Skeletal representations of orthogonal shapes

Abstract

In this paper we present two skeletal representations applied to orthogonal shapes of R n : the cube axis and a family of skeletal representations provided by the scale cube axis. Orthogonal shapes are a subset of polytopes, where the hyperplanes of the bounding facets are restricted to be axis aligned. Both skeletal representations rely on the L ∞ metric and are proven to be homotopically equivalent to its shape. The resulting skeleton is composed of n − 1 dimensional facets. We also provide an efficient and robust algorithm to compute the scale cube axis in R 2 and compare the resulting skeleton with other skeletal representations.

Compact Union of Disjoint Boxes: An Efficient Decomposition Model for Binary Volumes

“…Some 3D applications of OPP are: general computer graphics applications such as geometric transformations and Boolean operations [10,19], skeleton computation (instead of iterative peeling techniques) [18,37], and orthogonal hull computation [8,9]. OPP have been also used in theory of hybrid systems to model the solutions of reachable states [10,16].…”

A Practical and Robust Method to Compute the Boundary of Three-dimensional Axis-aligned Boxes

“…They have also been used to bound or approximate more complex shapes, with applications in spatial databases (Esperança and Samet, 1997), and motion planning (Albers et al, 1999). In particular, an efficient algorithm that obtains the union of boxes is required to compute the skeleton of orthogonal polyhedra (Martinez et al, 2013) The problem of computing the two-dimensional boundary of the union of a set of axis-aligned rectangles, among other properties, sparked interest in the 1980s (Jr. and Preparata, 1980;Wood, 1984;Güting, 1984). Several methods were proposed, and the problem has been optimally solved since then.…”

The three-dimensional cube and scale cube skeleton

“…The cube skeleton [33] is a novel skeletal representation of orthogonal polygons and polyhedra that rely on the L ∞ metric. The cube skeleton, which is composed of line segments and planar polygons with restricted orientation, refines the L ∞ Voronoi diagram (see Fig.…”

“…1b). The interior cube skeleton [33], which had only been defined for the two-dimensional case, is a homotopically equivalent subset of the cube skeleton whose segments that intersect the polygon boundary have been removed (see Fig. 1c).…”

“…A reliable computation of stable skeletal representations is still considered to be an open problem [8,37,43], despite the intense efforts of prior years. The scale cube skeleton [33], which is based upon the scale axis transform [25], tries to overcome these instability problems of the cube skeleton by defining a simplification scheme where shape features are ignored if they are small relative to their neighborhood.…”