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Thomas Batard - One of the best experts on this subject based on the ideXlab platform.
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CLIFFORD BUNDLES: A UNIFYING FrameWORK FOR IMAGES(VIDEOS), VECTOR FIELDS AND Orthonormal
2014Co-Authors: Frame Fields Regularization, Thomas BatardAbstract:Abstract. The aim of this paper is to present a new Framework for regularization by diffusion. The methods we develop in the sequel can be used to smooth nD images, nD videos, vector fields and Orthonormal Frame fields in any dimension.1 From a mathematical viewpoint, we deal with vector bundles over Riemannian manifolds and so-called generalized Laplacians. Sections are regularized from heat equations associated to generalized Laplacians, the solution being given as convolution by generalized heat kernels. The anisotropy of the diffusion is controlled by the metric of the base manifold and by the connection of the vector bundle. It finds applications to images and videos anisotropic regularization. The main topic of this paper is to show that this approach can be extended to other fields such as vector fields and Orthonormal Frame fields by considering the context of Clifford algebras. We introduce a Clifford-Hodge operator (and the corresponding Clifford-Hodge flow) as a generalized Laplacian on the Clifford bundle of a Riemannian manifold. Laplace-Beltrami diffusion appears as a particular case of Clifford-Hodge diffusion, namely diffusion for degree 0 sections (functions). Considering base manifolds of dimension 2 and 3, applications to multispectral images, multispectral videos, 2D and 3D vector fields and Orthonormal Frame fields regularization may be envisaged. Some of them are presented in this paper
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clifford bundles a common Framework for image vector field and Orthonormal Frame field regularization
Siam Journal on Imaging Sciences, 2010Co-Authors: Thomas BatardAbstract:The aim of this paper is to present a new Framework for regularization by diffusion. The methods we develop in what follows can be used to smooth multichannel images, multichannel image sequences (videos), vector fields, and Orthonormal Frame fields in any dimension. From a mathematical viewpoint, we deal with vector bundles over Riemannian manifolds and so-called generalized Laplacians. Sections are regularized from heat equations associated with generalized Laplacians, the solutions being approximated by convolutions with kernels. Then, the behavior of the diffusion is determined by the geometry of the vector bundle, i.e., by the metric of the base manifold and by a connection on the vector bundle. For instance, the heat equation associated with the Laplace-Beltrami operator can be considered from this point of view for applications to images and video regularization. The main topic of this paper is to show that this approach can be extended in several ways to vector fields and Orthonormal Frame fields by considering the context of Clifford algebras. We introduce Clifford-Beltrami and Clifford-Hodge operators as generalized Laplacians on Clifford bundles over Riemannian manifolds. Laplace-Beltrami diffusion appears as a particular case of diffusion for degree 0 sections (functions). Dealing with base manifolds of dimension 2, applications to multichannel image, two-dimensional vector field, and orientation field regularization are presented.
Mikjel Thorsrud - One of the best experts on this subject based on the ideXlab platform.
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bianchi models with a free massless scalar field invariant sets and higher symmetries
Classical and Quantum Gravity, 2019Co-Authors: Mikjel ThorsrudAbstract:We scrutinize the overall structure of the space of cosmological models of Bianchi type I-VII$_h$ that contain a free massless scalar field, with a spatially homogeneous gradient $\nabla_\mu \varphi$ that generally breaks isotropy, in addition to a standard perfect fluid. Specifically, state space is written as a union of disjoint invariant sets, each corresponding to a particular cosmological model that is classified with respect to the Bianchi type and the matter content. Subsets corresponding to higher-symmetry models, including all locally rotationally symmetric models and FLRW models, and models with a shear-free normal congruence, are also derived and classified. For each model the dimension ($d$) of the space of initial data is given, after fixing the orientation of the Orthonormal Frame uniquely relative to matter anisotropies and geometrical anisotropies.
Körpinar Z. - One of the best experts on this subject based on the ideXlab platform.
