基于小波框架的散乱数据重构及其在计算生物中的应用

In the past decades, signal restoration has become one of the most important research area in the fields of information theory, biologic science, engineering and medicine. Many mathematical methods have been successfully applied to investigate it. In a large number of practical applications, we need to reconstruct and interpret the analog signals based on minimal measurements, which are usually scattered data containing kinds of noises and degeneracy. This project will focus on practical problems in computational biology, such as coarse-grained molecular dynamic, free energy landscape, small angle X-ray scattering,etc., and investigate wavelet frame based scattered data reconstruction. According to the features of data in practical problems, first we formulate them as mathematical problems, then establish appropriate models and algorithms, and provide effective solutions to the practical problems. In the end, we extend our methods to other area, such as surface reconstruction. The project will also explore new theories and obtain more general results. Therefore, the research of this project is of great importance, for it not only enriches the theories of image restoration (2D regular grid data recovery) and compressed sensing (highly sparse discrete signal recovery), but also provides new ideas and methods to practical problems.

近十余年来,信号恢复问题被认为是信息科学、生命科学、工程领域以及医学领域的重要 研究内容,各种数学方法被成功地应用于该领域的研究中。在大量的实际问题中,人们需要通过尽可能少的观测值,通常为含有各种噪音和退化的散乱数据,来恢复、解释未知的连续信号。项目拟针对计算生物领域中的实际问题,如粗颗粒化分子动力学、自由能地貌图、小角X射线散射等,研究基于小波框架的散乱数据重构。首先我们针对实际问题中数据的特点将其从数学上进行抽象,再建立合适的模型和算法,为问题提供有效的解决方案。最后将数学方法进行拓展,并应用到其他领域,如曲面重构上。项目也将探索新的理论,获得更加一般的结果。这些研究内容不仅是对图像恢复(二维规则网格点数据恢复)、压缩感知(高度稀疏性离散信号恢复)等重要理论的实质性拓展,而且也为实际应用问题提供了新的思路和方法,具有重要的意义。

项目致力于研究基于小波框架的散乱数据重构理论及其在计算生物学等领域的应用。研究内容的第一部分以理论研究为主，根据散乱数据的物理性质，建立相应的函数逼近模型，分析模型解的逼近性质。研究内容的第二部分以算法和应用为主，主要考虑去噪算法以及如何逼近非光滑信号，保持其跳跃、间断等重要信息，并应用到计算生物、曲面重构等领域。研究工作按照计划有序展开，研究成果包括散乱数据函数逼近的理论研究和散乱数据去噪，建模，计算，分析等应用研究。.具体而言.1.首次将B样条小波框架系统应用于计算生物力拟合领域，建立了力拟合模型和重构算法。.2.提出基于小波框架的去噪和逼近模型，系统研究了模型解的逼近阶、模型偏差和降噪能力。.3.证明了随机采样数据重构平移不变空间函数的稳定性，并建立与重构算法矩阵条件数的联系。.这些成果推动了函数逼近，平移不变空间等数学理论的完善，也为相关的工业应用提供了理论基础和保障。

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