1,720,972 research outputs found
Automatic fidelity and regularization terms selection in variational image restoration
Full text linkThis paper addresses the study of a class of variational models for the image restoration inverse problem. The main assumption is that the additive noise model and the image gradient magnitudes follow a generalized normal (GN) distribution, whose very flexible probability density function (pdf) is characterized by two parameters—typically unknown in real world applications—determining its shape and scale. The unknown image and parameters, which are both modeled as random variables in light of the hierarchical Bayesian perspective adopted here, are jointly automatically estimated within a Maximum A Posteriori (MAP) framework. The hypermodels resulting from the selected prior, likelihood and hyperprior pdfs are minimized by means of an alternating scheme which benefits from a robust initialization based on the noise whiteness property. For the minimization problem with respect to the image, the Alternating Direction Method of Multipliers (ADMM) algorithm, which takes advantage of efficient procedures for the solution of proximal maps, is employed. Computed examples show that the proposed approach holds the potential to automatically detect the noise distribution, and it is also well-suited to process a wide range of images
Sparsity promoting hybrid solvers for hierarchical bayesian inverse problems
Full text linkThe recovery of sparse generative models from few noisy measurements is an important and challenging problem. Many deterministic algorithms rely on some form of l1-l2 minimization to combine the computational convenience of the l2 penalty and the sparsity promotion of the l1. It was recently shown within the Bayesian framework that sparsity promotion and computational efficiency can be attained with hierarchical models with conditionally Gaussian priors and gamma hyperpriors. The related Gibbs energy function is a convex functional, and its minimizer, which is the maximum a posteriori (MAP) estimate of the posterior, can be computed efficiently with the globally convergent Iterated Alternating Sequential (IAS) algorithm [D. Calvetti, E. Somersalo, and A. Strang, Inverse Problems, 35 (2019), 035003]. Generalization of the hyperpriors for these sparsity promoting hierarchical models to a generalized gamma family either yield globally convex Gibbs energy functionals or can exhibit local convexity for some choices for the hyperparameters [D. Calvetti et al., Inverse Problems, 36 (2020), 025010]. The main problem in computing the MAP solution for greedy hyperpriors that strongly promote sparsity is the presence of local minima. To overcome the premature stopping at a spurious local minimizer, we propose two hybrid algorithms that first exploit the global convergence associated with gamma hyperpriors to arrive in a neighborhood of the unique minimizer and then adopt a generalized gamma hyperprior that promotes sparsity more strongly. The performance of the two algorithms is illustrated with computed examples
Adaptive parameter selection for weighted-TV image reconstruction problems
Full text linkWe propose an efficient estimation technique for the automatic selection of locally-Adaptive Total Variation regularisation parameters based on an hybrid strategy which combines a local maximum-likelihood approach estimating space-variant image scales with a global discrepancy principle related to noise statistics. We verify the effectiveness of the proposed approach solving some exemplar image reconstruction problems and show its outperformance in comparison to state-of-The-Art parameter estimation strategies, the former weighting locally the fit with the data [4], the latter relying on a bilevel learning paradigm [8, 9]
A comparison of parameter choice rules for lp - lq minimization
Full text linkImages that have been contaminated by various kinds of blur and noise can be restored by the minimization of an lp-lq functional. The quality of the reconstruction depends on the choice of a regularization parameter. Several approaches to determine this parameter have been described in the literature. This work presents a numerical comparison of known approaches as well as of a new one
Residual Whiteness Principle for Automatic Parameter Selection in l2 - l2 Image Super-Resolution Problems
Full text linkWe propose an automatic parameter selection strategy for variational image super-resolution of blurred and down-sampled images corrupted by additive white Gaussian noise (AWGN) with unknown standard deviation. By exploiting particular properties of the operators describing the problem in the frequency domain, our strategy selects the optimal parameter as the one optimising a suitable residual whiteness measure. Numerical tests show the effectiveness of the proposed strategy for generalised l2 - l2 Tikhonov problems
Space-Adaptive Anisotropic Bivariate Laplacian Regularization for Image Restoration
