Image Processing Reference
In-Depth Information
1.
The binary strings being manipulated have a direct interpretation in terms of
real or complex numbers.
2.
The logical operations applied to the strings are restricted to those that carry out
equivalent linear operations, such as multiplication and addition of real or com-
plex numbers.
Nonlinear image processing is presented here as a generalization of the above oper-
ation by removing the linearity constraints. It seeks the optimum mapping imple-
mented directly in logic. The linear solution should be viewed as a special case of
the set of all logic-based solutions rather than as an alternative. Given this general-
ization, the optimum nonlinear solution will be either better or equivalent to the lin-
ear solution, but it should not be worse. This inequality holds regardless of the
problem or the criteria, provided that the training data is sufficient.
The above argument has lead to various researchers in this field issuing the pro-
vocative claim that “all image processing is nonlinear.” 1
The principal reason for adopting this strategy is to see if the other solutions
available through a logical approach are useful and offer advantages over linear so-
lutions. Linear solutions can be easy to compute. It is not difficult to derive the opti-
mum linear smoothing filter for an image with noise, but the result of applying this
filter is an image which is invariably blurred, causing a loss of signal information.
Here, a nonlinear solution such as the median filter gives much better results lead-
ing to noise removal and edge preservation without blurring, despite the fact that
the median filter takes no account of the image or noise statistics.
In removing the linear constraint, the process of finding the optimum solution
becomes much more difficult to compute. However, if the consequences of linear
processing are unacceptable results, we must try to do this.
The work in this area has focused on the design of filters. Many applications
are possible within this context such as noise reduction, shape, character and object
recognition, enhancement, restoration, texture classification, spatial and intensity
sampling, and rate conversion.
In practice, all filters are limited in some way. These limits are known as con-
straints. For example, the filter designed for a particular application may be con-
strained to lie in a particular class. The optimum filter is therefore the best filter
within that class. In this work, we seek the optimum filter from the class of filters
that have a logical implementation. This also includes morphological and rank-or-
der filters (which may be cast in the above context and therefore may provide solu-
tions that have an interpretation in terms of shape or numerical ordering).
Linear filters require little training data. In theory, only the same number of ex-
amples as the number of parameters is necessary to determine a solution. However,
for nonlinear filters, the training process amounts to the estimation of the condi-
tional output probabilities. In the most general case, each training example only
provides information about one specific combination of input variables. It is not
possible to infer anything about the behavior of the filter for other sets of inputs.
For a stochastic system, a sufficient number of observations of every input combi-
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