  This paper describes an interdisciplinary approach to geometry modeling of
geospatial boundaries. The objective is to extract surfaces from irregular
spatial patterns using differential geometry and obtain coherent directional
predictions along the boundary of extracted surfaces to enable more targeted
sampling and exploration. Specific difficulties of the data include sparsity,
incompleteness, causality and resolution disparity. Surface slopes are
estimated using only sparse samples from cross-sections within a geological
domain with no other information at intermediate depths. From boundary
detection to subsurface reconstruction, processes are automated in between. The
key problems to be solved are boundary extraction, region correspondence and
propagation of the boundaries via contour morphing. Established techniques from
computational physics, computer vision and signal processing are used with
appropriate modifications to address challenges in each area. To facilitate
boundary extraction, an edge map synthesis procedure is presented. This works
with connected component analysis, anisotropic diffusion and active contours to
convert unordered points into regularized boundaries. For region
correspondence, component relationships are handled via graphical
decomposition. FFT-based spatial alignment strategies are used in region
merging and splitting scenarios. Shape changes between aligned regions are
described by contour metamorphosis. Specifically, local spatial deformation is
modeled by PDE and computed using level-set methods. Directional predictions
are obtained using particle trajectories by following the evolving boundary.
However, when a branching point is encountered, particles may lose track of the
wavefront. To overcome this, a curvelet backtracking algorithm has been
proposed to recover information for boundary segments without particle coverage
to minimize shape distortion.
