RECONSTRUCTION OF A CURVE BY ITS POINT CLOUD USING GAUSSIAN POLYNOMIAL

Abstract

The paper describes the prospects of using a specific type of Gaussian polynomial function for the reconstruction of a continuous and regular surface by its point cloud.

         Restoring the curves and surfaces behind their point clouds is an important sub-problem for 3D scanning-based modeling. Also, a similar problem arises when converting surfaces from boundary representation to triangle mesh. For the latter, it is critically important to restore the orientation of the surface if the initial data does not contain such information, or contains false information.

         The Gaussian interpolation polynomial certainly goes to zero, while the argument of the function goes to plus- and minus-infinity at any configuration of the point cloud. Thus, if in a 3-dimensional space, we define the point cloud surface as an isosurface of some predefined positive value, for example, 1, and consider the surface oriented in the direction of isolines with a lower value, the surface constructed by a generalized interpolating Gaussian polynomial should always appear to be oriented correctly, i.e. the space bounded by the surface will always be closed, and the closure will contain the inner side of it. Moreover, the isosurface itself will be continuous and regular since the Gaussian functions that constitute the polynomial are infinitely differentiable.

         Despite these useful properties, the interpolation Gaussian polynomial has not been used in point cloud-based curve restoration. The purpose of this research is to assess the practical applicability of Gaussian interpolation in the context of restoring curves and surfaces by their point clouds.

         The challenge of such application lies not in obtaining the correct differential properties of the resulting curves and surfaces but in establishing their topological properties, namely connectivity, and their resulting shapes. The resulting shape of the curve or surface obtained by Gaussian interpolation should somehow correspond to the implied surface or curve behind the point cloud.

          Keywords: interpolation, Gaussian polynomial, regular surfaces, point cloud, mathematical optimization, geometrical modeling, modeling.