Bounded Variation Research Articles

Total generalized variation restoration with non-quadratic fidelity

Total variation (TV) based Models have been widely used in image restoration problems. However, these models are always accompanied by staircase effect due to the property of bounded variation (BV) space. In this paper, we present two high order variational models based on total generalized variation (TGV) with two common and important non-quadratic fidelity data terms for blurred images corrupted by impulsive and Poisson noises. Since the direct extension of alternative direction method of multipliers (ADMM) to solve three-block convex minimization problems is not necessarily convergent, we develop an efficient algorithm called Prediction–Correction ADMM to solve our models and also show the convergence of the proposed method. Moreover, we extend our models to deal with color images restoration. Numerical experiments demonstrate that the proposed high order models can reduce staircase effect while preserving edges and outperform classical TV based models in SNR and SSIM values.

The initial–boundary value problem for general non-local scalar conservation laws in one space dimension

We prove a global well-posedness result for a class of weak entropy solutions of bounded variation (BV) of scalar conservation laws with non-local flux on bounded domains, under suitable regularity assumptions on the flux function. In particular, existence is obtained by proving the convergence of an adapted Lax–Friedrichs algorithm. Lipschitz continuous dependence from initial and boundary data is derived applying Kružhkov’s doubling of variable technique.

Continuous dependence estimate for a degenerate parabolic–hyperbolic equation with Lévy noise

In this article, we are concerned with a multidimensional degenerate parabolic–hyperbolic equation driven by Levy processes. Using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities of the entropy solutions under the assumption that Levy noise depends only on the solution. This result is used to show the error estimate for the stochastic vanishing viscosity method. In addition, we establish fractional BV estimate for vanishing viscosity approximations in case the noise coefficients depend on both the solution and spatial variable.

On the global well-posedness of BV weak solutions to the Kuramoto–Sakaguchi equation

The Kuramoto model is a prototype phase model describing the synchronous behavior of weakly coupled limit-cycle oscillators. When the number of oscillators is sufficiently large, the dynamics of Kuramoto ensemble can be effectively approximated by the corresponding mean-field equation, namely “the Kuramoto–Sakaguchi (KS) equation”. This KS equation is a kind of scalar conservation law with a nonlocal flux function due to the mean-field interactions among oscillators. In this paper, we provide a unique global solvability of bounded variation (BV) weak solutions to the kinetic KS equation for identical oscillators using the method of front-tracking in hyperbolic conservation laws. Moreover, we also show that our BV weak solutions satisfy local-in-time L1-stability with respect to BV-initial data. For the ensemble of identical Kuramoto oscillators, we explicitly construct an exponentially growing BV weak solution generated from BV perturbation of incoherent state for any positive coupling strength. This implies the nonlinear instability of incoherent state in a positive coupling strength regime. We provide several numerical examples and compare them with our analytical results.

ITERATED FUNCTION SYSTEMS ON FUNCTIONS OF BOUNDED VARIATION

We show that under certain hypotheses, an iterated function system on mappings (IFSM) is a contraction on the complete space of functions of bounded variation (BV). It then possesses a unique attractor of BV. Some BV-based inverse problems based on the Collage Theorem for contraction maps are considered.

Weakly regular fluid flows with bounded variation on the domain of outer communication of a Schwarzschild black hole spacetime

We study the global dynamics of isothermal fluids evolving in the domain of outer communication of a Schwarzschild black hole. We first formulate the initial value problem within a class of weak solutions with bounded variation (BV), possibly containing shock waves. We then introduce a version of the random choice method and establish a global-in-time existence theory for the initial value problem within the proposed class of weakly regular fluid flows. The initial data may have arbitrary large bounded variation and can possibly blow up near the horizon of the black hole. Furthermore, we study the class of possibly discontinuous, equilibrium solutions and design a version of the random choice method in which these fluid equilibria are exactly preserved. This leads us to a nonlinear stability property for fluid equilibria under small perturbations with bounded variation. Furthermore, we can also encompass several limiting regimes (stiff matter, non-relativistic flows, extremal black hole) by letting the physical parameters (mass of the black hole, light speed, sound speed) reach extremal values.

