Data Conventions

Here we explain the various data conventions used in our repo.

Rotation Convention

We represent rotations with four dimensional vectors \(q = (w,x,y,z)\) such that the 3x3 \(SO(3)\) rotation matrix is defined by:

\[\begin{split}R = \begin{bmatrix} 1 - 2 \left( y^2 + z^2 \right) & 2 \left( x y - w z \right) & 2 \left( x z + w y \right) \\ 2 \left( x y + w z \right) & 1 - 2 \left( x^2 + z^2 \right) & 2 \left( y z - w x \right) \\ 2 \left( x z - w y \right) & 2 \left( y z + w x \right) & 1 - 2 \left( x^2 + y^2 \right) \\ \end{bmatrix}\end{split}\]

View Matrix

We refer to the view matrix \(W\) as the world to camera frame transformation (referred to as w2c in some sources) that maps 3D world points \((x,y,z)_{world}\) to 3D camera points \((x,y,z)_{cam}\) where \(z_{cam}\) is the relative depth to the camera center.