Haggren - 2 
Datums 
The datum is the basis of any coordinate system. For transformations between different coordinate systems 
we distinguish the internal and external datums. Each internal datum is defined for the purpose of the 
process use only, whereas an external datum is defined for the purpose of external use. For example, the 
image, camera, and stereo model coordinates relate primarily to internal datums, and the object model, 
design model and world coordinates relate to external datums. 
External Datum. The external datum is by default uniform. It is a 3-D coordinate system that is both 
orthogonal and of metric unity. The orthogonality relates to a rectangular coordinate system and ensures 
the geometric conformity as it regards the rotated geometries. The metric unity ensures that the unit vector 
is referred to the metric system. The local world coordinate system is usually orthogonal and metric. 
Similarly, the coordinate system of any design model is by intention orthogonal and metric. 
An external 3-D datum is usually definite by a minimum of seven coordinate values. These can be any 
coordinate combinations produced by at least three control points. 
Internal Datum. Opposite to the external datum the default for any internal coordinate system is that 
they are neither orthogonal nor metric unless they have been externally calibrated. However, this kind of 
projective 3-D datum is well defined with respect to that internal control function it is used for. For 
example, in a 2-D machine vision application the internal datum is very likely the original image 
coordinate system and all actions are controlled relative to that scene. Similarly, we are capable of 
controlling us more precisely in 3-D relative to our actual perception than by knowing an orthogonal 
datum. 
The internal datum can be either 2-D or 3-D. A 2-D projective datum becomes orthogonally definite by at 
least four XY-control points or by five control distances, and a 3-D projective datum by at least five 
control points or by nine control distances. In the case the datum is orthogonal, the minimum for a 2-D 
datum definition is two control points, and fpr a 3-D datum the seven coordinate values or the three 
control points. 
Orientation to Datum. In the case the object geometry defined in one datum should be transformed to 
any higher datum the transformation is solved by determining the orientation of said geometry. The 
orientation parameters usually include six parameters: three rotations around the coordinate axes and 
three shifts along them. This assumes both coordinate systems to be both orthogonal and of same scale. If 
that is not the case, the scale will increase the number of parameters by one, the orthogonality by eight. In 
the case the orthogonality is solved separately, this should be done first, as the three rotation parameters 
assume orthogonality of both coordinate systems. 
Orthogonality. The local coordinate systems are expectably both orthogonal and metric. A camera 
coordinate system is orthogonal by default only, if the transformation from image to camera coordinates 
is based on calibration certificate. The coordinate system according to which the fiducial coordinates and 
the camera constant have been calibrated should be de facto orthogonal. 
In the case the camera calibration will be performed on-the-job, the orthogonality can be solved under 
certain conditions. These may be: 
A minimum of three images with the same interior orientation are used for reconstruction. 
A minimum of eight distances of a common scale reference will recover the orthogonality. 
For external scaling, at least one external scale reference should be included to reconstruction. The ill- 
conditioned cases are the more likely, the closer the three cameras locate along one straight line. 
Similarly, the eight distances should be directed randomly and not parallel to each other.