The key to three-view geometry

摘要

In this article we describe a set of canonical transformations of the image spaces that make the description of three-view geometry very simple. The transformations depend on the three-view geometry and the canonically transformed trifocal tensor T' takes the form of a sparse array where 17 elements in well-defined positions are zero, it has a linear relation to the camera matrices and to two of the fundamental matrices, a third order relation to the third fundamental matrix, a second order relation to the other two trifocal tensors, and first order relations to the 10 three-view all-point matching constraints. In this canonical form, it is also simple to determine if the corresponding camera configuration is degenerate or co-linear. An important property of the three canonical transformations of the images spaces is that they are in SO(3). The 9 parameters needed to determine these transformations and the 9 parameters that determine the elements of T' together provide a minimal parameterization of the tensor. It does not have problems with multiple maps or multiple solutions that other parameterizations have, and is therefore simple to use. It also provides an implicit representation of the trifocal internal constraints: the sparse canonical representation of the trifocal tensor can be determined if and only if it is consistent with its internal constraints. In the non-ideal case, the canonical transformation can be determined by solving a minimization problem and a simple algorithm for determining the solution is provided. This allows us to extend the standard linear method for estimation of the trifocal tensor to include a constraint enforcement as a final step, similar to the constraint enforcement of the fundamental matrix. Experimental evaluation of this extended linear estimation method shows that it significantly reduces the geometric error of the resulting tensor, but on average the algebraic estimation method is even better. For a small percentage of cases, however, the extended linear method gives a smaller geometric error, implying that it can be used as a complement to the algebraic method for these cases.