hill, s/he will have difficulty in identifying the highest point.
However, the observer who stands some distance away from
the point can sees the peak clearly. Also when one views the
terrain surface from an airplane, small details disappear and the
main characteristics of the terrain variations become very clear.
It is a commonplace to photogrammetrists that the stereo-
models formed from high altitude photography are more
generalised than those formed from low altitude photography.
If one views the terrain surface from an satellite, then terrain
surfaces becomes very smooth. These phenomena can easily
checked by forming a stereo-model from a pair of satellite
images such as SPOT images or Spacelab Metric Camera
photography. These are just some out of many practical
examples illustrating the transformation in scale dimension,
which follows a natural principle.
4.2 The natural principle
The next question arising is “how these transformations are
achieved?”. In the case of human observation, it is due to the
limitation of eyes’ resolution. That is, all information within
the limitation of human resolution disappears. In the case of
stereo-models formed from images, it is due to the resolution of
images. That is, all information within the image resolution
(e.g. 10m per pixel in the case of SPOT images) disappears.
These examples underline a universal principle, a natural
principle as called by Li and Openshaw (1993), which states as
follows:
“for a given scale of interest, all details about the spatial
variations of geographic objects beyond certain limitation
are unable to be represented and can thus be neglected”.
In other words, by neglecting all information about spatial
variations within a given critical size (or limitation), the
transformation in scale dimension which is similar to the
generalization of natural phenomena can be achieved. Fig.5
illustrates how it works.
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Fig.5 By neglecting the detailed spatial variations within the
black square, the shape of the polygon is simplified..
More detailed discussion of this natural principle and more
practical illustrations can be found in the original paper by Li
and Openshaw (1993).
5. MATHEMATICAL BASIS FOR TRANSFORMATION IN SCALE
DIMENSION: MORPHOLOGICAL OPERATORS
After the introduction of this natural principle, it is the time to
examine how this principle can be realized mathematically.
5.1 Examples illustrating the mathematical basis
As has been discussed previously, the shapes and structures of
spatial objects are simplified when a transformation in scale
dimension is applied and such a transformation follows the
natural principle. Fig.6 and Fig.7 show examples which
illustrate how the shape of objects can be manipulated using
morphological operators in a way similar to the generalization
by the natural principle. In Fig.6, a process called erosion is
used. The natural principle can be best depicted by this
process. The size of the structuring element (see discussion
later) used in this process can mimic the critical size (within
which all spatial variations can be neglected) in the natural
principle However, this process does not work well in the case
when there are deep channels. In this case, a process called
closing should be proceeded. Fig.7 shows how such a
combination works.
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(a) Original image — (b) Shape simplified; (c) Further simplified
Fig.6 Shape simplified by erosion process
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(a) Original image; — (b) Channels closed; (c) Closed image simplified
Fig.7 A combination of closing and erosion works well for
even very complicated shape
5.2 The science of shape - mathematical morphology
It has been illustrated that the operators developed in
mathematical morphology has great potential for depicting the
digital-to-digital transformation of the generalization process.
Therefore, it seems pertinent to have a more detailed discussion
of mathematical morphology here.
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Mathematical morphology is a science of shape, form and
structure, based on set theory. It was developed by two French
456
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
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