Image Processing Reference
In-Depth Information
Fig. 17.3. (
Left
) The complement of
the
original image,
F
and (
right
) the complement of the
target to be used as (an erosion) filter
G
the operation of
G
it is im
por
tant when computing the complement
G
=1
−
G
.
Accordingly, our example
G
yields
⎛
⎞
11111
10
0
01
11111
⎝
⎠
G
=1
−
G
=
(17.9)
The binary image
F
and the morphological filter
G
are illustrated in Fig. 17.3.
This image, eroded with the shown filter, results in the binary image
R
, which
is shown in Fig. 17.4 (
left
).
3. Multiply pointwise the images
H
and
R
obtained in the steps above:
S
=
H
R
(17.10)
where
represents a pointwise multiplication. Note that pointwise multiplica-
tion of two matrices (which must have the same size) is different than multipli-
cation of two matrices, in that the former is achieved by multiplying two entries
from the same row and column from the input matrices and assigning it to the
corresponding entry of the resulting matrix. Because only ordinary multiplica-
tion between ones and zeros are involved, this step is equivalent to a logical AND
operation
1
between the pixel values of
H
and
R
. The step is illustrated by Fig.
17.4 (
right
), which is the result of applying
between Fig. 17.2 (
right
) and Fig.
17.4 (
left
). It shows the result of the hit-miss transform, which marks the found
target object (as defined by the filter) red.
For a detailed discussion of morphological operators and skeletonization we refer
to [32, 33, 35, 183-185, 197].
1
In some studies AND, alternatively
, is represented by
∩
.
