Recursive simplex stars

Abstract

This paper proposes a new method which builds a simplex based approximation of a $d-1$-dimensional manifold $M$ separating a $d$-dimensional compact set into two parts, and an efficient algorithm classifying points according to this approximation. In a first variant, the approximation is made of simplices that are defined in the cubes of a regular grid covering the compact set, from boundary points that approximate the intersection between $M$ and the edges of the cubes. All the simplices defined in a cube share the barycentre of the boundary points located in the cube and include simplices similarly defined in cube facets, and so on recursively. In a second variant, the Kuhn triangulation is used to break the cubes into simplices and the approximation is defined in these simplices from the boundary points computed on their edges, with the same principle. Both the approximation in cubes and in simplices define a separating surface on the whole grid and classifying a point on one side or the other of this surface requires only a small number (at most $d$) of simple tests. Under some conditions on the definition of the boundary points and on the reach of $M$, for both variants the Hausdorff distance between $M$ and its approximation decreases like $\mathcal{O}(d n_G^{-2})$, where $n_G$ is the number of points on each axis of the grid. The approximation in cubes requires computing less boundary points than the approximation in simplices but the latter is always a manifold and is more accurate for a given value of $n_G$. The paper reports tests of the method when varying $n_G$ and the dimensionality of the space (up to 9).