On the unique reconstruction of a signal from its unthresholded recurrence plot

Abstract

We address the information content of unthresholded recurrence plots, generated by the time-delay embedding method from scalar signals admitting a Fourier series representation (including periodic and sampled signals). This is important for making valid inferences from unthresholded recurrence plots. A graph theoretic framework is developed to give a complete analysis of the impact of the choice of time-delay and embedding dimension on information content. A distance measure for unthresholded recurrence plots is introduced to approach signal reconstruction and approximation by minimization, robust to inaccuracies and noise. Examples and an application from EEG analysis clarify the theoretical results and demonstrate their practical potential.