The Theoretical Reserch And Application On Haar Wavelet

The wavelet analysis theories and methods are in developing and far from maturation. Wavelet analysis and its application have great potentialities in many applied fields of natural science.From its mathematical theory,it is another canon after the Fourier transformation of the perfect combination of Pure Mathematics and Applied Mathematics,which is called the "mathematical microscope".From a purely mathematical point of view,wavelet analysis is the crystallization of the hard work of harmonic analysis in the past fifty years.Now the wavelet analysis theory has been a popular subject with a very wide range in scientific research and the application of engineering technology.Wavelet transformation is a partial transformation in space(time) and frequency,so it can extract the information from a signal efficiently.It can also precede Multi-Resolution analysis to a function or a signal by dilation and translation,Which solve many questions that cannot be solved by Fourier transformation.There are many wavelets which have been discovered and put into application,such as Haar wavelet,Symlets wavelet,Morlet wavelet and so on.For the example of Harr wavelet,this paper studied its Multi-Resolution analysis and the decomposition and reconstruction of non-uniform Haar wavelet,studied the a-scale Haar wavelet further.And then the author gave the definition and properties of non-uniform Haar wavelet in irregular region.Finally,the author got two improved algorithm on the digital watermark by the application of wavelet analysis.This paper consists of four chapters.Chapter 1 introduces the process of the rise and development of wavelet analysis,and the similarities and differences between wavelet analysis and Fourier transformation,then the author introduces the main content of the paper.Chapter 2 mainly lustrates the properties and theories of the wavelet analysis and Multi-Resolution analysis,and then we the author gets some improved theories.Chapter 3 discusses the definition and properties of Haar wavelet in irregular region,and studies their decomposition and reconstruction and obtains some meaningful results.Chapter 4 discusses the wavelet analysis and Multi-Resolution analysis theories.Using the modern information conceal technology,the author obtains some practical applicable results through processing the signal,and then precedes the security test.