Attribute Morphology
Mathematical morphology using shape attributes
Image filters that are used in mathematical morphology typically require a
structuring element or a series of structuring elements to define the bounds of the
filter. Examples include the classical forms of the erosion, dilation, opening and closing
filters, and cascades of such filters. However, openings and closings are defined by very
general properties that do not necessitate the use of a set of fixed structuring elements.
For example, attribute openings preserve only those connected components in an
image that satisfy a specified criterion based on some shape attribute. The advantage with
this approach is that the filter does not change the shape of the connected components in
the image; it either removes them or leaves them as they are. The figure below shows an
example. In Fig. 1(b) is the result of an attribute opening of the image in Fig. 1(a),
using the criterion: must have an area greater than 200 pixels . Fig. 1(c) shows
the result when the criterion is: must have a length greater than 50 pixels .
Figure 1: Binary attribute
openings
The notion of binary attribute openings can be extended to grey-scale
images. The resulting filter satisfies the three required properties of an opening
(idempotence, increasingness and anti-extensivity) but does not use a structuring element
to define the bounds of the filter. The figure below shows conceptually how the grey-scale
attribute opening works. The result of the opening is constructed from a maximum of
connected openings, where each connected opening works on a regional maxima in the image.
In Fig. 2(a) is shown the profile of a grey--scale image. The result for the connected
component opening for the middle regional maximum shown is in Fig. 2(b), using the
criterion: must have an area greater than 50 pixels (we have used this filter in
graph-based image processing). Here, we have descended down though the threshold sets of
the image that contain this middle regional maxima, removing those threshold sets that do
not satisfy the given criterion. Shown in Fig. 2(c) is the complete opening, obtained by
combining the results from the connected component openings working on the three regional
maxima in the image.
Figure 2: Grey-scale attribute
openings
By using criteria that are non-increasing we obtain attribute thinnings
(filters that are idempotent, anti-extensive, but not necessarily increasing). The use
of non-increasing criteria is seen as an important generalisation because it allows the
use of non-increasing shape descriptors such as compactness and eccentricity to be applied
to filter images. For example, in Fig. 3(a) is shown the Mona Lisa, from which we want to
distinguish the face from the background. In Fig. 3(b) is shown the difference between the
original image and a grey--scale thinning using the criterion: minor axis of the
best--fit ellipse must be greater than 200 pixels . The thinning component of this
filter allowed us to thin the brightness of the facial region in the image. Subtracting
this result from the original image then highlights the face, but other small features are
also retained, as shown in Fig. 3(b). In order to remove these, we finish by using a
grey--scale thinning with the non-increasing criterion: major axis must be greater than
250 pixels (using a lesser value than 250 would have retained the chest region in the
image). The result, shown in Fig. 3(c), shows the facial region distinguished from the
background of the image. Note that this result is rotationally invariant because the
criteria used are rotationally invariant.
Figure 3: Attribute
thinnings used
to highlight the face of Mona Lisa
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