DWT Lifting分解理论及其在图像压缩中的应用

Lifting Factorization Theory of DWT and Its Application to Image Compression

在形式化地分析DWT(Discrete Wavelet Transform)Lifting分解的基础上,提出了分解集合的分级结构概念与技术,其克服了求解问题的高复杂性所带来的实际计算与搜索困难;进一步地,从数值稳定性及计算代价两方面研究了分解的评价问题,给出了两种实用的分解稳定性准则;最后,结合分级结构技术提出了最优分解算法.这些方法应用在JPEG2000框架下的图像压缩系统中得到了一些深刻结论,如发现了LTD(Long Then Danger)现象等,可以为压缩系统中小波变换的快速实现提供新的理论依据与实用算法.实验结果表明了上述方法在求解效率与灵活性、寻优速度以及适用范围等方面的优越性.

This paper deals with lifting factorizations of DWTs（discrete wavelet transforms）. Firstly, definitions of several operators for Laurent polynomial division are given, and the GCD（greatest common divisor）solving process of the Euclidean algorithm is formulized and analyzed in details, based on which, a so-called multi-lewel structurization technology is developed so as to overcome the difficulties, due to exponential size of the whole set, in computing and searching factorizations. Secondly, the problem of how to evaluate a certain lifting factorization is investigated in terms of both numerical stability and computational cost, wherein authors present two practical stability measures. Thirdly, an algorithm for finding high performance factorizations is designed by combining the multi-level structurization method and a evaluation rule. Fourthly, authors discuss compression performance difference of factorizations in JPEG2000 framework, which results in the finding of a phenomenon named LTD（long then danger）, i.e., the longer the wavelet filters are, the more dangerous to choose one factorization for image compression it is. Lastly, a serial of experiments are conducted on both orthogonal and bi-orthogonal wavelet transforms of arbitrary length. The experimental results show that the methods proposed in this paper, compared with the known ones, obtain considerable improvements in efficiency and flexibility of solving, time for optimization, and application range.