Dice loss

Background

Dice loss optimises overlap directly, which makes it the default for image segmentation with imbalanced foreground/background. It is $1 - \text{Dice coefficient}$, where the Dice (F1) coefficient measures the overlap between predicted and ground-truth masks; unlike pixel-wise cross-entropy it isn't drowned out by the sea of easy background pixels.

Problem statement

Implement dice_loss(pred, target, smooth=1e-6):

$$\text{Dice} = \frac{2\sum_i p_i\,t_i + \text{smooth}}{\sum_i p_i + \sum_i t_i + \text{smooth}}, \qquad \mathcal{L} = 1 - \text{Dice}$$

Input

• pred — np.ndarray: predicted mask (probabilities in $[0, 1]$ or binary).
• target — np.ndarray: ground-truth mask (0/1), same shape.
• smooth — float: smoothing term to avoid division by zero.

Output

Returns a float in $[0, 1]$ (0 = perfect overlap).

Examples

Example 1

Input:  pred = [1, 1, 0, 0], target = [1, 0, 0, 0]
Output: 0.3333

Explanation: intersection $\sum p\,t = 1$; $\sum p = 2$, $\sum t = 1$. Dice $= 2(1)/(2+1) = 2/3$, so loss $= 1 - 2/3 = 0.3333$.

Constraints

• The Dice numerator is $2\times$ the elementwise-product sum; the denominator is the sum of both masks (with smooth added to both).
• Loss $= 1 - \text{Dice}$: perfect overlap → 0, no overlap → ~1.
• Works on flattened or full arrays; tests compare with atol=1e-4.

Notes

• The Dice coefficient is the pixel-wise F1 score: $2\text{TP}/(2\text{TP}+\text{FP}+\text{FN})$ for binary masks.
• The smoothing term keeps the loss well-defined (→ 0) when both masks are empty, where Dice would otherwise be $0/0$.