Purpose: To derive a multinomial probability function and quantitative measures of the data and epistemic uncertainty as direct output of a 3D U-Net segmentation network.

Approach: A set of T1 brain MRI images were downloaded from the Connectome Project and segmented using FMRIB’s FAST algorithm to be used as ground truth. A 3D U-Net neural network was trained with sample sizes of 200, 500, and 898 T1 brain images using a loss function defined as the negative logarithm of the likelihood based on a derivation of the definition of the multinomial probability function. From this definition, the epistemic and aleatoric uncertainty equations were derived and used to quantify maps of the uncertainty along with tissue segmentations.

Results: Maps of the tissue segmentation along with the epistemic and aleatoric uncertainty, per voxel, are presented. The uncertainty decreased based on the increasing number of training data used to train the neural network. The neural network trained with 898 volumes resulted in uncertainty maps that were high primarily in the tissue boundary regions. The epistemic and aleatoric uncertainty were averaged over all test data (connectome and tumor separately), and the epistemic uncertainty showed a decreasing trend, as expected, with increasing numbers of data used to train the model. The aleatoric uncertainty showed a similar trend which was also expected as the aleatoric uncertainty is not expected to be as dependent on the number of training data.

Conclusion: The derived uncertainty equations from a multinomial probability distribution were able to quantify the aleatoric and epistemic uncertainty per voxel and are applicable for all two-dimensional and three-dimensional neural networks.