超分辨率中的矩阵值算子学习问题

The key scientific issue in the field of example-based super-resolution is the issue of understanding the relationship between low- and high-resolution images with the help of computers. To deal with the issue, we plan to treat naturally all images as matrices and analyze the relationship between images by using matrix-value operators. All image information is preserved because the representation of all images is not changed, which will lead to more accurate understanding on the relationship between images. For verifying the feasibility and effectiveness of this proposal, we plan to conduct the following research: the issue of representing the relationship between low- and high-resolution images by matrix-value operators, including representing the globe and local information of training image pairs; the issue of developing example-based super-resolution based on the matrix-value operators, including matrix-value operator regression with globe information and its regularized version with the information of image similarity, boosting matrix-value operators according to the local information and its regularized version with sparse assumption, and novel example-based super-resolution algorithms developed according to the manifold information of matrix-value operators; and the convergence of the mentioned algorithms, including their generalization bounds and the rate of convergence. It could be expected that the proposal will enhance the research on example-based super-resolution and understanding of image pairs with computers.

实例型超分辨率研究中的关键科学问题是低-高分辨率图像对信息的计算机理解问题。针对该问题，本项目拟以图像数据的自然表示形式（矩阵）为基础，利用矩阵值算子工具直接分析图像间的对应关系。因不改变图像表示形式，不仅可更好的保持图像固有信息，而且还可使计算机更精确的理解图像间的对应关系。为验证该方案的可行性和有效性，拟开展如下研究：图像对信息的矩阵值算子表示问题，即低-高分辨率训练图像对全局及局部信息的矩阵值算子表示问题；基于矩阵值算子的超分辨率算法设计问题，即基于全局信息的矩阵值算子回归算法及带图像相似性约束的回归算法、基于局部信息的矩阵值算子集成算法及带稀疏约束的算子集成算法、基于矩阵值算子流形结构的超分辨率算法；超分辨率算法收敛性问题，即上述实例型超分辨率算法推广性能的界及收敛阶估计。此项研究的顺利开展将拓展实例型超分辨率的研究思路，丰富实例型超分辨率算法类型，发展图像对信息的计算机理解理论。

课题组围绕高-低分辨率图像对间对应关系的表示和提取问题开展了超分辨率算法设计、分析及其原理应用等三个方面的研究工作。根据既定研究方案，课题组以矩阵为主要表示工具，一方面利用矩阵表示图像块，另一方面利用矩阵表示图像对间对应关系。在此基础上，课题组得到了以局部回归关系的矩阵值算子表示为基础的一类局部化超分辨率算法；同时，以点态算子方法为基础建立了以矩阵的左右乘算子为基础的一类全局回归型超分辨率算法。通过实验数据和理论分析得到了上述两类超分辨率算法的收敛性结果。在此基础上，将矩阵值算子学习方法引入到多模态数据分析问题中，提出了异质流形概念，以便在统一模型下表示同质及异质数据间的相似关系。上述研究工作表明，矩阵值算子学习方法能有效的提炼及表示图像对间对应关系。对于研究图像间对应关系的问题，以上成果提供了有益的实验数据和初步的理论探讨。同时，该方法也可为相关领域研究提供研究思路和技术支持。