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.