Dice Loss

The problem, from first principles

In medical or satellite segmentation the object of interest often covers a tiny fraction of the pixels. A per-pixel cross-entropy loss is then dominated by the easy background, and a model that predicts "background everywhere" scores 99% accuracy and learns nothing. The Dice coefficient measures overlap instead: twice the intersection divided by the sum of the two areas. It ranges from 0 (no overlap) to 1 (perfect), and the background does not enter it at all. As a loss we use 1 - Dice computed on predicted probabilities (so it stays differentiable), and a small eps keeps the ratio defined when both masks are empty.

From theory to code

Implement dice_coefficient and dice_loss.

Constraints

• pred holds probabilities in [0, 1] and target holds 0/1 labels, both of shape (N, H, W).
• dice_coefficient(pred, target, eps) computes, per sample, (2 * sum(pred * target) + eps) / (sum(pred) + sum(target) + eps) over the spatial dimensions and returns the (N,) array.
• dice_loss(pred, target, eps=1e-6) is 1 - mean(dice_coefficient(...)) as a Python float.

Hints