The usual methods of interferometry make use of the Fourier transform relationship which holds between a radio-noise brightness distribution and the complex visibility function which is measured with a pair of antennas. The visibility function is a function of the distance or base line between the antennas. If it were known for all base lines, then the brightness distribution could be found by Fourier inversion. Unfortunately, the visibility function is not known for all base lines and the Fourier inversion is not unique. If the observer wishes to interpret his data by displaying a single possible brightness distribution, then he must choose from the infinite set of brightness distributions which could have produced his data. Previously, the author suggested that this be accomplished by representing the set of possible distributions as a statistical ensemble, and making the choice on a statistical basis so as to minimize the expected mean-square error.In the present communication, the results of the previous paper are presented for the two-dimensional case. The inversion formulas are worked out in detail for the cases of uniform point-source distributions in a square (or rectangle) and in a circular disk, and also for a point-source distribution with a Gaussian envelope taper. It is shown how to extend the point-source results to a distribution of nonpoint sources, and as an example the inversion equations are computed for the case of a distribution of Gaussian-shaped sources distributed with a Gaussian amplitude or density envelope. Finally, the appropriate inversion equations are derived for an observed visibility function which is contaminated with additive zero-mean Gaussian random noise, uncorrelated with the true visibility function.
