The Lifting Construction Of P-Band Multivariate Wavelet Bi-Frame Based On Neville Filter

With the advent of the information age, a large number of pictures and videos and other high-dimensional information have gradually penetrated into people's daily life. There is an urgent need for us to effective deal with those massive amounts of information. Although a large number of scientific researchers have made some achievements in this field, there are still challenges ahead. Wavelet frame is an effective tool to deal with high dimensional information, which has excellent properties of approximate translation, redundancy and time frequency localization analysis. Therefore, it is of great significance to construct the wavelet frame with good performance according to the application requirement.Most of the existing methods of constructing multivariate wavelet frame are special cases of special sampling matrix, and the computational complexity is high.While few research achievements of the general case of arbitrary sampling matrix are found in past years. That is to say, the research of the multivariate wavelet bi-frame has shown few results. It is difficult to fully reflect the advantages of multivariate wavelet frame in multi-dimensional signal processing. So it is very urgent and difficult to study the construction method of the multivariate wavelet bi-frame under the general sampling matrix.In order to get more multivariate dual wavelet frame for application requirements, this paper uses the lifting scheme to construct it. The wavelet construction method based on the lifting scheme has the following advantages: no necessity for spectral analysis tools, no dependence on the concept of translation and dilation, in-place calculation, faster implementation of wavelet transform. At present,the lifting scheme has been used to realize the construction of univariate and multivariate biorthogonal wavelet and univariate dual wavelet frame. But the lifting construction of the multivariate wavelet bi-frame has in its early stage, and there is no great improvement in this field. Hence, in the perspective of combination, in this paper, we take advantages of fast transformation of the multivariate dual wavelet frame and the basic principles of lifting, and then propose a new strategy for this problem. It is obtained for the first time that the lifting construction of the multivariate wavelet bi-frame can be obtained with any band and arbitrary sampling matrix, that is to say, the lifting construction of the multivariate wavelet bi-frame is equivalent to thesolutions of a series of lifting operators. Then, by studying the relationship between the lifting operator and the vanishing moments of the dual wavelet frames, we obtain the solution of the operator. The solution of the lifting operator is equivalent to the computation of the Neville filter. It is worth noting that only a portion of the lifting operator can pass the Neville filter solved directly, and another part of the lifting operator is obtained by combining the existing wavelet frames which satisfy the vanishing moment properties. Finally, the symmetry of them is analyzed in the case of construction, and a series of multivariate dual wavelet frames with elegant properties are obtained.The construction method of this paper has high generality, which not only can be used to construct the dual wavelet frame under multidimensional case, but also can be used to the case of one-dimensional. In this paper, we unify the construction method of univariate and multivariate dual wavelet bi-frames, and make the biorthogonal wavelet as the special case in our strategy.