The Impact Label Noise and Choice of Threshold has on Cross-Entropy and Soft-Dice in Image Segmentation

In image segmentation and specifically in medical image segmentation, the soft-Dice loss is often chosen instead of the more traditional cross-entropy loss to improve performance with respect to the Dice metric. Experimental work supporting this claim exists, but how and why the two loss functions lead to different predictions is not well understood. This paper explains the observed discrepancy as a consequence of how those loss functions are effected by label noise and what threshold is used to convert the predicted soft segmentation to predicted labels. In particular, it is shown (i) how the optimal solutions to the two loss functions diverge as the noise is increased, and (ii) how the optimal solutions to soft-Dice can be recovered by thresholding the solutions to cross-entropy with an a priori unknown but efficiently computable threshold. The theoretical results are supported by numerical experiments and it is concluded that cross-entropy with the alternative threshold yields the stability and informative label probability maps associated with cross-entropy without sacrificing the performance of soft-Dice.