Sampling multidimensional signals by a new class of quasi-random sequences

Abstract

A new method of equidistributed (EQD) sampling of nD signals is proposed and investigated. Equidistributed or quasi-random (QR) sequences are deterministic sequences, which cover nD space more evenly than uniformly distributed random sequences. We propose to generate quasi-random nD sample sites using space-filling curves, which propagate their measure and neighborhood preserving properties to sample points. It is proved that these points are not only equidistributed but also well distributed, which results in a good volume approximation from samples of 2D and 3D images. A general way of reconstructing output of a linear system from QR samples of its input nD signals is also proposed and proved to be convergent. As an example of applications of the theory, a double radial basis neural network is proposed, which allows for the reconstruction of 2D cross-sections from samples of 3D image or from a video sequence of 2D images.