Image Processing Reference
In-Depth Information
9
Appendices
9.1
Appendix 1: Homogeneous co-ordinate system
The homogeneous co-ordinate system is essentially the mathematics of how we relate
camera co-ordinates to ' real world ' co-ordinates: the relation between image and physical
space. Its major advantages are that it is linear , consistent and easy to use. Image
transformations become simple matrix operations, as opposed to geometric calculations. It
includes perspective (distance) and as such finds use in stereo and 3D vision applications
and in camera control. It is not mainstream to shape analysis, since in many applications
we use orthographic projections where spatial physical co-ordinates map directly to image
space co-ordinates ignoring projection. But there are occasions when perspective is extremely
important; as such it is necessary to have a co-ordinate system which can handle it. The
homogeneous co-ordinate system has proved popular for this task for many years.
It is common to represent position as a set of x , y and z co-ordinates where x and y
usually index spatial position and z is depth . By reference to the system arrangement
illustrated in the figure below, by triangulation, the image point co-ordinate y i is related to
the focal length f and the x , y , z co-ordinate of the physical point x p , y p , z p by
Image plane
World plane
Centre of projection (lens) = [0 0 f ] T
Image point
x i = [x i y i 0] T
x
Optical axis
y
z
f
World point
x p = [x p y p z p ] T
Co-ordinate system arrangement
 
 
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