Surfaces in three-dimensional digital images

Local configurations in discrete combinatorial surfaces

“…In [14], we see that Br m ′ (V) includes all types of points except for spherical points (type 3a) which are interior points of V and that there are relations between our discrete complexes C m constructed from V and Br m ′ (V) only for the pairs (m, m ′ ) = (6, 18), (6,26), (18,6), (26,6) [14]. Those relations indicate that boundary points of Br m ′ (V) do not have always semi-spherical stars, but also the other stars such as one-, two-and three-dimensional stars except for spherical stars depending on their local point configurations.…”

“…Morgenthaler et al defined discrete surfaces by using the point connectivity based on the Jordan surface theorem; any Jordan surface divides the space into two [18]. In [6], Couprie et al pointed out that, for the 26-neighborhood system, Morgenthaler's discrete surfaces have only 13 local configurations while their discrete surfaces, called simplicity surfaces, have 736 configurations.…”

Generalized Simple Surface Points

“…In this section we introduce a new notion of simple surface point that extends the original definition of Morgenthaler [8], and find some properties of subsets consisting entirely of voxels of this class. For this we start by recalling some basic definitions of the graph-theoretical approach to the digital topology of Z 3 .…”

“…Given S ⊆ Z 3 and an adjacency pair (k, k), k, k ∈ {6, 18, 26}, Morgenthaler's definition [8] states that σ ∈ S is a (simple) surface point if the following conditions hold: (a) exactly one k-component of N 26 (σ) ∩ S − {σ} is k-adjacent to σ; (b) S is k-thin at σ; and, (c) every voxel τ ∈ S k-adjacent to σ is k-adjacent to the k-components A σ and B σ of S σ . In fact, Kong and Roscoe [7] showed that property (a) is redundant for all sets consisting entirely of surface points.…”

“…It is commonly accepted that the first notion of a discrete surface, as a "thin" subset of voxels of the grid Z 3 , is due to Morgenthaler and Rosenfeld [8]. They define S ⊆ Z 3 to be a simple closed surface if all voxels in S are simple surface points, where such points are determined by three properties which are exclusively given in terms of the usual adjacency relations between voxels and subsets of voxels of Z 3 .…”