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Essential Matrix: Difference between revisions
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the essential matrix satisfies the equation <math>\mathbf{x}'^T \mathbf{E} \mathbf{x} = 0</math> | the essential matrix satisfies the equation <math>\mathbf{x}'^T \mathbf{E} \mathbf{x} = 0</math> | ||
==Derivation== | ==Background and Derivation== | ||
[[File: Epipolar_geometry.svg | link=Wikipedia | thumb | 400px | [[Wikipedia: Epipolar Geometriy]] ]] | |||
Much of this section is from <ref name="hartley">[http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.64.7518 An Investigation of the Essential Matrix] by Richard Hartley</ref> | |||
A pinhole camera with <math>3 \times 4</math> projection matrix <math>P = K(R | -RT)</math> takes points <math>\mathbf{x} = (x, y, z)^T</math> and projects them to <math>\mathbf{u} = (u, v, w)^T = \mathbf{R}(\mathbf{x} - \mathbf{t})</math>. | |||
We now consider two cameras: | |||
Camera 1 is at the origin of world space (or it's object space) <math>P = (I | 0)</math>. | |||
Camera 2 is displaced with some rotation <math>R</math> and translation <math>R</math>, <math>P' = (R | -RT)</math>.<br> | |||
Any point <math>\mathbf{u} = (u,v,w)^T</math> in camera 1 is represented by an epipolar line in camera 2.<br> | |||
Under camera 2, the position of camera 1 is <math>-RT</math> and <math>P' (u,v,w,0)^T = R\mathbf{u}</math> is somewhere on this epipolar line. | |||
Thus the line can be calculated by taking the cross product between the camera origin and <math>\mathbf{u}</math>. | |||
\[ | |||
(p,q,r)^T = RT \times R\mathbf{u} = R(T \times \mathbf{u}) = R[T]_{\times} \mathbf{u} | |||
\] | |||
Now the line is represented by <math>\{(u',v',w') \mid pu' + qv' + rw' = 0\}</math>, i.e. all points orthogonal to <math>(p,q,r)^T</math>. | |||
Given a vector <math>\mathbf{t}</math>, the matrix form of its cross product is:<br> | |||
<math> | |||
[\mathbf{t}]_{\times} = | |||
\begin{bmatrix} | |||
0 & -t_z & t_y\\ | |||
t_z & 0 & -t_x\\ | |||
-t_y & t_x & 0 | |||
\end{bmatrix} | |||
</math> | |||
* <math>[\mathbf{t}]_{\times} \mathbf{u} = \mathbf{t} \times \mathbf{u}</math> | |||
* This matrix is skew-symmetric. I.e. <math>[\mathbf{t}]^T_{\times} = -[\mathbf{t}]_{\times}</math> | |||
Given feature points <math>\mathbf{x}</math> and <math>\mathbf{x'}</math> from two images, | Given feature points <math>\mathbf{x}</math> and <math>\mathbf{x'}</math> from two images, | ||
we can relate them with a rotation <math>\mathbf{R}</math> and a translation <math>\mathbf{t}</math> | we can relate them with a rotation <math>\mathbf{R}</math> and a translation <math>\mathbf{t}</math> | ||