Adaptive Wavelet Research

Wavelet analysis has attracted many researchers' attentions in recent twenty years. It becomes one of the important tools in harmonic analysis and signal analysis fields. From the point of view of mathematics, wavelet is in fact in accordance with specific space called wavelet basis function (usually have a distinct physical meaning) on the mathematical expression of the start and approximation. As a fast, efficient, accurate approximation methods, promote the wavelet analysis in the field of harmonic analysis Fourier analysis of the important development. And Fourier transform from a triangular base function compared wavelet function for the most rapid attenuation, fully smooth, energy mainly concentrated in a local function. The wavelet analysis of the "adaptive nature" and "nature of mathematical microscope", is widely used in basic science, applied science, particularly information science and the analysis of the signal in all aspects. It has not only become a mathematician to study hot spots, but it is also a cause of physicists, biologists, engineers and other workers in the field of science and extensive attention. Wavelet analysis of the theoretical research and practical application of the depth and scope is rapidly expanding.For the adaptive wavelet, it is expected that the wavelet is limited suppoting, so that the Mallat algorithm more fast; It is expected that the wavelet is smooth so that we can simulate and analyze the signals more precisely; it also is expected that the wavelet which is localized well in the time-domain and frequency-domain so that it can approximate the signals more precisely. Daubechies did outstanding contribution for these.It is expected, when we process the signals using wavelets, we can find the best wavelet to do it for any signals or a class of signals, so that it can be used to represent signals better.An alternative approach is to seek a signal-adapted wavelet that is match to a particular signal or to a narrow class of signals. In recent years, several authors have developed systematic and efficient design algorithms for signal-adapted filterbanks. In particular, for a given signal of interest, we will address the problem of finding the wavelet (or, more precisely, the scaling function) that minimizes the squared error between the signal and some finite resolution wavelet representation of itself.In this paper, we develop and improve an efficient method for selecting the optimal orthonormal wavelet that is matched to a given signal in the sense that the squared error between the signal and some finite resolution wavelet representation of it is minimized.This paper is composed of four parts: The chapter 1 is an introduction which summarizes the emergence, development of wavelet analysis, the research status about the wavelet transform technology in the image coding and simply introduces the adaptive wavelet theory.The chapter 2 firstly presents the wavelet analysis and the basic nature of multiresolution analysis, which is raised by Mallat in 1998 and also named Multiscale analysis. It is one of important concepts in wavelet analysis. It researchs the function or multiscale description of the signals from the function spaces. The role of Multisolution analysis is decomposed the signals into the part of different spaces. Otherwise, it also supplies a unity frame wavelet and supply the fast algorithm of decomposition and reconstruction of the wavelets. Then, the second part introduces the algorithm and the basic nature about it in detail. In the third part, some natures of approximation are involved and do the work for the adaptive wavelet algorithm.The chapter 3 firstly presents the basic knowledge which can implement the algorithm of this paper. Then the part 2 develops the design algorithm. The original algorithm about the design algorithm of the adaptive wavelet is involved, but we found it existing insurmountable difficult. So we transform it into the simply one. In the end, the optimal orthnormal wavelet can be get using the optimal algorithm.The first part of Chapter 4 we cited two examples of one-dimensional signal comparison with other methods; second part we use the optimal choice from the two-dimensional image wavelet decomposition and reconstruction. Finally, we have this way of a summary.