Lior Wolf and Amnon
Shashua
School of Computer Science and Engineering,
The Hebrew University,
Jerusalem 91904, Israel
The quadrifocal tensor which connects image measurements along 4 views
is not yet well understood as its counterparts the fundamental matrix
and the trifocal tensor. This paper establishes the structure of the
tensor as an ``epipole-homography'' pairing
$$Q^{ijkl}=v'^jH^{ikl} - v''^k H^{ijl} +
v'''^lH^{ijk}$$
where $v',v'',v'''$ are the epipoles in views 2,3,4, $H$ is the
``homography tensor'' the 3-view analogue
of the homography matrix,
and the indices $i,j,k,l$ are attached to views 1,2,3,4 respectively
--- i.e., $H^{ikl}$ is the homography
tensor of views 1,3,4.
In the coarse of deriving the structure $Q^{ijkl}$ we show that Linear
Line Complex (LLC) mappings are the basic building block in the
process. We also introduce a complete break-down of the tensor slices:
3x3x3 slices are homography tensors, and 3x3 slices are LLC
mappings. Furthermore, we present a closed-form formula of the
quadrifocal tensor described by the trifocal tensor and fundamental
matrix, and also show how to generally recover projection matrices
from the quadrifocal tensor, and we describe the form of the 51
non-linear constraints a quadrifocal tensor must adhere to. Taken
together, we bring the quadrifocal tensor to a level of
structural description (both geometric and algebraic) and property
break-down comparable to that of the trifocal tensor and fundamental
matrix.