  The Hausdorff Distance (HD) is widely used in evaluating medical image
segmentation methods. However, existing segmentation methods do not attempt to
reduce HD directly. In this paper, we present novel loss functions for training
convolutional neural network (CNN)-based segmentation methods with the goal of
reducing HD directly. We propose three methods to estimate HD from the
segmentation probability map produced by a CNN. One method makes use of the
distance transform of the segmentation boundary. Another method is based on
applying morphological erosion on the difference between the true and estimated
segmentation maps. The third method works by applying circular/spherical
convolution kernels of different radii on the segmentation probability maps.
Based on these three methods for estimating HD, we suggest three loss functions
that can be used for training to reduce HD. We use these loss functions to
train CNNs for segmentation of the prostate, liver, and pancreas in ultrasound,
magnetic resonance, and computed tomography images and compare the results with
commonly-used loss functions. Our results show that the proposed loss functions
can lead to approximately 18-45 % reduction in HD without degrading other
segmentation performance criteria such as the Dice similarity coefficient. The
proposed loss functions can be used for training medical image segmentation
methods in order to reduce the large segmentation errors.
