On the regularization of inverse problems in imaging radiometry by aperture synthesis.

Synthetic aperture imaging radiometers are potentially powerful instruments for high-resolution observation of the Earth at low microwave frequencies. Interferometric measurements, also called complex visibilities, are obtained by cross-correlating the signals collected by pairs of antennae which have overlapping fields of view. They are related to the radiometric brightness temperature of the scene under observation by a modeling operator which is a spatial Fourier-like integral. The corresponding inverse problem, which aims at inverting the modeling operator to retrieve the radiometric brightness temperature map from the complex visibilities, is often ill-posed unless a regularizing constraint is introduced in order to provide a unique and stable solution to the problem. Synthetic aperture imaging radiometers belong to the family of band-limited imaging devices because the finite physical size of a synthetic antenna results in a truncation of the visibility samples into the so-called experimental frequency coverage. Such a physical property should certainly be taken into account in the regularization of the imaging problem. However, other regularizing methods could lead to the same results, even if their physical meaning is somewhere hidden by the mathematical foundations. This contribution makes a detailed review of standard methods for the regularization of inverse problems in imaging radiometry by aperture synthesis: the regularized solutions in the sense of Tikhonov, the solutions with minimal energy and those with band-limited properties are analyzed. The links between their physical and mathematical meanings are established. It is shown that the threshold of the truncated singular values decomposition used to approximate the solutions with minimal energy is closely related to the regularization parameter of Tikhonov. Moreover, the number of spatial frequencies characterizing the solutions with band-limited properties in the experimental frequency coverage is equal to the number of singular values kept in the inversion of the modeling operator, while the number of singular values discarded prior to inversion is equal to the number of redundant visibilities. The stability of each reconstruction process is studied in depth, including the influence of the windowing function on the systematic error as well as the propagation of random errors from the complex visibilities to the reconstructed radiometric brightness temperature map. To support the theory and to illustrate the performances of these methods, in terms of accuracy and computational time, numerical simulations are carried out within the frame of the SMOS space mission, a project led by the European Space Agency and devoted to the remote sensing of soil moisture and ocean salinity from a low orbit platform.