Quantitative decay rates for dispersive solutions to the Einstein-scalar field system in spherical symmetry

In this paper, we study the future causally geodesically complete solutions of the spherically symmetric Einstein-scalar field system. Under the a priori assumption that the scalar field $\phi$ scatters locally in the scale-invariant bounded-variation (BV) norm, we prove that $\phi$ and its derivatives decay polynomially. Moreover, we show that the decay rates are sharp. In particular, we obtain sharp quantitative decay for the class of global solutions with small BV norms constructed by Christodoulou. As a consequence of our results, for every future causally geodesically complete solution with sufficiently regular initial data, we show the dichotomy that either the sharp power law tail holds or that the spacetime blows up at infinity in the sense that some scale invariant spacetime norms blow up.

Continuous dependence estimate for conservation laws with Lévy noise

We are concerned with multidimensional stochastic balance laws driven by Lévy processes. Using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities of the entropy solutions under the assumption that Lévy noise only depends on the solution. This result is used to show the error estimate for the stochastic vanishing viscosity method. In addition, we establish fractional BV estimate for vanishing viscosity approximations in case the noise coefficient depends on both the solution and spatial variable.

Convergence of a level‐set algorithm in scalar conservation laws

This article is concerned with the convergence of the level‐set algorithm introduced by Aslam (J Comput Phys 167 (2001), 413–438) for tracking the discontinuities in scalar conservation laws in the case of linear or strictly convex flux function. The numerical method is deduced by the level‐set representation of the entropy solution: the zero of a level‐set function is used as an indicator of the discontinuity curves and two auxiliary states, which are assumed continuous through the discontinuities, are introduced. We rewrite the numerical level‐set algorithm as a procedure consisting of three big steps: (a) initialization, (b) evolution, and (c) reconstruction. In (a), we choose an entropy admissible level‐set representation of the initial condition. In (b), for each iteration step, we solve an uncoupled system of three equations and select the entropy admissible level‐set representation of the solution profile at the end of the time iteration. In (c), we reconstruct the entropy solution using the level‐set representation. We prove the convergence of the numerical solution to the entropy solution in for every , using ‐weak bounded variation (BV) estimates and a cell entropy inequality. In addition, some numerical examples focused on the elementary wave interaction are presented. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1310–1343, 2015

Mixed spatially varying L 2-BV regularization of inverse ill-posed problems

Abstract Several generalizations of the traditional Tikhonov–Phillips regularization method have been proposed during the last two decades. Many of these generalizations are based upon inducing stability throughout the use of different penalizers which allow the capturing of diverse properties of the exact solution (e.g. edges, discontinuities, borders, etc.). However, in some problems in which it is known that the regularity of the exact solution is heterogeneous and/or anisotropic, it is reasonable to think that a much better option could be the simultaneous use of two or more penalizers of different nature. Such is the case, for instance, in some image restoration problems in which preservation of edges, borders or discontinuities is an important matter. In this work we present some results on the simultaneous use of penalizers of L 2 and of bounded variation (BV) type. For particular cases, existence and uniqueness results are proved. Open problems are discussed and results to signal restoration problems are presented.

Finer regularity of an entropy solution for 1-d scalar conservation laws with non uniform convex flux

Consider a scalar conservation law in one space dimension with initial data in L^\infty. If the flux \kern 1pt f is in C^2 and locally uniformly convex, then for all t> 0 , the entropy solution is locally in BV (functions of bounded variation) in space variable. In this case it was shown in [5], that for all most every t > 0 , locally, the solution is in SBV (Special functions of bounded variations). Furthermore it was shown with an example that for almost everywhere in t> 0 cannot be removed. This paper deals with the regularity of the entropy solutions of the strict convex C^1 flux f which need not be in C^2 and locally uniformly convex. In this case, the entropy solution need not be locally in BV in space variable, but the composition with the derivative of the flux function is locally in BV. Here we prove that, this composition is locally is in SBV on all most every t> 0 . Furthermore we show that this is optimal.

