Key Concepts

• calibration
• perspective

Briefing

World frame and camera frame

\[\mathbf{X} = R\mathbf{X}_0 + T\in\mathbb{R}^3\]

Projection

\[ Z \begin{bmatrix}x\\y\\1\end{bmatrix} = \begin{bmatrix}f 0 0 0 \\0 f 0 0 \\0 0 1 0\end{bmatrix} \cdot \begin{bmatrix}X\\Y\\Z\\1\end{bmatrix}\]

Note that \(Z\) is typically unknown. We write \(\lambda (=Z)\) for this unknown constant.

\[ \lambda \begin{bmatrix}x\\y\\1\end{bmatrix} = \begin{bmatrix}f 0 0 \\0 f 0 \\0 0 1 \end{bmatrix} \cdot \begin{bmatrix}1 0 0 0 \\0 1 0 0 \\0 0 1 0\end{bmatrix} \cdot \begin{bmatrix}R T \\0 1\end{bmatrix} \cdot \begin{bmatrix}X\\Y\\Z\\1\end{bmatrix}\]

\[ \lambda \textbf{x} = K_f\Pi_0\mathbf{X} = K_f\Pi_0g\mathbf{X}_0\]

Ideal Camera Projection

Exercises

Debrief