Shape from Motion
The Constraint A moving camera images a plane. The orientation of the plane is defined by its normal N. In the plane there are (texture) lines with orientation vector L, whose projections are observed in the images at two instances of time as shown in Figure 1. The image lines are represented by the vector l_{1}, perpendicular to the plane through the image center O_{1} and the first projection, and by the vector l_{2}, perpendicular to the plane through the image center O_{2} and the second projection. Figure 1 (l_{1}×R^{T} l_{2}) Since the surface normal is perpendicular to the texture line we obtain the linear constraint N·(l_{1}×R^{T} l_{2}) = 0 and we solve for N minimizing the deviation from this constraint for all texture lines using least squares minimization.Assuming there is noise in the orientation of the observed image lines and there is noise in the estimation of the rotation, the minimization gives a biased estimate for the surface normal. The bias in general depends on the motion parameters, the orientation of the image lines (that is the texture of the plane), the surface normal and the errors in the observed image lines and estimated rotation. As a parameterization for the surface normal often the angles slant (σ) and tilt (τ) are used. The slant denotes the angle between N and the negative optical axis (0^{°} slant corresponds to a plane parallel to the image plane). The tilt is the angle between the parallel projection of N on the image plane and the image xaxis. Psychophysical experiments have found experimentally an underestimation of slant [[22],[23]]. This is predicted by the bias model described here.
