Reconstruction of multidimensional stochastic fields from discrete measurements of amplitude and gradient

Optimum weighting functions are derived for least-mean-square reconstruction of N-dimensional stochastic fields from discrete sample measurements of amplitude and gradient. Exact interpolation can be achieved when the intensity spectrum of the field is wave-number-limited and the sampling lattice is periodic, provided that the spectral images induced by sampling overlap nowhere more than (N + 1) times. Compared with conventional (amplitude-only) sampling, a network (N + 1) times less dense is thus required.Examples include a rederivation of the one-dimensional bandlimited case, and the calculation of weighting functions and reconstruction errors for a second-order “Butterworth” process under various postulated sampling schemes. Weighting functions are derived for a two-dimensional isotropic wave-number-limited process, sampled to yield a 3:1 spectral overlap