Image-Dependent Calibration Maps for Compound Segmentation Losses

In many domains, a segmentation network has to provide more than a binary decision: a calibrated probability map. For example, in radiotherapy planning, treatment margins and dose can be defined based on the probability that a specific area (i.e., voxel in the image) belongs to a tumor. However, models trained with compound losses, such as cross-entropy plus soft Dice, encode different posterior probabilities into the same output. This shift is caused by the optimum forced by the compound loss. By balancing the opposing loss gradients at a stationary point, we derive a calibration map ${\Psi }_{f,D}$ in closed form, parameterized by two properties of the image, the foreground fraction $f$ and the attained Dice $D$. We describe this map for the example of the cross-entropy plus soft Dice compound loss, and show that no beta, Platt, or temperature map can represent ${\Psi }_{f,D}$, and no map fitted once per model can recover the true posterior of multiple images. In addition to ${\Psi }_{f,D}$, we propose a foreground-conditioned version of isotonic regression that has sufficient capacity to model ${\Psi }_{f,D}$ and improves on standard isotonic regression. Without any held-out labels or retraining, correcting with ${\Psi }_{f,D}$ alone consistently reduces NLL and ECE scores compared to raw outputs. Furthermore, applying a beta or isotonic map fitted on held-out data in conjunction with ${\Psi }_{f,D}$ is more effective than the respective fitted map alone. Both ${\Psi }_{f,D}$ and the foreground-conditioned isotonic regression improve the placement of uncertainty margins across three tumor segmentation tasks, measured by the sample-wise IoU of the 0.25–0.75 probability margin.