Commit fcb537ba authored by catree's avatar catree

Add a figure and some formulas to illustrate the PnP pose estimation principle.

parent cac4a7e5
......@@ -567,7 +567,97 @@ focal length.
function requires exactly four object and image points.
The function estimates the object pose given a set of object points, their corresponding image
projections, as well as the camera matrix and the distortion coefficients.
projections, as well as the camera matrix and the distortion coefficients, see the figure below
(more precisely, the X-axis of the camera frame is pointing to the right, the Y-axis downward
and the Z-axis forward).
![](pnp.jpg)
Points expressed in the world frame \f$ \bf{X}_w \f$ are projected into the image plane \f$ \left[ u, v \right] \f$
using the perspective projection model \f$ \Pi \f$ and the camera intrinsic parameters matrix \f$ \bf{A} \f$:
\f[
\begin{align*}
\begin{bmatrix}
u \\
v \\
1
\end{bmatrix} &=
\bf{A} \hspace{0.1em} \Pi \hspace{0.2em} ^{c}\bf{M}_w
\begin{bmatrix}
X_{w} \\
Y_{w} \\
Z_{w} \\
1
\end{bmatrix} \\
\begin{bmatrix}
u \\
v \\
1
\end{bmatrix} &=
\begin{bmatrix}
f_x & 0 & c_x \\
0 & f_y & c_y \\
0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0
\end{bmatrix}
\begin{bmatrix}
r_{11} & r_{12} & r_{13} & t_x \\
r_{21} & r_{22} & r_{23} & t_y \\
r_{31} & r_{32} & r_{33} & t_z \\
0 & 0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
X_{w} \\
Y_{w} \\
Z_{w} \\
1
\end{bmatrix}
\end{align*}
\f]
The estimated pose is thus the rotation (`rvec`) and the translation (`tvec`) vectors that allow to transform
a 3D point expressed in the world frame into the camera frame:
\f[
\begin{align*}
\begin{bmatrix}
X_c \\
Y_c \\
Z_c \\
1
\end{bmatrix} &=
\hspace{0.2em} ^{c}\bf{M}_w
\begin{bmatrix}
X_{w} \\
Y_{w} \\
Z_{w} \\
1
\end{bmatrix} \\
\begin{bmatrix}
X_c \\
Y_c \\
Z_c \\
1
\end{bmatrix} &=
\begin{bmatrix}
r_{11} & r_{12} & r_{13} & t_x \\
r_{21} & r_{22} & r_{23} & t_y \\
r_{31} & r_{32} & r_{33} & t_z \\
0 & 0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
X_{w} \\
Y_{w} \\
Z_{w} \\
1
\end{bmatrix}
\end{align*}
\f]
@note
- An example of how to use solvePnP for planar augmented reality can be found at
......
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