Problems in Topology

1. Background In computer graphics and image processing a scene is often represented as an array of 0's and 1's. The set of 1's represents the object or objects in the scene and the set of 0's represents the background. The array is usually two-dimensional, but three-dimensional image arrays are produced by reconstruction from projections in applications such as computer tomography and electron microscopy (see Rosenfeld and Kak [1982, Chapter 11]). The array elements are called pixels in the 2D case and voxels in the 3D case. We identify each array element with the lattice point in the plane or 3-space whose coordinates are the array indices of the element. The lattice points that correspond to array elements with value 1 are called black points and the other lattice points are called white points. Let S be the set of black points. In pattern recognition one sometimes wants to reduce the black point set to a " skeleton " S ⊆ S with the property that the inclusion of S in S is " topology-preserving ". This is called thinning. Figure 1 shows what effect a thinning algorithm might have on a digitized '6'. In Figure 1 the large black dots represent points in S, and the boxed black dots represent points in the skeleton S ⊆ S. In this paper we are mainly concerned with the requirement that a thinning algorithm must preserve topology. However, it has to be pointed out that a thinning algorithm must satisfy certain non-topological conditions as well. (For example, the skeleton produced by thinning the digitized '6' in Figure 1 must look like a '6', which means that the 'arm' of the 6 must not be shortened too much.) The non-topological requirements of thinning are hard to specify precisely 1 and are beyond the scope of this paper. For n = 2 or 3 write E n for n-dimensional Euclidean space and write Z n for the set of lattice points in E n. The topological requirements of two-dimensional thinning are well understood. Let S ⊇ S be finite subsets of Z 2. In this section we define what it means for the inclusion of S in S to preserve topology. In fact we shall give three different but equivalent definitions. Given any set T ⊆ Z 2 we can construct a plane polyhedron C(T) ⊆ E 2 from T as follows. For each …