Image Processing Reference
In-Depth Information
called an AND function. Matheron made the observation that every increasing,
translation-invariant set operator may be represented as a union of erosions. To an
electronics engineer this means that all operators can be implemented as a sum of
products (and they do not require complementation). The building blocks of mathe-
matical morphology such as erosion, dilation, opening, closing, and their repeti-
tions under unions and intersections all have straightforward implementations in
digital logic.
The second historical line came from the field of rank-order-based filters.
These are inherently grayscale in nature and have at their core the ordering of the
variables within an input window into their rank order. Trivial examples are the
maximum and minimum but the success story of these filters is the median. It pos-
sesses powerful noise-removal properties and requires no knowledge of signal and
noise distributions. It can be shown to be the optimum estimator of samples in unbi-
ased noise for an MAE criteria.
The final strand of nonlinear filtering is stack filters which are based on
Boolean logic operations applied within a finite window. They process grayscale
signals by thresholding them at a number of levels and filtering the resultant stack
of binary signals with a logic function.
The three types of filters have the following relationship:
Order-statistic filters
⊂
stack filters
⊂
morphological filters
In other words, morphological filters are the most general of the three, stack filters
are a subset within morphological filters, and order-statistic filters are a subset
within stack filters.
The literature describing the above methods tends to be quite academic and
mathematical. It is the purpose of this topic to bring these methods together and ex-
plain them in terms of logical operations. The objective is to bring these techniques
to a whole new community, namely electronic engineers and computer scientists.
The text assumes a basic knowledge of logic minimization such as could be
achieved through simple K maps. It also uses very basic statistics to identify the op-
timum filters in the examples given.
The remainder of the topic is structured as follows:
Chapter 2 introduces the concept of logic-based image processing through a
document restoration example. Chapter 3 considers methods of evaluating the er-
rors in filtering and gives more examples of document processing including resolu-
tion changing, edge noise, and optical character recognition. Chapter 4 looks at
filter training and the trade-off between the different types of errors. Chapter 5 de-
velops the relationship between logic-based image processing and mathematical
morphology and introduces increasing filters. Chapter 6 establishes the link be-
tween logic-based image processing and certain classes of order-statistic filters in-
volving variations on the median. Chapter 7 extends these concepts to grayscale
through the model of computational morphology. Chapter 8 describes how each of
