The Vese-Chan model for multiphase image segmentation uses m binary label functions to construct 2^m characteristic functions for different phases/regions systematically; the terms in this model have moderate degrees comparing with other schemes of multiphase segmentation. However, if the number of desired regions is less than 2^m, there exist some empty phases which need costly parameter estimation for segmentation purpose. In this paper, we propose an automatic construction method for characteristic functions via transformation between a natural number and its binary expression, and thus, the characteristic functions of empty phases can be written and recognized naturally. In order to avoid the redundant parameter estimations of these regions, we add area constraints in the original model to replace the corresponding region terms to preserve its systematic form and achieve high efficiency. Additionally, we design the alternating direction method of multipliers (ADMM) for the proposed modified model to decompose it into some simple sub-problems of optimization, which can be solved using Gauss-Seidel iterative method or generalized soft thresholding formulas. Some numerical examples for gray images and color images are presented finally to demonstrate that the proposed model has the same or better segmentation effects as the original one, and it reduces the estimation of redundant parameters and improves the segmentation efficiency.

中文翻译：

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无冗余参数估计的Vese-Chan模型用于多相图像分割

用于多相图像分割的Vese-Chan模型使用m个二元标签函数针对不同的相位/区域系统地构建2 ^m个特征函数；与其他多阶段分割方案相比，该模型中的术语具有中等程度。但是，如果所需区域的数量小于2 ^m，存在一些空白阶段，需要昂贵的参数估计来进行细分。在本文中，我们提出了一种通过自然数与其二进制表达式之间的转换来自动构建特征函数的方法，从而可以自然地编写和识别空相的特征函数。为了避免这些区域的冗余参数估计，我们在原始模型中添加了区域约束以替换相应的区域项，以保留其系统形式并实现高效率。此外，我们针对提出的改进模型设计了乘数交替方向方法（ADMM），以将其分解为一些简单的优化子问题，可以使用Gauss-Seidel迭代方法或广义软阈值公式进行求解。