An Algorithm for a 3D Simplicity Test

According to Kong [4], a thinning algorithm T is connectivity preserving if and only if every set deleted by T can Every thinning (or shrinking) algorithm can be stated as a list of deletion configurations. If a neighborhood of a given be ordered in such a sequence that every point is simple image point matches one of these configurations, the point will after all previous points in the sequence are deleted. To be deleted. In order to make design of a (3D) thinning (or prove a parallel thinning algorithm preserves connectivity, shrinking) algorithm easy to handle, and therefore the list of one must show it preserves connectivity for all possible configurations as short as possible, every configuration is deimages. But it could be difficult to check all images, since scribed in three colors, black, white, and ‘‘don’t-care,’’ where there are too many possible images. Ronsenfeld [11], the don’t-care ‘‘color’’ matches either a black or white point in Ronse [10], and Hall [3], proposed different techniques to a given image. A thinning algorithm is connectivity preserving if simplify the proof that a 2D thinning algorithm preserves and only if every set deleted by this algorithm can be ordered connectivity. Hall [2] and Ma [6] proposed corresponding in such a sequence that every point is simple after all previous 3D techniques, where Hall’s 3D results were extended points in the sequence are deleted. Therefore, it is very important to determine whether a deleted point is simple or not. from his 2D results, and Ma’s 3D results were based on We present the first 3D algorithm for determining the simplicity Ronse’s 2D results. All these techniques require that if a of any three-color deletion configuration. This algorithm is black point is deleted, the deletion of such a point does memory efficient and fast. Hence it can be a very useful tool for not change the connectivity of the image. Therefore, while designing 3D connectivity preserving thinning (or shrinking) designing a 3D thinning (or shrinking) algorithm, one has algorithms.  1996 Academic Press, Inc. to take care that every point deleted by the algorithm is simple. Every deletion condition can be stated as a list of dele