Key Concepts

• calibration
• perspective

Briefing

Image Projection

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.

Ideal Camera Projection

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

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

From meters to pixels

• Same units for world frame and image frame, i.e. meter
• Meaningless - images are measured in pixels

TODO complete

Distortion

Calibration

• Camera Calibration is a transformation in the ideal model
• In addition we need calibration to compensate for distortion = imperfection in the camera
• Two types
  • Radial Distortion - typical for wide angle lenses
  • Tangential Distortion - when image plane and lense are not parallel

Radial Distortion

• Artifact of wide angle lenses (wide field of view).
• Simplest effective model:

\[ \begin{align} x &= x_d(1 + a_1r^2 + a_2r^4) \\ y &= y_d(1 + a_1r^2 + a_2r^4) \\ r &= x_d^2+y_d^2 \end{align} \]

• \((x,y)\) are the true co-ordinate of some point
• \((x_d,y_d)\) are the co-ordinates in the distorted image

• This is easy to automate, so we can disregard distortion for analysis

• OpenCV uses at sixth order model

\[ \begin{align} x_d &= x(1 + a_1r^2 + a_2r^4 + a_3r^6) \\ y_d &= y(1 + a_1r^2 + a_2r^4 + a_3r^6) \end{align} \]

Tangential Distortion

Not discussed in the text book

• Occurs when the lens and image plane are not parallel

\[ \begin{align} x_d &= x+ [2p_1xy+p_2(r^2+2x^2)] y_d &= y+[p_1(r^2+2y^2)+2p_2xy] \end{align} \]