MRA-frame High-pass Filter Multiplier Matrix

Wavelet frame can overcome the shortcomings of orthogonal wavelet, and increases a appropriate redundancy in it. Wavelet frame not only retains all the property of orthogonal wavelet, such as time-frequency localization and shift invariant, except orthogonality, but also it can put the smoothness, compact support, symmetry (or antisymmetry) together in practice. Wavelet frame has a better stability than orthogonal wavelet in signal reconstruction. And wavelet frame is easier to design than orthogonal wavelet.This paper first introduces the multi-resolution analysis and wavelet frame, and then introduces the frame(MRA-frame) constructed by MRA, whose decomposition and reconstruction algorithm is very similar to that of the wavelet (MRA-wavelet) constructed by MRA. This simple algorithm is just a layered iteration (similar to Mallat algorithm). The frame multiplier is introduced in this paper, including frame multiplier corresponding to frame with single generator and Fourier multiplier matrix corresponding to frame with some generator, which all can construct a new wavelet frame from an existing one.Based on the advantage of MRA-frame and the thought of matrix Fourier multiplier, limiting matrix Fourier multiplier within the scope of MRA-frame, obtains the concept of matrix high-pass filter multiplier. Matrix high-pass filter multiplier can get new MRA-frame from an existing one. And we obtain the sufficient condition of matrix high-pass filter multiplier. We obtain the condition which can make the results of addition and multiplication of different symmetry types Laurent polynomial is still symmetric (or antisymmetric) Laurent polynomial. According to this condition of Laurent polynomial, we can construct matrix high-pass filter multiplier with particular symmetry types, which can get new symmetric (or antisymmetric) MRA-frame from an existing one, which is the content of algorithm 1. Then we give a few examples by using algorithm 1, which are all about constructing new symmetric (or antisymmetric) MRA-frame with two (or three) high-pass filters from an existing one. Finally, using the simplest image processing-image denoising, illustrates that the MRA-frame constructed by matrix high-pass filter multiplier has certain use value in signal processing.