Medial axis lookup table and test neighborhood computation for 3D chamfer norms

Abstract

Abstract. Chamfer distances are discrete distances based on the propagation of local distances, or weights, defined in a mask. The medial axis, i.e. the centers of maximal balls (balls which are not contained in any other ball), is a powerful tool for shape representation and analysis. The extraction of maximal disks is performed in the general case by testing the inclusion of a ball in a local neighborhood with covering relations usually represented by lookup tables. The proposed method determines if a mask induces a norm and in this case, computes the lookup tables and the test neighborhood based on geometric properties of the balls of chamfer norms, represented as H-polytopes. The method does not need to repeatedly scan the image space, and improves the computation time of both the test neighborhood detection and the lookup AbstractChamfer distances are discrete distances based on the propagation of local distances, or weights, defined in a mask. The medial axis, i.e. the centers of maximal balls (balls which are not contained in any other ball), is a powerful tool for shape representation and analysis. The extraction of maximal disks is performed in the general case by testing the inclusion of a ball in a local neighborhood with covering relations usually represented by lookup tables. The proposed method determines if a mask induces a norm and in this case, computes the lookup tables and the test neighborhood based on geometric properties of the balls of chamfer norms, represented as H-polytopes. The method does not need to repeatedly scan the image space, and improves the computation time of both the test neighborhood detection and the lookup table computation.

Metric Bases for Polyhedral Gauges

“…Given a convex polyhedron C in R n , an hyperplane h i is a supporting hyperplane if h i contains a facet f i of C. We name supporting half-space H i the half-space bounded by h i and containing C. The intersection ∩ i H i of the supporting half-spaces is equal to C. In R n , the decomposition of C in supporting half-spaces is unique. Note that this is generally not the case in Z n , see [15].…”

2D Subquadratic Separable Distance Transformation for Path-Based Norms

“…We define a wedge as a pair (v k , v k+1 ) of vectors. To each wedge is associated a row A k in the minimal H-representation of A (A k can also be seen as a -non-unitary-normal vector to B R facets [19]). Using similar notations, [17,6] demonstrate that the distance evaluation of point a can be obtained in two steps: First, we compute the wedge (v k , v k+1 ) a belongs to.…”

“…Definition 4 (Rational ball, minimal H-representation [17,19]). Given a Chamfer norm M, the rational ball associated with M is the polytope…”

“…Proposition 1 (Direct Distance Computation [19]). Given a chamfer mask M induced by a norm and (A, y) its minimal parameter H-representation, then for any a ∈ Z n , the chamfer distance of a from the origin is…”

Appearance Radii in Medial Axis Test Mask for Small Planar Chamfer Norms

“…In recent papers, Normand andÉvenou have proposed a faster method for chamfer norms in 2D and 3D based on a polytope representation of chamfer balls [12,13].…”