Framework For Development Of The Theory And Wavelet Applications In Signal Processing

Wavelet analysis is a useful signal time-frequent tool after Fourier transform,it has time-frequent localizing and multiresolution analysis ability.The nature of wavelet analysis is to choose several coefficient of transformation after the transform,on which the compression ratio and calculation speed can be improved greatly.Fame theory derives from the signal processing.In 1952, Duffin and Shaffer introduced the concept of frame for Hilbert spaces in order to study some deep problem in non-harmonic Fourier series.Because the redundancy of the frame,the signal can be precision synthesis.So it become the hot problem of the learner.In 2004,weighted frames were introduced by I.Bogdanova and P.Vandergheynst to get a numerically more efficient approximation algorithm.Since this concept was used just as a tool for spherical wavelets,it was not discussed in full detail there.In 2006,Peter Balazs discuss some of its related properties,today these theories have been widely used in wavelet analysis,signal analysis,image processing,numerical calculation, Banach space theory and other theories in the field of application.This paper mainly talks about the wavelet transform theory,frame theory,weighted frame theory and a method for image edge detection based on wavelet transformation and simple singularities.On the basis of the results,this paper generalizes some results and gives some new results.This paper is composed of four parts:The chapter 1 is an introduction which summarizes the emergence,development of wavelet analysis and frames theory.The second chapter describes several important wavelet transform and multiresolution analysis theory.The third chapter divides into three parts,the first and second parts introduced the definition of the Bessel sequence, frame and their basic nature.The frame is a special Bessel sequence,which extends the Parseval equality of orthonormal basis in Hilbert space to the two-sided inequality in general sequence.The definition,basic nature of weighted frame were discussed in the third part,its stability was introduced by specific examples,and then give the contact between frame operator and frame multipliers.Chapter 4 presents the applications of wavelet transform in image edge detection.Along with the rapid development of digital image gathering and processing technology,image has become an important way of people to gain information.Image edge information reflects the most valuable information of the image.The Edge detecting is one of the most important and classic topic in the image processing and computer vision.As wavelet transform has goodlocal quality and multi-scale identity,it can satisfy the need of edge detection in multi-scales.Detecting edge using wavelet transform is recognized an efficient way.This chapter combine the wavelet transform and WISDOW method to detect the edge of image.Simulation results show that it is effect to extract the medical image edge.