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.

引入概率坐标场（PCFs）和 PCF-Net，作为图像对应问题的新几何和不变坐标表示法和概率网络，通过联合优化坐标场和它们的置信度条件来实现的，可以在各种特征描述符中定量地表达 PCFs 的可靠性，并且可以作为现有对应依赖方法的插件，其可靠的座标表示对匹配的准确性有所帮助，这些得到的结果即可以被用于改善优化匹配的任务或用于引入到纹理转换等多个新应用程序中。