Cumulative chord piecewise-quartics for length and curve estimation

We discuss the problem of estimating an arbitrary regular parameterized curve and its length from an ordered sample of interpolation points in n-dimensional Euclidean space. The corresponding tabular parameters are assumed to be unknown. In this paper the convergence rates for estimating both curve and its length with cumulative chord piecewise-quartics are established for different types of unparameterized data including epsilon-uniform samplings. The latter extends previous results on cumulative chord piecewise-quadratics and piecewise-cubics. The numerical experiments carried out for planar and space curves confirm sharpness of the derived asymptotics. A high quality approximation property of piecewise-quartic cumulative chords is also experimentally verified on sporadic data. Our results may be of interest in computer vision (e.g. in edge and range image segmentation or in tracking), digital image processing, computer graphics, approximation and complexity theory or digital and computational geometry.