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Image formation

Image formation involves the projection of points in \ensuremath{{\cal P}^3}(the world) to points in \ensuremath{{\cal P}^2} (the image plane). The perspective projection equations with which we are familiar,

\begin{eqnarray*}x = -f\frac{X}{Z} \\
y = -f\frac{Y}{Z}
\end{eqnarray*}


where the point (X,Y,Z) in the world is projected to the point (x,y)on the image plane, are inherently nonlinear. Converting to homogeneous coordinates, however, makes them linear:

\begin{displaymath}\ensuremath{{\bf p}} ' = T_{perspective}\ensuremath{{\bf p}} ,
\end{displaymath}

where $\ensuremath{{\bf p}} ' = [x,y,w]^T$, $\ensuremath{{\bf p}} = [X,Y,Z,W]^T$, and the perspective projection matrix T is given by:

\begin{displaymath}T_{perspective} = \left[\matrix{-f & 0 & 0 & 0 \cr
0 & -f & 0 & 0 \cr
0 & 0 & 1 & 0}\right].
\end{displaymath}

The entire image formation process includes perspective projection, along with matrices for internal and external calibration:

  \begin{eqnarray*}\ensuremath{{\tilde P}} = T_{internal} T_{perspective} T_{exter...
...atrix{R & \ensuremath{{\bf t}} }\right] \nonumber \\
& = & A D,
\end{eqnarray*} (4)


where $\alpha_u$ and $\alpha_v$ are the scale factors of the image plane (in units of the focal length f), $\theta$ is the skew ( $\theta=\pi/2$ for most real cameras), the point (u0,v0) is the principal point, R is the $3 \times 3$ rotation matrix, and $\ensuremath{{\bf t}} $ is the $3 \times 1$ translation vector. The matrix A contains the internal parameters and perspective projection, while D contains the external parameters.

It is sometimes convenient to decompose the $3 \times 4$ projection matrix $\ensuremath{{\tilde P}} $ into a $3 \times 3$ matrix P and a $3 \times 1$ vector p

\begin{displaymath}\ensuremath{{\tilde P}} = \left[\matrix{P & p}\right]
\end{displaymath}

so that

 \begin{displaymath}P = AR \; \; \mbox{and} \; \; p = A \ensuremath{{\bf t}} .
\end{displaymath} (5)


next up previous
Next: Essential and fundamental matrices Up: Projective Geometry Applied to Previous: Projective Geometry Applied to
Stanley Birchfield
1998-04-23