Detection of the non-topology preservation of Ma and Sonka’s algorithm, by the use of P-simple points

Abstract

In 1996, Ma and Sonka proposed a thinning algorithm which yields curve skeletons for 3D binary images [C. Ma, M. Sonka, A fully parallel 3D thinning algorithm and its applications, Comput. Vis. Image Underst. 64 (3) (1996) 420–433]. This algorithm is one of the most referred thinning algorithms in the context of digital topology: either by its use in medical applications or for comparisons with other thinning algorithms. In 2007, Wang and Basu [T. Wang, A. Basu, A note on ‘a fully parallel 3D thinning algorithm and its applications’, Pattern Recognit. Lett. 28 (4) (2007) 501–506] wrote a paper in which they claim that Ma and Sonka’s 3D thinning algorithm does not preserve topology. As they highlight in their paper, a counter-example was given in 2001, in Lohou’s thesis [C. Lohou, Contribution à l’analyse topologique des images: étude d’algorithmes de squelettisation pour images 2D et 3D selon une approche topologie digitale ou topologie discrète. Ph.D. thesis, University of Marne-la-Vallée, France, 2001]. In this paper, it is shown how P-simple points have guided the author towards a proof that Ma and Sonka’s algorithm does not always preserve topology. Moreover, the reasoning being very general, it could be reused for such a purpose, i.e., to simplify the proof on the non-topology preservation.