Exact bias correction and covariance estimation for stereo vision

We present an approach for correcting the bias in 3D reconstruction of points imaged by a calibrated stereo rig. Our analysis is based on the observation that, due to quantization error, a 3D point reconstructed by triangulation essentially represents an entire region in space. The true location of the world point that generated the triangulated point could be anywhere in this region. We argue that the reconstructed point, if it is to represent this region in space without bias, should be located at the centroid of this region, which is not what has been done in the literature. We derive the exact geometry of these regions in space, which we call 3D cells, and we show how they can be viewed as uniform distributions of possible pre-images of the pair of corresponding pixels. By assuming a uniform distribution of points in 3D, as opposed to a uniform distribution of the projections of these 3D points on the images, we arrive at a fast and exact computation of the triangulation bias in each cell. In addition, we derive the exact covariance matrices of the 3D cells. We validate our approach in a variety of simulations ranging from 3D reconstruction to camera localization and relative motion estimation. In all cases, we are able to demonstrate a marked improvement compared to conventional techniques for small disparity values, for which bias is significant and the required corrections are large.