Counting Gaps in Binary Pictures Valentin E. Brimkov1 , Angelo Maimone2 , and Giorgio Nordo2 1
Mathematics Department, SUNY Buffalo State College, Buffalo, NY 14222, USA
[email protected] 2 Dipartimento di Matematica, Universit` a di Messina, 98166 Messina, Italy
[email protected],
[email protected] Abstract. An important concept in combinatorial image analysis is that of gap. In this paper we derive a simple formula for the number of gaps in a 2D binary picture. Our approach is based on introducing the notions of free vertex and free edge and studying their properties from point of view of combinatorial topology. The number of gaps characterizes the topological structure of a binary picture and is of potential interest in property-based image analysis. Keywords: digital geometry, 2D binary picture, gap, gap-freeness.
1
Introduction
An important concept in combinatorial image analysis is that of gap. Intuitively, gaps are locations in a digital picture (that is any finite set of pixels/voxels in 2D/3D) through which a “discrete path” can penetrate. Gaps play an important role in rendering pixelized/voxelized scenes by casting digital rays from the image to the scene [8, 9]. Thus it is useful to know if a digital picture is gap-free or it has gaps of certain type. This is particularly interesting when dealing with digital curves or surfaces. It may also be helpful to have an estimation for the number of gaps (if any) in the considered object, possibly as a function of other object characteristics. Such kind of information may help better understand the topological structure of a binary picture and is of potential interest in propertybased image analysis. Results of this sort belong to combinatorial topology, but are also of interest in several other disciplines, such as digital geometry, combinatorial image analysis, and theory of computer graphics. A classical result is the famous Descartes-Euler formula v − e + f = 2 that relates the number of vertices (v), edges (e), and facets (f ) of a polytope. For various applications of this last formula and other similar results to image analysis and digital geometry, see Chapters 4 and 6 of [10]. In particular, digital picture gap-freeness appears to be equivalent to the notion of well-composedness of a set of pixels proposed by Latecki, Eckhardt, and Rosenfeld [13]. This last paper demonstrates the wealth of using well-composed (i.e., gap-free) sets in image analysis. A recent work [7] provided the formula g = v − 2(p + c − h) + b, where g is the number of gaps, v the number of vertices, p the number of pixels, h the number of holes, c the number of connected components, and b the number of U. Eckardt et al. (Eds.): IWCIA 2006, LNCS 4040, pp. 16–24, 2006. c Springer-Verlag Berlin Heidelberg 2006
Counting Gaps in Binary Pictures
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2 × 2 grid squares in a digital picture. In the present paper we obtain a simpler (and computationally more relevant) formula that expresses the number of gaps in a generic 2D digital picture in terms of the new notions of free vertex and free edge. We achieve this by certain combinatorial considerations within a digital topology framework. In the next section we introduce some basic notions and notations of digital topology. In Section 3 we present our main results, and we conclude in Section 4.
2 2.1
Preliminaries Some Basic Notions of Digital Topology
In this section we introduce some basic notions of digital topology to be used in the sequel. We conform to the terminology used in [10]. See also [11, 15, 18] for further details. All considerations take place in the grid cell model that consists of the grid cells of Z2 , together with the related topology. In the grid cell model we represent pixels as squares, called 2-cells. Their edges and vertices are called 1-cells and 0-cells, respectively. (i) For every i = 0, 1, 2 the set of all cells of size i (i-cells) is denoted by C2 . 2 (i) Further, we define the space C2 = i=0 C2 . We say that two 2-cells are 0(0) (1) adjacent (1-adjacent) if e ∩ e ∈ C2 (e ∩ e ∈ C2 ). The relation of 0-adjacency (resp., 1-adjacency) is denoted by A0 (resp., A1 ). Given a 2-cell p, by A0 (p) and A1 (p) we denote the A0 and A1 neighborhoods of p, respectively, that are the sets of all 2-cells which are 0-adjacent (resp. 1-adjacent) to p. (These are also called 0/1-neighbors of p.) We can also consider the grid cell model as an incidence structure, i.e., as a triple (C2 , I, dim) where I is an incidence relation defined as follows. For every pair of cells e, e ∈ C2 , we have eIe if and only if e is adjacent to e or e is adjacent to e and dim is a mapping from C2 to the set {0, 1, 2}. Note that the incidence relation I is reflexive and symmetric while the adjacency relations A0 and A1 are irreflexive and symmetric. The grid cell model can also be considered as an abstract cell complex (C2 ,
