We introduce a multi-dimensional generalization of the Euclidean algorithm and show how it is related to digital geometry and particularly to the generation and recognition of digital planes. We show how to associate ...
详细信息
We introduce a multi-dimensional generalization of the Euclidean algorithm and show how it is related to digital geometry and particularly to the generation and recognition of digital planes. We show how to associate with the steps of the algorithm geometrical extensions of substitutions, i.e., rules that replace faces by unions of faces, to build finite sets called patterns. We examine several of their combinatorial, geometrical and topological properties. This work is a first step toward the incremental computation of patterns that locally fit a digital surface for the accurate approximation of tangent planes.
When processing the geometry of digital surfaces (boundaries of voxel sets), linear local structures such as pieces of digital planes play an important role. To capture such geometrical features, planeprobing algorith...
详细信息
When processing the geometry of digital surfaces (boundaries of voxel sets), linear local structures such as pieces of digital planes play an important role. To capture such geometrical features, planeprobingalgorithms have demonstrated their strength: starting from an initial triangle, the digital structure is locally probed to expand the triangle approximating the plane parameters more and more precisely (converging to the exact parameters for infinite digital planes). Among the different plane-probing algorithms, the L-algorithm is a plane-probing algorithm variant which takes into account a generally larger neighborhood of points for its update process. We show in this paper that this algorithm has the advantage to guarantee the so-called Delaunay property of the set of probing points, which has interesting consequences: it provides a minimal basis of the plane and guarantees an as-local-as-possible computation.
We show that the plane-probing algorithms introduced in Lachaud et al. (J. Math. Imaging Vis., 59, 1, 23-39, 2017), which compute the normal vector of a digital plane from a starting point and a set-membership predica...
详细信息
ISBN:
(纸本)9783031198960;9783031198977
We show that the plane-probing algorithms introduced in Lachaud et al. (J. Math. Imaging Vis., 59, 1, 23-39, 2017), which compute the normal vector of a digital plane from a starting point and a set-membership predicate, are closely related to a three-dimensional generalization of the Euclidean algorithm. In addition, we show how to associate with the steps of these algorithms generalized substitutions, i.e., rules that replace square faces by unions of square faces, to build finite sets of elements that periodically generate digital planes. This work is a first step towards the incremental computation of a hierarchy of pieces of digital plane that locally fit a digital surface.
暂无评论