Homogeneous Coordinates

Throwing out the last coordinate: $$(X', Y') = (-d \frac{X}{Z}, -d \frac{Y}{Z})$$

Homogeneous Coordinates

Previous equation is not linear

Trick: add one more coordinate:

2D

$$(x, y) \rightarrow \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}$$

3D

$$(x, y, z) \rightarrow \begin{bmatrix} x \\ y \\ z \\ 1 \end{bmatrix}$$

Homogeneous coordinates are invariant under scaling

If you scale the homogeneous coordinate by some $a$, the coordinate in pixel space will be unaffected because of the division by a $w$.

Convert from Homogeneous Coordinates

$$\begin{bmatrix} x \\ y \\ w \end{bmatrix} \rightarrow (x / w, y / w)$$

$$\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix} \rightarrow (x / w, y / w, z / w)$$

Perspective Projection

Projection is a matrix multiply using homogeneous coordinates.

3D -> 2D $$\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \frac{1}{f} & 0 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ 1 \end{bmatrix} = \begin{bmatrix} x \\ y \\ \frac{z}{f} \\ \end{bmatrix} \rightarrow (f \frac{x}{z} , f \frac{y}{z}) \rightarrow (u, v)$$ This multiplication is a linear transformation.

Here $f$ is the focal length, which is the distance from the center of projection to the projection plane ($d$ in the previous figure)