结构化解析字典学习及其在压缩感知图像重构中的应用研究

Dictionary learning is to learn a dictionary adaptively from given signals, which has been widely used in signal reconstruction, representation, classification and so on. Dictionary learning based on the analysis model has the superior capability of representing the signal sparsely, therefore it could process the inverse problem well. Furthermore, analysis dictionary learning with the intrinsic structure can represent the intrinsic geometric structure of the data, which has been widely applied to medical image processing, pattern recognition and so on. However, analysis dictionary learning with the intrinsic structure faces some challenging issues: first, the analysis sparse coding is NP-hard problem; second, the analysis dictionary has the trivial solutions; third, it is difficult to capture the intrinsic structure of data for the analysis dictionary. To solve the above challenges, we plan to investigate a novel analysis dictionary learning algorithm keeping the intrinsic structure of data and its application to compressed sensing image reconstruction in this project. First, we will choose the nonlinear functions as sparsity measure, and impose an orthonormality constraint on the dictionary to avoid the trivial solutions. Second, we will employ the proximal operator and the manifold optimization based on Cayley transform, and propose the analysis dictionary learning algorithm which can keep the intrinsic structure of data. At last, the proposed algorithm will be applied to compressed sensing image reconstruction with the intrinsic structure.

字典学习通常是根据信号本身特点自适应训练字典，通过学习获得的字典能够与信号本身结构匹配，已被广泛应用于信号重构、表示及分类等方面。解析稀疏模型下的字典学习即解析字典学习具有丰富灵活的稀疏表示能力，在处理逆问题时具有更大优势。结构化解析字典具有对数据内在结构属性表达的能力，在医学成像、模式识别等方面具有重要应用价值。针对结构化解析字典学习中解析稀疏编码NP-hard问题、解析字典易出现平凡解及对数据内在几何结构表达能力不足等问题，本项目拟重点研究结构化解析字典学习及其在压缩感知图像重构中的应用。首先，选取非线性函数作为稀疏度量，对解析字典施加正交约束以消除平凡解，建立解析字典学习模型；其次，采用邻近算子技术和基于Cayley变换的流形优化方法，开发高效的结构化解析字典学习算法；最后，将所开发的结构化解析字典学习算法应用于需考虑数据内在结构特性的压缩感知图像重构。

结构化解析字典具有丰富灵活的稀疏表示能力，且具有对数据内在结构属性表达的能力，在处理逆问题例如医学成像等方面具有重要应用价值。然而结构化解析字典学习仍然存在解析稀疏编码NP-hard问题、解析字典易出现平凡解及对数据内在几何结构表达能力不足等问题。因此，本项目首先针对解析稀疏编码NP-hard问题，采用非线性稀疏函数作为稀疏正则项进行解析稀疏编码。为克服非线性函数的非凸性，采用凸差分方法将非线性稀疏正则子分解为两个凸函数的差分，进而优化其子问题，开发有效解析稀疏编码算法。其次，针对字典易出现平凡解且对数据内在几何结构表达能力不足的问题，本项目对解析字典施加正交约束，正交约束构成流形空间，从流形空间角度直接求解解析字典，同时满足正交约束，进而避免对求解后字典进行额外操作以消除平凡解。最后，本项目开发的非线性稀疏正则化的解析稀疏编码和基于流形优化的解析字典学习算法已应用于自然图像和MR图像去噪及融合，可获得比已有方法更高的峰值信噪比。三年来，申请人围绕着本项目的研究内容以第一作者及通讯作者在IEEE Trans.Cybernetics、 IEEE Trans.NNLS等国际权威期刊发表论文6篇和EI检索会议论文4篇，发明国家专利2件。课题组培养毕业研究生2名，在读硕士研究生4名。项目负责人及指导的学生多次参加国内国际相关会议，与国内同行紧密交流研讨，取得了一定的国际国内学术影响力。

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