Computing rigorous bounds to the accuracy of calibrated stereo reconstruction

The authors deal with the problem of computing rigorous bounds on the position of 3-D points obtained by stereo triangulation when both the camera matrix and the coordinates of image points are affected by measurement errors. By ‘rigorous bounds’ it is meant that the true unknown 3-D points are guaranteed to lie within the intervals computed by the method, with mathematical certainty. To this end, the calibration process is first modelled by assuming a bounded error in the localisation of the reference points in the image, then narrow intervals are computed for the entries of the camera matrix using numerical methods based on interval analysis. Finally, triangulation is applied to obtain cuboids that bound point coordinates. Two state-of-the-art methods were employed for the solution of linear systems of interval equations, namely Rump’s and Shary’s methods. It is concluded that a careful selection of numerical techniques allows the use of interval analysis as a tool for obtaining realistic bounds on the output error, even in the presence of significant errors in the input data.