Nonlinear Techniques for Color Image Processing - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 472

(a) (b)

FIGURE 12.6
In the digital path approach to first and last (DPAF and DPAL) filters, the weights are assigned
to the pixels surrounding the central pixel and are determined in different ways. In the DPAF
approach (a), the weights in Equation 12.74 are calculated exploring all digital paths starting
from the central pixel and crossing its nearest neighbors; then a weighted average of the nearest
neighbors of the central pixel is calculated (Equation 12.75). In the DPAL approach, the weights
are obtained by exploring all digital paths leading from the central pixel to the pixels contained in
the filtering window (b) and then a weighted average of all pixels from that window is calculated
(Equation 12.81).

The new filter class based on digital paths and connection cost can be seen as
a powerful generalization of the multichannel AD presented in Section 12.3,
and as an extension of the fuzzy adaptive filters described in Section 12.4.2.
The filters discussed there are shown in this section to be a special case of the
new filtering scheme, when a digital path is degenerated to a step of length 1.
The path connection costs evaluated over all possible digital paths are used
to derive fuzzy membership functions that quantify the similarity between
vectorial inputs. The proposed filtering structure then uses the function out-
puts to appropriately weight input contributions in order to determine the
filtering result. The proposed filtering schemes parallelize the structure of the
adaptive multichannel filter introduced in Reference 115, and they can suc-
cessfully eliminate Gaussian, impulsive, as well as mixed-type noise. How-
ever, thanks to the introduction of the digital paths in their supporting ele-
ment, the new filters not only preserve edges and fine image details, but also
can act as image-sharpening operators.

12.5.1 Connection Cost Defined over Digital Paths

To perform operations based on the distances, we first need to precisely define
the notion of a topological distance. The concept of a topological distance
between image points is of extreme importance in many applications based