Image segmentation with a volume constraint is an important prior for many real applications. In this work, we present a novel volume preserving image segmentation algorithm, which is based on the entropy and Total Variation (TV) regularized optimal transport theory. The volume and classification constraints can be regarded as two measures preserving constraints in the optimal transport. By studying the dual problem, we develop a simple but efficient dual algorithm for our model. Moreover, to be different from many variational based image segmentation algorithms, the proposed algorithm can be directly unrolled to a new Volume Preserving and TV regularized softmax (VPTV-softmax) layer for semantic segmentation in the popular Deep Convolution Neural Network (DCNN). The experiment results show that our proposed model is very competitive and can improve the performance of many semantic segmentation networks such as the popular U-net and DeepLabv3+.

具有体积约束的图像分割对于许多实际应用而言是重要的先决条件。在这项工作中，我们提出了一种新颖的体积保留图像分割算法，该算法基于熵和总变化（TV）正则化最优输运理论。数量和分类约束可以看作是在最佳运输中保持约束的两种措施。通过研究对偶问题，我们为模型开发了一种简单而有效的对偶算法。此外，与许多基于变分的图像分割算法不同，该算法可以直接展开到新的“体积保留和电视规则化” softmax（VPTV-softmax）层，以在流行的深度卷积神经网络（DCNN）中进行语义分割。