Estimation of Hidden Frequencies for 2D Stationary Processes

Abstract

We study a stationary random field model that is composed of a signal of an unknown number of sine and cosine functions, and a coloured noise. This model has been used in image analysis and modelling spatial data, and is useful for signal extraction in the presence of coloured noise. The problem is to estimate the number of unknown frequencies and the unknown frequencies. The analogous time series model and related problems have been extensively studied. Our approach is based on some analytic properties of periodograms of stationary random fields that we establish in the paper. In particular, we show that the periodogram of a stationary random field of a moving average has a uniform upper bound of O (ln( N 2 )) where N 2 is the sample size, and that the periodogram of the observed process has a magnitude of the order N 2 uniformly in a neighbourhood of any hidden frequency, and much smaller outside.