Image Processing Reference
In-Depth Information
involves a constraint that is achieved by restricting the possible logic functions that
may be used.
The techniques described may be extended to grayscale images through a num-
ber of approaches. The issue of training and estimation error is further compounded
by this extension to grayscale since each pixel would have at least 8 bits. The most
straightforward way to extend these techniques to grayscale is by using threshold
decomposition in a technique known as stack filtering. The input signal is split into
a “stack” of binary signals, each of which may be processed by a binary filter. This
binary filter may be estimated from a training set. The output from the binary filters
is then restacked to produce a grayscale output signal. Many useful operators in-
cluding rank-order, median, and some linear filters fall into the class of stack filters,
as well as grayscale morphology with flat structuring elements.
More complex grayscale operations may be implemented in a framework
known as computational morphology. This is inherently suited to digital imple-
mentation in a fixed number of bits. Its structure is similar to the stack filter because
it produces a series of stacked binary outputs. However, the filtering operation is
more complex and is based on a technique known as elemental erosion. Computa-
tional morphology is capable of implementing any operation, linear or nonlinear
within the window size chosen. The result of designing a filter in this framework is
a kernel of structuring elements. In the most general case, these are unrelated, other
than the fact that they must observe an ordering to avoid violating the stacking
property. For grayscale morphology and stack filters, the kernels are related in a
simple way.
The design of filters based on computational morphology is difficult because
they are so general in nature and their search space is very large, requiring unrealis-
tic amounts of training data.
A simplification of computational morphology is known as the aperture filter.
This is based on a windowing process in the amplitude domain of the signal, as well
as the spatial domain. Signal points lying outside of the window are simply clipped.
As a result of their reduced dynamic range, aperture filters may be designed with
training sets of realistic size. They have been successfully used in many applica-
tions including deblurring and object recognition. Aperture filters have been ex-
tended to multiscale applications. The most difficult problem remains that of
aperture placement.
A useful technique that reduces gross errors in nonlinear filtering is called en-
velope filtering. In this case, the output of a filter is forced to be contained between
and upper and lower limits of a bounding envelope. This has the advantage that the
error is never larger than the envelope's range.
Designing image processing operators in terms of digital logic results in solu-
tions that can be transferred straight to hardware without further mapping or trunca-
tion. Frequently the implementations resulting directly from the design methods
described here are cumbersome. For example, in the case of stack filters it is expen-
sive to duplicate the processing hardware for each of the 256 levels present in an
8-bit grayscale image. Fortunately, other techniques may be used to reduce the