No full textIn this paper we present a new regularization term for variational image restoration which can be regarded as a space-variant anistropic extension of the classical Total Variation (TV) regularizer. The proposed regularizer comes from the statistical assumption that the gradients of the unknown target image distribute locally according to space-variant bivariate Laplacian distributions. The high flexibility of the proposed regularizer holds the potential for the effective modelling of local image properties, in particular driving in an adaptive manner the strength and the directionality of non-linear TV-diffusion. The free parameters of the regularizer are automatically set - and, eventually, updated - based on a robust Maximumum Likelihood estimation procedure. A minimization algorithm based on the Alternating Direction Method of Multipliers is presented for the efficient numerical solution of the proposed variational model. Some experimental results are reported. They demonstrate the high-quality of restorations achievable by the proposed model, in particular with respect to classical TV-regularized models
Residual whiteness principle for parameter-free image restoration. ETNA - Electronic Transactions on Numerical Analysis
Full text linkSelecting the regularization parameter in the image restoration variational framework is of crucial importance, since it can highly influence the quality of the final restoration. In this paper, we propose a parameter-free approach for automatically selecting the regularization parameter when the blur is space-invariant and known and the noise is additive white Gaussian with unknown standard deviation, based on the so-called residual whiteness principle. More precisely, the regularization parameter is required to minimize the residual whiteness function, namely the normalized auto-correlation of the residual image of the restoration. The proposed method can be applied to a wide class of variational models, such as those including in their formulation regularizers of Tikhonov and Total Variation type. For non-quadratic regularizers, the residual whiteness principle is nested in an iterative optimization scheme based on the alternating direction method of multipliers. The effectiveness of the proposed approach is verified by solving some test examples and performing a comparison with other parameter estimation state-of-the-art strategies, such as the discrepancy principle
A flexible space-variant anisotropic regularization for image restoration with automated parameter selection
No full textWe propose a new space-variant anisotropic regular ization term for variational image restoration, based on the statistical assumption that the gradients of the target image distribute locally according to a bivariate generalized Gaussian distribution. The highly flexible variational structure of the corresponding regularizer encodes several free parameters which hold the potential for faithfully modeling 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. A suitable preliminary variable splitting together with a novel result in multivariate nonconvex proximal calculus yield a very efficient minimization 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
A general framework for whiteness-based parameters selection in variational models
No full textIn this work, we extend the residual whiteness principle, originally proposed in (Lanza et al. in Electron Trans Numer Anal 53:329–352 2020) for the selection of a single regularization parameter in variational models for inverse problems under additive white noise corruption, to much broader scenarios. More specifically, we address the problem of estimating multiple parameters for imaging inverse problems subject to both white and non-white but whitenable noise corruptions, thus covering most of the application cases. The proposed parameter selection criterion, referred to as generalized whiteness principle, is formulated as a bilevel optimization problem. To circumvent the non-smoothness of the variational models typically employed in imaging problems—the non-smoothness representing a bottleneck in the bilevel set-up—we propose to adopt a derivative-free minimization algorithm for the solution of the designed bilevel problem. We refer to this novel numerical solution paradigm as bilevel derivative-free approach. Numerical tests highlight both the ability of the proposed generalized whiteness principle to effectively select multiple parameters and the significant advantages, in terms of computational cost, of the bilevel derivative-free numerical solution framework
ADMM-Based Residual Whiteness Principle for Automatic Parameter Selection in Single Image Super-Resolution Problems
No full textWe propose an automatic parameter selection strategy for the single image super-resolution problem for images corrupted by blur and additive white Gaussian noise with unknown standard deviation. The proposed approach exploits the structure of both the down-sampling and the blur operators in the frequency domain and computes the optimal regularisation parameter as the one optimising a suitably defined residual whiteness measure. Computationally, the proposed strategy relies on the fast solution of generalised Tikhonov l2–l2 problems as proposed in Zhao et al. (IEEE Trans Image Process 25:3683–3697, 2016). These problems naturally appear as substeps of the Alternating Direction Method of Multipliers used to solve single image super-resolution problems with non-quadratic, non-smooth, sparsity-promoting regularisers both in convex and in non-convex regimes. After detailing the theoretical properties allowing to express the whiteness functional in a compact way, we report an exhaustive list of numerical experiments proving the effectiveness of the proposed approach for different type of problems, in comparison with well-known parameter selection strategies such as, e.g., the discrepancy principle
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