The triangulation tensor

Abstract

This article presents a computationally efficient approach to the triangulation of 3D points from their projections in two views. The homogenous coordinates of a 3D point is given as a multi-linear mapping on its homogeneous image coordinates, a computation of low computational complexity. The multi-linear mapping is a tensor, and an element of a projective space, that can be computed directly from the camera matrices and some parameters. These parameters imply that the tensor is not unique: for a given camera pair the subspace K of triangulation tensors is six-dimensional. The triangulation tensor is 3D projective covariant and satisfies a set of internal constraints. Reconstruction of 3D points using the proposed tensor is studied for the non-ideal case, when the image coordinates are perturbed by noise and the epipolar constraint exactly is not satisfied exactly. A particular tensor of K is then the optimal choice for a simple reduction of 3D errors, and we present a computationally efficient approach for determining this tensor. This approach implies that normalizing image coordinate transformations are important for obtaining small 3D errors. In addition to computing the tensor from the cameras, we also investigate how it can be further optimized relative to error measures in the 3D and 2D spaces. This optimization is evaluated for sets of real 3D + 2D + 2D data by comparing the reconstruction to some of the triangulation methods found in the literature, in particular the so-called optimal method that minimizes 2D L2 errors. The general conclusion is that, depending on the choice of error measure and the optimization implementation, it is possible to find a tensor that produces smaller 3D errors (both L1 and L2) but slightly larger 2D errors than the optimal method does. Alternatively, we may find a tensor that gives approximately comparable results to the optimal method in terms of both 3D and 2D errors. This means that the proposed tensor based method of triangulation is both computationally efficient and can be calibrated to produce small reconstruction or reprojection errors for a given data set. © 2009 Elsevier Inc. All rights reserved.