In this paper, we propose a general method to devise maximum likelihood
penalized (regularized) algorithms with positivity constraints. Moreover, we
explain how to obtain ‘product forms’ of these algorithms. The algorithmic
method is based on Kuhn–Tucker first-order optimality conditions. Its application
domain is not restricted to the cases considered in this paper, but it can be
applied to any convex objective function with linear constraints. It is
specially adapted to the case of objective functions with a bounded domain,
which completely encloses the domain of the (linear) constraints. The
Poisson noise case typical of this last situation and the Gaussian additive
noise case are considered and they are associated with various forms of
regularization functions, mainly quadratic and entropy terms. The algorithms are
applied to the deconvolution of synthetic images blurred by a realistic point
spread function similar to that of Hubble Space Telescope operating in
the far-ultraviolet and corrupted by noise. The effect of the relaxation
on the convergence speed of the algorithms is analysed. The particular
behaviour of the algorithms corresponding to different forms of regularization
functions is described. We show that the ‘prior’ image is a key point in
the regularization and that the best results are obtained with Tikhonov
regularization with a Laplacian operator. The analysis of the Poisson process and
of a Gaussian additive noise leads to similar conclusions. We bring to
the fore the close relationship between Tikhonov regularization using
derivative operators, and regularization by a distance to a ‘default image’
introduced by Horne (Horne K 1985 Mon. Not. R. Astron. Soc. 213 129–41).
International audienceIn this work we investigate a generic method able to extract information on molecular organization in biological samples from polarized second harmonic generation (SHG) microscopy, without the need to infer an a priori model for the molecular orientational distribution. The mean orientation of this distribution, as well as its first and third orders of symmetry, are estimated by monitoring SHG intensity signals under a varying incident polarization. We introduce, in particular, a reduction of the problem to a two-dimensional approach appropriate to the microscopy geometry. This method allows us to retrieve determining information which is not available in the traditional model-oriented methods, as illustrated in molecular-order imaging in collagen fibrils. The precision of the parameters estimation is evaluated by a Monte Carlo analysis, based on the Poisson noise statistics of the measured signal
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