Projection Matrix

Definition

The projection matrix $M$ describes the whole forward projection, from world coordinates to pixel coordinates, as a single matrix product in homogeneous coordinates.

flowchart LR
  W["World<br/>(U, V, W)"] -- "M_ext" --> C["Camera<br/>(X, Y, Z)"] -- "M_proj" --> F["Film<br/>(x, y)"] -- "M_aff" --> P["Pixel<br/>(u, v)"]

Formula

$$\tilde{u}=M{\tilde{P}}_W,M=M_{int}{M}_{ext}\in {\mathbb{R}}^{3\times 4}$$

$M_{ext}$: rotation and translation (extrinsics). $M_{int}=M_{aff}{M}_{proj}$: focal length, pixel sizes, principal point (intrinsics). Pixel coordinates come from dividing by the third homogeneous coordinate.

Trap

Only the step camera → film loses information (depth). Computer vision is mostly about going backwards (backward projection) with stereo, motion or learned priors.

Appears in

• Lecture 2.1, Imaging geometry: the four coordinate systems
• Lecture 2.1, Summary: forward projection

Related concepts

• Homogeneous Coordinates
• Pinhole Camera Model
• Intrinsic Parameters
• Extrinsic Parameters