37 research outputs found
Space adaptive and hierarchical Bayesian variational models for image restoration
The main contribution of this thesis is the proposal of novel space-variant regularization or penalty terms motivated by a strong statistical rational. In light of the connection between the classical variational framework and the Bayesian formulation, we will focus on the design of highly flexible priors characterized by a large number of unknown parameters. The latter will be automatically estimated by setting up a hierarchical modeling framework, i.e. introducing informative or non-informative hyperpriors depending on the information at hand on the parameters.
More specifically, in the first part of the thesis we will focus on the restoration of natural images, by introducing highly parametrized distribution to model the local behavior of the gradients in the image. The resulting regularizers hold the potential to adapt to the local smoothness, directionality and sparsity in the data. The estimation of the unknown parameters will be addressed by means of non-informative hyperpriors, namely uniform distributions over the parameter domain, thus leading to the classical Maximum Likelihood approach.
In the second part of the thesis, we will address the problem of designing suitable penalty terms for the recovery of sparse signals. The space-variance in the proposed penalties, corresponding to a family of informative hyperpriors, namely generalized gamma hyperpriors, will follow directly from the assumption of the independence of the components in the signal. The study of the properties of the resulting energy functionals will thus lead to the introduction of two hybrid algorithms, aimed at combining the strong sparsity promotion characterizing non-convex penalty terms with the desirable guarantees of convex optimization
A flexible space-variant anisotropic regularisation for image restoration with automated parameter selection
We propose a new space-variant anisotropic regularisation term for
variational image restoration, based on the statistical assumption that the
gradients of the target image distribute locally according to a bivariate
generalised Gaussian distribution. The highly flexible variational structure of
the corresponding regulariser encodes several free parameters which hold the
potential for faithfully modelling the local geometry in the image and
describing local orientation preferences. For an automatic estimation of such
parameters, we design a robust maximum likelihood approach and report results
on its reliability on synthetic data and natural images. For the numerical
solution of the corresponding image restoration model, we use an iterative
algorithm based on the Alternating Direction Method of Multipliers (ADMM). A
suitable preliminary variable splitting together with a novel result in
multivariate non-convex proximal calculus yield a very efficient minimisation
algorithm. Several numerical results showing significant quality-improvement of
the proposed model with respect to some related state-of-the-art competitors
are reported, in particular in terms of texture and detail preservation
Masked unbiased principles for parameter selection in variational image restoration under Poisson noise
In this paper we address the problem of automatically selecting the regularization parameter in variational models for the restoration of images corrupted by Poisson noise. More specifically, we first review relevant existing unmasked selection criteria which fully exploit the acquired data by considering all pixels in the selection procedure. Then, based on an idea originally proposed by Carlavan and Blanc-Feraud to effectively deal with dark backgrounds and/or low photon-counting regimes, we introduce and discuss the masked versions—some of them already existing—of the considered unmasked selection principles formulated by simply discarding the pixels measuring zero photons. However, we prove that such a blind masking strategy yields a bias in the resulting principles that can be overcome by introducing a novel positive Poisson distribution correctly modeling the statistical properties of the undiscarded noisy data. Such distribution is at the core of newly proposed masked unbiased counterparts of the discussed strategies. All the unmasked, masked biased and masked unbiased principles are extensively compared on the restoration of different images in a wide range of photon-counting regimes. Our tests allow to conclude that the novel masked unbiased selection strategies, on average, compare favorably with unmasked and masked biased counterparts
Space-variant Generalized Gaussian Regularization for Image Restoration
We propose a new space-variant regularization term for variational image
restoration based on the assumption that the gradient magnitudes of the target
image distribute locally according to a half-Generalized Gaussian distribution.
This leads to a highly flexible regularizer characterized by two per-pixel free
parameters, which are automatically estimated from the observed image. The
proposed regularizer is coupled with either the or the fidelity
terms, in order to effectively deal with additive white Gaussian noise or
impulsive noises such as, e.g, additive white Laplace and salt and pepper
noise. The restored image is efficiently computed by means of an iterative
numerical algorithm based on the alternating direction method of multipliers.
Numerical examples indicate that the proposed regularizer holds the potential
for achieving high quality restorations for a wide range of target images
characterized by different gradient distributions and for the different types
of noise considered
Bilevel learning of regularization models and their discretization for image deblurring and super-resolution
Bilevel learning is a powerful optimization technique that has extensively
been employed in recent years to bridge the world of model-driven variational
approaches with data-driven methods. Upon suitable parametrization of the
desired quantities of interest (e.g., regularization terms or discretization
filters), such approach computes optimal parameter values by solving a nested
optimization problem where the variational model acts as a constraint. In this
work, we consider two different use cases of bilevel learning for the problem
of image restoration. First, we focus on learning scalar weights and
convolutional filters defining a Field of Experts regularizer to restore
natural images degraded by blur and noise. For improving the practical
performance, the lower-level problem is solved by means of a gradient descent
scheme combined with a line-search strategy based on the Barzilai-Borwein rule.
As a second application, the bilevel setup is employed for learning a
discretization of the popular total variation regularizer for solving image
restoration problems (in particular, deblurring and super-resolution).
Numerical results show the effectiveness of the approach and their
generalization to multiple tasks.Comment: Acknowledgments correcte