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Summary: Abstract
We compute the sign of Gaussian curvature using a
purely geometric definition. Consider a point p on a
smooth surface S and a closed curve on S which encloses
p. The image of on the unit normal Gaussian sphere is a
new curve . The Gaussian curvature at p is defined as the
ratio of the area enclosed by over the area enclosed by
as contracts to p. The sign of Gaussian curvature at p is
determined by the relative orientations of the closed curves
and . We directly compute the relative orientation of two
such curves from intensity data. We employ three unknown
illumination conditions to create a photometric scatter
plot. This plot is in one-to-one correspondence with the
subset of the unit Gaussian sphere containing the mutually
illuminated surface normals. This permits direct computa-
tion of the sign of Gaussian curvature without the recovery
of surface normals. Our method is albedo invariant. We
assume diffuse reflectance, but the nature of the diffuse
reflectance can be general and unknown. Simulations, as
well as empirical results, demonstrate the accuracy of our
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