  In many medical imaging and classical computer vision tasks, the Dice score
and Jaccard index are used to evaluate the segmentation performance. Despite
the existence and great empirical success of metric-sensitive losses, i.e.
relaxations of these metrics such as soft Dice, soft Jaccard and
Lovasz-Softmax, many researchers still use per-pixel losses, such as (weighted)
cross-entropy to train CNNs for segmentation. Therefore, the target metric is
in many cases not directly optimized. We investigate from a theoretical
perspective, the relation within the group of metric-sensitive loss functions
and question the existence of an optimal weighting scheme for weighted
cross-entropy to optimize the Dice score and Jaccard index at test time. We
find that the Dice score and Jaccard index approximate each other relatively
and absolutely, but we find no such approximation for a weighted Hamming
similarity. For the Tversky loss, the approximation gets monotonically worse
when deviating from the trivial weight setting where soft Tversky equals soft
Dice. We verify these results empirically in an extensive validation on six
medical segmentation tasks and can confirm that metric-sensitive losses are
superior to cross-entropy based loss functions in case of evaluation with Dice
Score or Jaccard Index. This further holds in a multi-class setting, and across
different object sizes and foreground/background ratios. These results
encourage a wider adoption of metric-sensitive loss functions for medical
segmentation tasks where the performance measure of interest is the Dice score
or Jaccard index.
