多元对偶小波框架的提升构造及其在图像去噪中的应用

Wavelet frame is a powerful tool in signal processing. Most of its application fields are in the case of high dimensions. The multi-dimensional data analysis is one of the challenging problems in wavelets . Construction of multi-variate wavelet systems with good properties is the key issue to the success of its application. The lifting scheme is a new technique for designing wavelets frame. According to the entirely spatial-domain interpretation, it is worthwhile to simplify the multi-variate wavelet construction to some extent. In this work, based on the lifting scheme, we will focus on the construction of multi-dimensional dual wavelet frame and its application in image denoising. The relationships between the lifting operators and the important properties of dual wavelets frame, such as the symmetry, vanishing moment, approximation property and the optimum time–frequency resolution, will be systematically studied. The proposal aims to construct multi-variate dual wavelet frame with the given desired properties. According to some properties such as high order vanishing moments and the optimum time–frequency resolution not satisfied at the same time, the obtained dual wavelet frame will be used for different image denoising.Moreover,the optimal trading off between some properties will be analyzed. The study will provide a guideline for constructing multivariate wavelet frame in image denoising. It is valuable in both wavelet frame theory and its 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，构造具有优美性质的多元小波是其成功应用的关键保障。鉴于冗余小波在信号处理中优势，为了满足不同应用的需求，本项目拟开展多元对偶小波框架的构造及其在图像去噪中的应用研究,重点研究多元对偶小波框架的提升构造方法，分析紧支、对称、消失矩、逼近阶、时频特性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对偶小波框架等价数学描述；为评估构造的小波框架性能，还将展开小波框架在图像去噪处理中的应用研究，研究紧支、对称、消失矩、逼近阶、时频优化特性与提升算子关联对不同应用图像的影响。本研究对完善小波框架理论、推动小波框架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。

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，构造具有优美性质的多元小波是其成功应用的关键保障。鉴于冗余小波在信号处理中优势，为了满足不同应用的需求，项目组开展了多元对偶小波框架的构造及其在图像去噪中的应用研究。项目组从多元对偶小波框架快速变换的角度出发，结合提升格式的基本原理，首次得到任意采样矩阵下的任意带多元对偶小波框架的提升格式，即多元对偶小波框架的提升构造等效为一系列提升算子的求解；通过分析提升算子和对偶小波框架的消失矩性质之间的关系，得到了提升算子的求解方法，即提升算子的求解转换为Neville滤波器的计算；从构造实例中分析其对称性，得到了一系列性质优美的多元对偶小波框架；针对以上研究，已在国际权威SCI期刊和IEEE国际会议上发表标注的相关论文6篇，指导撰写硕士论文《基于Neville滤波器的P带多元对偶小波框架提升构造研究》。应用方面重点研究小波框架的图像去噪性能，指导撰写硕士论文《基于紧小波框架的多幅盲运动图像修复算法研究》。此外，我们开展了与课题研究内容密切相关的应用研究，主要关注图像分割及图像二值化算法的优化应用研究，项目课题组申请发明专利1项。