单分量模型与时频分析领域中若干关键问题的研究

In the past decade, with the rapid development of the information technology, a great number of scientific problems have been attributed to the analysis and processing of large scale data. The adapitive sparse representation based on the time-frequency structure of the signal can efficiently extract principal features of the signal and has the ability of dimensional reduction and feature extraction, which makes it have great prospects for application. In this project, we intend to study the modeling of mono-component signals and several key problems in time-frequency analysis and do the tasks as followed: (1) Based on the fact that the essence of the mono-component signal is the relative rate of change between the amplitude and phase of the signal, we will break through the shackle of the traditional analytic condition proposed by Gabor and model the mono-component signal through a stretch transformation; (2) We will use the theory and techniques developed in the area of compressive sensing and sparse approximation to study the adaptive sparse representation for multi-component signals and obtain their time-frequency distributions, then analyze their internal relations with the independent component analysis and multilayer neural network model; (3) As an application, we will use the adapitive sparse representation to detect QRS waves and P,T waves in ECG signals. This project intends to establish a reasonable and solid theoretical foundation for the time-frequency analysis of non-stationary siganls, and to some extent promote the development of algorithms and applications in this field.

近十多年来，随着信息技术的迅速发展，大量的科学问题都归结为对大规模数据的分析和处理。基于信号时频结构的自适应稀疏表示能有效地提取信号的主要特征，具有降维和特征提取的能力，因而有重大的应用前景。本项目将对单分量模型与时频分析领域中若干关键问题进行研究，开展如下工作：（1）单分量信号的本质是振幅和相位的相对变化率，我们将突破传统Gabor解析性条件的束缚，通过拉伸变换建立单分量模型；（2）我们将利用压缩感知和稀疏逼近的理论和方法，给出多分量信号的自适应稀疏表示和时频分布，并研究结果与独立成分分析、多层神经网络模型所得结果之间的内在关联；（3）作为应用，我们拟将自适应稀疏表示方法应用于心电信号处理，检测心电图中的QRS波和P、T波。本项目将为非平稳信号之时频分析建立合理而坚实的理论基础，并在一定程度上推进该领域的算法和应用发展。

时频分析是信号处理的核心研究领域。本项目对时频分析中若干关键问题进行研究，包括单分量信号模型与自适应信号分解算法、图上信号的时频分析与处理、模式识别与机器学习的理论与应用，取得了丰硕的成果：（1）研究了epsilon-单分量信号的构造及自适应信号分解算法，提出了单频率模型，建立了Bedrosian等式的统一理论框架；（2）提出了有限带宽图信号的迭代重构算法，提出了基于L1范数变分极小化问题的图上Fourier变换的新定义，提出了图信号的投影最小二乘重构算法；（3）研究了深度卷积网络在遥感图像中的应用，提出了基于典型相关分析的ECG信号分类算法，提出了改进的并行细化算法，研究了受限玻尔兹曼机的极大似然估计算法，分析了马尔科夫域上随机性算法的收敛性，提出了基于增量非负矩阵分解的自适应背景模型。本项目推进了时频分析领域中理论与算法的发展。