Summary: Journal of Mathematical Imaging and Vision, 4, 291-302 (1994).
© Kluwer Academic Publishers. Manufactured in The Netherlands.
Convergence of Fuzzy-Pyramid Algorithms*
Department of Computer Science, 431 State Hail, Wayne State University,Detroit, MI 48202
Siemens Corporate Research, 755 CollegeRoad East, Princeton, NJ 08540
Computer and Vision Research Center, Department of Electrical and Computer Engineering, Universityof
Texas at Austin, Austin, TX 78712
Abstract. 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.
Key words, pyramid segmentation, image processing, multiresolution segmentation, pyramid