Studies on Signals and Images via the Fourier Transform

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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.