Image filters 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 [6, 7]. However, such filters are defined by very general properties that do not necessitate the use of a set of fixed structuring elements; for example, Vincent [8] introduces an opening filter that satisfies the three required properties of an opening (idempotence, increasingness and anti-extensivity) but removes information from the image on the basis of area. Breen and Jones [9, 10, 11] described an attribute-based approach to mathematical morphology openings. The use of non-increasing-shape attributes is advocated because they allow the use of shape descriptors such as compactness and eccentricity to be applied to filter grey scale images.
For binary images, attribute openings preserve only those connected
components that satisfy a specified criterion [6].
For grey-scale images, an attribute opening 
is given by:
where g is a grey-scale image, T is an increasing criterion ( 
), 
is the set of positions of
regional maxima in the image and 
is a connected
opening. The criterion T for area opening is an increasing
criterion. The connected opening of a set X at a point x is: (i)
the connected component of X that contains x if x 
X;
and (ii) the empty set if x 
X.
The attribute opening can then be formed as a point-wise maximum
of trivial openings, using only a set of regional
maxima point . Each trivial opening
can be implemented by descending down through the thresholds
from the regional maximum until a threshold set
is reached that satisfy criterion T.