On Quasi-Isometry of Threshold-Based Sampling

The problem of isometry for threshold-based sampling such as integrate-and-fire or send-on-delta is addressed. While for uniform sampling the Parseval theorem provides isometry and makes the Euclidean metric canonical, there is no analogy for threshold-based sampling. The relaxation of the isometric postulate to quasi-isometry, however, allows the discovery of the underlying metric structure of threshold-based sampling. This paper characterizes this metric structure, making Hermann Weyl's discrepancy measure canonical for threshold-based sampling. The usefulness of the approach is demonstrated by means of a novel adaptive threshold-based sampling scheme, which, applied on electrocardiography signals, allows a substantial reduction of the number of samples while maintaining nearly the same SNR after reconstruction.