Total Variation Flow Research Articles

On the behavior in time of the solutions to total variation flow

Abstract In this paper we study the regularity and the behavior in time of the solutions to the Dirichlet problem for the total variation flow in a bounded domain. We show that the regularity of the initial datum influences the behavior and can produce extinction in finite time and “immediately boundedness of the solutions”.

Adaptive Fractional-Order Multi-Scale Optimization TV-L1 Optical Flow Algorithm

We propose an adaptive fractional multi-scale optimization optical flow algorithm, which for the first time improves the over-smoothing of optical flow estimation under the total variation model from the perspective of global feature and local texture balance, and solves the problem that the convergence of fractional optical flow algorithms depends on the order parameter. Specifically, a fractional-order discrete L1-regularization Total Variational Optical Flow model is constructed. On this basis, the Ant Lion algorithm is innovatively used to realize the iterative calculation of the optical flow equation, and the fractional order is dynamically adjusted to obtain an adaptive optimization algorithm with strong search accuracy and high efficiency. In this paper, the flexibility of optical flow estimation in weak gradient texture scenes is increased, and the optical flow extraction rate of target features at multiple scales is greatly improved. We show excellent recognition performance and stability under the MPI_Sintel and Middlebury benchmarks.

Spectral Total-variation Processing of Shapes—Theory and Applications

We present a comprehensive analysis of total variation (TV) on non-Euclidean domains and its eigenfunctions. We specifically address parameterized surfaces, a natural representation of the shapes used in 3D graphics. Our work sheds new light on the celebrated Beltrami and Anisotropic TV flows and explains experimental findings from recent years on shape spectral TV [Fumero et al. 2020 ] and adaptive anisotropic spectral TV [Biton and Gilboa 2022 ]. A new notion of convexity on surfaces is derived by characterizing structures that are stable throughout the TV flow, performed on surfaces. We establish and numerically demonstrate quantitative relationships between TV, area, eigenvalue, and eigenfunctions of the TV operator on surfaces. Moreover, we expand the shape spectral TV toolkit to include zero-homogeneous flows, leading to efficient and versatile shape processing methods. These methods are exemplified through applications in smoothing, enhancement, and exaggeration filters. We introduce a novel method that, for the first time, addresses the shape deformation task using TV. This deformation technique is characterized by the concentration of deformation along geometrical bottlenecks, shown to coincide with the discontinuities of eigenfunctions. Overall, our findings elucidate recent experimental observations in spectral TV, provide a diverse framework for shape filtering, and present the first TV-based approach to shape deformation.

Doubly nonlinear equations for the 1-Laplacian

This paper is concerned with the Neumann problem for a class of doubly nonlinear equations for the 1-Laplacian, ∂v∂t-Δ1u∋0in(0,∞)×Ω,v∈γ(u),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\frac{\\partial v}{\\partial t} - \\Delta _1 u \ i 0 \\hbox { in } (0, \\infty ) \ imes \\Omega , \\quad v\\in \\gamma (u), \\end{aligned}$$\\end{document}and initial data in L1(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1(\\Omega )$$\\end{document}, where Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} is a bounded smooth domain in RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^N$$\\end{document} and γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document} is a maximal monotone graph in R×R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}\ imes {\\mathbb {R}}$$\\end{document}. We prove that, under certain assumptions on the graph γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document}, there is existence and uniqueness of solutions. Moreover, we proof that these solutions coincide with the ones of the Neumann problem for the total variational flow. We show that such assumptions are necessary.

Numerical approximation of probabilistically weak and strong solutions of the stochastic total variation flow

We propose a fully practical numerical scheme for the simulation of the stochastic total variation flow (STVF). The approximation is based on a stable time-implicit finite element space-time approximation of a regularized STVF equation. The approximation also involves a finite dimensional discretization of the noise that makes the scheme fully implementable on physical hardware. We show that the proposed numerical scheme converges in law to a solution that is defined in the sense of stochastic variational inequalities (SVIs). Under strengthened assumptions the convergence can be show to holds even in probability. As a by product of our convergence analysis we provide a generalization of the concept of probabilistically weak solutions of stochastic partial differential equation (SPDEs) to the setting of SVIs. We also prove convergence of the numerical scheme to a probabilistically strong solution in probability if pathwise uniqueness holds. We perform numerical simulations to illustrate the behavior of the proposed numerical scheme as well as its non-conforming variant in the context of image denoising.