Image restoration with a half-quadratic approach to mixed weighted smooth and anisotropic bounded variation regularization

The problem of restoring a signal or image is often tantamount to approximating the solution of a linear inverse ill-posed problem. In order to find such an approximation one might regularize the problem by penalizing variations on the estimated solution. Among the wide variety of methods available to perform this penalization, the most commonly used is the Tikhonov-Phillips regularization, which is appropriate when the sought signal or image is expected to be smooth, but it results unsuitable whenever preservation of discontinuities and edges is an important matter. Nonetheless, there are other methods with edge preserving properties, such as bounded variation (BV ) regularization. However, these methods tend to produce piecewise constant solutions showing the so called “staircasing effect” and their numerical implementations entail great computational effort and cost. In order to overcome these obstacles, we consider a mixed weighted Tikhonov and anisotropic BV regularization method to obtain improved restorations and we use a half-quadratic approach to construct highly efficient numerical algorithms. Several numerical results in signal and image restoration problems are presented.

Optimal Viterbi Based Total Variation Sequence Detection (TVSD) For Robust Image/Video Decoding In Wireless Sensor Networks

In this letter, we propose a novel scheme for robust reconstruction, based on total variation regularization, towards image/video communication in multimedia wireless sensor networks. We derive the optimal joint source channel decoder as the combination of a maximum likelihood cost function and an anisotropic total variation norm based regularization factor. The proposed scheme exploits the bounded variation (BV) property of images and is thus ideally suited for reconstruction. Subsequently, it is demonstrated that the trellis based Viterbi decoder can be employed for robust image reconstruction using modified total variation state and branch metrics. Simulation results for BTC and JPEG encoded image transmission demonstrate a superior reconstruction performance for the proposed scheme in comparison to conventional methods.

Local and Nonlocal Regularization to Image Interpolation

This paper presents an image interpolation model with local and nonlocal regularization. A nonlocal bounded variation (BV) regularizer is formulated by an exponential function including gradient. It acts as the Perona-Malik equation. Thus our nonlocal BV regularizer possesses the properties of the anisotropic diffusion equation and nonlocal functional. The local total variation (TV) regularizer dissipates image energy along the orthogonal direction to the gradient to avoid blurring image edges. The derived model efficiently reconstructs the real image, leading to a natural interpolation which reduces blurring and staircase artifacts. We present experimental results that prove the potential and efficacy of the method.

A general framework for integration of bioluminescence tomography and diffuse optical tomography

Diffuse optical tomography (DOT) and bioluminescence tomography (BLT) are different but complementary modalities in optical molecular imaging. The goal of DOT is to reconstruct optical parameters of a medium from surface measurements induced by external sources, whereas that of BLT is to reconstruct running a bioluminescent source distribution in the medium from surface measurements induced by internal bioluminescent sources. A pre-requisite for BLT reconstruction is knowledge on the distribution of optical parameters within the medium, which is the output of DOT. In this paper, we propose a mathematical model integrating BLT and DOT at the fundamental level; that is, performing the two types of reconstructions simultaneously instead of doing them sequentially. The model is introduced through minimizing the difference between predicted quantities and boundary measurements, as well as incorporating regularization terms. In practice, the optical parameters are assumed to be piecewise constants or piecewise smooth functions. Hence, we seek the optical parameters in the space of functions with bounded variation (BV). For the BV-regularized data-fitting functional and its discretized approximations, we demonstrate the existence of minima and show convergence of the minima of the discretized functionals to that of the non-discretized one. We also present numerical results to illustrate the strength of our proposed model.

