Riemannian Geometry

Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques

doi:10.1007/978-3-642-18855-8

Cited by 394 publications

(390 citation statements)

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“…However, we do have a necessary condition for the sectional curvature to be nonpositive. Indeed, it is mentioned in [10] that Riemannian manifolds that are hypersurfaces of R n , n ≥ 4, can not have strictly negative sectional curvature. Hence, because the manifolds (RGB, h β2 ) are hypersurfaces of R 4 (see construction in Sect.…”

Section: The Case β 2 >mentioning

confidence: 99%

Harmonic Flow for Histogram Matching

Batard

Bertalmío

2014

Image and Video Technology – PSIVT 2013 Workshops

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Abstract. We present a method to perform histogram matching between two color images based on the concept of harmonic mapping between Riemannian manifolds. The key idea is to associate the histogram of a color image to a Riemannian manifold. In this context, the energy of the matching between the two images is measured by the Dirichlet energy of the mapping between the Riemannian manifolds. Then, we assimilate optimal matchings to critical points of the Dirichlet energy. Such points are called harmonic maps. As there is no explicit expression for harmonic maps in general, we use a gradient descent flow with boundary condition to reach them, that we call harmonic flow. We present an application to color transfer, however many others applications can be envisaged using this general framework.

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Section: The Case β 2 >mentioning

confidence: 99%

Harmonic Flow for Histogram Matching

Batard

Bertalmío

2014

Image and Video Technology – PSIVT 2013 Workshops

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“…where Γ ik j denotes the Christoffel's symbol of the second kind and the symbol of summation with respect to i and j in (5) is omitted by obeying the Einstein's rule in differential geometry [18,19]. Equation (5) is also expressed equivalently in the formq …”

Section: Riemannian Manifold and Euler's Equationmentioning

confidence: 99%

Iterative Learning without Reinforcement or Reward for Multijoint Movements: A Revisit of Bernstein's DOF Problem on Dexterity

Arimoto

Sekimoto

Tahara

2010

Journal of Robotics

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A robot designed to mimic a human becomes kinematically redundant and its total degrees of freedom becomes larger than the number of physical variables required for describing a given task. Kinematic redundancy may contribute to enhancement of dexterity and versatility but it incurs a problem of ill-posedness of inverse kinematics from the task space to the joint space. This ill-posedness was originally found by Bernstein, who tried to unveil the secret of the central nervous system and how nicely it coordinates a skeletomotor system with many DOFs interacting in complex ways. In the history of robotics research, such illposedness has not yet been resolved directly but circumvented by introducing an artificial performance index and determining uniquely an inverse kinematics solution by minimization. This paper tackles such Bernstein's problem and proposes a new method for resolving the ill-posedness in a natural way without invoking any artificial index. First, given a curve on a horizontal plane for a redundant robot arm whose endpoint is imposed to trace the curve, the existence of a unique ideal joint trajectory is proved. Second, such a uniquely determined motion can be acquired eventually as a joint control signal through iterative learning without reinforcement or reward.

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“…It is also interesting to see that the torus in Figure 3 is made to be topologically coincident with the set of all arm endpoints P = (x, y, z). As discussed in detail in mathematical text books [5,6], the topological manifold (M, p) of such a torus can be regarded as a differentiable manifold of class C ∞ . Now, it is necessary to define a tangent vector to an abstract differentiable manifold M at p ∈ M. Let I be an interval (− , ) and define a curve c(t) by a mapping c : depend on choice of the coordinate chart at p, as discussed in text books [5,6].…”

Section: Riemannian Manifold: a Set Of All Posturesmentioning

confidence: 99%

“…As discussed in detail in mathematical text books [5,6], the topological manifold (M, p) of such a torus can be regarded as a differentiable manifold of class C ∞ . Now, it is necessary to define a tangent vector to an abstract differentiable manifold M at p ∈ M. Let I be an interval (− , ) and define a curve c(t) by a mapping c : depend on choice of the coordinate chart at p, as discussed in text books [5,6]. Let us denote the set of all tangent vectors to M at p by T p M and call it the tangent space at p ∈ M. It has an n-dimensional linear space structure like R n .…”

Section: Riemannian Manifold: a Set Of All Posturesmentioning

confidence: 99%

A Riemannian-Geometry Approach for Modeling and Control of Dynamics of Object Manipulation under Constraints

Arimoto

Yoshida

Sekimoto

et al. 2009

Journal of Robotics

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A Riemannian-geometry approach for modeling and control of dynamics of object manipulation under holonomic or nonholonomic constraints is presented. First, position/force hybrid control of an endeffector of a multijoint redundant (or nonredundant) robot under a holonomic constraint is reinterpreted in terms of "submersion" in Riemannian geometry. A force control signal constructed in the image space of the constraint gradient is regarded as a lifting (or pressing) in the direction orthogonal to the kernel space. By means of the Riemannian distance on the constraint submanifold, stability of position control under holonomic constraints is discussed. Second, modeling and control of two-dimensional object grasping by a pair of multijoint robot fingers are challenged, when the object is of arbitrary shape. It is shown that rolling contact constraints induce the Euler equation of motion, in which constraint forces appear as wrench vectors affecting the object. The Riemannian metric is introduced on a constraint submanifold characterized with arclength parameters. An explicit form of the quotient dynamics is expressed in the kernel space with accompaniment of a pair of first-order differential equations concerning the arclength parameters. An extension of Dirichlet-Lagrange's stability theorem to redundant systems under constraints is suggested by introducing a Morse-Lyapunov function.

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Riemannian Geometry

Abstract: The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Cited by 394 publications

References 0 publications

Harmonic Flow for Histogram Matching

Harmonic Flow for Histogram Matching

Iterative Learning without Reinforcement or Reward for Multijoint Movements: A Revisit of Bernstein's DOF Problem on Dexterity

A Riemannian-Geometry Approach for Modeling and Control of Dynamics of Object Manipulation under Constraints

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