Complex Wavelet Framework, The M-scale Complex Wavelet And The Dual Tree Complex Wavelet Constructed

Since the 1980s, wavelet analysis theory has drawn many attentions at home and abroad, and became a popular subject in scientific research. Now, Wavelet analysis has widely used in signal processing,image processing,pattern recognition,seismic exploration,CT imaging and many multitudinous discipline and related technology research. Although classic wavelet theory has developed very perfect in the early 1990s, it still exists defects in the practical application:(1) The direction selective poor. Only in horizontal, vertical, diagonal line direction, but the detail can't be recognized in positive and negative direction of the 45°. (2) Real discrete wavelet transform lacks of shift invariance in specific sampling process. Signal small displace-ment can cause coefficient unknown change. In order to make up the insufficiency of discrete real wavelet, N. G. Kingsbury forwards the dual tree complex wavelet (DT CWT) concept in 1998, it can overcome DWT defects.Focusing on complex wavelet transform in studied, based on deep research to complex wavelet theory, we construct complex wavelet frame, compactly supported M scale symmetric orthogonal complex wavelet and dual tree complex wavelet filters. Soms of the real wavelet excellent properties spread to complex wavelet, so that the excellent properties of complex wavelet get more widely application.This thesis consists of four chapters, and the main contents of each chapter are as follows:In Chapter 1, we outline the development histories of wavelet analysis and current research situations of wavelet theory at home and abroad. Main results of this thesis will be introduced.In Chapter 2, we mainly give the basic theoretical and useful lemma of the frame, M scale complex wavelet and the dual tree complex waveletIn Chapter 3, based on existing real wavelet tight frame theory, we give com-plex wavelet frames with 2 scale. Because the bandwidth of M scale wavelet is more fine, the energy is more focused, has symmetry, smoothness, and other unique ad-vantage, so this chapter gives a methods for constructing a class M scale orthogonal symmetric complex wavelet.In Chapter 4, Mainly analyzes the principle of dual tree complex wavelet trans-form, a method for the design of the dual tree Q-shift complex wavelet filter is proposed in this paper.The filters are constructed using this method satisfy the per- fect reconstruction conditions with good directional selectivity and approximately shift invariance properties.