Two-dimensional linear prediction and spectral estimation on a polar raster

A zero-mean homogeneous random field is defined on a discrete polar raster. Given sample values inside a disk of finite radius, the authors wish to estimate the field´s power spectral density using linear prediction. Issues arising include estimation of covariance lags and extendibility of a finite set of lag estimates into a positive semidefinite covariance extension (required for a meaningful spectral density). The authors give a generalized autocorrelation procedure that guarantees positive semidefinite covariance estimates. It first interpolates the data using Gaussians, computes its Radon transform, and applies familiar 1D techniques to each slice. Some numerical examples are provided to justify the validity of the proposed procedure. The authors also propose a correlation-matching covariance extension procedure that uses the Radon transform to extend a given set of covariance lags to the entire plane, when this is possible, and discuss circumstances for which this is impossible