Möbius Invariants of Shapes and Images

Marsland, Stephen; McLachlan, Robert I.

doi:10.3842/sigma.2016.080

Cited by 7 publications

(6 citation statements)

References 41 publications

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“…This completes our derivation of the moving frame based on the cross-section (8). The right equivariant moving frame map ρ : J 3 \ V → PSL(3) is obtained by combining the preceding normalisation formulas (9,10,13,15,21), producing fairly long formulas for all the group parameters in terms of the third-order jet coordinates, which we will not write out in detail.…”

Section: Remarkmentioning

confidence: 73%

“…Returning to our moving frame calculation, we now recall the unimodularity constraint (2) Under the normalisations accumulated so far (9,10,13,15), this becomes…”

Section: Remarkmentioning

confidence: 99%

“…To obtain their explicit formulas, we merely substitute the moving frame normalisations (9,10,13,15,21) into the formulas for the unnormalised transformed third-order jet coordinates U XXX , U XXY , U XYY , respectively, to eliminate all the group parameters. The resulting expressions are the third-order differential invariants.…”

Section: Remarkmentioning

confidence: 99%

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Projective invariants of images

Olver

2022

Eur. J. Appl. Math

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The method of equivariant moving frames is employed to construct and completely classify the differential invariants for the action of the projective group on functions defined on the two-dimensional projective plane. While there are four independent differential invariants of order $\leq 3$ , it is proved that the algebra of differential invariants is generated by just two of them through invariant differentiation. The projective differential invariants are, in particular, of importance in image processing applications.

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Section: Remarkmentioning

confidence: 73%

“…Returning to our moving frame calculation, we now recall the unimodularity constraint (2) Under the normalisations accumulated so far (9,10,13,15), this becomes…”

Section: Remarkmentioning

confidence: 99%

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Projective invariants of images

Olver

2022

Eur. J. Appl. Math

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“…(In the case of curves in Euclidean space, the signature curve was introduced earlier by Bruce and Giblin, [1], under the name "Monge-Taylor map".) See, for example, [2,7,8,10,16,25] for various applications of the differential invariant signature to object recognition in digital images.…”

Section: Plane Curvesmentioning

confidence: 99%

Normal Forms for Submanifolds Under Group Actions

Olver

2018

Springer Proceedings in Mathematics &Amp; Statistics

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We describe computational algorithms for constructing the explicit power series expansions for normal forms of submanifolds under transformation groups. The procedure used to derive the coefficients relies on the recurrence formulae for differential invariants provided by the method of equivariant moving frames. 1. Introduction. The equivariant method of moving frames, introduced in [4], provides a powerful computational tool for investigating the equivalence and symmetry properties of submanifolds under general Lie group actions (and, more generally, infinite-dimensional Lie pseudo-groups, [23, 24]), and determining the required differential invariants. The main new tool is the recurrence relations, which completely prescribe the structure of the noncommutative differential algebra they generate through the process of invariant differentiation. Remarkably, these relations and the consequent differential algebraic structure can be completely and straightforwardly constructed, requiring only basic linear algebra, † Supported in part by NSF Grant DMS 11-08894.

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“…Logarithmic spiral crop up in the computer science literature ( [3,4] provide examples). A logarithmic spiral is also conjectured to provide a solution to a problem of interest in computer science, dubbed Search for the shore: finding the optimal trajectory to reach a straight line from a point on the plane when neither its distance nor its orientation is known.…”

Section: Introductionmentioning

confidence: 99%

Conformal Mechanics of Planar Curves

Guven,

Manrique

2019

Preprint

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It is possible to associate curvature-dependent Möbius invariant energies with planar curves. They are of particular interest because their tension-free stationary states, lacking a length scale, form self-similar curves. As such one would expect these energies to show up in self-similar growth processes. The simplest among them is the conformal arc-length. Its tension-free states are logarithmic spirals characterized by the rate of growth; in general its stationary states have constant conformal curvature: but any such state can be generated from a logarithmic spirals by applying an appropriate Möbius transformation to it. In the process, tension is introduced and with it a length scale. Generically these Möbius descendants form double spirals, characterized up to similarity by a second parameter, the distance between the two poles. In this paper, we will show how this invariant system can be approached from a mechanical point of view. Central to this approach is the identification of the four currents conserved along stationary curves: the tension and torque associated with Euclidean invariance; in addition scalar and vector currents reflect the invariance under scaling and special conformal transformations respectively. It will be shown that the conformal Casimir invariant, proportional to the conformal curvature, can be decomposed in terms of these currents. This approach permits the familiar geometrical properties of logarithmic spirals to be reinterpreted in mechanical terms, while suggesting new ones. In logarithmic spirals, not just the tension but also the special conformal current vanish with respect to a reference frame centered on the spiral apex. The torque M and the scaling current S are constrained to satisfy 4M S = 1 in these states. The tension generated in a double spiral is evaluated. While it is, as expected, inversely proportional to the distance between its poles, it is not directed along the line connecting these two points; the non-vanishing angle it makes with this direction is the angle defining the logarithmic spiral, independent of the inter-polar distance. An explicit construction of these states using the conserved currents will be presented. In higher order Möbius invariant theories, logarithmic spirals generally get decorated by self-similar internal structure.

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Möbius Invariants of Shapes and Images

Cited by 7 publications

References 41 publications

Projective invariants of images

Projective invariants of images

Normal Forms for Submanifolds Under Group Actions

Conformal Mechanics of Planar Curves

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