Nonuniform Haar Wavelet

Wavelet is an offshoot of applied mathematic which develops for overcoming the limitation of Fourier Transform. Since it perfect in math and widely using, it rapidly developed in application. Fourier Transform react the whole character of the signal and function. But in practice, people pay more attention to the characters in part time area of the signal. Although Gabor transform improve the lack of time-frequency analysis in Fourier transform, the width of its window can not be changed if it is chosed.As the development of Fourier analysis, wavelet analysis is not only save the advantage of Fourier analysis,but also make up for the lack of Fourier analysis by multiscale analysis.Wavelet at present has get rapid development, but it often appear that wavelet and its corresponding multiresolution analysis are given on uniform partitions. In this paper we study nonuniform Haar wavelet on bounded interval and rectangle region. This study consists of five chapters, and the main contents of each chapter are as follows.In Chapter 1, we simply summarize the main idea of multiresolution analysis.Outline the process of direct and inverse wavelet transform.In Chapter 2, we introduce uniform Haar scaling function and wavelet function. we also give procedure decomposition and reconstruction.In Chapter 3, we define nonuniform Haar scaling function and nonuniform wavelet on bounded interval.Research one-dimensinal nonuniform multiresolution analysis. We get the algorithm of direct and inverse nonuniform Haar wavelet transformIn Chapter 4,using tensor product we bivariate separable nonuniform Haar scaling function and Haar wavelets on rectangle region.We study it's multiresolution analysis.the algorithm of decomposition and reconstruction are discussed.In Chapter 5,we construct 2D nonuniform Haar wavelets on rectangle region and investigate nonuniform multiresolution analysis. We also obtain the algorithm of direct and inverse nonuniform Haar wavelet transform.