Convergence of fuzzy-pyramid algorithms

Pyramid linking is an important technique for segmenting images and has many applications in image processing and computer vision. The algorithm is closely related to the ISODATA clustering algorithm and shares some of its properties. This paper investigates this relationship and presents a proof of convergence for the pyramid linking algorithm. The convergence of the hard-pyramid linking algorithm has been shown in the past; however, there has been no proof of the convergence of fuzzy-pyramid linking algorithms. The proof of convergence is based on Zangwill's theorem, which describes the convergence of an iterative algorithm in terms of a “descent function” of the algorithm. We show the existence of such a descent function of the pyramid algorithm and, further, show that all the conditions of Zangwill's theorem are met; hence the algorithm converges.