On the Structure and Properties of the Quadrifocal Tensor


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