  Recent state-of-the-art image segmentation algorithms are mostly based on
deep neural networks, thanks to their high performance and fast computation
time. However, these methods are usually trained in a supervised manner, which
requires large number of high quality ground-truth segmentation masks. On the
other hand, classical image segmentation approaches such as level-set methods
are formulated in a self-supervised manner by minimizing energy functions such
as Mumford-Shah functional, so they are still useful to help generation of
segmentation masks without labels. Unfortunately, these algorithms are usually
computationally expensive and often have limitation in semantic segmentation.
In this paper, we propose a novel loss function based on Mumford-Shah
functional that can be used in deep-learning based image segmentation without
or with small labeled data. This loss function is based on the observation that
the softmax layer of deep neural networks has striking similarity to the
characteristic function in the Mumford-Shah functional. We show that the new
loss function enables semi-supervised and unsupervised segmentation. In
addition, our loss function can be also used as a regularized function to
enhance supervised semantic segmentation algorithms. Experimental results on
multiple datasets demonstrate the effectiveness of the proposed method.
