1,721,008 research outputs found
Analysis and automatic parameter selection of a variational model for mixed Gaussian and salt-and-pepper noise removal
No full textWe analyse a variational regularisation problem for mixed noise removal that has been recently proposed in Calatroni et al (2017 SIAM J. Imaging Sci. 10 1196-233). The data discrepancy term of the model combines L 1 and L 2 terms in an infimal convolution fashion and it is appropriate for the joint removal of Gaussian and Salt & Pepper noise. Here, we perform a fine analysis of the model that emphasises the balancing effect of the two parameters appearing in the discrepancy term. Namely, we study the asymptotic behaviour of the model for large and small values of these parameters and we compare its solutions to the ones of the corresponding variational models with L 1 and L 2 data fidelity. Extensions to the general linear inverse problems setting are also discussed. Furthermore, we compute exact solutions to the denoising problem, for simple data functions in the case of total variation regularisation. Using these theoretical results, we then analytically study a bilevel optimisation strategy for the automatic selection of model parameters by means of a training set. Finally, we report some numerical results which confirm the validity of our analysis and the use of popular data models in the case of 'blind' optimal noise model selection
Backtracking strategies for accelerated descent methods with smooth composite objectives
No full textWe present and analyze a backtracking strategy for a general fast iterative shrinkage/thresholding algorithm proposed by Chambolle and Pock [Acta Numer., 25 (2016), pp. 161–319] for strongly convex composite objective functions. Unlike classical Armijo-type line searching, our backtracking rule allows for local increasing and decreasing of the descent step size (i.e., proximal parameter) along the iterations. We prove accelerated convergence rates and show numerical results for some exemplar problems
Efficient l Gradient-Based Super-Resolution for Simplified Image Segmentation
Full text linkWe consider a variational model for single-image super-resolution based on the assumption that the gradient of the target image is sparse. We enforce this assumption by considering both an isotropic and an anisotropic l 0 regularisation on the image gradient combined with a quadratic data fidelity, similarly as studied in [1] for signal recovery problems. For the numerical realisation of the model, we propose a novel efficient ADMM splitting algorithm whose substeps solutions are computed efficiently by means of hard-thresholding and standard conjugate-gradient solvers. We test our model on highly-degraded synthetic and real-world data and quantitatively compare our results with several sparsity-promoting variational approaches as well as with state-of-the-art deep-learning techniques. Our experiments show that thanks to the l 0 smoothing on the gradient, the super-resolved images can be used to improve the accuracy of standard segmentation algorithms for applications like QR codes and cell detection and land-cover classification problems
ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing
No full textWe present directional operator splitting schemes for the numerical solution of a fourth-order, nonlinear partial differential evolution equation which arises in image processing. This equation constitutes the H -1-gradient flow of the total variation and represents a prototype of higher-order equations of similar type which are popular in imaging for denoising, deblurring and inpainting problems. The efficient numerical solution of this equation is very challenging due to the stiffness of most numerical schemes. We show that the combination of directional splitting schemes with implicit time-stepping provides a stable and computationally cheap numerical realisation of the equation
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]
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
A Unified Surface Geometric Framework for Feature-Aware Denoising, Hole Filling and Context-Aware Completion
Full text linkTechnologies for 3D data acquisition and 3D printing have enormously developed in the past few years, and, consequently, the demand for 3D virtual twins of the original scanned objects has increased. In this context, feature-aware denoising, hole filling and context-aware completion are three essential (but far from trivial) tasks. In this work, they are integrated within a geometric framework and realized through a unified variational model aiming at recovering triangulated surfaces from scanned, damaged and possibly incomplete noisy observations. The underlying non-convex optimization problem incorporates two regularisation terms: a discrete approximation of the Willmore energy forcing local sphericity and suited for the recovery of rounded features, and an approximation of the l(0) pseudo-norm penalty favouring sparsity in the normal variation. The proposed numerical method solving the model is parameterization-free, avoids expensive implicit volumebased computations and based on the efficient use of the Alternating Direction Method of Multipliers. Experiments show how the proposed framework can provide a robust and elegant solution suited for accurate restorations even in the presence of severe random noise and large damaged areas
Whiteness-based bilevel learning of regularization parameters in imaging
Full text linkWe consider an unsupervised bilevel optimization strategy for learning regularization parameters in the context of imaging inverse problems in the presence of additive white Gaussian noise. Compared to supervised and weakly-supervised metrics relying either on the prior knowledge of reference data and/or on some (partial) knowledge on the noise statistics, the proposed approach optimizes the whiteness of the residual between the observed data and the observation model with no need of ground-truth data. We validate the approach on standard Total Variation-regularized image deconvolution problems which show that the proposed quality metric provides estimates close to the mean-square error oracle and to discrepancy-based principles
On and Beyond Total Variation Regularization in Imaging: The Role of Space Variance
Full text linkOver the last 30 years a plethora of variational regularization models for image reconstruction have been proposed and thoroughly inspected by the applied mathematics community. Among them, the pioneering prototype often taught and learned in basic courses in mathematical image processing is the celebrated Rudin-Osher-Fatemi (ROF) model [L. I. Rudin, S. Osher, and E. Fatemi, Phys. D, 60 (1992), pp. 259-268], which relies on the minimization of the edge-preserving total variation (TV) seminorm as a regularization term. Despite its (often limiting) simplicity, this model is still very much employed in many applications and used as a benchmark for assessing the performance of modern learning-based image reconstruction approaches, thanks to its thorough analytical and numerical understanding. Among the many extensions to TV proposed over the years, a large class is based on the concept of space variance. Space-variant models can indeed overcome the intrinsic inability of TV to describe local features (strength, sharpness, directionality) by means of an adaptive mathematical modeling which accommodates local regularization weighting, variable smoothness, and anisotropy. Those ideas can further be cast in the flexible Bayesian framework of generalized Gaussian distributions and combined with maximum likelihood and hierarchical optimization approaches for efficient hyperparameter estimation. In this work, we review and connect the major contributions in the field of space-variant TV-type image reconstruction models, focusing, in particular, on their Bayesian interpretation which paves the way to new exciting and unexplored research directions
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