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The minimization of a functional consisting of a combined L ¹/L ²-data-fidelity term and a total variation term, named L ¹-L ²-TV model, is considered to remove a mixture of Gaussian and impulse noise in images, which are possibly additionally deformed by some convolution operator. We investigate analytically the stability of this model with respec...
Citations
... Small values of and lead to an oversmoothed reconstruction, which eliminates both noise and detail in the image. In contrast, large values of and retain noise [15]. An improvement of the 1 2 / model has been proposed in [16], where ‖ ‖ 1 replaces the TV. ...
Inspired by the ROF model and the $ {L}^{1}/TV $ image denoising model, we propose a combined model to remove Gaussian noise and salt-and-pepper noise simultaneously. This model combines the $ {L}^{1} $ -data fidelity term, $ {L}^{2} $ -data fidelity term and a fractional-order total variation regularization term, and is termed the $ {L}^{1}{L}^{2}/{TV}^{\alpha } $ model. We have used the proximity algorithm to solve the proposed model. Through this method, the non-differentiable term is solved by using the fixed-point equations of the proximity operator. The numerical experiments show that the proposed model can effectively remove Gaussian noise and salt and pepper noise through implementation of the proximity algorithm. As we varied the fractional order $ \alpha $ from 0.8 to 1.9 in increments of 0.1, we observed that different images correspond to different optimal values of α.
... It is demonstrated in [36,43,45] that optimization problem (1) is well suited to the task of removing a mixture of Gaussian and impulse noise. Moreover it is easy to see that (1) is a generalization of two well-known total variation models. ...
Based on the Fenchel duality we build a primal-dual framework for minimizing a general functional consisting of a combined L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document} and L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} data-fidelity term and a scalar or vectorial total variation regularisation term. The minimization is performed over the space of functions of bounded variations and appropriate discrete subspaces. We analyze the existence and uniqueness of solutions of the respective minimization problems. For computing a numerical solution we derive a semi-smooth Newton method on finite element spaces and highlight applications in denoising, inpainting and optical flow estimation.
... Moreover, the combination of data-fidelity terms has been considered. Specifically, a linear combination of the L 1 and L 2 data-fidelity terms was considered [50,51]. The removal of mixed Poisson and Gaussian (MPG) has also been extensively studied. ...
In this article, we propose a novel variational model for restoring images in the presence of the mixture of Cauchy and Gaussian noise. The model involves a novel data-fidelity term that features the mixed noise as an infimal convolution of two noise distributions and total variation regularization. This data-fidelity term contributes to suitable separation of Cauchy noise and Gaussian noise components, facilitating simultaneous removal of the mixed noise. Besides, the total variation regularization enables adequate denoising in homogeneous regions while conserving edges. Despite the nonconvexity of the model, the existence of a solution is proven. By employing an alternating minimization approach and the alternating direction method of multipliers, we present an iterative algorithm for solving the proposed model. Experimental results validate the effectiveness of the proposed model compared to other existing models according to both visual quality and some image quality measurements.
... Although these approaches have demonstrated their effectiveness in numerous fields of appli-cation, the treatment of these two noises leads to the loss of the image characteristics. Recently, in [16], a statistical characterization of the mixed Gaussian-impulse noise is used to perform a fully automated parameter algorithm for the combined L 1 -L 2 norm. This approach has proven its robustness in removing the mixture of Gaussian and impulse noise keeping safe the image edges. ...
In this paper, we are interested in the mathematical and simulation study of a new non-convex constrained PDE to remove the mixture of Gaussian–impulse noise densities. The model incorporates a non-convex data-fidelity term with a fractional constrained PDE. In addition, we adopt a non-smooth primal-dual algorithm to resolve the obtained proximal linearized minimization problem. The non-convex fidelity term is used to handle the high-frequency of the impulse noise component, while the fractional operator enables the efficient denoising of smooth areas, avoiding also the staircasing effect that appears on the relevant variational denoising models. Moreover, the proposed primal-dual algorithm helps in preserving fine structures and texture with good convergence rate. Numerical experiments, including ultrasound images, show that the proposed non-convex constrained PDE produces better denoising results compared to the state-of-the-art denoising models.
... The total variation (TV) functional is popular as a regularizer in imaging and inverse problems; see for instance [11,51,110,139] and [157,Chapter 8]. For a real-valued function u ∈ W 1,1 (Ω) on a bounded domain Ω in R 2 , the total variation seminorm is (4.1) ...
... Automatic parameter selection strategies can clearly be applied here as well, but this is out of the scope of the present thesis. We refer the reader to [64,110] for examples of such strategies. As is expected and well known, the use of surface area regularization leads to results in which the identified inclusion Γ 1 is smoothed out. ...
This thesis is concerned with applying the total variation (TV) regularizer to surfaces and different types of shape optimization problems. The resulting problems are challenging since they suffer from the non-differentiability of the TV-seminorm, but unlike most other priors it favors piecewise constant solutions, which results in piecewise flat geometries for shape optimization problems.The first part of this thesis deals with an analogue of the TV image reconstruction approach [Rudin, Osher, Fatemi (Physica D, 1992)] for images on smooth surfaces. A rigorous analytical framework is developed for this model and its Fenchel predual, which is a quadratic optimization problem with pointwise inequality constraints on the surface. A function space interior point method is proposed to solve it. Afterwards, a discrete variant (DTV) based on a nodal quadrature formula is defined for piecewise polynomial, globally discontinuous and continuous finite element functions on triangulated surface meshes. DTV has favorable properties, which include a convenient dual representation. Next, an analogue of the total variation prior for the normal vector field along the boundary of smooth shapes in 3D is introduced. Its analysis is based on a differential geometric setting in which the unit normal vector is viewed as an element of the two-dimensional sphere manifold. Shape calculus is used to characterize the relevant derivatives and an variant of the split Bregman method for manifold valued functions is proposed. This is followed by an extension of the total variation prior for the normal vector field for piecewise flat surfaces and the previous variant of split Bregman method is adapted. Numerical experiments confirm that the new prior favours polyhedral shapes.
