Geodesic-based manifold learning for parameterization of triangular meshes

Abstract

Reverse Engineering (RE) requires representing with free forms (NURBS, Spline, Bézier) a real surface which has been pointsampled -- To serve this purpose, we have implemented an algorithm that minimizes the accumulated distance between the free form and the (noisy) point sample -- We use a dualdistance calculation point to / from surfaces, which discourages the forming of outliers and artifacts -- This algorithm seeks a minimum in a function that represents the fitting error, by using as tuning variable the control polyhedron for the free form -- The topology (rows, columns) and geometry of the control polyhedron are determined by alternative geodesicbased dimensionality reduction methods: (a) graphapproximated geodesics (Isomap), or (b) PL orthogonal geodesic grids -- We assume the existence of a triangular mesh of the point sample (a reasonable expectation in current RE) -- A bijective composition mapping allows to estimate a size of the control polyhedrons favorable to uniformspeed parameterizations -- Our results show that orthogonal geodesic grids is a direct and intuitive parameterization method, which requires more exploration for irregular triangle meshes -- Isomap gives a usable initial parameterization whenever the graph approximation of geodesics on be faithful -- These initial guesses, in turn, produce efficient free form optimization processes with minimal errors -- Future work is required in further exploiting the usual triangular mesh underlying the point sample for (a) enhancing the segmentation of the point set into faces, and (b) using a more accurate approximation of the geodesic distances within , which would benefit its dimensionality reduction