Abstract
We consider the problem of restoring astronomical images acquired with charge coupled device cameras. The astronomical object is first blurred by the point spread function of the instrument-atmosphere set. The resulting convolved image is corrupted by a Poissonian noise due to low light intensity, then, a Gaussian white noise is added during the electronic read-out operation. We show first that the split gradient method (SGM) previously proposed can be used to obtain maximum likelihood (ML) iterative algorithms adapted in such noise combinations. However, when ML algorithms are used for image restoration, whatever the noise process is, instabilities due to noise amplification appear when the iteration number increases. To avoid this drawback and to obtain physically meaningful solutions, we introduce various classical penalization-regularization terms to impose a smoothness property on the solution. We show that the SGM can be extended to such penalized ML objective functions, allowing us to obtain new algorithms leading to maximum a posteriori stable solutions. The proposed algorithms are checked on typical astronomical images and the choice of the penalty function is discussed following the kind of object.
Highlights
The image restoration problem and image deconvolution, is an inverse problem, ill posed in the sense of Hadamard, whose solution is unstable when the data is corrupted by noise
The first one is an additive Gaussian noise appearing in high intensity measurements; in this case, the maximum likelihood estimator (MLE) under positivity constraint is obtained, for example, using the ISRA multiplicative iterative algorithm [1]
We will show that the corresponding MLE iterative algorithm can be obtained using the split gradient method (SGM) we have previously proposed for Poisson or Gaussian noise [10, 11]
Summary
The image restoration problem and image deconvolution, is an inverse problem, ill posed in the sense of Hadamard, whose solution is unstable when the data is corrupted by noise. In a more realistic but less used model, both noise processes must be taken into account simultaneously We will describe this model previously analyzed by Snyder et al [3, 4, 5, 6] and by Llacer and Nunez [7, 8, 9]. In all the ML methods, whatever the noise process considered, only the adequacy of the solution to the data is taken into account; so, when iterative algorithms are used, instabilities appear due to the noise amplification when the iteration number increases. In this context, to obtain physically satisfactory solutions, the iterative process must be interrupted before instabilities appears. Another way to avoid this drawback is to perform an explicit regularization of the problem, that is, to introduce an a priori knowledge to impose, for example, a smoothness property on the solution, a maximum a posteriori (MAP) solution is searched for
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