A medial-surface oriented 3-d two-sub®eld thinning algorithm Cherng-Min Ma * , Shu-Yen Wan Department of Information Management, Chang Gung University, 259, Wen-Hwa 1st Rd., Kwei-Shan, Tao-Yuan 333, Taiwan, ROC Received 27 November 2000; received in revised form 15 March 2001 Abstract A new thinning algorithm for extracting medial surfaces on 3-d binary images is proposed. It works in cubic grids where the 26-adjacency relation is used in the set of 1-voxels and the 6-adjacency relation is used in the set of 0-voxels. The new thinning algorithm is a two-sub®eld algorithm, i.e., a 3-d image is separated into two isometric sub®elds and the algorithm works on one sub®eld at a time. For extracting medial-surface skeletons, the algorithm preserves ``edge voxels''. The thinning algorithm is proved to preserve connectivity. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Connectivity preservation; Thinning; 2-Sub®eld; 3-d Image; Medial surface 1. Background Thinning is a fundamental preprocess of many image processing and pattern recognition opera- tions. The purpose of thinning is to remove un- necessary information so that the remaining information is sucient to allow topological analysis Kong, 1989). The resulting images of thinning can be applied to OCR or to medical informatics, etc. There are two kinds of skeletons that can be extracted by 3-d thinning algorithms: medial curves and medial surfaces. A thinning algorithm should preserve connec- tivity Kong, 1995; Ma, 1994; Ronse, 1986, 1988) that implies, for example, no object component can be vanished completely or be split to two or more object components. In this paper, a 3-d two- sub®eld parallel thinning algorithm for extracting skeletons as medial surfaces is proposed where the set of voxels of a 3-d image is classi®ed into two isometric sub®elds. The thinning algorithm is applied to voxels in each sub®eld alternatively. Several test images and their medial-surface skeletons are provided. The thinning algorithm is proved to preserve connectivity shown in Appendix A). 2. Basic notations There are only two kinds of voxels, 0's and 1's, in 3-d binary) images. See Kong, 1989) for de®- nitions of k-adjacencies, k-neighbors, k-connected- ness for k 6, 18 or 26. Also see Kong, 1989) for the de®nitions of 0's and 1's components, denoted 0- and 1-components, respectively. www.elsevier.com/locate/patrec Pattern Recognition Letters 22 2001) 1439±1446 * Corresponding author. Tel.: +886-33283016; fax: +886- 33271304. E-mail addresses: minma@mail.cgu.edu.tw C.-M. Ma), sywan@mail.cgu.edu.tw S.-Y. Wan). 0167-8655/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII:S0167-865501)00083-6