CHAPTER 11: COORDINATE LOGIC FILTERS 333
sorting, and their CLOs of dilation and erosion result in signal values that may
not be included in the initial set of input signal values; therefore, CL filters do not
overlap with any class of OS filters.
A methodology has been developed for implementing morphological filters us-
ing CLOs. If the image gray levels are mapped to an appropriately selected set
of decimal numbers, the CLOs of erosion and dilation act on this specific set ex-
actly as the MIN and MAX operators of morphological filtering do, using very sim-
ple hardware structures. The only drawback of this approach is the large binary
word lengths required for the assignment of the image gray quantization levels
to this specific set of numbers, which is alleviated by exploiting the properties of
CLOs and developing decomposition techniques that operate with smaller binary
lengths, and with less demanding hardware structures. This approach may be
extended for the implementation of any nonlinear filter that falls in the general
class of OS filters, using CLOs.
A class of nonlinear filters that is based on Boolean operators is that of the
generalized stack (GS) filters [Lin90, Mar87]. The difference between GS and CL
filters is that the stack filters operate on signal levels, whereas CL filters operate
on binary representations. The definition of GS filters is based on threshold de-
composition and Boolean operators. Threshold decomposition maps the 2n-level
input signal (where n is the word length) into 2 n - i binary signals by threshold-
ing the original signal at each of the allowable levels. The set of 2 n - 1 signals is
then filtered by 2 n - 1 Boolean operators that are constrained to have the stacking
property. The multilevel output is finally obtained as the sum of the 2 n - i binary
output signals. In contrast, CL filters decompose the signal into n binary signals
that operate in parallel and achieve the desired processing by executing only direct
logic operations among the binary values of the given signal.
The CL operators are defined and their properties are derived in Sec. 11.2, and
Sec. 11.3 presents the basic CL filters that execute the dilation, erosion, opening,
and closing operations as well as their filter structures. Section 11.4 reviews the
properties of duality for dilation-erosion and opening-closing, of idempotency
for opening and closing, and of extensivity for the dilation, erosion, opening, and
closing of CL filters. Section 11.5 introduces the proposed scheme for performing
morphological operations using CLOs and presents a simple hardware implemen-
tation of the basic MIN and MAX operations using this scheme. Section 11.6 covers
image analysis and feature extraction applications, including edge extraction, cal-
culation of the pattern spectrum, noise removal, fractal transformation, and the
design of cellular automata. Concluding remarks are given in Sec. 11.7.
1 1.2 Coordinate Logic Operations on Digital Signals
The underlying idea in coordinate logic image processing is the execution of CLOs
among gray-level pixels. These operations are executed among the corresponding
binary bits of equal position of the considered pixels, without counting the carry

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