空间约束下多字典学习的形态学成分分析

Usually a complex system is consisted of many simple systems. It will be helpful if we can decompose a complex system into its components. For example, in the image processing fields, a digital image is constructed by cantoon part and texture part.It will be more efficient to deal with these two parts separetily than to deal with the image as a whole..Morphological component analysis(MCA) is the frontier in the image processing field. It is a very important and useful method both in the theory and in the engineering aspects. In fact, MCA decompose a given image into different morphological parts based on the basis constructed before, such as wavelet and DCT..On the other hand, in the field of numerical harmonic analysis, the image decomposition is made by the use of different spaces. .Based on the dictionary learning, in this project, we plan to combine the two methods above together and construct a novel model. Dictionary learning is more adaptive than the dictionary analitically constructed. To the decomposition problem, the dictionary learning should be guided by the proper spaces which were studied in the harmonic analysis field. Furthor more, in order to learn multiple dictionaries, the constraints are needed to ensure the existence of the non-trivial solutions, which makes the law-rank constraint dictionary learning model. Put all these together, we obtain a new MCA based on the dictionary learning under the constraint of spaces and law-rank.. The project will study the model analytically, choose the proper guided spaces, optimize the dictionary constraints, and design the efficient algorithm. The end of the project is to develop a new model and a fast algorithm to the problem of image decomposition, which will make the MCA more efficient.

数字图像处理中的形态学成分分析方法（MCA）利用解析构造的基函数为字典，如小波、DCT等，实现图像的结构和纹理成分的分解。本项目针对MCA的两个基本问题展开研究。（1）MCA中基函数的选择是先验的。本项目计划通过空间引导方法，利用计算调和分析中的分解空间作为约束，通过自适应学习方法，建立基于字典学习和数据驱动的新的更有效分解模型和算法。（2）MCA中的字典是单一的。针对图像不同区域的特征，本项目计划建立相应约束下的多字典学习方法。本项目将要研究的主要问题包括：模型的建立和理论分析、合适的引导空间的选择、字典约束的优化形式和相应快速算法的设计等。在发展MCA理论的基础上，我们将给出新方法在图像恢复（去噪、插值、修补）、边缘检测、盲源分离等问题上的应用。本课题预期在理论上有突破，方法和技术上有创新，为该方法的实际应用奠定理论和技术基础。

本项目包括如下几个部分：分解空间的刻画，空间约束下的（多）字典学习方法，上述理论与方法在图像恢复（去噪、插值、修补）、边缘检测、盲源分离等问题上的应用。通过项目的完成，我们在理论上有所突破，方法和技术上有创新。我们建立了一个新的图像分解变分模型和由扩散流引导的字典学习算法；通过对自相似性质的深入分析, 提出了描述自相似性质的两个原则: “两方向” 原则和“求同存异” 原则；在这两个原则基础上，我们给出了非凸稀疏正则化方法，稀疏半迭代阈值方法，两方向非局部稀疏表示方法，以及求同存异的非局部方法。共发表论文30余篇，其中在国际重要期刊或重要国际会议以及国内一级期刊上发表论文16篇，完成项目研究报告，培养博士研究生5人。