Trilinearity of three perspective views and its associated tensor

It has been established that certain trilinear forms of three perspective views give rise to a tensor of 27 intrinsic coefficients [11]. We show in this paper that a permutation of the trilinear coefficients produces three homography matrices (projective transformations of planes) of three distinct intrinsic planes, respectively. This, in turn, yields the result that 3D invariants are recovered directly - simply by appropriate arrangement of the tensor's coefficients. On a secondary level, we show new relations between fundamental matrix, epipoles, Euclidean structure and the trilinear tensor. On the practical side, the new results extend the existing envelope of methods of 3D recovery from 2D views - for example, new linear methods that cut through the epipolar geometry, and new methods for computing epipolar geometry using redundancy available across many views.