... Such fidelities were studied e.g. in [43] and allow to simultaneously handle data from different modalities. Furthermore, in [44][45][46] fidelites of L 1 + L 2 -type were analysed and used for image restoration in the presence of mixed Gaussian and impulse noise. If H 1 and H 2 are proper, it holds Furthermore, under the hypothesis that H 1 is coercive, H 2 is bounded from below, and both are weakly- * lower semicontinuous convex functions, it holds that H is weakly- * lower semicontinuous, proper, and exact (see [49] for the statement and [50] for a proof on Hilbert spaces which generalises to Banach spaces). ...
We study variational regularisation methods for inverse problems with imperfect forward operators whose errors can be modelled by order intervals in a partial order of a Banach lattice. We carry out analysis with respect to existence and convex duality for general data fidelity terms and regularisation functionals. Both for a priori and a posteriori parameter choice rules, we obtain convergence rates of the regularised solutions in terms of Bregman distances. Our results apply to fidelity terms such as Wasserstein distances, phi-divergences, norms, as well as sums and infimal convolutions of those.
... With these developments, researchers of different fields are likely to be attracted to use data learning in their works. In the context of inverse problems, in addition to data usage for informing and validating inverse problems solutions [6,9,29], applying learning ability from training data to inform model selection in inverse problems has more recently been given real attention in the community [14,15,20,21,23,32,34,35]. Among these challenges, the papers [20,34] are pioneer works in signal denoising by solving a bi-level optimization problem that arises from the merge between learning and inverse problems. ...
By the recent advances in computer technology leading to the invention of more powerful processors, the importance of creating models using data training is even greater than ever. Given the significance of this issue, this theory tries to establish a connection among Machine Learning, Inverse Problems, and Applied Harmonic Analysis. Inspired by methods introduced in [12, 17, 22, 30], which are connections between Wavelet and Inverse Problems, we offer a model with the capability of learning in terms of an application in signal processing. In order to reach this model, a bi-level optimization problem will have to be faced. For solving this, a sequence of step functions is presented that its convergence to the solution will be proved. Each of these step functions derives from several constrained optimization problems on $\mathbb{R^n}$ that will be introduced here.
... Such fidelities were studied e.g. in [43] and allow to simultaneously handle data from different modalities. Furthermore, in [44,45,46] fidelites of L 1 + L 2 -type were analysed and used for image restoration in the presence of mixed Gaussian and impulse noise. If H 1 and H 2 are proper, it holds (3.26) where the term on the right hand side is the so-called infimal convolution of H 1 and H 2 . ...
We study variational regularisation methods for inverse problems with imperfect forward operators whose errors can be modelled by order intervals in a partial order of a Banach lattice. We carry out analysis with respect to existence and convex duality for general data fidelity terms and regularisation functionals. Both for a-priori and a-posteriori parameter choice rules, we obtain convergence rates of the regularized solutions in terms of Bregman distances. Our results apply to fidelity terms such as Wasserstein distances, f-divergences, norms, as well as sums and infimal convolutions of those.
... Automatic parameter selection strategies can clearly be applied here as well, but this is out of the scope of the present paper. We refer the reader to De los Reyes et al (2017) and Langer (2017) for examples of such strategies. ...
An analogue of the total variation prior for the normal vector field along the boundary of piecewise flat shapes in 3D is introduced. A major class of examples are triangulated surfaces as they occur for instance in finite element computations. The analysis of the functional is based on a differential geometric setting in which the unit normal vector is viewed as an element of the two-dimensional sphere manifold. It is found to agree with the discrete total mean curvature known in discrete differential geometry. A split Bregman iteration is proposed for the solution of discretized shape optimization problems, in which the total variation of the normal appears as a regularizer. Unlike most other priors, such as surface area, the new functional allows for piecewise flat shapes. As two applications, a mesh denoising and a geometric inverse problem of inclusion detection type involving a partial differential equation are considered. Numerical experiments confirm that polyhedral shapes can be identified quite accurately.
... The total variation (TV) functional is popular as a regularizer in imaging and inverse problems; see for instance Rudin et al (1992), Chan et al (1999), Vogel (2002, Chapter 8), Bachmayr and Burger (2009) and Langer (2017). For a real-valued function u ∈ W 1,1 (Ω) on a bounded domain Ω in R 2 , the total variation seminorm is defined as |u| TV(Ω) := Ω |∇u| 2 d x = Ω |(Du) e 1 | 2 + |(Du) e 2 | 2 1/2 . ...
An analogue of the total variation prior for the normal vector field along the boundary of smooth shapes in 3D is introduced. The analysis of the total variation of the normal vector field is based on a differential geometric setting in which the unit normal vector is viewed as an element of the two-dimensional sphere manifold. It is shown that spheres are stationary points when the total variation of the normal is minimized under an area constraint. Shape calculus is used to characterize the relevant derivatives. Since the total variation functional is non-differentiable whenever the boundary contains flat regions, an extension of the split Bregman method to manifold valued functions is proposed.
