The total variation (TV)-seminorm is considered for piecewise polynomial, globally discontinuous (DG) and continuous (CG) finite element functions on simplicial meshes. A novel, discrete variant (DTV) based on a nodal quadrature formula is defined. DTV has favorable properties, compared to the original TV-seminorm for finite element functions. These include a convenient dual representation in terms of the supremum over the space of Raviart-Thomas finite element functions, subject to a set of simple constraints. It can therefore be shown that a variety of algorithms for classical image reconstruction problems, including TV-L 2 and TV-L 1 , can be implemented in low and higher-order finite element spaces with the same efficiency as their counterparts originally developed for images on Cartesian grids.1. Introduction. The total-variation (TV)-seminorm | · | T V is ubiquitous as a regularizing functional in image analysis and related applications; see for instance [50,28,17,14]. When Ω ⊂ R 2 is a bounded domain, this seminorm is defined aswhere s ∈ [1, ∞], s * = s s−1 denotes the conjugate of s and | · | s * is the usual s * -norm of vectors in R 2 . Frequent choices include s = 2 (the isotropic case) and s = 1, see Figure 1.1.It has been observed in [21] that "the rigorous definition of the TV for discrete images has received little attention." In this paper we propose and analyze a discrete analogue of (1.1) for functions u belonging to a space DG r (Ω) or CG r (Ω) of globally discontinuous or continuous finite element functions of polynomial degree 1 0 ≤ r ≤ 4 on a geometrically conforming, simplicial triangulation of Ω, consisting of triangles T and interior edges E. 2 * This work was supported by DFG grants HE 6077/10-1 and SCHM 3248/2-1 within the Priority Program SPP 1962 (Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization), which is gratefully acknowledged.
Background: The vacuum in the light-front representation of quantum field theory is trivial while vacuum in the equivalent canonical representation of the same theory is non-trivial. Purpose: Understand the relation between the vacuum in light-front and canonical representations of quantum field theory and the role of zero-modes in this relation. Method: Vacuua are defined as linear functionals on an algebra of field operators. The role of the algebra in the definition of the vacuum is exploited to understand this relation. Results: The vacuum functional can be extended from the light-front Fock algebra to an algebra of local observables. The extension to the algebra of local observables is responsible for the inequivalence. The extension defines a unitary mapping between the physical representation of the local algebra and a sub-algebra of the light-front Fock algebra. Conclusion: There is a unitary mapping from the physical representation of the algebra of local observables to a sub-algebra of the light-front Fock algebra with the free light-front Fock vacuum. The dynamics appears in the mapping and the structure of the sub-algebra. This correspondence provides a formulation of locality and Poincaré invariance on the light-front Fock space.
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 twodimensional 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.Notice that we restrict the discussion to the isotropic case here, i.e., | · | 2 denotes the Euclidean norm. Moreover, Du is the derivative of u and {e 1 , e 2 } denotes the standard Euclidean basis. The seminorm (1.1) extends to less regular, so-called BV functions (bounded variation), whose distributional gradient exists only in the sense of measures. We refer the reader to Giusti, 1984;Attouch, Buttazzo, Michaille, 2006 for an extensive discussion of BV functions. The utility of (1.1) as a regularizer, or prior, lies in the fact that it favors piecewise constant solutions.In this paper, we introduce a novel regularizer based on the total variation, which can be used, for instance, in shape optimization applications as well as geometric inverse problems. In the latter class, the unknown, which one seeks to recover, is a shape Ω ⊂ R 3 , which might represent the location of a source or inclusion inside a given, larger domain, or the geometry of an inclusion or a scatterer. The boundary of Ω will be denoted by Γ.The novel functional, which we term the total variation of the normal field along a smooth surface Γ, is defined by |n| T V (Γ) := Γ |(D Γ n) ξ 1 | 2 g + |(D Γ n) ξ 2 | 2 g 1/2 ds (1.2)Date: August 22, 2019.
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.
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