Weak solutions to gradient flows in metric measure spaces

AbstractDue to the fact that in a metric space there is (in general) no notion of directional derivatives, the definition of solutions to gradient flows in metric measure spaces necessarily avoid their direct use. The most classical example is the heat flow, or more generally the p‐Laplacian evolution equation, which has been studied as the gradient flow in L2 of the p‐Cheeger energy. Typically, solutions are defined using the semigroup theory through a subdifferential of the energy. Other popular approaches include variational solutions and the weighted energy‐dissipation formalism. In this paper, using the first‐order differential structure on a metric measure space introduced by Gigli, we characterize the subdifferential in L2 of the p‐Cheeger energy. This gives rise to a new notion of solutions to the p‐Laplacian evolution equation in metric measure spaces. In the case p = 1, we introduce a metric analogue of the Anzellotti pairing and obtain a Green‐Gauss formula, which is then used in place of Gigli's structure to characterise the 1‐Laplacian operator and study the total variation flow.

The fourth-order total variation flow in $ \mathbb{R}^n $

<abstract><p>We define rigorously a solution to the fourth-order total variation flow equation in $ \mathbb{R}^n $. If $ n\geq3 $, it can be understood as a gradient flow of the total variation energy in $ D^{-1} $, the dual space of $ D^1_0 $, which is the completion of the space of compactly supported smooth functions in the Dirichlet norm. However, in the low dimensional case $ n\leq2 $, the space $ D^{-1} $ does not contain characteristic functions of sets of positive measure, so we extend the notion of solution to a larger space. We characterize the solution in terms of what is called the Cahn-Hoffman vector field, based on a duality argument. This argument relies on an approximation lemma which itself is interesting. We introduce a notion of calibrability of a set in our fourth-order setting. This notion is related to whether a characteristic function preserves its form throughout the evolution. It turns out that all balls are calibrable. However, unlike in the second-order total variation flow, the outside of a ball is calibrable if and only if $ n\neq2 $. If $ n\neq2 $, all annuli are calibrable, while in the case $ n = 2 $, if an annulus is too thick, it is not calibrable. We compute explicitly the solution emanating from the characteristic function of a ball. We also provide a description of the solution emanating from any piecewise constant, radially symmetric datum in terms of a system of ODEs.</p></abstract>

Some qualitative properties for the Kirchhoff total variation flow

The first goal of this paper is to establish a result on the existence and uniqueness of solution to an initial-boundary value problem for parabolic equations of Kirchhoff type involving the 1-Laplace operator. The second goal is to discuss some qualitative properties, such as asymptotic behaviour and the extinction of solutions for the considered problem.

A Posteriori Estimates for the Stochastic Total Variation Flow

We derive a posteriori error estimates for a fully discrete time-implicit finite element approximation of the stochastic total variaton flow (STVF) with additive space time noise. The estimates are first derived for an implementable fully discrete approximation of a regularized stochastic total variation flow. We then show that the derived a posteriori estimates remain valid for the unregularized flow up to a perturbation term that can be controlled by the regularization parameter. Based on the derived a posteriori estimates we propose a pathwise algorithm for the adaptive space-time refinement and perform numerical simulation for the regularized STVF to demonstrate the behavior of the proposed algorithm.

On the p-Laplacian evolution equation in metric measure spaces

The p-Laplacian evolution equation in metric measure spaces has been studied as the gradient flow in L2 of the p-Cheeger energy (for 1<p<∞). In this paper, using the first-order differential structure on a metric measure space introduced by Gigli, we characterise the subdifferential in L2 of the p-Cheeger energy. This gives rise to a new definition of the p-Laplacian operator in metric measure spaces, which allows us to work with this operator in more detail. In this way, we introduce a new notion of solutions to the p-Laplacian evolution equation in metric measure spaces. For p=1, we obtain a Green-Gauss formula similar to the one by Anzellotti for Euclidean spaces, and use it to characterise the 1-Laplacian operator and study the total variation flow. We also study the asymptotic behaviour of the solutions of the p-Laplacian evolution equation, showing that for 1≤p<2 we have finite extinction time.

