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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004
2 SCALE-SPACE BASED GENERALIZATION
2.1 Scale-Spaces and Image Analysis
Every object has a certain scale range, in which it can be
represented. For a wood another scale is appropriate than for a
single tree or for the leaves of a branch. For object recognition,
it is often reasonable to analyse the image at different scales. In
many cases a one-parameter-family of derived signals is
generated, where the different representations depend only on
the current scale described by the scale-parameter o. There are
several means to generate a scale-space.
[n image processing often the Linear scale-space is used. It is
obtained by convolving an image with a Gaussian kernel. It
combines causality, isotropy, and inhomogeneity. The most
important constraint is causality. This means, that each feature
in a coarse scale has to have a reason in fine scale (Koenderink
1984). In Figure 2 it can be seen, that after the convolution (and
the additional application of a threshold on the right hand side)
the original object is split at one scale and merged at another.
Because they can cause these so-called scale-space events,
scale-spaces are suited for generalization, especially for
simplification and aggregation (Mayer 1998). Scale-space
events can be external events such as the split or merge of
objects, or internal events, that affect only topologically related
object parts, e.g., the climination of a small protrusion. For the
generalization of buildings the linear scale-space has the
disadvantage, that corners are rounded and straight lines get lost
(cf. Fig. 2).
zs
JE ®%
a) b)
Figure 2: a) An object is convolved with a Gaussian kernel of
different sizes, b) After applying a threshold a split and a merge
become visible (Mayer 1998).
A scale-space with different characteristics is mathematical
morphology (Serra 1982). The two basic transformations
dilation and erosion and the two combined transformations
opening and closing are formulated as follows for n-
dimensional binary images (Haralick and Shapiro 1992):
Dilation: A®B={ceE"|c=a+b for some acA and beB} (1)
Erosion: A@B={xeE"| x+beA for all be B} (2)
Opening: A°B =(AGB) ®B (3)
Closing: AeB =(A®B) OB (4)
A is the original binary image to be processed and B is called
structure or structuring element (Serra 1982, Su et al. 1997). By
varying the size of a usually square or circular structuring
element, a scale-space with a predictable behavior over the
scales (causality) is obtained. The basic operations erosion and
dilation are combined to opening or closing in order to *reset"
an object to its original range of size. Opening eliminates object
parts, which are smaller than the structuring element. Closing
fills small gaps (cf. Fig.3).
195
Original Structure Element
Opening Closing
Figure 3: Mathematical morphology for a binary image with a
square structuring element. Opening eliminates small parts,
closing fills gaps (Su et al. 1997).
A scale-space that combines the two complementary scale-
spaces mathematical morphology and linear scale-space is the
reaction-diffusion-space of (Kimia et al. 1985). The reaction
part is similar to mathematical morphology, whereas the
diffusion part is for a small c equivalent to linear scale-space.
For a large o it diverges in this respect that only parts with high
curvature are eliminated.
2.2 Scale-Spaces for 2D Ground Plans
(Li 1996) and (Mayer 1998) showed, that scale-space theory
together with scale-space events are suitable for generalization.
For the simplification of 2D vector data, particularly building
ground plans, i.e., objects that consist mostly of straight lines
arranged in right angles, (Mayer 1998) proposes a process
similar to the reaction-diffusion-space. Reaction part and
diffusion part are applied sequentially.
The reaction part, i.e., mathematical morphology, is realized by
incrementally shifting all segments inwards or outwards,
intersecting the segments to preserve the corners. Objects can
be split or merged and small protrusions or notches can be
eliminated or filled (cf. Fig.4).
(t
Erosion
— P-
EXC ERN | Di lati on
Figure 4: Mathematical morphology applied to vector data -
split (top) for erosion and merge for dilation (bottom)
