Optimal knots selection for sparse reduced data

Abstract

We discuss an interpolation scheme (based on optimization) to fit a given ordered sample of reduced data Qm in arbitrary Euclidean space. Here the corresponding knots are not given and need to be first somehow guessed. This is accomplished by solving an appropriate optimization problem, where the missing knots minimize the cost function measuring the total squared norm of acceleration of the interpolant (here a natural spline). The initial infinite dimensional optimization (set to minimize an acceleration within the class of admissible curves) is reduced to the finite dimensional problem, for which the unknown optimal interpolation knots are to be found. The latter introduces a highly non-linear optimization task, both difficult for theoretical analysis and in derivation of computationally feasible optimization scheme (in particular handling medium and large number of data points). The experiments to compare the interpolants based either on optimal knots or on the so-called cumulative chords are performed for 2D and 3D data. The problem of interpolating or approximating reduced data is applicable in computer vision (image segmentation), in computer graphics (curve modeling in computer aided geometrical design) or in engineering and physics (trajectory modeling).