TY - JOUR

T1 - Estimating the orientation of planar surfaces

T2 - algorithms and bounds

AU - Permuter, Haim

AU - Francos, Joseph M.

N1 - Funding Information:
Manuscrpit received August 31, 1999; revised May 3, 2000. This work was supported in part by the Israel Ministry of Science, the Eshkol Fellowship Program in Applied Mathematics under Grant 0616196, in part by the Israel Ministry of Science and the French Ministry of Research and Technology under Grant 8814297, and in part by the Israel Ministry of Science under Grant 8635297.

PY - 2000/8/1

Y1 - 2000/8/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0034248249&partnerID=8YFLogxK

U2 - 10.1109/18.857800

DO - 10.1109/18.857800

M3 - Article

AN - SCOPUS:0034248249

SN - 0018-9448

VL - 46

SP - 1908

EP - 1920

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 5

ER -