We ran tests to verify that the algorithm performed rectification correctly, and also to check that the accuracy of the 3-D reconstruction did not decrease when performed from the rectified images directly.
Figure:
Reconstruction error vs noise levels in the image coordinates (left) and
calibration parameters (right) for a synthetic stereo rig with camera
translation
mm and rotation angles roll=
pitch=
and yaw=
. Crosses refer to reconstruction from rectified images, circles to
reconstruction from unrectified images.
The tests used both synthetic and real data. Each set of synthetic data consisted of a cloud of 3-D points and a pair of projection matrices; both points and matrices were chosen randomly, to cover a range of very different stereo geometries and camera parameters. For reasons of space, we do not show examples. Real-data experiments used several calibrated stereo pairs available from the INRIA-Syntim WWW site, which include the cameras' projection matrices [ 12 ]. Figure 2 shows an example. The right image of each pair shows three epipolar lines corresponding to the points marked by a cross in the left image. The output images are cropped to the size of the input images for display purposes only; the pixel coordinates of the rectified images are not constrained to lie in any special part of the image plane.
In order to evaluate the errors introduced by rectification on reconstruction, we compared the accuracy of 3-D reconstruction computed from original and rectified images. We used synthetic, noisy images of random clouds of 3-D points and random stereo geometries, as explained in the previous subsection. Imaging errors were simulated by perturbing the image coordinates, and calibration errors by perturbing the intrinsic and extrinsic parameters, both with additive, Gaussian noise. As an example, Figure 1 shows the average (over the set of points) relative error measured on 3-D point position, plotted against noise, for one of the several synthetic stereo rigs used for testing. Each point plotted is an average over 100 independent trials. The abscissa is the standard deviation of the relative error on coordinates of image point or calibration parameters. It can be seen that the accuracy does not suffer when reconstructing directly from the rectified images.
Figure 2:
Original ``Color'' stereo pair (top) and rectified pair (bottom). The
left pictures plot the epipolar lines corresponding to the point marked
in the right pictures.
Adrian F Clark