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68 lines
2.4 KiB
TeX
68 lines
2.4 KiB
TeX
\subsection{Cameras}
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\shortdefinition[Pinhole projection]
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$\begin{bmatrix}
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u & v
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\end{bmatrix}^\top
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= \frac{f}{z}
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\begin{bmatrix}
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x & y
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\end{bmatrix}^\top$
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with $f$ the distance to the lens and $z$ the full distance
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\newpage
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$u = c_u + f \cdot x'$ and $v = c_v + f \cdot y'$ where $x' = t_x \div t_z$ and $y' = t_y \div t_z$
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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,
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and $\vec{{_C}\vec{t}_P} = [t_x, t_y, t_z]^\top$
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The full proj: $\vec{u} = \begin{bmatrix}
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\lambda u \\
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\lambda v \\
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\lambda
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\end{bmatrix}
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=
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\begin{bmatrix}
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f & 0 & c_u \\
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0 & f & c_v \\
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0 & 0 & 1
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\end{bmatrix}
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\begin{bmatrix}
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t_x \\t_y\\t_z
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\end{bmatrix}
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= \mat{K}\; {_C}\vec{t}_P$
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If p. in diff frame ${_W} \vec{t}_P$, then $\vec{u} = \mat{K}[\mat{R}_{CW}\; {_C}\vec{t}_{CW}] = {_W}\vec{t}_P$
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\subsubsection{Pinhole Camera Projection with distortion}
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\shortdefinition Model: $\vec{u} = \vec{k}(\vec{d}(\vec{p}({_C}\vec{t}_P)))$, with:
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(Projection) $\vec{x'} = \vec{p}({_C}\vec{t}_P) = t_z^{-1} \cdot [t_x, t_y]^\top$
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(Distortion model, $r^2 = x'^2 + y'^2$, $\vec{x''} = \vec{d}(\vec{x'})$\\
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$\vec{x''} = \! \frac{1 + k_1 r^2 + k_2 r^4 + k_3 r^6}{1 + k_4 r^2 + k_5 r^4 + k_6 r^6} \vec{x'}
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+ \begin{bmatrix}
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2p_1 x' y' + p_2(r^2 + 2x'^2) \\
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p_1 (r^2 + 2y'^2) + 2p_2 x'y'
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\end{bmatrix}$
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(Scale and Centre) $\vec{u} = \vec{k}(\vec{x''}) = \text{diag}([f_u, f_v]) \cdot \vec{x''} + \vec{x}$
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All with $k_i$ radial distortion params, optional for $i > 2$, $p_i$ tang. dist. param, $f_u, f_v$ focal length in pixels
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\shade{green}{Inverse} ${_C}\vec{r} = [\vec{d}^{-1}(\vec{k^{-1}(\vec{u})}), 1]^\top$\\
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(To unit plane)
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$\vec{x''} = \vec{k}^{-1}(\vec{u}) = [f_u^{-1}, f_v^{-1}]^\top (\vec{u} - \vec{c})$
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(Un-distort)
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$\vec{x'} = \vec{d}^{-1}(\vec{x''})$ (usually comp. numerically)
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(Compute ray)
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${_C}\vec{r} = [\vec{x'}, 1]^\top$
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\subsubsection{Undestorting a whole image}
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$\vec{u}_i = \vec{k}(\vec{d}(\vec{k_{\text{new}}}^{-1}(\vec{u}_{i, \text{new}})))$
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where $\vec{u}_{i, \text{new}}$ is the place of pixel in output, $\vec{u}_i$ is the input
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\bi{Omnidir. Cam} undistortion model with $f(u, v) = \sum_{i = 0}^{N} a_i \rho^i$
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with $\rho = \sqrt{(u - c_u)^2 + (v - c_v)^2}$, $N = 4$ accurately describes it for most fisheye and catadioptric cameras
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