Homography Tensors: On Algebraic Entities That Represent Three Views of Static or Moving Planar Points

*Lior Wolf and Amnon
Shashua *

*School of Computer Science and Engineering**, *

*The Hebrew University**, *

*Jerusalem 91904, Israel*

We introduce a 3 x 3 x 3 tensor $H^{ijk}$ and its dual

$H_{ijk}$ which represent the 2D projective mapping of points across

three projections (views). The tensor $H^{ijk}$ is a generalization of

the well known 2D collineation matrix (homography matrix) and it

concatenates two homography matrices to represent the joint mapping

across three views. The dual tensor $H_{ijk}$ concatenates two dual

homography matrices (mappings of line space) and is responsible for

representing the mapping associated with {\bf moving} points along

straight-line paths, i.e., $H_{ijk}$ can be recovered from

line-of-sight measurements only.

The paper includes a detailed exposition of these tensors and their

properties. We present two applications for these tensors, one related

to ``plane stabilization'' across image sequences and the other related

to handling planar scenes rich with (unsegmented) static and moving points.

Keywords: Structure from motion, Multiple views geometry.