Extension to Grayscale - Logic-based Nonlinear Image Processing - page 82

In practice, a more structured form of processing is required. The most power-
ful and flexible approach devised to date is computational morphology.
This is a general structure which can implement any filter defined within a
given window, be it linear, nonlinear, increasing, etc. The examples presented here
use 1D signals, but the concepts extend to images in a straightforward way.
Whereas grayscale morphology is defined in the continuous domain and re-
quires a signal range extending to -

, computational morphology works with dis-
crete data over a fixed range. It is therefore ideal for signal and image processing
where the data is sampled to a fixed number of bits.
The implementation can be carried out directly using either discrete logic or
comparators and does not require multipliers.
Special cases of computational morphology include implementation of grayscale
morphology, aperture, and stack filters.
On first viewing computational morphology, its structure appears to be very sim-
ilar to that of stack filters and has three main components: thresholding, elemental
erosion, and stacking. Stacking and thresholding have already been described as part
of the stack filter description but elemental erosion is a new concept unique to com-
putational morphology.

∞

7.4 Elemental Erosion

is a grayscale-to-binary operation with two grayscale inputs (one
waveform I and one structuring element B i ) resulting in a single binary output T i , i.e.,

An elemental erosion e

T i =I e B i .

It is similar to a standard morphological grayscale erosion in that it probes whether
the structuring element B i “fits” beneath the waveform I and returns a 1 at the loca-
tions where it fits and a 0 where it does not. Hence, a binary signal T i is produced. In
the same way as standard morphology, the structuring element B i has a single refer-
ence point which indicates the precise location where the output is affected.
The main difference between the elemental and standard grayscale erosion is
that the structuring element is only allowed to move horizontally. It cannot move
vertically and is “anchored” to the x axis. It thus produces a binary rather than
grayscale output. Figure 7.10 illustrates elemental erosion.
In practice, elemental erosion is carried out over a set of structuring elements B i
known as a kernel. An increasing grayscale-to-binary filter, based on elemental
erosion by a kernel of structuring elements B i can be formed as a maximum of ele-
mental erosions (sum of products),

T = I e B 1 + I e B 2 +…………... I e B N , (7.8)

where + represents the logical OR operator.