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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