Estimating the orientation of planar surfaces using the phase differencing algorithm

This paper presents a parametric solution to the problem of estimating the orientation in space of a planar textured surface, from a single observed image of it. The coordinate transformation from surface to image coordinates, due to the perspective projection, transforms each homogeneous sinusoidal component of the surface texture into a sinusoid whose frequency is a function of location. It is shown in this paper that the phase of each of the sinusoids can be expressed as a linear function of some constants that are related, in a rather simple form, to the surface tilt and slant angles. Using the phase differencing algorithm we fit a polynomial phase model to a sinusoidal component of the observed texture. Substituting in the derived linear relation, the unknown phase with the one estimated using the phase differencing algorithm, we obtain a closed form, analytic, and computationally efficient solution to the problem of estimating the tilt and slant angles. The algorithm is shown to produce estimates that are close to the Cramer-Rao bound, at computational complexity which is considerably lower than that of any existing algorithm