Modelling and Filtering of Physiological Oscillations in Near-Infrared Spectroscopy by Time-Varying Fourier Series

Abstract

Raw near-infrared spectroscopy signals contain oscillatory components, namely low frequency oscillations (Mayer waves), breathing and the heartbeat. We propose an approach to model and estimate them from noisy measurements assuming that they are linearly superposed. Estimating them is important, since they pose disturbing effects, but they are also of scientific interest. These components are not strictly periodic; we characterize them as “almost periodic.” The model of an almost periodic signal is a Fourier series where the Fourier coefficients and the fundamental frequency are allowed to (slowly) change over time. This model can be represented by factor graphs which we use to derive message passing algorithms to estimate the time-dependent model parameters from a measured signal. An implementation of the proposed algorithm processes a 100 s long measurement in 2 s (on a modern PC) which is ∼10× faster than a comparable previous implementation. Thus, real-time applications, for example, online monitoring, could be realized using slower, inexpensive or power-saving hardware. The increase in speed was achieved by using a different parameterisation of the model which allows Gaussian message passing (with only two parameters: mean and variance), whereas previously some messages were digitized. In the previous implementation, the number of harmonics in the model is chosen manually (for each subject and data channel). In this chapter, we show an intuitive procedure to estimate this number from the measured signal. In conclusion, the proposed algorithm is able to separate the heartbeat and, in contrast to the previous implementation, the low frequency oscillation effectively and in real time.