Weighted Myriad Filters - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 175

large values), the myriad structure assures that they are deemphasized due to
the outlier rejection property of K . The center weight W c is chosen balancing
between outlier rejection and detail preservation. It should be large enough to
emphasize the center sample and preserve signal details, but not large enough
to let through impulsive noise.
It can also be shown that the CWMy smoother with K and W c defined as
above has the capability of completely rejecting “pepper” type noise (having
values close to 0). 29 This can be seen as follows. For a single “pepper” outlier
sample, the cost function (Equation 5.52) evaluated at β = K will always

be smaller than that at 0. This means that pepper noise will never go

through the smoother as the output given that the parameters K and W c are
correctly chosen. Denote X as the corrupted image, Y the output smoothed
image, and CWMy smoother operation. A special two-pass CWMy smoother
can be defined as follows:

β =

Y = 1 − CWMy ( 1 − CWMy ( X )). (5.53)

Figure 5.10 depicts results of the algorithm defined in Equation 5.53.
Figure 5.10a is a noise-free image with 256 gray levels. Pixel values are nor-
malized to be in [0, 1], with 0 representing the brightest and 1 the dark-
est intensity. Figure 5.10b is Figure 5.10a corrupted by 5% salt and pepper
noise. The impulses occur randomly and were generated by MATLAB's im-
noise function. Figure 5.10c is the output of a 5 × 5 CWM smoother with

W c = 15, and Figure 5.10d that of a CWMy smoother with W c = 10 , 000 and

= (

X ( 21 ) + X ( 5 ) )/

2. The superior performance of the CWMy smoother can
be readily seen in this figure. The CWMy smoother preserves the original
image features significantly better than the CWM smoother (notice the hair
details). The mean square error of the CWMy output is consistently less than
half of that of the CWM output for this particular image.

K

5.7.2 Myriadization

The linear property indicates that for very large values of K , the weighted
myriad smoother reduces to a constrained linear FIR smoother. The mean-
ing of K suggests that a linear smoother can be provided with resistance to
impulsive noise by simply reducing the linearity parameter from K

=∞
to a finite value. This would transform the linear smoother into a myriad
smoother with the same weights. In the same way as the term linearization is
commonly used to denote the transformation of an operator into a linear one,
the above transformation is referred to as myriadization .
Myriadization is a simple but powerful technique that brings impulse re-
sistance to constrained linear filters. It also provides a simple methodology
to design suboptimal myriad smoothers in impulsive environments. Basi-
cally, a constrained linear smoother can be designed for Gaussian or noiseless