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40 lines
1.3 KiB
TeX
40 lines
1.3 KiB
TeX
\subsection{Bootstrapping}
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\bi{PnP Problem} {\scriptsize Persp. $n$-P.}
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Find sol. for camera pose \textit{directly}
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\bi{RANSAC} {\scriptsize RANdom SAmpling Consensus} for find. outliers \& correct
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% TODO: Do we need to know this algorithm / be able to implement it?
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% -> likely not
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\bi{Stereo Triang.} Given two rays (known poses for points in 2D).
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Find good point in 3D. Fast sol: \bi{Midpoint Method}:
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\bi{1} Find p. along ray w/ min. dist (Lin. Least Squares)
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\[
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\vec{\lambda}\! =\! [\lambda_1 \; \lambda_2]^\top\! = \! \argmin{} ||({_W}\vec{t}_{C_2} + \lambda_2 {_W}\vec{e}_2) - ({_W}\vec{t}_{C_1} + \lambda_1 {_W}\vec{e}_2)||^2
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\]
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\rmvspace
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\bi{2} Solve normal equation $\mat{A} \vec{\lambda} = \vec{b}$ with $\vec{q} = -{_W}\vec{e}^\top_1 {_W}\vec{e}_2$:
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\[
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\mat{A} =
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{\scriptsize
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\begin{bmatrix}
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1 & \vec{q} \\
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\vec{q} & 1
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\end{bmatrix}
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}
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\quad
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\vec{b} =
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{\scriptsize
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\begin{bmatrix}
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\vec{e}_1^\top \cdot ({_W}\vec{t}_{C_2} - {_W}\vec{t}_{C_1}) \\
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-\vec{e}_2^\top \cdot ({_W}\vec{t}_{C_2} - {_W}\vec{t}_{C_1})
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\end{bmatrix}
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}
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\quad C_i \text{ cam}
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\]
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\bi{3} Pick midp. ${_W}\vec{t}_P \! = \! 0.5(\tau_1 \! + \! \tau_2)$; $\tau_n \! = \! {_W}\vec{t}_{C_n} + \lambda_n{_W}\vec{e}_n)$
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