A general theory of image segmentation: level set segmentation in the fuzzy connectedness framework

Abstract

In the current vast image segmentation literature, there is a serious lack of methods that would allow theoretical comparison of the algorithms introduced by using different mathematical methodologies. The main goal of this article is to introduce a general theoretical framework for image segmentation that would allow such comparison. The framework is based on the formal definitions designed to answer the following fundamental questions: What is the relation between an idealized image and its digital representation? What properties a segmentation algorithm must satisfy to be acknowledged as acceptable? What does it mean that a digital image segmentation algorithm truly approximates an idealized segmentation model? We use the formulated framework to analyze the front propagation (FP) level set algorithm of Malladi, Sethian, and Vemuri and compare it with the fuzzy connectedness family of algorithms. In particular, we prove that the FP algorithm is weakly model-equivalent with the absolute fuzzy connectedness algorithm of Udupa and Samarasekera used with gradient based affinity. Experimental evidence of this equivalence is also provided. The presented theoretical framework can be used to analyze any arbitrary segmentation algorithm. This line of investigation is a subject of our forthcoming work.