minima (Figure 1). The operations erosion and dilation can be composed into two
more basic operations—opening and closing. The opening with a SE, denoted by E,
is expressed as
I ◦ E ¼ I ⊖ E
ðÞ ⊕ E
(1)
The closing with a SE is expressed as
I • E ¼ I ⊕ E
ðÞ ⊖ E
(2)
The opening operation removes the objects, which are covered by E, while the
closing, by duality, removes object’s complement (i.e., holes in objects), which are
covered by E. The so-developed theory is topological in nature because it does not
depend explicitly on the concept of size but only on covering and inclusion. Classically, the MM theory was developed for uniform homothetic scaling of the SEs, but
it can be extended to nonhomogeneous groups of scaling transformations. The
scaling can be interpreted as generating a system of neighborhoods of every given
point, thus reinforcing the topological interpretation. This gives rise to partial
differential equation interpretation of the MM theory [4].
The multiscale aspects of the theory are due to the scaling of the structure
elements. For example, the seed SE can be rescaled homothetically and then applied
to the image. Such series of successive openings provides a measure of the prevalence of objects of a given size and is called granulometry (Figure 2). Granulometry
can be used also to segment compact bright objects by means of a top-hat transform, where from the primitive image its opened version is subtracted:
T E I
ðÞ¼I � E ◦ I, G U, L I
ðÞ¼L ◦ I � U ◦ I
(3)
The second equation represents the granulometric filtering operation, which can
extract bright objects of a specific size range from an image [5, 6].
Homogeneous scaling, that is, homothety, can be varied with the metric, which
is induced on the SE. This can be box-like, circular, diamond, etc.
Figure 1.
Fundamental morphological operations. On the first row, an image of cell nuclei stained with DAPI (left)
eroded (center) or dilated (right) with a disk of radius 10. On the second row, the same image is opened (left),
closed (center), or granulometrically filtered. The inscribed numbers denote SE radii.
50
Advances in Neural Signal Processing
more basic operations—opening and closing. The opening with a SE, denoted by E,
is expressed as
I ◦ E ¼ I ⊖ E
ðÞ ⊕ E
(1)
The closing with a SE is expressed as
I • E ¼ I ⊕ E
ðÞ ⊖ E
(2)
The opening operation removes the objects, which are covered by E, while the
closing, by duality, removes object’s complement (i.e., holes in objects), which are
covered by E. The so-developed theory is topological in nature because it does not
depend explicitly on the concept of size but only on covering and inclusion. Classically, the MM theory was developed for uniform homothetic scaling of the SEs, but
it can be extended to nonhomogeneous groups of scaling transformations. The
scaling can be interpreted as generating a system of neighborhoods of every given
point, thus reinforcing the topological interpretation. This gives rise to partial
differential equation interpretation of the MM theory [4].
The multiscale aspects of the theory are due to the scaling of the structure
elements. For example, the seed SE can be rescaled homothetically and then applied
to the image. Such series of successive openings provides a measure of the prevalence of objects of a given size and is called granulometry (Figure 2). Granulometry
can be used also to segment compact bright objects by means of a top-hat transform, where from the primitive image its opened version is subtracted:
T E I
ðÞ¼I � E ◦ I, G U, L I
ðÞ¼L ◦ I � U ◦ I
(3)
The second equation represents the granulometric filtering operation, which can
extract bright objects of a specific size range from an image [5, 6].
Homogeneous scaling, that is, homothety, can be varied with the metric, which
is induced on the SE. This can be box-like, circular, diamond, etc.
Figure 1.
Fundamental morphological operations. On the first row, an image of cell nuclei stained with DAPI (left)
eroded (center) or dilated (right) with a disk of radius 10. On the second row, the same image is opened (left),
closed (center), or granulometrically filtered. The inscribed numbers denote SE radii.
50
Advances in Neural Signal Processing