Solving inverse problems with overcomplete transforms and convex optimization techniques
Résumé
Many algorithms have been proposed during the last decade in order to deal with inverse problems. Of particular interest are convex optimization approaches that consist of minimizing a criteria generally composed of two terms: a data fidelity (linked to noise) term and a prior (regularization) term. As image properties are often easier to extract in a transform domain, frame representations may be fruitful. Potential functions are then chosen as priors to fit as well as possible empirical coefficient distributions. As a consequence, the minimization problem can be considered from two viewpoints : a minimization along the coefficients or along the image pixels directly. Some recently proposed iterative optimization algorithms can be easily implemented when the frame representation reduces to an orthonormal basis. Furthermore, it can be noticed that in this particular case, it is equivalent to minimize the criterion in the transform domain or in the image domain. However, much attention should be paid when an overcomplete representation is considered. In that case, there is no longer equivalence between coefficient and image domain minimization. This point will be developed throughout this paper. Moreover, we will discuss how the choice of the transform may influence parameters and operators necessary to implement algorithms.
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