  Explicit representations of the global match distributions of pixel-wise
correspondences between pairs of images are desirable for uncertainty
estimation and downstream applications. However, the computation of the match
density for each pixel may be prohibitively expensive due to the large number
of candidates. In this paper, we propose Hierarchical Discrete Distribution
Decomposition (HD^3), a framework suitable for learning probabilistic pixel
correspondences in both optical flow and stereo matching. We decompose the full
match density into multiple scales hierarchically, and estimate the local
matching distributions at each scale conditioned on the matching and warping at
coarser scales. The local distributions can then be composed together to form
the global match density. Despite its simplicity, our probabilistic method
achieves state-of-the-art results for both optical flow and stereo matching on
established benchmarks. We also find the estimated uncertainty is a good
indication of the reliability of the predicted correspondences.
