On Boundaries and Boundary Crack-Codes of Multidimensional Digital Images

Abstract

An asymmetric, anisotropic relation on the boundary elements of a binary image, due to Gordon and Udupa (Gordon, D., Udupa, J.K. (1989). Fast surface tracking in three-dimensional binary images, Computer Vision, Graphics and Image Processing 45, pp. 196–214.), is used to generalize the 2-D concept of a difference crack-code to crack-codes that represent boundaries in higher-dimensional binary images. For an n-dimensional binary image where n ≥ 3, it is shown how each connected component of the boundary (with respect to Gordon and Udupa’s relation) can be represented by a “crack code” consisting of a single pair of sequences. It is also shown that the amount of memory required to store such a crack code for each component of the boundary does not exceed (4 + [log(n − 1)])(1−1/n) bits per boundary element. In particular, the memory requirement is no more than 3 1/3 bits per boundary element for 3-D images, and no more than 4 1/2 bits per boundary element for 4-D images.

A part of the work reported in this paper was done while the author held a visiting appointment at the Medical Image Processing Group, Department of Radiology, University of Pennsylvania, Philadelphia, PA 19104, USA. The author has enjoyed many useful and stimulating discussions with Dr G. T. Herman and Dr J. K. Udupa on the subject of boundaries in multidimensional digital images.