A Method For Implementing Wavelet Transform On Computer

Wavelet transform is the improvement of Fourier transform, which has been widely used in engineering fields. This thesis discusses a method for implementing wavelet transform on computer. Our main work includes:(1) Introduce Fourier transform and short time Fourier transform with their drawbacks so as to put forward the concept of wavelet transform. We also briefly introduce the self-adaptability of wavelet transform in time-frequency analysis and some wavelets that are commonly used.(2) Study the implementation of "continuous wavelet transform" on computer. First,we introduce a method that has some flaws,then we put forward a better method based on "pre-sampling" and "integral values of wavelet". We also illustrate the "zero-hole phenomenon" during the process of computing.(3) Due to the redundancy of "continuous wavelet transform", we discuss the discretization of wavelet transform and bring up the concept of "wavelet transform on discrete dyadic grids".(4) Based on Multiresolution Analysis, we deduce Mallat's Algorithm for decomposition and reconstruction, providing theoretic basis for implementing "wavelet transform on discrete dyadic grids".(5) Express Mallat's Algorithm in the form of discrete convolution by deduction so as to connect wavelet transform with the concept of filter banks. As an application of Mallat's Algorithm, we discuss the way of computing the values of scaling function and wavelet function on discrete dyadic points.(6) Discuss the problem of "initialization" when implementing "wavelet transform on discrete dyadic grids" with filter banks, and the result proves our solution's advantage.(7) Introduce "discrete sequence wavelet transform" and compare its computational complexity with "continuous wavelet transform".