A framework for comparing different image segmentation methods and its use in studying equivalences between level set and fuzzy connectedness frameworks

In the current vast image segmentation literature, there seems to be considerable redundancy among algorithms, while there is a serious lack of methods that would allow their theoretical comparison to establish their similarity, equivalence, or distinctness. In this paper, we make an attempt to fill this gap. To accomplish this goal, we argue that: (1) every digital segmentation algorithm [Formula: see text] should have a well defined continuous counterpart [Formula: see text], referred to as its model, which constitutes an asymptotic of [Formula: see text] when image resolution goes to infinity; (2) the equality of two such models [Formula: see text] and [Formula: see text] establishes a theoretical (asymptotic) equivalence of their digital counterparts [Formula: see text] and [Formula: see text]. Such a comparison is of full theoretical value only when, for each involved algorithm [Formula: see text], its model [Formula: see text] is proved to be an asymptotic of [Formula: see text]. So far, such proofs do not appear anywhere in the literature, even in the case of algorithms introduced as digitizations of continuous models, like level set segmentation algorithms.The main goal of this article is to explore a line of investigation for formally pairing the digital segmentation algorithms with their asymptotic models, justifying such relations with mathematical proofs, and using the results to compare the segmentation algorithms in this general theoretical framework. As a first step towards this general goal, we prove here that the gradient based thresholding model [Formula: see text] is the asymptotic for the fuzzy connectedness Udupa and Samarasekera segmentation algorithm used with gradient based affinity [Formula: see text]. We also argue that, in a sense, [Formula: see text] is the asymptotic for the original front propagation level set algorithm of Malladi, Sethian, and Vemuri, thus establishing a theoretical equivalence between these two specific algorithms. Experimental evidence of this last equivalence is also provided.