  A learning-based posterior distribution estimation method, Probabilistic
Dipole Inversion (PDI), is proposed to solve the quantitative susceptibility
mapping (QSM) inverse problem in MRI with uncertainty estimation. In PDI, a
deep convolutional neural network (CNN) is used to represent the multivariate
Gaussian distribution as the approximate posterior distribution of
susceptibility given the input measured field. Such CNN is first trained on
healthy subjects via posterior density estimation, where the training dataset
contains samples from the true posterior distribution. Domain adaptations are
then deployed on patient datasets with new pathologies not included in
pre-training, where PDI updates the pre-trained CNN's weights in an
unsupervised fashion by minimizing the Kullback-Leibler divergence between the
approximate posterior distribution represented by CNN and the true posterior
distribution from the likelihood distribution of a known physical model and
pre-defined prior distribution. Based on our experiments, PDI provides
additional uncertainty estimation compared to the conventional MAP approach,
while addressing the potential issue of the pre-trained CNN when test data
deviates from training.
