Geometric Calibration (deprecated)

Homogeneous Coordinates

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Requirements Software operation Defining Calibration Tasks Definitions and Theory

Supported Systems

Supported Targets

Required Evidence

Test Setup

Calibration Procedure

User Interface

Use in Imatest IT

Module settings

Module outputs

Defining a Device

Defining Distortion

Defining the System of Devices

Defining the Target

Defining a Test Capture

Defining a Test Image

Homogenous Coordinates

Projective Camera Model

Multi-Camera Systems

Distortion Models

Coordinate Systems

Rotations and Translations



Homogeneous coordinates are a set of coordinates with useful properties for perspective geometry:

  • Infinity may be represented with a finite value.
  • Rotations and translations may be represented by a single matrix operation.
  • Homogeneous coordinates may be used for a space with arbitrary dimension including 2D (image) and 3D (world) coordinates.

A “standard” coordinate is referred to as inhomogeneous. 


  • Iff the last coordinate of a homogeneous coordinate is 0, then it is at infinity.
  • Iff the last coordinate of a homogeneous coordinate is not 0, then it is at a finite location.
  • Two homogeneous points are the same iff there exists a non-zero scalar multiplier between them, i.e., \(\mathbf{x}=k\cdot\mathbf{y}\).

Inhomogeneous to Homogeneous

The simplest way to convert from an inhomogeneous coordinate to a homogeneous one is to append a 1 to the end of the coordinate. 



The general conversion is to append 1 and multiply by any non-zero real number.

\(\begin{bmatrix}x\\y\end{bmatrix}\rightarrow\begin{bmatrix}k\cdot x\\k\cdot y\\k\end{bmatrix}\)

\(\begin{bmatrix}X\\Y\\Z\end{bmatrix}\rightarrow\begin{bmatrix}k\cdot X\\k\cdot Y\\k\cdot Z\\k\end{bmatrix}\)

Homogeneous to Inhomogeneous

To convert from a homogeneous coordinate to an inhomogeneous one, divide all of the components by the last one, which is discarded.