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Measurement of the Gaussian and principal curvatures at points on smooth object surfaces is important to robotic vision applications for determining surface regions which are convex, concave or planar. Accurate measurement of the directions of principal curvature can determine the orientation of objects such as cylinders and cones. Unfortunately the accurate measurement of Gaussian and principal curvature and directions of principal curvature is very difficult in the presence of noise, using conventional methods which determine depth and local surface orientation at points. Determination of the Gaussian and principal curvatures of a smooth surface, parametrized by image coordinates and height above the image plane, requires the measurement of the second order variations of height with respect to image coordinates. Noise inherent to depth measurements obtained from a laser range finder will be extremely compounded when computing second order derivatives. Surface normals (i.e. first order variations) obtained from photometric stereo techniques are not determined accurately enough for a low error measurement of their rate of change. This paper expounds upon an idea first presented in [Woodham 1978] of using a reflectance map to determine the viewer-centered curvature matrix. Accurate measurement of this matrix, which consists of the second order variations of object height with respect to image coordinates, enables accurate measurement of Gaussian curvature and the principal directions of curvature. The method presented by Woodham to obtain the viewer-centered curvature matrix yields an underconstrained solution and requires auxiliary assumptions about the curvature of the object surface for a unique determination. Presented here is a technique using multiple light sources which not only uniquely determines the viewer-centered curvature matrix in the absence of auxiliary constraints, but overconstrains the solution for accurate measurement in the presence of noise. Other insights are given into obtaining more accuracy using directional derivatives of an image function, and positioning the light sources so as to generate a more static reflectance gradient vector field.
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