Apodization functions for 2-D hexagonally sampled synthetic aperture Imaging radiometers

It is now well established that synthetic aperture imaging radiometers promise to be powerful sensors for high-resolution observations of the earth at low microwave frequencies. Within this context, the European Space Agency is currently developing the Soil Moisture and Ocean Salinity (SMOS) mission. The Y-shaped array selected for SMOS is fitted with equally spaced antennae and leads to a natural hexagonal sampling of the Fourier plane. This paper deals with the choice of the apodization function to be applied to the complex visibilities. The aim of this function is to reduce the Gibbs phenomenon produced by the finite extent of the star-shaped frequency coverage and the resulting sharp frequency cut-off. A large number of windows are introduced. A comparison of these in terms of their spatial domain properties is given, according to criteria relevant for remote sensing of the earth's surface. This paper also describes how discrete Fourier transform calculations over hexagonal grids can be performed using a simple algorithm. Actually, standard fast Fourier transform algorithms designed for Cartesian grids and which have a long track record of optimization can be reused. Finally, an interpolation formula is given for resampling data from hexagonal grids without introducing any aliasing artifacts in the resampled data.