The Neumann and Dirichlet problems for the total variation flow in metric measure spaces

Abstract We study the Neumann and Dirichlet problems for the total variation flow in doubling metric measure spaces supporting a weak Poincaré inequality. We prove existence and uniqueness of weak solutions and study their asymptotic behavior. Furthermore, in the Neumann problem we provide a notion of solutions which is valid for L 1 {L^{1}} initial data, as well as prove their existence and uniqueness. Our main tools are the first-order linear differential structure due to Gigli and a version of the Gauss–Green formula.

Variational solutions to the total variation flow on metric measure spaces

We discuss a purely variational approach to the total variation flow on metric measure spaces with a doubling measure and a Poincaré inequality. We apply the concept of parabolic De Giorgi classes together with upper gradients, Newtonian spaces and functions of bounded variation to prove a necessary and sufficient condition for a variational solution to be continuous at a given point.

On the definition of solution to the total variation flow

We show that the notions of weak solution to the total variation flow based on the Anzellotti pairing and the variational inequality coincide under some restrictions on the boundary data. The key ingredient in the argument is a duality result for the total variation functional, which is based on an approximation of the total variation by area-type functionals.

Existence of parabolic minimizers to the total variation flow on metric measure spaces

We give an existence proof for variational solutions u associated to the total variation flow. Here, the functions being considered are defined on a metric measure space ({mathcal {X}}, d, mu ) satisfying a doubling condition and supporting a Poincaré inequality. For such parabolic minimizers that coincide with a time-independent Cauchy–Dirichlet datum u_0 on the parabolic boundary of a space-time-cylinder Omega times (0, T) with Omega subset {mathcal {X}} an open set and T > 0, we prove existence in the weak parabolic function space L^1_w(0, T; mathrm {BV}(Omega )). In this paper, we generalize results from a previous work by Bögelein, Duzaar and Marcellini by introducing a more abstract notion for mathrm {BV}-valued parabolic function spaces. We argue completely on a variational level.

The total variation flow in metric graphs

&lt;abstract&gt;&lt;p&gt;Our aim is to study the total variation flow in metric graphs. First, we define the functions of bounded variation in metric graphs and their total variation, we also give an integration by parts formula. We prove existence and uniqueness of solutions and that the solutions reach the mean of the initial data in finite time. Moreover, we obtain explicit solutions.&lt;/p&gt;&lt;/abstract&gt;

Adaptive Extraction of Oil Painting Texture Features Based on Reaction Diffusion Equation

The oil painting retrieval technology based on the reaction diffusion equation has attracted widespread attention in the fields of oil painting processing and pattern recognition. The description and extraction of oil painting information and the classification method of oil paintings are two important processes in content-based oil painting retrieval. Inspired by the restoration and decomposition functional model of equal oil painting, we propose a reaction diffusion equation model. The new model contains two reaction diffusion equations with different principal parts. One principal part is total variation diffusion, which is used to remove noise. The other main part is thermal diffusion, which is used to modify the source term of the denoising reaction-diffusion equation to achieve the effect of protecting the texture of the oil painting. The interaction of the two reaction-diffusion equations finally achieves denoising while maintaining the boundaries and textures. Under the framework of the above reaction diffusion equation model, we introduce Laplace flow to replace the original total variation flow, so that the new denoising reaction diffusion equation combines the isotropic diffusion and total variation flow of the thermal reaction diffusion equation to achieve the effect of adaptive theoretical research. Using regularization methods and methods, we, respectively, get the well-posedness of the two model solutions, which provides the necessary preparation for numerical calculations. Based on the statistical theory and classification principles of support vector machines, combined with the characteristics of oil painting classification, the research and analysis are carried out from the three important aspects of kernel function, training algorithm, and multiclass classifier algorithm that affect the classification effect and speed. Numerical experiments show that the given filter model has a better processing effect on images with different types and different degrees of noise pollution. On this basis, an oil painting classification system based on texture features is designed, combined with an improved gray-level cooccurrence matrix algorithm and a multiclass support vector machine classification model, to extract, train, and classify oil paintings. Experiments with three types of oil paintings prove that the system can achieve a good oil painting classification effect. Different from the original model, the new model is based on the framework of reaction-diffusion equations. In addition, the new model has good effects in removing step effects, maintaining boundaries and denoising, especially in maintaining texture.

