research background for wavelet analysis and frame theory is narrated,and the important research results having been achieved on frame theory are summarized.The basic concepts and properties for the typical frames such as Gabor frames,wavelet frames and shearlet frames are outlined.The research background and achievement for digital watermarking are overviewed.Secondly,according to the basic theories for the frame,the characteristic of vector-valued Gabor frames in Hilbert space is investigated.A necessary and sufficient condition for the family of vector-valued functions,{E_mbT_nag}_m,n?Z^d constituting vector-valued Gabor frames for L²?R^d,C^s? and vector-valued Gabor tight frames for L²?R^d,C^s? are provided.The sufficient condition for a family of functions {E_mblT_nal?_l,m,n?Z,l=0,···,r-1}to be an irregular multi-generator Gabor frames is proposed.Thirdly,vectorvalued frame wavelets are introduced.The concept for interval minimum-energy vector-valued frame wavelets are introduced.A sufficient and necessary condition for interval minimum-energy vector-valued frame wavelets is established.A constructive method for a pair of frame wavelets that originate from vector-valued scaling functions is proposed by using the mixed oblique extension principle.Finally,based on visual cryptography algorithms and singular-valued decomposition method for matrice combining with compactly supported orthogonal wavelets possessing good characteristics,a zero-watermarking algorithm associated with M-band orthogonal wavelets is provided.By virtue of fully scrambling of the original images and implementing singular-valued decomposition and M-band orthogonal wavelet transform,we obtain image characteristic information.The zero-watermarks are generated by combining with the image characteristic information and meaningful guests' watermarks.