### Image Reconstruction via Classification Using Fourier Synthesis and Prior Information in Bayesian Analysis

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#### Abstract

Fourier Synthesis inverse problem consists in reconstructing an image from the measured data which correspond to partial and noisy information of its Fourier Transform. This inverse problem is known to be nonlinear and ill-posed. It then needs to be regularized by introducing prior information. In this paper we propose two priors information. In the first prior information, we assume that the original image is composed by homogeneous regions, so in this case we propose the Hidden Markov Modeling dedicated to classification which is the most appropriate distribution for the image labels in a Bayesian framework. In the second prior information we assume that the noise is a Gaussian centered and in order to improve the quality of image reconstruction we introduce a total variation algorithm with a Bayesian analysis to regularize the solution.

Appropriate Markov Chain Monte Carlo algorithms are proposed to implement our approach. This method is applied on synthetics and real images

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