Table of Contents:

Transformation Classes

sheer, affine, etc.

Spatial Transformer

Homography

When variation in depth is comparable to distance to object then the affine transformation is not a good model. Here we need the homography. [3]

The motion of a plane can be described by a homography. This is true for a moving plane imaged by a static camera. It is true for a static plane imaged by a moving camera [2].

video stabilization F-matrix is special case

3x3 Homography Parameterization

The simplest way to parameterize a homography is with a 3x3 matrix and a fixed scale. The homography maps \([u, v]\), the pixels in the left image, to \([u^{\prime}, v^{\prime}]\), the pixels in the right image, and is defined up to scale:

\[\begin{bmatrix} u^{\prime} \\ v^{\prime} \\1 \end{bmatrix} = \begin{bmatrix} H_{11} & H_{12} & H_{13} \\ H_{21} & H_{22} & H_{23} \\ H_{31} & H_{32} & H_{33} \\ \end{bmatrix} \begin{bmatrix} u \\ v \\ 1 \end{bmatrix}\]

the submatrix \(\begin{bmatrix} H_{11} & H_{12} \\ H_{21} & H_{22} \end{bmatrix}\), represents the rotational terms in the homography, while the vector \(\begin{bmatrix}H_{13} \\ H_{23} \end{bmatrix}\) is the translational offset.

4-Point Homography Parameterization

Baker et al. [2] dicuss the 4pt parameterization.

examples

[1 0 0] [0 1 0] [0 0 1]

[s 0 0 [0 s 0] [0 0 1]

[1 0 tx] [0 1 ty] [0 0 1]

[R 0] [ 0 [0 0 1]

Infinite Homography

http://robots.engin.umich.edu/~rwolcott/uploads/rwolcott-2014a.pdf ui′= KRK−1ui.

References

[1] Detone, Malisiewicz, Rabinovich. Deep Image Homography Estimation.

[2] Simon Baker, Ankur Datta, and Takeo Kanade. Parameterizing homographies. Technical Report CMU-RI-TR06-11, Robotics Institute, Pittsburgh, PA, March 2006.

[3] Simon Lucey. http://16623.courses.cs.cmu.edu/slides/Lecture_6.pdf