Riemannian Geometry

Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques

doi:10.1007/978-3-642-97242-3

Cited by 292 publications

(238 citation statements)

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“…In particular, the stress tensor with respect to E 2 is [8] S(ϕ, g) = 1 2 |dϕ| 2 g − ϕ * h. The goal of this section is to give sufficient and necessary conditions for a given map to be restricted harmonic, that is, to compute the first variation formula for E ϕ : SDiff 0 (M, g) → R. We will make frequent use of some standard facts about Lie derivatives, which we now summarize (see [16] for details).…”

Section: Definitionmentioning

confidence: 99%

Near BPS skyrmions and restricted harmonic maps

Speight

2015

Journal of Geometry and Physics

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Motivated by a class of near BPS Skyrme models introduced by Adam, Sánchez-Guillén and Wereszczyński, the following variant of the harmonic map problem is introduced: a map ϕ : (M, g) → (N, h) between Riemannian manifolds is restricted harmonic if it locally extremizes E 2 on its SDiff(M ) orbit, where SDiff(M ) denotes the group of volume preserving diffeomorphisms of (M, g), and E 2 denotes the Dirichlet energy. It is conjectured that near BPS skyrmions tend to restricted harmonic maps in the BPS limit. It is shown that ϕ is restricted harmonic if and only if ϕ * h has exact divergence, and a linear stability theory of restricted harmonic maps is developed, from which it follows that all weakly conformal maps are stable restricted harmonic. Examples of restricted harmonic maps in every degree class R 3 → SU (2) and R 2 → S 2 are constructed. It is shown that the axially symmetric BPS skyrmions on which all previous analytic studies of near BPS Skyrme models have been based, are not restricted harmonic, casting doubt on the phenomenological predictions of such studies. The problem of minimizing E 2 for ϕ : R k → N over all linear volume preserving diffeomorphisms is solved explicitly, and a deformed axially symmetric family of Skyrme fields constructed which are candidates for approximate near BPS skyrmions at low baryon number. The notion of restricted harmonicity is generalized to restricted F -criticality where F is any functional on maps (M, g) → (N, h) which is, in a precise sense, geometrically natural. The case where F is a linear combination of E 2 and E 4 , the usual Skyrme term, is studied in detail, and it is shown that inverse stereographic projection R 3 → S 3 ≡ SU (2) is stable restricted F -critical for every such F .

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Section: Definitionmentioning

confidence: 99%

Near BPS skyrmions and restricted harmonic maps

Speight

2015

Journal of Geometry and Physics

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“…There is no loss in imposing a strict inequality such as a < 1 in preference to the weaker inequality a ≤ 1, because the points of A(2) omitted from T by imposing strict inequalities have a five dimensional volume equal to zero. The canonical density τ (θ) defined on T by the Riemannian metric K(θ) (Chavel, 1996;Gallot et al, 1990) is…”

Section: The Parameter Manifold For Ellipsesmentioning

confidence: 99%

“…Let u(θ, 4), u(θ, 5) be the resulting vectors in Tan θ T . Recall that the exponential map exp θ is defined on an open neighbourhood of the origin in Tan θ T and that if the open neighbourhood is sufficiently small, then the neighbourhood is mapped diffeomorphicly onto its image in T (Chavel, 1996;Gallot et al, 1990). The manifold T is sampled at the points…”

Section: For U(1) U(4)mentioning

confidence: 99%

Application of the Fisher-Rao Metric to Ellipse Detection

Maybank

2006

Int J Comput Vision

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Abstract. The parameter space for the ellipses in a two dimensional image is a five dimensional manifold, where each point of the manifold corresponds to an ellipse in the image. The parameter space becomes a Riemannian manifold under a Fisher-Rao metric, which is derived from a Gaussian model for the blurring of ellipses in the image. Two points in the parameter space are close together under the Fisher-Rao metric if the corresponding ellipses are close together in the image. The Fisher-Rao metric is accurately approximated by a simpler metric under the assumption that the blurring is small compared with the sizes of the ellipses under consideration. It is shown that the parameter space for the ellipses in the image has a finite volume under the approximation to the Fisher-Rao metric. As a consequence the parameter space can be replaced, for the purpose of ellipse detection, by a finite set of points sampled from it. An efficient algorithm for sampling the parameter space is described. The algorithm uses the fact that the approximating metric is flat, and therefore locally Euclidean, on each three dimensional family of ellipses with a fixed orientation and a fixed eccentricity. Once the sample points have been obtained, ellipses are detected in a given image by checking each sample point in turn to see if the corresponding ellipse is supported by the nearby image pixel values. The resulting algorithm for ellipse detection is implemented. A multiresolution version of the algorithm is also implemented. The experimental results suggest that ellipses can be reliably detected in a given low resolution image and that the number of false detections can be reduced using the multiresolution algorithm.

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“…If two points θ (1), θ (2) of T are close together under J, then the corresponding structures are difficult to distinguish given only a single measurement. The second key role of J is that it defines the volume [6,11] The two main advantages of this sample based approach are firstly that it is possible to detect structures in the presence of very large numbers of outliers and secondly any structure that is present in the image is detected. The accuracy with which a structure is located depends on the density of the sample points θ (i).…”

Section: P(x|θ) Is Obtained By Integrating P(x|x) Over M (θ) With Weimentioning

confidence: 99%

“…The superscript notation θ i for the components of θ is usual in the context of Riemannian geometry [6,11]. The matrix J(θ) is the Fisher information at θ for a single measurement…”

Section: Definition Of the Fisher-rao Metricmentioning

confidence: 99%

Application of the Fisher-Rao Metric to Structure Detection

Maybank

2006

J Math Imaging Vis

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-Certain structure detection problems can be solved by sampling a parameter space for the different structures at a finite number of points and checking each point to see if the corresponding structure has a sufficient number of inlying measurements. The measurement space is a Riemannian manifold and the measurements relevant to a given structure are near to or on a submanifold which constitutes the structure. The probability density function for the errors in the measurements is described using a generalisation of the Gaussian density to Riemannian manifolds. The conditional probability density function for the measurements yields the Fisher information which defines a metric, known as the Fisher-Rao metric, on the parameter space. The main result is a derivation of an asymptotic approximation to the Fisher-Rao metric, under the assumption that the measurement noise is small. Using this approximation to the Fisher-Rao metric, the parameter space is sampled, such that each point of the parameter space is near to at least one sample point, to within the level of accuracy allowed by the measurement errors. The probability of a false detection of a structure is estimated. The feasibility of this ap-1 proach to structure detection is tested experimentally using the example of line detection in digital images.

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Riemannian Geometry

Cited by 292 publications

References 0 publications

Near BPS skyrmions and restricted harmonic maps

Near BPS skyrmions and restricted harmonic maps

Application of the Fisher-Rao Metric to Ellipse Detection

Application of the Fisher-Rao Metric to Structure Detection

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