Reconstruction of parametric curves through a probabilistic approach

Abstract

This work deals with the reconstruction of parametric curves using probability density functions to generate a sampling of points in the curve domain and approximate it over a specific interval. There are several approaches for sampling parametric curves, which allow it to be in accordance with the curvature or arc length. In general, these techniques are based on heuristics, and fail to provide optimal solutions. In this article, we intend to use a probabilistic approach, in the way that the resulting point sampling is in accordance with some density function defined in the curve domain, placing more points where this density is greater. As it is more general, this approach includes the cases mentioned above as particular cases. In general, approximating flat curves based on the Uniform Distribution proved to be more efficient than taking the Exponential Distribution as a reference. The value of lambda in the Exponential Distribution interferes in the approximation of some curves, being necessary to find a value of lambda suitable to get a good approximation of the curve. Approaching the curve based on its curvature is a method used when it is wanted to generate more samples where the curvature is greater. However, this method can only be used in special cases since the integral of the curvature function, in most cases, is very difficult to be calculated.