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Soliton propagation of electromagnetic field vectors of polarized light ray traveling along with coiled optical fiber on the unit 2-sphere S2
'Sociedad Mexicana de Fisica A C', 2019Co-Authors: Körpinar T., Demirkol R.c., Körpinar Z.Abstract:In this paper, we relate the evolution equation of the electric field and magnetic field vectors of the polarized light ray traveling along with a coiled optical fiber on the unit 2-sphere S2 into the nonlinear Schrodingers equation, by proposing new kinds of binormal motions and new kinds of Hasimoto functions, in addition to commonly known formula of the binormal motion and Hasimoto function. All these operations have been conducted by using the Orthonormal Frame of spherical equations, that is defined along with the coiled optical fiber lying on the unit 2-sphere S2. We also propose perturbed solutions of the nonlinear Schrodingers evolution equation that governs the propagation of solitons through the electric field (E) and magnetic field (M) vectors: Finally, we provide some numerical simulations to supplement the analytical outcomes. © 2019 Sociedad Mexicana de Fisica
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Soliton propagation of electromagnetic field vectors of polarized light ray traveling in a coiled optical fiber in the ordinary space
'World Scientific Pub Co Pte Lt', 2019Co-Authors: Körpinar T., Demirkol R.c., Körpinar Z.Abstract:In this paper, we relate the evolution equations of the electric field and magnetic field vectors of the polarized light ray traveling in a coiled optical fiber in the ordinary space into the nonlinear Schrödinger's equation by proposing new kinds of binormal motions and new kinds of Hasimoto functions in addition to commonly known formula of the binormal motion and Hasimoto function. All these operations have been conducted by using the Orthonormal Frame of Bishop equations that is defined along with the coiled optical fiber. We also propose perturbed solutions of the nonlinear Schrödinger's evolution equation that governs the propagation of solitons through the electric field (E) and magnetic field (M) vectors. Finally, we provide some numerical simulations to supplement the analytical outcomes. © 2019 World Scientific Publishing Company
Rt Farouki - One of the best experts on this subject based on the ideXlab platform.
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Rational minimal-twist motions on curves with rotation-minimizing Euler–Rodrigues Frames
eScholarship University of California, 2019Co-Authors: Rt Farouki, Giannelli C, Sestini AAbstract:© 2018 Elsevier B.V. A minimal twist Frame (f1(ξ),f2(ξ),f3(ξ)) on a polynomial space curve r(ξ), ξ∈[0,1] is an Orthonormal Frame, where f1(ξ) is the tangent and the normal-plane vectors f2(ξ),f3(ξ) have the least variation between given initial and final instances f2(0),f3(0) and f2(1),f3(1). Namely, if ω=ω1f1+ω2f2+ω3f3 is the Frame angular velocity, the component ω1 does not change sign, and its arc length integral has the smallest value consistent with the boundary conditions. We consider construction of curves with rational minimal twist Frames, based on the Pythagorean-hodograph curves of degree 7 that have rational rotation-minimizing Euler–Rodrigues Frames (e1(ξ),e2(ξ),e3(ξ)) — i.e., the normal-plane vectors e2(ξ),e3(ξ) have no rotation about the tangent e1(ξ). A set of equations that govern the construction of such curves with prescribed initial/final points and tangents, and total arc length, is derived. For the resulting curves f2(ξ),f3(ξ) are then obtained from e2(ξ),e3(ξ) by a monotone rational normal-plane rotation, subject to the boundary conditions. A selection of computed examples is included to illustrate the construction, and it is shown that the desirable feature of a uniform rotation rate (i.e., ω1=constant) can be accurately approximated
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Rational minimal-twist motions on curves with rotation-minimizing Euler–Rodrigues Frames