Space–time adaptive splitting scheme for the numerical simulation of polycrystallization

We consider a space–time adaptive splitting scheme for polycrystallization processes described by a two-field phase field model. The phase field model consists of a coupled system of evolutionary processes for the local degree of crystallinity ϕ and the orientation angle Θ one of them being of first order total variation flow type. The splitting scheme is based on an implicit discretization in time which allows a decoupling of the system in the sense that at each time step minimization problems in ϕ and Θ have to be solved successively. The discretization in space is taken care of by a standard finite element approximation for the problem in ϕ and an Interior Penalty Discontinuous Galerkin (IPDG) approximation for the one in Θ. The adaptivity in space relies on equilibrated a posteriori error estimators for the discretization errors in ϕ and Θ in terms of primal and dual energy functionals associated with the respective minimization problems. The adaptive time stepping is dictated by the convergence of a semismooth Newton method for the numerical solution of the nonlinear problem in Θ. Numerical results illustrate the performance of the adaptive space–time splitting scheme for two representative polycrystallization processes.

Gradient flows in metric random walk spaces

Recently, motivated by problems in image processing, by the analysis of the peridynamic formulation of the continuous mechanic and by the study of Markov jump processes, there has been an increasing interest in the research of nonlocal partial differential equations. In the last years and with these problems in mind, we have studied some gradient flows in the general framework of a metric random walk space, that is, a Polish metric space (X, d) together with a probability measure assigned to each xin X, which encode the jumps of a Markov process. In this way, we have unified into a broad framework the study of partial differential equations in weighted discrete graphs and in other nonlocal models of interest. Our aim here is to provide a summary of the results that we have obtained for the heat flow and the total variational flow in metric random walk spaces. Moreover, some of our results on other problems related to the diffusion operators involved in such processes are also included, like the ones for evolution problems of p-Laplacian type with nonhomogeneous Neumann boundary conditions.

Image Details Enhancement Method via Spectrum Total Variation

In order to enhance the texture details of images and preserve the complete structure of images, we propose a novel image detail enhancement method based on spectral total variation, which combines the total variation method and spectral analysis method. Firstly, the total variation flow, as a non-ideal high-pass filter, decomposes image and obtains the spectral characteristics of different time scales in the transform domain. The transform domain can be regarded as a frequency domain. Then, we construct a special filter in the transform domain, and the spectrum features are filtered and enhanced by the filter. Finally, according to the proposed reconstruction method, the filtering results in the transform domain are inversely converted to the spatial domain, that is, the final enhanced image is obtained. The experimental results show that the proposed spectral total variation method can effectively use the spectral information of the image to enhance the local details, so that the enhanced image can obtain more prominent texture information, while maintaining the integrity of the image structure.

Image decomposition based on nonlinear reaction–diffusion systems

The separation of image content into semantic parts plays a vital role in applications such as compression, enhancement, restoration, and more. In recent years, several pioneering works suggested such a separation be based on variational formulation and others using independent component analysis and sparsity. This paper presents a novel method for separating images into texture and piecewise smooth (cartoon) parts. The new model is based on a class of degenerate and singular reaction–diffusion systems coupled by the minimizing total variation flow and the p-Laplace flow. The existence and uniqueness of entropy solutions to the system with BV initial data and Neumann boundary conditions are established by the regularization method. Experimental results illustrate that the new model can preserve textures better than other methods.