Generalized Eigenvalue Decomposition Applied to Estimation of Spatial rPPG Distribution of Skin

Remote photoplethysmography (rPPG) has been at the forefront recently, thanks to its capacity in estimating non-contact physiological parameters such as heart rate and heart rate variability (Wang et al. in FBB 6:33, 2018). rPPG signals are typically extracted from facial videos by performing spatial averaging to obtain temporal RGB traces. Although this spatial averaging simplifies computation, it is accompanied by loss of essential spatial information which might reveal interesting relationships between signals from different spatial regions. In this article, we present a novel algorithm adapted from generalized eigenvalue decomposition (GEVD) to estimate this spatial rPPG distribution. GEVD is an extremely versatile algorithm that finds uses in signal and image processing and analytical problems such as principal component analysis and Fisher discriminant analysis (Ghojogh et al. in Tutorial 2: 1–8, 2019)(Han and Clemmensen in PR 49:43-54, 2016). It is performed using the QZ algorithm (Moler and Stewart in JNA 10(2):241–256, 2010), which in turn uses Householder transformations (Householder in JACM 5(4):339–342, 1958) to extract generalized eigenvectors of a pair of matrices. We adapt the QZ algorithm for the domain of spatio-temporal biomedical signals such as remote photoplethysmography (rPPG), electrocardiography and electroencephalography signals. We call this algorithm Temporal-QZ, which employs vectorization techniques to extract generalized eigenvectors over spatial data points simultaneously. We validate this extension in the domain of remote photoplethysmography (rPPG) measurement, for the estimation of spatial rPPG distribution of skin.