A semi-analytic method of determining stereo camera geometry from matched points in a pair of images: Coincident meridional planes, exact or noisy data

We describe a novel semi-analytic method to determine the parameters of stereo camera geometry from pairs of matched points under the restriction of coincident meridional lanes. By a judicious choice of variables to represent rotation and translation, the noise-free problem reduces to solving two quadratics—one in each variable. With noissy data, the least squares problem reduces to a fifth-degree polynomial in a single variable, all the solutions of which can be (numerically) exactly computed along with an estimate of the standard deviation. The problem of locating the global minimum of the error function in two variables (which generally admits an unpredictable number of local extrema) in this instance becomes that of comparing five numbers. The global minimum can therefore be guaranteed. The algorithm does not break down whether or not the translation vanishes. In fact, we propose an effective signature to detect vanishing translation in the presence of noise. The general algorithm also handles the degenerate case of all imaged points lying in a vertical plane which is known to admit two solutions. Both solutions are found.