eScholarship University of California, 2019Co-Authors: Rt Farouki, Giannelli C, Sestini AAbstract:A minimal twist Frame (f (ξ),f (ξ),f (ξ)) on a polynomial space curve r(ξ), ξ∈[0,1] is an Orthonormal Frame, where f (ξ) is the tangent and the normal-plane vectors f (ξ),f (ξ) have the least variation between given initial and final instances f (0),f (0) and f (1),f (1). Namely, if ω=ω f +ω f +ω f is the Frame angular velocity, the component ω does not change sign, and its arc length integral has the smallest value consistent with the boundary conditions. We consider construction of curves with rational minimal twist Frames, based on the Pythagorean-hodograph curves of degree 7 that have rational rotation-minimizing Euler–Rodrigues Frames (e (ξ),e (ξ),e (ξ)) — i.e., the normal-plane vectors e (ξ),e (ξ) have no rotation about the tangent e (ξ). A set of equations that govern the construction of such curves with prescribed initial/final points and tangents, and total arc length, is derived. For the resulting curves f (ξ),f (ξ) are then obtained from e (ξ),e (ξ) by a monotone rational normal-plane rotation, subject to the boundary conditions. A selection of computed examples is included to illustrate the construction, and it is shown that the desirable feature of a uniform rotation rate (i.e., ω =constant) can be accurately approximated. 1 2 3 1 2 3 2 3 2 3 1 1 2 2 3 3 1 1 2 3 2 3 1 2 3 2 3
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Rational Frames of minimal twist along space curves under specified boundary conditions
eScholarship University of California, 2018Co-Authors: Rt Farouki, Hp MoonAbstract:An adapted Orthonormal Frame (f1(ξ),f2(ξ),f3(ξ)) on a space curve r(ξ), ξ ∈ [0, 1] comprises the curve tangent f1(ξ) = r′(ξ) / | r′(ξ) | and two unit vectors f2(ξ),f3(ξ) that span the normal plane. The variation of this Frame is specified by its angular velocity Ω = Ω1f1 + Ω2f2 + Ω3f3, and the twist of the Framed curve is the integral of the component Ω1 with respect to arc length. A minimal twist Frame (MTF) has the least possible twist value, subject to prescribed initial and final orientations f2(0),f3(0) and f2(1),f3(1) of the normal–plane vectors. Employing the Euler–Rodrigues Frame (ERF) — a rational adapted Frame defined on spatial Pythagorean–hodograph curves — as an intermediary, an exact expression for an MTF with Ω1 = constant is derived. However, since this involves rather complicated transcendental terms, a construction of rational MTFs is proposed by the imposition of a rational rotation on the ERF normal–plane vectors. For spatial PH quintics, it is shown that rational MTFs compatible with the boundary conditions can be constructed, with only modest deviations of Ω1 about the mean value, by a rational quartic normal–plane rotation of the ERF. If necessary, subdivision methods can be invoked to ensure that the rational MTF is free of inflections, or to more accurately approximate a constant Ω1. The procedure is summarized by an algorithm outline, and illustrated by a representative selection of computed examples
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Rational Frames of minimal twist along space curves under specified boundary conditions
eScholarship University of California, 2018Co-Authors: Rt Farouki, Hp MoonAbstract:© 2018, Springer Science+Business Media, LLC, part of Springer Nature. An adapted Orthonormal Frame (f1(ξ),f2(ξ),f3(ξ)) on a space curve r(ξ), ξ ∈ [0, 1] comprises the curve tangent f1(ξ) = r′(ξ) / | r′(ξ) | and two unit vectors f2(ξ),f3(ξ) that span the normal plane. The variation of this Frame is specified by its angular velocity Ω = Ω1f1 + Ω2f2 + Ω3f3, and the twist of the Framed curve is the integral of the component Ω1 with respect to arc length. A minimal twist Frame (MTF) has the least possible twist value, subject to prescribed initial and final orientations f2(0),f3(0) and f2(1),f3(1) of the normal–plane vectors. Employing the Euler–Rodrigues Frame (ERF) — a rational adapted Frame defined on spatial Pythagorean–hodograph curves — as an intermediary, an exact expression for an MTF with Ω1 = constant is derived. However, since this involves rather complicated transcendental terms, a construction of rational MTFs is proposed by the imposition of a rational rotation on the ERF normal–plane vectors. For spatial PH quintics, it is shown that rational MTFs compatible with the boundary conditions can be constructed, with only modest deviations of Ω1 about the mean value, by a rational quartic normal–plane rotation of the ERF. If necessary, subdivision methods can be invoked to ensure that the rational MTF is free of inflections, or to more accurately approximate a constant Ω1. The procedure is summarized by an algorithm outline, and illustrated by a representative selection of computed examples
Ugo Boscain - One of the best experts on this subject based on the ideXlab platform.
