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Homography is a concept in the mathematical science of geometry. It is defined as a relation between two figures, such that any given point in one figure corresponds to one and only one point in the other, and vice versa.
### Computer Vision Applications

In the field of computer vision, a homography is defined in 2 dimensional space as a mapping between a point on a ground plane as seen from one camera, to the same point on the ground plane as seen from a second camera. This has many practical applications, most notably it provides a method for compositing 2D or 3D objects into an image or video with the correct pose. The homography matrix is sometimes known as a homograph, a term which has a different meaning in linguistics.
#### 3D plane to plane equation

### Mathematical definition

Homogeneous coordinates are used, because matrix multiplication cannot directly perform the division required for perspective projection.### 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 $h\_\{31\}=h\_\{32\}=0,\; ;\; h\_\{33\}=1$.
## See also

## External links

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

Passing the projections of $P\_i$ from $\{\}^bp\_i$ in b to a point $\{\}^ap\_i$ in a:

$\{\}^ap\_i\; =\; K\_a\; cdot\; H\_\{ba\}\; cdot\; K\_b^\{-1\}\; cdot\; \{\}^bp\_i$

where $H\_\{ba\}$ 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.

Given:

- $p\_\{a\}\; =\; begin\{bmatrix\}\; x\_\{a\}y\_\{a\}1end\{bmatrix\},\; p^\{prime\}\_\{b\}\; =\; begin\{bmatrix\}\; w^\{prime\}x\_\{b\}w^\{prime\}y\_\{b\}w^\{prime\}end\{bmatrix\},\; mathbf\{H\}\_\{ab\}\; =\; begin\{bmatrix\}\; h\_\{11\}\&h\_\{12\}\&h\_\{13\}h\_\{21\}\&h\_\{22\}\&h\_\{23\}h\_\{31\}\&h\_\{32\}\&h\_\{33\}\; end\{bmatrix\}$

- $p^\{prime\}\_\{b\}\; =\; mathbf\{H\}\_\{ab\}p\_\{a\}$

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

- $p\_\{b\}\; =\; p^\{prime\}\_\{b\}/w^\{prime\}\; =\; begin\{bmatrix\}\; x\_\{b\}y\_\{b\}1end\{bmatrix\}$

- 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.
- 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

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Last updated on Monday October 06, 2008 at 13:58:13 PDT (GMT -0700)

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Last updated on Monday October 06, 2008 at 13:58:13 PDT (GMT -0700)

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