  Albeit the Dice loss is one of the dominant loss functions in medical image
segmentation, most research omits a closer look at its derivative, i.e. the
real motor of the optimization when using gradient descent. In this paper, we
highlight the peculiar action of the Dice loss in the presence of missing or
empty labels. First, we formulate a theoretical basis that gives a general
description of the Dice loss and its derivative. It turns out that the choice
of the reduction dimensions $\Phi$ and the smoothing term $\epsilon$ is
non-trivial and greatly influences its behavior. We find and propose heuristic
combinations of $\Phi$ and $\epsilon$ that work in a segmentation setting with
either missing or empty labels. Second, we empirically validate these findings
in a binary and multiclass segmentation setting using two publicly available
datasets. We confirm that the choice of $\Phi$ and $\epsilon$ is indeed
pivotal. With $\Phi$ chosen such that the reductions happen over a single batch
(and class) element and with a negligible $\epsilon$, the Dice loss deals with
missing labels naturally and performs similarly compared to recent adaptations
specific for missing labels. With $\Phi$ chosen such that the reductions happen
over multiple batch elements or with a heuristic value for $\epsilon$, the Dice
loss handles empty labels correctly. We believe that this work highlights some
essential perspectives and hope that it encourages researchers to better
describe their exact implementation of the Dice loss in future work.
