Image Processing Reference
In-Depth Information
Figure 7.1
Threshold decomposition.
The binary signals x t are known as a stack . They may be processed using a binary
filter
to produce a series of binary output signals y t ,
( x t ) = y t .
(7.2)
These may be summed (or stacked ) to give a grayscale output signal Y as follows:
=
y t
Y
.
(7.3)
t
An example of stacking a set of binary signals to give a grayscale signal is shown in
Fig. 7.2. For a certain class of filters, the grayscale output signal Y resulting from
the process of thresholding followed by binary filtering and then summation is pre-
cisely the same as that which results from filtering the grayscale signal X with the
grayscale version of the filter . The class of filters for which this holds includes
many types of filters such as WOS, including the median, weighted median, and
rank-order filters.
The ability to decompose a grayscale function into a series of binary operations
can be a valuable one. As shown in the previous chapter, the binary median filter
may be implemented as a counting operation rather than sorting. The stack filter al-
lows this property to be utilized even for grayscale signals. The decomposition may
be useful for proving theorems and characterizing filters. It is rarely, however, of
direct use in implementation.
An example of a 3-point running median (with three levels) implemented as a
grayscale operation and via a stack filter is shown in Fig. 7.3. The grayscale signal
is median filtered by two routes. The first is by applying threshold decomposition
to process a stack of binary signals. These are individually filtered using a 3-point
binary running median. The binary median consists of a simple counting operation.
The resulting binary outputs are then stacked to produce the grayscale output sig-
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