A BOUNDED VARIATION ESTIMATE FOR THE FORCE SCHEME APPLIED TO STRICTLY HYPERBOLIC SYSTEMS OF CONSERVATION LAWS IN THE TEMPLE CLASS

Bressan and Jenssen established a uniform bounded variation (BV) estimate for the Godunov scheme for Temple-type strictly hyperbolic systems of conservation laws and gave a proof based on the probability theory of random walks. In this paper, we provide a different proof which is simpler and does not use any probability theory. Applying our theory, we establish a uniform BV estimate for the Force scheme for the same class of hyperbolic systems, under the assumption of small total variation of initial data.

Total variation time flow with quantile regression for image restoration

We analyze the time flow of a modification of the standard Rudin–Osher–Fatemi (ROF) model for image restoration using the bounded variation semi-norm. The solution of the minimization problem is a restored image in the space of bounded variation (BV) recovered from a corrupted one, and the functional may include a more general term S(x,u) rather than the L2 penalty term as used in the original ROF model. These new models were introduced in Osher and Scherzer (2004) [31]. In particular, we will examine the non-differentiable quantile regression penalty. We will define an appropriate weak solution of the time flow and in particular prove the existence in L∞([0,∞);BV) and L∞ stability.

Mixed Tikhonov regularization in Banach spaces based on domain decomposition

We analyze Tikhonov regularization where the forward operator is defined on a direct sum U of several Banach spaces Ui,i=1,…,m. The regularization term on U is a sum of different regularizations on each Ui. The theoretical framework, known for the case m=1, can be easily reformulated to the case m>1. Under certain assumptions on weak topologies and on the forward operator it is possible to extend the results on the well-posedness of the minimization process, on its stability with respect to the data, and on the corresponding existence and convergence results, to the general case m⩾1.We consider two particular regularizations, the total or the so-called bounded variation (BV) regularization and the smooth (H1) regularization. Assume that the domain Ω is decomposed into two non-overlapping subdomains Ω1,Ω2. The direct sum of Banach spaces for m=2 will be U=H1(Ω1)⊕BV(Ω2) and the underlying regularization will be mixed H1–BV regularization. We define proper weak topology on U and prove that the assumptions for the above mentioned framework are fulfilled. In such a way we get the theoretical results for a broad range of H1–BV regularizations including the image denoising for the identity forward operator, the deconvolution problems for the convolution forward operator, or any other inverse problem for the continuous nonlinear forward operators.We demonstrate how the mixed regularization can be applied in practice. We suggest a denoising algorithm and compare its behavior with the BV regularization and the H1 regularization. We discuss the results of the denoising on several real world images justifying the usefulness of the mixed H1–BV regularization.

Γ-convergence, Sobolev norms, and BV functions

We prove that the family of functionals (Iδ) defined by Iδ(g)=∫∫RN×RN|g(x)-g(y)|>δδp|x-y|N+pdxdy, ∀g∈Lp(RN), for p≥1 and δ>0, Γ-converges in Lp(RN), as δ goes to zero, when p≥1. Hereafter | | denotes the Euclidean norm of RN. We also introduce a characterization for bounded variation (BV) functions which has some advantages in comparison with the classic one based on the notion of essential variation on almost every line.

An Upwind Finite-Difference Method for Total Variation–Based Image Smoothing

In this paper we study finite-difference approximations to the variational problem using the bounded variation (BV) smoothness penalty that was introduced in an image smoothing context by Rudin, Osher, and Fatemi. We give a dual formulation for an upwind finite-difference approximation for the BV seminorm; this formulation is in the same spirit as one popularized by the first author for a simpler, less isotropic, finite-difference approximation to the (isotropic) BV seminorm. We introduce a multiscale method for speeding up the approximation of both Chambolle's original method and of the new formulation of the upwind scheme. We demonstrate numerically that the multiscale method is effective, and we provide numerical examples that illustrate both the qualitative and quantitative behavior of the solutions of the numerical formulations.