Two border following algorithms are proposed for the topological analysis of digitized binary images. The first one determines the surroundness relations among the borders of a binary image. Since the outer borders and the hole borders have a one-to-one correspondence to the connected components of 1-pixels and to the holes, respectively, the proposed algorithm yields a representation of a binary image, from which one can extract some sort of features without reconstructing the image. The second algorithm, which is a modified version of the first, follows only the outermost borders (i.e., the outer borders which are not surrounded by holes). These algorithms can be effectively used in component counting, shrinking, and topological structural analysis of binary images, when a sequential digital computer is used.

Topological Structural Analysis of Digitized Binary Images by Border Following

Digital topology deals with the topological properties of digital images: or, more generally, of discrete arrays in two or more dimensions. It provides the theoretical foundations for important image processing operations such as connected component labeling and counting, border following, contour filling, and thinning—and their generalizations to three- (or higher-) dimensional “images.” This paper reviews the fundamental concepts of digital topology and surveys the major theoretical results in the field. A bibliography of almost 140 references is included.

Digital topology: introduction and survey

Digital geometry is the study of geometrical properties of subsets of digital images. If the digitization is sufficiently fine-grained, such properties can be regarded as approximations to the corresponding properties of the "real" sets that gave rise, by digitization, to the digital sets; but it is also important to define how the properties can be computed for the digital sets themselves. Questions of particular interest include how images and image subsets are digitized; how geometric properties are defined for digitized sets; the computational complexity of computing them--in particular, whether they can be computed using simple (e.g., local) operations; characterizing image operations that preserve them; and characterizing digital objects that could be the digitizations of real objects that have given geometric properties. Concepts that have been extensively studied include topological properties (connected components, boundaries); curves and surfaces; straightness, curvature, convexity, and elongatedness; distance, extent, length, area, surface area, volume, and moments; shape description, similarity, symmetry, and relative position; shape simplification and skeletonization.

Digital geometry

Neighborhood coding of binary images for fast contour following and general binary array processing

An algorithm for connected component labeling of binary patterns using SIMD mesh connected computers is presented. The algorithm consists of three major steps: identifying exactly one point (seed point) within each connected component (region), assigning a unique label to each seed point, and expanding the labels to fill all pixels in the respective regions. Two approaches are given for identifying seed points. The first approach is based on shrinking and the second on the iterative replacement of equivalent labels with local minima or maxima. The shrinking algorithm reduces simply connected regions into single pixels, but multiply connected regions form rings around the holes contained in the regions. A parallel algorithm is developed to break each such ring at a single point. The broken rings are then reduced to single pixels by reshrinking. With iterations consisting of shrinking, breaking rings, if any, and reshrinking, each pattern (of any complexity) is reduced to isolated points within itself. In the second approach every region pixel in the image is initially given a unique label equal to its address in the image. Every 3 × 3 neighborhood in the image is then examined in parallel to replace the central label with the maximum (or minimum) of the labels assigned to the set of region pixels in the neighborhood. This is done iteratively until there is no further change. The seed points are then the locations where the pixel addresses match their converged labels. A parallel sorting method is used for assigning a consecutive set of numbers as labels to the seed points. Parallel expansion up to the boundaries of the original patterns then completes the connected component labeling. The computational complexities of the algorithm are discussed.

B Prasada

Papers

A parallel shrinking algorithm for binary patterns