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Quantum confinement for the curvature Laplacian $− \frac{1}{2} \Delta + cK$ on 2D-almost-Riemannian manifolds
2020Co-Authors: Ivan Beschastnyi, Ugo Boscain, Eugenio PozzoliAbstract:Two-dimension almost-Riemannian structures of step 2 are natural generalizations of the Grushin plane. They are generalized Riemannian structures for which the vectors of a local Orthonormal Frame can become parallel. Under the 2-step assumption the singular set Z, where the structure is not Riemannian, is a 1D embedded submanifold. While approaching the singular set, all Riemannian quantities diverge. A remarkable property of these structure is that the geodesics can cross the singular set without singularities, but the heat and the solution of the Schrödinger equation (with the Laplace-Beltrami operator ∆) cannot. This is due to the fact that (under a natural compactness hypothesis), the Laplace-Beltrami operator is essentially self-adjoint on a connected component of the manifold without the singular set. In the literature such phenomenon is called quantum confinement. In this paper we study the self-adjointness of the curvature Laplacian, namely $− 1 2 ∆ + cK$, for c > 0 (here K is the Gaussian curvature), which originates in coordinate free quantization procedures (as for instance in path-integral or covariant Weyl quantization). We prove that there is no quantum confinement for this type of operators.
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the laplace beltrami operator in almost riemannian geometry
Annales de l'Institut Fourier, 2013Co-Authors: Ugo Boscain, Camille LaurentAbstract:Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local Orthonormal Frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, the singular set is an embedded one dimensional manifold and there are three type of points: Riemannian points where the two vector fields are linearly independent, Grushin points where the two vector fields are collinear but their Lie bracket is not and tangency points where the two vector fields and their Lie bracket are collinear and the missing direction is obtained with one more bracket. Generically tangency points are isolated. In this paper we study the Laplace-Beltrami operator on such a structure. In the case of a compact orientable surface without tangency points, we prove that the Laplace-Beltrami operator is essentially self-adjoint and has discrete spectrum. As a consequence a quantum particle in such a structure cannot cross the singular set and the heat cannot flow through the singularity. This is an interesting phenomenon since when approaching the singular set (i.e. where the vector fields become collinear), all Riemannian quantities explode, but geodesics are still well defined and can cross the singular set without singularities. This phenomenon appears also in sub-Riemannian structure which are not equiregular i.e. in which the grow vector depends on the point. We show this fact by analyzing the Martinet
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the laplace beltrami operator in almost riemannian geometry
arXiv: Spectral Theory, 2011Co-Authors: Ugo Boscain, Camille LaurentAbstract:Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local Orthonormal Frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, the singular set is an embedded one dimensional manifold and there are three type of points: Riemannian points where the two vector fields are linearly independent, Grushin points where the two vector fields are collinear but their Lie bracket is not and tangency points where the two vector fields and their Lie bracket are collinear and the missing direction is obtained with one more bracket. Generically tangency points are isolated. In this paper we study the Laplace-Beltrami operator on such a structure. In the case of a compact orientable surface without tangency points, we prove that the Laplace-Beltrami operator is essentially self-adjoint and has discrete spectrum. As a consequence a quantum particle in such a structure cannot cross the singular set and the heat cannot flow through the singularity. This is an interesting phenomenon since when approaching the singular set (i.e. where the vector fields become collinear), all Riemannian quantities explode, but geodesics are still well defined and can cross the singular set without singularities. This phenomenon appears also in sub-Riemannian structure which are not equiregular i.e. in which the grow vector depends on the point. We show this fact by analyzing the Martinet case.
