Reconstruction of piecewise homogeneous images from partial knowledge of their Fourier Transform

. Fourier synthesis (FS) inverse problem,consists in reconstructing a multi-variable function from the measured data which correspond to partial and uncertain knowledge of its Fourier Transform (FT). By partial knowledge,we mean,either partial support and/or the knowledge of only the module,and by uncertain we mean,both uncertainty of the model,and noisy data. This inverse problem arises in many applications such as : optical imaging, radio astronomy, magnetic resonance imaging (MRI) and diffraction scattering (ultrasounds or microwave imaging). Most classical methods,of inversion are based on interpolation of the data and fast inverse FT. But when the data do not ll uniformly the Fourier domain or when the phase of the signal is lacking as in optical interferometry, the results obtained by such methods are not satisfactory, because these inverse problems are ill-posed. The Bayesian estimation approach, via an appropriate modeling of the unknown functions gives the possibility of compensating the lack of information in the data, thus giving satisfactory results. In this paper we study the case where the observations are a part of the FT modulus,of objects which are composed,of a few number,of homogeneous,materials. To model such objects we use a Hierarchical Hidden Markov Modeling (HMM) and propose a Bayesian inversion method,using appropriate Markov Chain Monte Carlo (MCMC) algorithms. Key words : Fourier Synthesis, Inverse Problem, Potts Markov Random Field, HMM, Gibbs