On the Structure and Properties of the Quadrifocal Tensor

Abstract

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^′j H^ikl - v′k H ijl + v‴l Hijk 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 course 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: 3 × 3 × 3 slices are homography tensors, and 3 × 3 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 recover projection matrices from the quadrifocal tensor. We also describe the form of the 51 non-linear constraints a quadrifocal tensor must adhere to.