The Construction Of Wavelet Frame Based On Multiresolution Algorithms

When using wavelet transform in engineering applications, people often avoid the process of constructing wavelet bases, but use the existing ones in multiresolution algorithms, so the construction of wavelet bases becomes people’s pursuits in a period of time.This paper mainly studies the constructions of uniform symmetric bi-frames, and the main purpose is to use this type of filter banks to solve the problem of special points in surface multiresolution processing. For the whole construction process, the parametric representation of filter banks plays a key role in the whole construction process, this paper uses template form to realize the multiresolution algorithms, and starts by the representation of the most simple template, and complexes the simple template form by alternately using lifting scheme, and the more complexity of template, the greater probability of getting ideal filter banks. This paper gets the parametric representations of filter banks through deriving the template to the form of multiresolution algorithms. According to conditions of the mixed unitary extension principle, we could construct a series of equations associated with filter banks, and get examples of parameters through solving these equations, but for bi-frames with three generators, the structure of equations may be very complex, so that elimination method can’t directly solve them, in order to solve this problems, this paper firstly instantiates a part of parameters, and then simplify the equations by elimination method to simplify the equations, so repeatedly until all the parameters are instantiated, which reduces the amount of calculation in the process to a certain extent.For the part of constructing multiresolution template, the paper is divided into three kinds of situations to discuss, so as to genetating three types of bi-frames, and gets the according symmetric proof for each type of bi-frames, and obtains an instance of each type. At last, this paper also gives a brief summary of the split bregman algorithm based on framelet transform, and shows the according simulation experiments of using the algorithm into image deblurring, and compares the experimental results of each type of filter banks.