Fairing of Discrete Planar Curves by Discrete Eulers Elasticae

After characterizing the integrable discrete analogue of the Eulers elastica, we focus our attention on the problem of approximating a given discrete planar curve by an appropriate discrete Eulers elastica. We carry out the fairing process via a $L^2!$-distance minimization to avoid the numerical instabilities. The optimization problem is solved via a gradient-driven optimization method (IPOPT). This problem is non-convex and the result strongly depends on the initial guess, so that we use a discrete analogue of the algorithm provided by Brander et al., which gives an initial guess to the optimization method.