  Image segmentation is a fundamental topic in image processing and has been
studied for many decades. Deep learning-based supervised segmentation models
have achieved state-of-the-art performance but most of them are limited by
using pixel-wise loss functions for training without geometrical constraints.
Inspired by Euler's Elastica model and recent active contour models introduced
into the field of deep learning, we propose a novel active contour with
elastica (ACE) loss function incorporating Elastica (curvature and length) and
region information as geometrically-natural constraints for the image
segmentation tasks. We introduce the mean curvature i.e. the average of all
principal curvatures, as a more effective image prior to representing curvature
in our ACE loss function. Furthermore, based on the definition of the mean
curvature, we propose a fast solution to approximate the ACE loss in
three-dimensional (3D) by using Laplace operators for 3D image segmentation. We
evaluate our ACE loss function on four 2D and 3D natural and biomedical image
datasets. Our results show that the proposed loss function outperforms other
mainstream loss functions on different segmentation networks. Our source code
is available at https://github.com/HiLab-git/ACELoss.
