Equivalent definitions of BV space and of total variation on metric measure spaces

Ambrosio, Luigi; Marino, Simone Di

doi:10.1016/j.jfa.2014.02.002

Cited by 120 publications

(185 citation statements)

References 18 publications

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“…We denote by L the σ-essential supremum of the Lipschitz constants of the curves in Γ. Notice that for 9) and that for every nonnegative Borel function f one has…”

Section: Remark 82mentioning

confidence: 99%

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On the duality between $p$-modulus and probability measures

2015

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Motivated by recent developments on calculus in metric measure spaces (X, d, m), we prove a general duality principle between Fuglede's notion [15] of p-modulus for families of finite Borel measures in (X, d) and probability measures with barycenter in L q (X, m), with q dual exponent of p ∈ (1, ∞). We apply this general duality principle to study null sets for families of parametric and non-parametric curves in X. In the final part of the paper we provide a new proof, independent of optimal transportation, of the equivalence of notions of weak upper gradient based on ppodulus ([21], [23]) and suitable probability measures in the space of curves ([6], [7]).

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“…We denote by L the σ-essential supremum of the Lipschitz constants of the curves in Γ. Notice that for 9) and that for every nonnegative Borel function f one has…”

Section: Remark 82mentioning

confidence: 99%

“…The following notions have already been used in [6] (q = 2) and [7] (in connection with the Sobolev spaces with gradient in L p (X, m), with q = p ′ ; see also [9] in connection with the BV theory).…”

Section: Test Plans and Their Null Setsmentioning

confidence: 99%

On the duality between $p$-modulus and probability measures

2015

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“…In fact, one can easily see that in (4) the sub-Riemannian metric gradient (X 1 u, X 2 u) of u appears, and it is composed with the one-dimensional function Φ. This observation can be used to make the connection to the Hopf-Lax type formula (5), where the solution appears in terms of the sub-Riemannian metric of the Heisenberg group. The above observation was generalized to more general sub-Riemannian geometries defined in terms of Hörmander vector fields by Dragoni in [20].…”

Section: Introductionmentioning

confidence: 98%

“…Hamilton-Jacobi equations play a major role in the theory of optimal mass transportation and logarithmic Sobolev inequalities and have been studied recently in the general context of geodesic metric measure spaces by Lott-Villani [28], Balogh-Engulatov-Hunziker-Maasalo [6], Ambrosio-Gigli-Savaré [3], [4] and Ambrosio and Di Marino [5] . In these works it was shown that even in the general setting of geodesic spaces the solution to the HamiltonJacobi equation can be expressed as an inf-convolution akin to the classical Hopf-Lax formula [26,27].…”

Section: Introductionmentioning

confidence: 99%

The Hopf–Lax formula in Carnot groups: a control theoretic approach

Balogh¹,

Calogero²,

Pini³

2013

Calc. Var.

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The purpose of this paper is to bring a new light on the state-dependent HamiltonJacobi equation and its connection with the Hopf-Lax formula in the framework of a Carnot group (G, •). The equation we shall consider is of the formwhere X 1 , . . . , X m are a basis of the first layer of the Lie algebra of the group G, and Ψ : R m → R is a superlinear, convex function. The main result shows that the unique viscosity solution of the Hamilton-Jacobi equation can be given by the Hopf-Lax formula u(x, t) = infwhere Ψ G : G → R is the G-Legendre-Fenchel transform of Ψ, defined by a control theoretical approach. We recover, as special cases some known results: the classical Hopf-Lax formula in the Euclidean spaces R n showing that Ψ R n is the Legendre-Fenchel trasnsform Ψ * of Ψ; moreover, we recover the result by Manfredi and Stroffolini in the Heisenberg group for particular Hamiltonian function Ψ. In this paper we follow an optimal control problem approach and we obtain several properties for the value functions u and Ψ G : in particular we prove a precise estimate for the horizontal gradient of the solution u, two existence results of the optimal control for the optimal problems and we show that Ψ G is convex.

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“…Compared with these models, TV is much more efficient to be solved, making TV-based methods remain active in image and vision studies [17][18][19][20][21][22][23][24]. Moreover, TV may be complementary with the other models, and thus proper combination of them can lead to better performance [25,26].…”

Section: Introductionmentioning

confidence: 99%

A Coordinate Descent Method for Total Variation Minimization

Deng

Ren

Xiao³

et al. 2017

Mathematical Problems in Engineering

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Total variation (TV) is a well-known image model with extensive applications in various images and vision tasks, for example, denoising, deblurring, superresolution, inpainting, and compressed sensing. In this paper, we systematically study the coordinate descent (CoD) method for solving general total variation (TV) minimization problems. Based on multidirectional gradients representation, the proposed CoD method provides a unified solution for both anisotropic and isotropic TV-based denoising (CoDenoise). With sequential sweeping and small random perturbations, CoDenoise is efficient in denoising and empirically converges to optimal solution. Moreover, CoDenoise also delivers new perspective on understanding recursive weighted median filtering. By incorporating with the Augmented Lagrangian Method (ALM), CoD was further extended to TV-based image deblurring (ALMCD). The results on denoising and deblurring validate the efficiency and effectiveness of the CoD-based methods.

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Equivalent definitions of BV space and of total variation on metric measure spaces

Cited by 120 publications

References 18 publications

On the duality between $p$-modulus and probability measures

On the duality between $p$-modulus and probability measures

The Hopf–Lax formula in Carnot groups: a control theoretic approach

A Coordinate Descent Method for Total Variation Minimization

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