图函数的时频分析及其应用

信号、图像和视频都是定义在欧几里得空间上的函数，信号处理和分析的基本方法是变换或分解。Fourier分析、小波分析以及非平稳信号处理的时频分析方法是传统信号分析和处理的重要工具。然而，在现代科学技术中，大量的实际问题中的数据并非是定义在欧几里得空间上的函数，它们常常是不规则离散点集上的函数，比如高维流形数据、复杂网络、点云数据、疾病传播网络、城市交通网络、人类迁徙网络等。这些数据可以模型化为加权图上的函数，由于图不具备自然的代数和拓扑结构， 因此传统的信号处理方法不再适应。为了对这类数据进行有效的分析和处理，需要建立相关的数据处理理论和方法。本项目将研究图上函数的时频分析及其应用，包括图函数Fourier频谱的图结构及其频谱性质研究；图函数的时频原子与函数分解；图的分层结构及其相关的图小波；从图函数的样本数据反求图权重等问题及其应用。

Signals, images and videos are functions defined on Euclidean spaces. The basic methods for signal processing and analysis are transformations or decompositions. Fourier analysis, wavelet analysis and time-frequency analysis for non-stationary signal processing are important tools for traditional signal analysis and processing. However, in the modern science and technology, the data from a large number of practical problems are not defined on Euclidean spaces. They are often defined on irregular discrete point sets, such as high dimensional manifold, complex network, point cloud, disease transmission network, city traffic network, human migration network etc.. These data can be modeled as functions on the weighted graphs. Since the graph does not have a natural algebraic and topological structure as the classical Euclidean space, the traditional signal processing methods are no longer suitable for them. In order to analyze and process these data effectively, it is necessary to establish new theory and methods. This project will study the time-frequency analysis for functions defined on graphs and its applications, including the graph structure of the Fourier spectra of functions on graphs; the time-frequency atoms and function decomposition on graphs; the hierarchical structure of graph and the construction of graph wavelet; the estimation of weights of the graph from sample data of a graph function.