RESUMEN
A weak notion of elastic energy for (not necessarily regular) rectifiable curves in any space dimension is proposed. Our [Formula: see text]-energy is defined through a relaxation process, where a suitable [Formula: see text]-rotation of inscribed polygons is adopted. The discrete [Formula: see text]-rotation we choose has a geometric flavour: a polygon is viewed as an approximation to a smooth curve, and hence its discrete curvature is spread out into a smooth density. For any exponent [Formula: see text] greater than 1, the [Formula: see text]-energy is finite if and only if the arc-length parametrization of the curve has a second-order summability with the same growth exponent. In that case, moreover, the energy agrees with the natural extension of the integral of the [Formula: see text]th power of the scalar curvature. Finally, a comparison with other definitions of discrete curvature is provided. This article is part of the theme issue 'Foundational issues, analysis and geometry in continuum mechanics'.
RESUMEN
It is an urgent problem to know how to quickly and accurately measure the length of irregular curves in complex background images. To solve the problem, we first proposed a quasi-bimodal threshold segmentation (QBTS) algorithm, which transforms the multimodal histogram into a quasi-bimodal histogram to achieve a faster and more accurate segmentation of the target curve. Then, we proposed a single-pixel skeleton length measurement (SPSLM) algorithm based on the 8-neighborhood model, which used the 8-neighborhood feature to measure the length for the first time, and achieved a more accurate measurement of the curve length. Finally, the two algorithms were tested and analyzed in terms of accuracy and speed on the two original datasets of this paper. The experimental results show that the algorithms proposed in this paper can quickly and accurately segment the target curve from the neon design rendering with complex background interference and measure its length.