多元二代小波构造及其自适应选择机制研究

Wavelet analysis is a powerful tool in signal processing. Most of its application fields are in the case of high dimensions. One of the challenging problems in wavelets is the high-dimensional data analysis. Construction of multivariate wavelet systems with good properties is the key issue to the success of its application. The lifting scheme is a new technique for both designing wavelets and performing the adaptive wavelet transform. The resulted wavelet is referred as the second generation wavelet. According to the entirely spatial-domain interpretation, it worthwhile to simplify the multivariate wavelet construction to some extent. In this work, based on the lifting scheme, we will focus on the multidimensional wavelets construction and its adaptive selection schemes for image processing. Multivariate wavelets include 2-band wavelets, multiband wavelets, scalar wavelets, multiwavelets and their corresponding prefilters all will be concerned in this work. The relationships between the lifting operators and the important properties of wavelets, such as the symmetry, vanishing moment, compact support and interpolation, will also be systematically studied. The proposal aims to achieve a general framework for constructing multivariate wavelet with the given desired properties. According to some properties such as high order vanishing moments and short support length are not satisfied at the same time, the obtained wavelets will be used for different image processing, the optimal trade off between some properties will be analyzed. Based on image local features, an adaptive multivariate wavelet selection algorithm will be obtained. The study will provide a guideline for constructing and selecting multivariate wavelet in image processing. It is valuable in both theory and applications.

高维数据处理已成为当前小波研究领域所面临的一大挑战，构造具有各种重要性质的多元小波是其成功应用的关键保障。提升格式是构造第二代小波、实现自适应小波变换最为有效的方法之一。为了满足不同应用的需要，本项目拟开展第二代多元小波的设计及其在图像处理中的自适应机制研究,重点研究各类多元小波的对称、消失矩、紧支、插值等重要性质与提升算子之间的关系，建立满足各种重要性质的多元二代小波一般性构造算法；同时将构造得到的小波应用于图像处理，研究多元小波的对称性、消失矩阶数、支撑长度对不同应用图像的影响，设计一种基于图像特征的小波自适应选取方案，为多元小波在图像处理中的实际应用提供科学设计指南。本研究对完善小波分析理论、推动小波在高维信号处理和大规模科学计算等领域的应用具有重要的理论价值。

随着信息技术的发展，人们对数据获取的能力大大增强。小波因其良好的时频分析性能，成为信息处理的重要手段，因此小波在高维海量数据中的应用倍受关注。基于提升方法的小波构造、自适应小波变换受到研究者们的青睐。本项目按照研究计划，围绕多元、多带、多重等复杂情形下的小波构造及其应用进行研究，主要包括：（1）研究提升算子与滤波器性质之间的关系，为自适应小波变换及其应用提供基础，提出了任意维任意带任意重小波的提升构造框架；（2）利用小波的多分辨分析性质，将构造得到的小波应用到图像和生物信息等高维数据处理；（3）研究基于小波的稀疏字典设计，根据图像特征设计小波稀疏字典，并将其应用到高分辨图像重建。在项目研究过程中，在IEEE汇刊, Journal of Software等重要学术期刊上发表标注资助学术论文7篇，培养毕业硕士研究生5名；开展了广泛的学术交流，扩大了影响，也为以后的研究积累了丰富的经验。