Connectivity-preserving morphological image transformations

Methods for thinning connected components of an image differ in the size of support, type of connectivity preserved, degrees of parallelism and pipelining, and smoothness and fidelity to structure of the results. A unifying framework is presented, using image morphology, of all 4- and 8-connectivity-preserving (CP) transformations that use a 3 X 3 basis of support on binary images discretized on a square lattice. Two types of atomic CP transformations are defined: weak CP neither breaks nor joins components and strong CP additionally preserves the number of connected components. It is shown that out of thousands of possible 3 X 3 hit-miss structuring elements (SEs), in their most general form there are only four SEs (and their rotational isomorphs) for each of the two sets (4- and 8-connectivity) that satisfy strong CP for atomic operations. Simple symmetry properties exist between elements of each set, and duality relations exist between these sets of SEs under reversal of foreground/background and thinning/thickening operations. The atomic morphological operations, that use one SE, are intrinsically parallel and translationally invariant, and the best thinned skeletons are produced by sequences of operations that use multiple SEs in parallel. A subset of SEs that preserve both 4- and 8-connectivity and have a high degree of symmetry can be used in the most parallel fashion without breaking connectivity and produce very smooth skeletons. For thickening operations, foreground components either self-limit on convex or expand indefinitely. The self-limited convex hulls are formed either by horizontal and vertical lines, or by lines of slope +/- 1. Four types of boundary contours can result for thickening operations that expand indefinitely. Thickened text images result in a variety of typographically interesting forms.