Estimating the orientation of planar surfaces: algorithms and bounds

This paper presents a computationally and statistically efficient parametric solution to the problem of estimating the orientation in space of a planar textured surface from a single, noisy, 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. The functional dependence of the sinusoid phase in location is uniquely determined by the tilt and slant angles of the surface. From the physical model of the perspective projection, we derive the Cramer-Rao lower bound on the error variance of estimating the tilt and slant of the observed surface in the presence of observation noise. It is shown in this paper that the phase of each of the sinusoids can be expressed as a linear function of some variables that are related 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 performance is shown to be close to the Cramer-Rao bound, even for low signal-to-noise ratios, at computational complexity which is considerably lower than that of any existing algorithm.