Studies on Signals and Images via the Fourier Transform

该项目的目标是开发新的统计方法，以解决当前在分析神经成像中经常遇到的时空数据方面的一些挑战。该项目的一个主要应用是识别大脑信号的特征，这些特征可以将健康个体与患有神经或精神疾病的患者区分开来。第二个应用是识别认知处理过程中大脑信号发生的变化（例如，当人类学习一项新的运动技能时，或者当老鼠在对照实验中学习风险和奖励时）。第三个应用是识别大脑信号中的生物标志物，这些生物标志物可以预测中风患者恢复运动功能丧失的能力。解决这些问题的方法需要对这些大脑信号的振荡模式进行研究。在这些实际问题的激励下，基于离散傅里叶变换（DFT）的统计方法得到了发展。DFT给出了时间序列中方差分解的指示。在平稳性下，DFT的协方差是稀疏的，因此偏离稀疏性表示非平稳性。此外，DFT的协方差可以用作信号类别之间的鉴别器。利用离散傅里叶变换的特性，提出了基于离散傅里叶变换稀疏性的时间序列变化点检测方法，以及基于离散傅里叶变换协方差的时间序列分类和判别方法。DFT还将用于估计谱函数的方差，并检验非线性时间序列的序列相关性和平稳性。使用二维DFT验证平稳空间过程和非平稳空间过程也将被开发。

The goal of this project is to develop novel statistical methods that address some of the current challenges in analyzing spatio-temporal data frequently encountered in neuroimaging. One major application of this project is to identify features in brain signals that could differentiate healthy individuals from patients with neurological or mental diseases. The second application is to identify changes that take place in a brain signal during cognitive processing (e.g., while a human learns a new motor skill or while a rat learns risks and rewards in a controlled experiment). The third application is to identify biomarkers in brain signals that could predict a stroke patient's ability to recover loss of motor functionality. The approach used to solve these problems requires a study of the oscillatory patterns in these brain signals. Motivated by these practical problems, statistical methods based on the discrete Fourier transform (DFT) are developed. The DFT gives an indication of the decomposition of variance in the time series. Under stationarity, the covariance of the DFT is sparse and thus a departure from sparsity is an indication of non-stationarity. Moreover, the covariance of the DFT can be utilized as a discriminator between classes of signals. Using the properties of the DFT, novel methods for (1) change-point detection in time series based on sparsity of the DFT, and (2) discrimination and classification of classes of time series based on the properties of the covariance of the DFT will be developed. The DFT will also be used to estimate the variance of functionals of the spectrum and test for serial correlation and stationarity in nonlinear time series. Validation for stationary spatial processes and non-stationary spatial processes using the two-dimensional DFT will also be developed.