  Ambiguity is inevitable in medical images, which often results in different
image interpretations (e.g. object boundaries or segmentation maps) from
different human experts. Thus, a model that learns the ambiguity and outputs a
probability distribution of the target, would be valuable for medical
applications to assess the uncertainty of diagnosis. In this paper, we propose
a powerful generative model to learn a representation of ambiguity and to
generate probabilistic outputs. Our model, named Coordinate Quantization
Variational Autoencoder (CQ-VAE) employs a discrete latent space with an
internal discrete probability distribution by quantizing the coordinates of a
continuous latent space. As a result, the output distribution from CQ-VAE is
discrete. During training, Gumbel-Softmax sampling is used to enable
backpropagation through the discrete latent space. A matching algorithm is used
to establish the correspondence between model-generated samples and
"ground-truth" samples, which makes a trade-off between the ability to generate
new samples and the ability to represent training samples. Besides these
probabilistic components to generate possible outputs, our model has a
deterministic path to output the best estimation. We demonstrated our method on
a lumbar disk image dataset, and the results show that our CQ-VAE can learn
lumbar disk shape variation and uncertainty.
