Image Processing Reference
In-Depth Information
It is possible to generalize the filter such that all locations in the filter window
may be allocated individual weights. It should be remembered that each generaliza-
tion will bring an associated increase in the amount of training data required. The
details of these methods are beyond the scope of this text but further details may be
found in Marshall.
7
Before closing the chapter, it is worth saying a little more about the value of
differencing filters in image processing. Theoretically, the direct and differencing
representation of a filter are equivalent. They give an identical result in much the
same way as a sum of products and product of sums are equivalent. However, in
practice the differencing filter can possess certain advantages.
In image restoration problems, for example, it is typically the case that only
10-20% of the pixels are corrupted and therefore require correction. This means
80-90% of the pixels should remain unchanged by filtering. The differencing filter is
therefore a relatively inactive filter—it identifies a small percentage of patterns and
corrects them. This means that hardware implementations of differencing filters for
these type of problems can require much fewer resources than direct implementations.
It also has advantages when extended to practical filters designed by training.
For image patterns where the number of training examples observed is zero or too
low to be statistically significant, the differencing filter can simply give a value of 0
and leave the pixel unchanged. For further discussion of differencing filters, see
Dougherty and Lotufo.
8
6.8 Summary
This chapter has described two variations on the median filter. They both attempt to
make it more flexible either for the rejection of noise or the preservation of image
detail. They involve allowing the filter weights and the threshold parameter to vary.
Design methods have been presented for these filters based on the weight-
monotonic property. The differencing filter has been introduced to ensure that the
weighted median filters are self-dual.
Optimum design of both the weighted order statistic (WOS) filters and the
weighted median filters (WMF) are not restricted to binary images and may be ex-
tended to grayscale
9
processing via the threshold decomposition theorem. This is
explained in the next chapter.
References
1
M. Gabbouj, E. Coyle, and N. Gallagher, Jr., “An overview of median and
stack filtering,”
Circuits, Systems, and Signal Processing
,
11
(1), 7-45 (1992).
2
P. Maragos and R. W. Schafer, “Morphological filters—Part I: Their relations to
medians, order statistics, and stack filters,”
IEEE Trans. Acoustics, Speech, and
Signal Processing
,
35
, 1153-1169 (1987).
