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Linearised Euclidean Shortening Flow of Curve Geometry

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Abstract

The geometry of a space curve is described in terms of a Euclidean invariant frame field, metric, connection, torsion and curvature. Here the torsion and curvature of the connection quantify the curve geometry. In order to retain a stable and reproducible description of that geometry, such that it is slightly affected by non-uniform protrusions of the curve, a linearised Euclidean shortening flow is proposed. (Semi)-discretised versions of the flow subsequently physically realise a concise and exact (semi-)discrete curve geometry. Imposing special ordering relations the torsion and curvature in the curve geometry can be retrieved on a multi-scale basis not only for simply closed planar curves but also for open, branching, intersecting and space curves of non-trivial knot type. In the context of the shortening flows we revisit the maximum principle, the semi-group property and the comparison principle normally required in scale-space theories. We show that our linearised flow satisfies an adapted maximum principle, and that its Green's functions possess a semi-group property. We argue that the comparison principle in the case of knots can obstruct topological changes being in contradiction with the required curve simplification principle. Our linearised flow paradigm is not hampered by this drawback; all non-symmetric knots tend to trivial ones being infinitely small circles in a plane. Finally, the differential and integral geometry of the multi-scale representation of the curve geometry under the flow is quantified by endowing the scale-space of curves with an appropriate connection, and calculating related torsion and curvature aspects. This multi-scale modern geometric analysis forms therewith an alternative for curve description methods based on entropy scale-space theories.

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  1. Imaging Sciences Institute, Utrecht University Hospital, Room E.01.334, Heidelberglaan 100, 3584 CX, Utrecht, the Netherlands

    Alfons H. Salden, Bart M. ter Haar Romeny & Max A. Viergever

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  1. Alfons H. Salden
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Salden, A.H., ter Haar Romeny, B.M. & Viergever, M.A. Linearised Euclidean Shortening Flow of Curve Geometry. International Journal of Computer Vision 34, 29–67 (1999). https://doi.org/10.1023/A:1008172320786

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