Hans Georg Schaathun
NTNU, Noregs Teknisk-Naturvitskaplege Universitet
September 2023
Point $P$ has co-ordinates
Seek mappings $$\vec{x}_w\mapsto \vec{X}\mapsto \vec{x}\mapsto (i,j)$$
Change of basis | $$\mathbf{X} = R\mathbf{x}_w + T_{O_w} \in\mathbb{R}^3$$ |
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where | $R = \begin{bmatrix} \mathbf{e}_{w,1} \mathbf{e}_{w,2} \mathbf{e}_{w,3} \end{bmatrix}$ |
or with homogeneous co-ordinates | $$\tilde{\mathbf{X}} = \begin{bmatrix} R & T_{O_w} \\ 0 & 1 \end{bmatrix}\cdot \tilde{\mathbf{x}}_w$$ |
ideal perspective | $$\mathbf{x} = \begin{bmatrix}x\\y\end{bmatrix} = \frac fZ\begin{bmatrix}X\\Y\end{bmatrix}$$ |
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homogeneous | $$ Z \begin{bmatrix}x\\y\\1\end{bmatrix} = \begin{bmatrix}f & 0 & 0 & 0 \\0 & f & 0 & 0 & \\0 & 0 & 1 & 0 \end{bmatrix} \cdot \begin{bmatrix}X\\Y\\Z\\1\end{bmatrix}$$ |
$Z$ is unknown. Write $\lambda (=Z)$ | $$ \lambda \begin{bmatrix}x\\y\\1\end{bmatrix} = \begin{bmatrix}f & 0 & 0 & 0 \\0 & f & 0 & 0 \\0 & 0 & 1 & 0 \end{bmatrix} \cdot \begin{bmatrix}X\\Y\\Z\\1\end{bmatrix}$$ |
$$ \lambda \begin{bmatrix}x\\y\\1\end{bmatrix} $$ | $$ = $$ | $$\begin{bmatrix}f & 0 & 0 \\0 & f & 0 \\0 & 0 & 1 \end{bmatrix}$$ | $$\cdot$$ | $$\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & 0\end{bmatrix} $$ | $$\cdot$$ | $$ \begin{bmatrix}R & T \\0 & 1\end{bmatrix}$$ | $$\cdot$$ | $$\begin{bmatrix}X\\Y\\Z\\1\end{bmatrix}$$ |
$$ \lambda \textbf{x} $$ | $$ = $$ | $$K_f$$ | $$\cdot$$ | $$\Pi_0$$ | $$\cdot$$ | $$g$$ | $$\cdot$$ | $$\mathbf{X}_0$$ |
pixels per metric unit | $s_x\times s_y$ |
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axes scaled differently | $$\begin{bmatrix} x_s\\y_s\end{bmatrix} = \begin{bmatrix} s_x & 0\\ 0 & s_y\end{bmatrix} \cdot \begin{bmatrix} x\\y\end{bmatrix} $$ |
translate the origin | $$\begin{align}i &= x_s + o_x \\ j &= y_s + o_y\end{align} $$ |
homogeneous co-ordinates | $$\begin{bmatrix} i\\j\end{bmatrix} = \begin{bmatrix} s_x & 0 & o_x \\ 0 & s_y & o_y \\ 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} x\\y\\1\end{bmatrix} $$ |
$$\begin{bmatrix} i\\j\\1\end{bmatrix} = \begin{bmatrix} s_x & s_\theta & o_x \\ 0 & s_y & o_y \\ 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} x\\y\\1\end{bmatrix} $$
$$ \lambda \begin{bmatrix}x\\y\\1\end{bmatrix} $$ | $$ = $$ | $$\begin{bmatrix} s_x & s_\theta & o_x \\ 0 & s_y & o_y \\ 0 & 0 & 1 \end{bmatrix}$$ | $$\cdot$$ | $$\begin{bmatrix}f & 0 & 0 \\0 & f & 0 \\0 & 0 & 1 \end{bmatrix}$$ | $$\cdot$$ | $$\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & 0\end{bmatrix} $$ | $$\cdot$$ | $$\begin{bmatrix}X\\Y\\Z\\1\end{bmatrix}$$ |
$$ \lambda \textbf{x} $$ | $$ = $$ | $$K_s$$ | $$\cdot$$ | $$K_f$$ | $$\cdot$$ | $$\Pi_0$$ | $$\cdot$$ | $$\mathbf{X}_0$$ |
Camera Intrinsic Parameter Matrix | $$K=K_sK_f= \begin{bmatrix} fs_x & fs_\theta & o_x \\ 0 & fs_y & o_y \\ 0 & 0 & 1 \end{bmatrix}$$ |
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