\subsection{Cameras}
\shortdefinition[Pinhole projection]
$\begin{bmatrix}
        u & v
    \end{bmatrix}^\top
    = \frac{f}{z}
    \begin{bmatrix}
        x & y
    \end{bmatrix}^\top$
with $f$ the distance to the lens and $z$ the full distance

\newpage
$u = c_u + f \cdot x'$ and $v = c_v + f \cdot y'$ where $x' = t_x \div t_z$ and $y' = t_y \div t_z$
where $u, v$ are the pixel $x, y$ coords, $\vec{c} = [c_u, c_v]^\top$ is optical centre of cam in pixel coords, $f$ scale factor,
and $\vec{{_C}\vec{t}_P} = [t_x, t_y, t_z]^\top$

The full proj: $\vec{u} = \begin{bmatrix}
        \lambda u \\
        \lambda v \\
        \lambda
    \end{bmatrix}
    =
    \begin{bmatrix}
        f & 0 & c_u \\
        0 & f & c_v \\
        0 & 0 & 1
    \end{bmatrix}
    \begin{bmatrix}
        t_x \\t_y\\t_z
    \end{bmatrix}
    = \mat{K}\; {_C}\vec{t}_P$

If p. in diff frame ${_W} \vec{t}_P$, then $\vec{u} = \mat{K}[\mat{R}_{CW}\; {_C}\vec{t}_{CW}] = {_W}\vec{t}_P$

\subsubsection{Pinhole Camera Projection with distortion}
\shortdefinition Model: $\vec{u} = \vec{k}(\vec{d}(\vec{p}({_C}\vec{t}_P)))$, with:

(Projection) $\vec{x'} = \vec{p}({_C}\vec{t}_P) = t_z^{-1} \cdot [t_x, t_y]^\top$

(Distortion model, $r^2 = x'^2 + y'^2$, $\vec{x''} = \vec{d}(\vec{x'})$\\
$\vec{x''} = \! \frac{1 + k_1 r^2 + k_2 r^4 + k_3 r^6}{1 + k_4 r^2 + k_5 r^4 + k_6 r^6} \vec{x'}
    + \begin{bmatrix}
        2p_1 x' y' + p_2(r^2 + 2x'^2) \\
        p_1 (r^2 + 2y'^2) + 2p_2 x'y'
    \end{bmatrix}$

(Scale and Centre) $\vec{u} = \vec{k}(\vec{x''}) = \text{diag}([f_u, f_v]) \cdot \vec{x''} + \vec{x}$

All with $k_i$ radial distortion params, optional for $i > 2$, $p_i$ tang. dist. param, $f_u, f_v$ focal length in pixels

\shade{green}{Inverse} ${_C}\vec{r} = [\vec{d}^{-1}(\vec{k^{-1}(\vec{u})}), 1]^\top$\\
(To unit plane)
$\vec{x''} = \vec{k}^{-1}(\vec{u}) = [f_u^{-1}, f_v^{-1}]^\top (\vec{u} - \vec{c})$

(Un-distort)
$\vec{x'} = \vec{d}^{-1}(\vec{x''})$ (usually comp. numerically)

(Compute ray)
${_C}\vec{r} = [\vec{x'}, 1]^\top$


\subsubsection{Undestorting a whole image}
$\vec{u}_i = \vec{k}(\vec{d}(\vec{k_{\text{new}}}^{-1}(\vec{u}_{i, \text{new}})))$
where $\vec{u}_{i, \text{new}}$ is the place of pixel in output, $\vec{u}_i$ is the input

\bi{Omnidir. Cam} undistortion model with $f(u, v) = \sum_{i = 0}^{N} a_i \rho^i$
with $\rho = \sqrt{(u - c_u)^2 + (v - c_v)^2}$, $N = 4$ accurately describes it for most fisheye and catadioptric cameras