  Signed distance fields (SDFs) are a widely used implicit surface
representation, with broad applications in computer graphics, computer vision,
and applied mathematics. To reconstruct an explicit triangle mesh surface
corresponding to an SDF, traditional isosurfacing methods, such as Marching
Cubes and and its variants, are typically used. However, these methods overlook
fundamental properties of SDFs, resulting in reconstructions that exhibit
severe oversmoothing and feature loss. To address this shortcoming, we propose
a novel method based on a key insight: each SDF sample corresponds to a
spherical region that must lie fully inside or outside the surface, depending
on its sign, and that must be tangent to the surface at some point. Leveraging
this understanding, we formulate an energy that gauges the degree of violation
of tangency constraints by a proposed surface. We then employ a gradient flow
that minimizes our energy, starting from an initial triangle mesh that
encapsulates the surface. This algorithm yields superior reconstructions to
previous methods, even with sparsely sampled SDFs. Our approach provides a more
nuanced understanding of SDFs and offers significant improvements in surface
reconstruction.
