Homography: Information from Answers.com

For the concept in linguistics also called homography, see heterography and homography.

Homography is a concept in the mathematical science of geometry. A homography is an invertible transformation from the real projective plane to the projective plane that maps straight lines to straight lines. Synonyms are collineation, projective transformation, and projectivity,^[1] though "collineation" is also used more generally.

Formally, a projective transformation is a transformation used in projective geometry: it is the composition of a pair of perspective projections. It describes what happens to the perceived positions of observed objects when the point of view of the observer changes. Projective transformations do not preserve sizes or angles but do preserve incidence and cross-ratio: two properties which are important in projective geometry. Projectivities form a group.^[1]

For more general projective spaces – of different dimensions or over different fields – "homography" means a projective linear transformation (an invertible transformation induced by a linear transformation of the associated vector space), while "collineation" (meaning "maps lines to lines") is more general, and includes both homographies and automorphic collineations (collineations induced by a field automorphism), as well as combinations of these.

1 Computer vision applications
- 1.1 3D plane to plane equation
2 Mathematical definition
3 Affine homography
4 See also
5 References
6 External links

Computer vision applications

In the field of computer vision, any two images of the same planar surface in space are related by a homography (assuming a pinhole camera model). This has many practical applications, such as image rectification, image registration, or computation of camera motion—rotation and translation—between two images. Once camera rotation and translation have been extracted from an estimated homography matrix, this information may be used for navigation, or to insert models of 3D objects into an image or video, so that they are rendered with the correct perspective and appear to have been part of the original scene. (This is called Augmented Reality.)

If the camera motion between two images is pure rotation, with no translation, then the two images are related by a homography (assuming a pinhole camera model).

3D plane to plane equation

We have two cameras a and b, looking at points $P i$ in a plane.

Passing the projections of $P i$ from $b p i$ in b to a point $a p i$ in a:

${}^ap_i = K_a \cdot H_{ba} \cdot K_b^{-1} \cdot {}^bp_i$

where $H b a$ is

$H_{ba} = R - \frac{t n^T}{d}$

$R$ is the rotation matrix by which b is rotated in relation to a; t is the translation vector from a to b; n and d are the normal vector of the plane and the distance to the plane respectively.

K_a and K_b are the cameras' intrinsic parameter matrices.

The figure shows camera b looking at the plane at distance d.

Mathematical definition

Homogeneous coordinates are used, because matrix multiplication cannot directly perform the division required for perspective projection.

Given:

$p_a = \begin{bmatrix} x_a\\y_a\\1\end{bmatrix}, p^\prime_b = \begin{bmatrix} w^{\prime}x_b\\w^{\prime}y_{b}\\w^{\prime}\end{bmatrix}, \mathbf{H}_{ab} = \begin{bmatrix} h_{11}&h_{12}&h_{13}\\h_{21}&h_{22}&h_{23}\\h_{31}&h_{32}&h_{33} \end{bmatrix}$

Then:

$p^\prime_b = \mathbf{H}_{ab}p_a \,$

where:

$\mathbf{H}_{ba} = \mathbf{H}_{ab}^{-1}.$

Also:

$p_b = p^\prime_b/w^\prime = \begin{bmatrix} x_b\\y_b\\ 1\end{bmatrix}$

Affine homography

When the image region in which the homography is computed is small or the image has been acquired with a large focal length, an affine homography is a more appropriate model of image displacements. An affine homography is a special type of a general homography whose last row is fixed to

$h_{31}=h_{32}=0, \; h_{33}=1.$

References

^ ^a ^b Richard Hartley and Andrew Zisserman (2003). Multiple View Geometry in computer vision. Cambridge University Press. pp. 32–33. ISBN 0-521-54051-8.

O. Chum and T. Pajdla and P. Sturm (2005). "The Geometric Error for Homographies". Computer Vision and Image Understanding 97(1): 86–102. doi:10.1016/j.cviu.2004.03.004.

This article needs additional citations for verification.
Please help improve this article by adding reliable references. Unsourced material may be challenged and removed. (December 2009)

External links

M. Lourakis' homest is a GPL C/C++ library for robust, non-linear (based on the Levenberg-Marquardt algorithm) homography estimation from matched point pairs. homest can estimate fully projective and affine homographies with a variety of objective functions.
OpenCV is a complete (open and free) computer vision software library that has many routines related to homography estimation (search for cvFindHomography in ref) and re-projection (search for cvPerspectiveTransform in ref). Download and documentation information is on the OpenCV Wiki.
Computing the plane to plane homography
How to compute a homography
MATLAB Functions for Multiple View Geometry Matlab functions for calculating a homography and the fundamental matrix

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Donate to Wikimedia

	homographic
	homography
	Reciprocity (projective geometry)

Homography

Contents