  Given a 3D surface defined by an elevation function on a 2D grid as well as
non-spatial features observed at each pixel, the problem of surface
segmentation aims to classify pixels into contiguous classes based on both
non-spatial features and surface topology. The problem has important
applications in hydrology, planetary science, and biochemistry but is uniquely
challenging for several reasons. First, the spatial extent of class segments
follows surface contours in the topological space, regardless of their spatial
shapes and directions. Second, the topological structure exists in multiple
spatial scales based on different surface resolutions. Existing widely
successful deep learning models for image segmentation are often not applicable
due to their reliance on convolution and pooling operations to learn regular
structural patterns on a grid. In contrast, we propose to represent surface
topological structure by a contour tree skeleton, which is a polytree capturing
the evolution of surface contours at different elevation levels. We further
design a graph neural network based on the contour tree hierarchy to model
surface topological structure at different spatial scales. Experimental
evaluations based on real-world hydrological datasets show that our model
outperforms several baseline methods in classification accuracy.
