Surface Thinning in 3D Cubical Complexes

Abstract

Abstract. We introduce a parallel thinning algorithm with directional substeps based on the collapse operation, which is guaranteed to preserve topology and to provide a thin result. Then, we propose two variants of a surface-preserving thinning scheme, based on this parallel directional thinning algorithm. Finally, we propose a methodology to produce filtered surface skeletons, based on the above thinning methods and the recently introduced discrete λ-medial axis.

Collapses and Watersheds in Pseudomanifolds of Arbitrary Dimension

“…Note that if X is not a subcomplex of an n-pseudomanifold, the previous property is, in general, not true. Note also that this property appears in [15] in the case where the space is the cubical complex F n , which is a discrete manifold whose faces are made of points in Z n . Before proving Property 6, let us first state a remark.…”

Thin structure filtering framework with non-local means, Gaussian derivatives and spatially-variant mathematical morphology

“…For this, the vesselness image I ves is thresholded so that most of the vessellike objects are preserved and results in an image I t ves . Then, we obtain a morphological skeleton of I t ves , from which we extract the endpoints [12]. By identifying the endpoints of this skeleton [13], we ensure that we use the tubular objects as markers for direction field propagation and further reconnection only between disconnected objects.…”

SurfCut: Surfaces of Minimal Paths from Topological Structures

“…Our novel algorithms use concepts from cubical complex theory. In contrast to [28], our method is designed to robustly extract ridges of a function or data defined on a surface (defined by Fast Marching), rather than geometrical properties of a surface.…”

“…Since the algorithms we define require the extraction of lower dimensional structures (ridge curves from surfaces, and valley surfaces from volumes), it is important that the algorithms are guaranteed to produce lower dimensional structures with correct topology. The theory of cubical complexes (e.g., [27], [28]) guarantees such lower dimensional structures are generated with homotopy equivalence to the original data.…”