Camera Parameters

Camera Parameters

Intrinsics

The is the projection matrix which turns camera coordinates to image coordinates.

It consists of the following:

• Focal Length \(f\) - this determines the field of view.
• Image Center \(\mathbf{o} = (o_x, o_y)\) (also known as principal point)
• Size of pixels \(\mathbf{s} = (s_x, s_y)\) (based on the resolution)
• Axis skew \(s\) typically 0

The formula for this matrix is: \[ \begin{equation} M_{int} = \begin{bmatrix} f/s_x & s & o_x\\ 0 & f/s_x & o_y\\ 0 & 0 & 1 \end{bmatrix} \end{equation} \]

E.g. if your camera has a 90 deg FOV on each side and outputs a resolution of \(\displaystyle 256 \times 256\), then the intrinsic matrix should project \(\displaystyle (1,0,1)\) to \(\displaystyle (256, 0)\): \[ \begin{equation} M_{int} = \begin{bmatrix} 128/256 & 0 & 128/256\\ 0 & 128/256 & 128/256\\ 0 & 0 & 1 \end{bmatrix} \end{equation} \]

Note you can also write \(\displaystyle f/s_x\) as \(\displaystyle f_x\), and similar for \(\displaystyle f_y\).

Extrinsics

This is the view matrix which encodes the camera's position and rotation.

Suppose the camera position is \(\mathbf{C}\) and rotation \(\mathbf{R}_c\).

\[ \begin{equation} M_{ext}= [\mathbf{R} | \mathbf{t}] = [\mathbf{R}_c^T | -\mathbf{R}_c^T \mathbf{C}] \end{equation} \]

Resources

• Camera Models and Parameters slides by Pablo Sala, University of Toronto
• Dissecting the intrinsic matrix blog post by Kyle Simek
  • Part 1: Extrinsic/Intrinsic Decomposition
  • Part 2: The Extrinsic Matrix
  • Part 3: The Intrinsic Matrix