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Summary: 1056 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 20, NO. 10, OCTOBER 1998
Sign of Gaussian Curvature From Curve
Orientation in Photometric Space
Elli Angelopoulou and Lawrence B. Wolff
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 g 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 computation 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. Error analysis on simulated images shows the accuracy of our technique. We also
demonstrate the performance of this methodology on empirical data.
Index Terms--Gaussian curvature, differential geometry, photometric invariant, photometric data, shape recovery, curve orientation.
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1 INTRODUCTION
URFACE curvature provides a unique three-dimensional,
viewpoint-invariant description of local surface shape.
Thus, curvature is a useful tool for scene analysis, feature
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