Detecting spontaneous brain activity in functional magnetic resonance imaging using finite rate of innovation

Several methods have been developed for the sampling and reconstruction of specific classes of signals known as signals with finite rate of innovation (FRI). It is possible to recover the innovations of the signals from very low-rate samples by using adequate exponential reproduction sampling kernels. Recently, the FRI theory has been extended to arbitrary sampling kernels that reproduce approximate exponentials. In this paper, we develop the method for the detection of spontaneous brain activity in functional magnetic resonance imaging (fMRI) data. We model the fMRI timecourse for every voxel as a convolution between the innovation signal - a stream of Diracs- and the hemodynamic response function (HRF). Relaxing the exact exponential reproduction constraint given by Strang-Fix condition, we design an adequate FRI sampling kernel using the canonical HRF model that allows us to retrieve the innovation instants in continuous domain. We illustrate the feasibility of our method by detecting spontaneous brain activity on the simulated and degraded fMRI data using an iterative denoising scheme.