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 framework of entropic regularized optimal transport theory. The classical Total Variation (TV) regularizer and volume preserving are integrated into a regularized optimal transport model, and the volume and classification constraints can be regarded as two measures preserving constraints in the optimal transport problem. By studying the dual problem, we develop a simple and 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 nets such as the popular U-net.
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