Ciesielski Krzysztof Chris, Udupa Jayaram K
Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310.
Comput Vis Image Underst. 2011 Jun 1;115(6):721-734. doi: 10.1016/j.cviu.2011.01.003.
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.
在当前大量的图像分割文献中，算法之间似乎存在相当多的冗余，而严重缺乏能够对它们进行理论比较以确定其相似性、等效性或差异性的方法。在本文中，我们试图填补这一空白。为实现这一目标，我们认为：（1）每个数字分割算法[公式：见原文]都应有一个定义明确的连续对应物[公式：见原文]，称为其模型，当图像分辨率趋于无穷大时，该模型构成[公式：见原文]的渐近形式；（2）两个这样的模型[公式：见原文]和[公式：见原文]相等，确立了它们数字对应物[公式：见原文]和[公式：见原文]的理论（渐近）等效性。只有当对于每个涉及的算法[公式：见原文]，其模型[公式：见原文]被证明是[公式：见原文]的渐近形式时，这种比较才具有完整的理论价值。到目前为止，即使在作为连续模型数字化引入的算法（如实值水平集分割算法）的情况下，此类证明在文献中也未出现。本文的主要目标是探索一条研究路线，用于将数字分割算法与其渐近模型进行形式上的配对，用数学证明来证明这种关系，并利用结果在这个一般理论框架内比较分割算法。作为朝着这个总体目标迈出的第一步，我们在此证明基于梯度的阈值模型[公式：见原文]是使用基于梯度的亲和力[公式：见原文]的模糊连通性Udupa和Samarasekera分割算法的渐近形式。我们还认为，从某种意义上说，[公式：见原文]是Malladi、Sethian和Vemuri的原始前沿传播水平集算法的渐近形式，从而确立了这两种特定算法之间的理论等效性。还提供了这最后一等效性的实验证据。
