Global Optimization Of The Design Of Complex Wavelet And Its Application

Wavelet is a signal processing tool widely employed to analyze non-stationary signal. It has both significant theoretical value and wide practical applications. However, the traditional discrete wavelet transform has some drawbacks that undermine its application, in which the shift sensitivity is one of the most troublesome one. As a result, Kingsbury has proposed the dual-tree complex wavelet transform with a lot of good properties, such as shift-insensitivity, phase information, and direction selectivity. Nevertheless, there are still many problems left unsolved in the complex wavelet domain. First of all, the performance of the complex wavelet is highly dependent on the design of the proper discrete complex wavelet, which is considerably more difficult than the design of the traditional discrete wavelet. Second, due to its desirable properties, the complex wavelet is expected to be employed in a wider range of areas. In order to solve these problems, this thesis focuses on the design of discrete complex wavelet and the applications. The main contents of our thesis are:In this thesis, the approximate FIR realization of the Hilbert transform pairs is formulated as an optimization problem in the sense of the lP(p=1,2, or infinite) norm minimization on the approximate error of the magnitude and phase conditions of the scaling filters. The orthogonality and regularity conditions of the CQF bank pairs are taken as the constraints of such an optimization problem. Whereafter the branch and bound technique is employed to obtain the globally optimal solution of the resulting bilinear program optimization problem. Since the orthogonality and regularity conditions are explicitly taken as the constraints of our optimization problem, the attained solution is an approximate Hilbert transform pair satisfying these conditions exactly. Some orthogonal wavelet bases designed herein demonstrate that our design scheme is superior to those that have been reported in the literature. Moreover, the designed orthogonal wavelet bases show that minimizing the l1 norm of the approximate error should be advocated for obtaining better approximated Hilbert pairs.With the designed complex wavelet, this thesis proposes a similarity measure based on the complex wavelet image quality index. Due to the shift insensitivity property of the complex wavelet, the proposed measure is more robust to local deformations than the mutual information. The experimental results show that using the proposed measure can produce better registration results in Digital Subtraction Angiography (DSA) image registration. In addition, the registration result is iterative ly refined with the help of the extracted vessel information. The experimental results show that the iterative scheme help increase the registration precision.Finally, we propose a multi-spectral and panchromatic images fusion algorithm using the wavelet-domain hidden Markov tree model. Our algorithm exploits the wavelet-domain hidden Markov tree model obtained from the high-resolution panchromatic image to perform super resolution to the low-resolution multi-spectral image. In this way, the desired high-resolution multispectral image is obtained. The experimental results show that the proposed algorithm can produce sharper images as well as retaining good color. Moreover, as a result of the insensitivity of the wavelet coefficients'statistical information to the noises, our algorithm exhibit stronger robustness to the additive noises.