  We introduce Probabilistic Coordinate Fields (PCFs), a novel
geometric-invariant coordinate representation for image correspondence
problems. In contrast to standard Cartesian coordinates, PCFs encode
coordinates in correspondence-specific barycentric coordinate systems (BCS)
with affine invariance. To know \textit{when and where to trust} the encoded
coordinates, we implement PCFs in a probabilistic network termed PCF-Net, which
parameterizes the distribution of coordinate fields as Gaussian mixture models.
By jointly optimizing coordinate fields and their confidence conditioned on
dense flows, PCF-Net can work with various feature descriptors when quantifying
the reliability of PCFs by confidence maps. An interesting observation of this
work is that the learned confidence map converges to geometrically coherent and
semantically consistent regions, which facilitates robust coordinate
representation. By delivering the confident coordinates to keypoint/feature
descriptors, we show that PCF-Net can be used as a plug-in to existing
correspondence-dependent approaches. Extensive experiments on both indoor and
outdoor datasets suggest that accurate geometric invariant coordinates help to
achieve the state of the art in several correspondence problems, such as sparse
feature matching, dense image registration, camera pose estimation, and
consistency filtering. Further, the interpretable confidence map predicted by
PCF-Net can also be leveraged to other novel applications from texture transfer
to multi-homography classification.
