Deep convolutional networks recently made many breakthroughs in medical image segmentation. Still, some anatomical artefacts may be observed in the segmentation results, with holes or inaccuracies near the object boundaries. To address these issues, loss functions that incorporate constraints, such as spatial information or prior knowledge, have been introduced. An example of such prior losses are the contour-based losses, which exploit distance maps to conduct point-by-point optimization between ground-truth and predicted contours. However, such losses may be computationally expensive or susceptible to trivial local solutions and vanishing gradient problems. Moreover, they depend on distance maps which tend to underestimate the contour-to-contour distances. We propose a novel loss constraint that optimizes the perimeter length of the segmented object relative to the ground-truth segmentation. The novelty lies in computing the perimeter with a soft approximation of the contour of the probability map via specialized non-trainable layers in the network. Moreover, we optimize the mean squared error between the predicted perimeter length and ground-truth perimeter length. This soft optimization of contour boundaries allows the network to take into consideration border irregularities within organs while still being efficient. Our experiments on three public datasets (spleen, hippocampus and cardiac structures) show that the proposed method outperforms state-of-the-art boundary losses for both single and multi-organ segmentation.

% as the difference between dilation and erosion of predicted segmentations carried out