单分量框架与信号自适应稀疏表示

小波分析具有多分辨率的、自适应的时频局部分析能力，对正则信号具有紧凑和稀疏的表示，这为基于信号时频特征的数据分析提供了强有力的工具。但小波分析不具有波形和时频结构的自适应性，而且几乎所有的小波均不是单分量的，因而其对于非线性非平稳信号的时频刻画存在明显的缺陷。近十多年来，随着信息技术的迅猛发展，大量科学问题归结为对巨量高维数据的有效分析和处理。信号表示的时频局部性、稀疏性和自适应性成为当代信号处理和数据理解的关键技术瓶颈，同时也为应用与计算调和分析以及时频分析提出了新的研究课题。本项目中，我们拟对单分量信号的模型、频谱刻画（瞬时频率），以及由单分量信号所构成的函数系的框架条件、最优框架的选择等问题给予研究，并利用压缩感知和稀疏逼近的最新理论和方法，给出多分量信号的自适应稀疏表示和时频分布。作为应用，我们将研究其对fMRI数据进行分析和处理，研究抑郁症的时频图谱，探索抑郁症疾病诊断的新途径。

Wavelet can characterize signals locally and adaptively with multi-resolution on both time and frequency domains. It provides sparse representation for regular signals, which make it be a very strong tool for time-frequency analysis of signals. However, wavelet cannot match waves and time-frequency structures of the signal adaptively, and almost all the wavelets are not mono-component signals themselves, which limit its efficient applications for nonlinear and non-stationary signal processing. In the past decade, with the rapid development of information technology, a large number of scientific problems have been attributed to the efficient analysis and processing of high-dimensional and large scale data. In contemporary signal processing and data understanding, it has become the key technical bottleneck to represent signals sparsely, adaptively and locally on both time and frequency domains. This also proposes new research topics for applied and computational harmonic analysis and time-frequency analysis. In this project, we intend to study the modeling of mono-component signals and instantaneous frequency, as well as the conditions that a family of mono-component signals constitutes a frame, and algorithms for constructing optimal frames. We also use the theory and techniques developed in the area of compressive sensing and sparse approximation to study the adaptive decomposition for multi-component signals. As an application, we will analyze and process the fMRI data, especially those of depression diseases, explore the time-frequency distribution and develop new techniques for the diagnosis of depression diseases.