Another useful realization is the morphological gradient operation, which is the
difference between an opening and a closure δ E I
ðÞ¼I ◦ E � I • E. It can be used to
extract connected shapes by subsequent thresholding.
4. Geometrical image features
Mathematically, images can be represented as surfaces in the three-dimensional
Euclidean space, where the elevation represents the signal intensity. In this sense,
the intensity at a certain point in the direction x + r can be interpolated from its local
neighborhood as
Ixþ r
ð
Þ¼Ix ð Þþr � ∇I þ
1
2
r
T � IH I � r þ O r
T r
ÀÁ
(4)
Components of the gradient are given by ∇u ¼ u x ,u y
ÀÁ
. The Hessian tensor is
given by the matrix
IH u ¼
u xx u xy
u yx u yy
(5)
where for smooth signals the partial derivatives commute u xy ¼ u yx . This picture
is a part of the so-called jet space—a higher dimensional differential descriptor
Figure 2.
Granulometry of cell nuclei. An image of cell nuclei stained with DAPI is opened with an increasing sequence of
disk-shaped kernels. Note the eventual disappearance of the central bright object. The inscribed numbers denote
SE radii.
51
Multiscale Segmentation of Microscopic Images
DOI: http://dx.doi.org/10.5772/intechopen.89003