Fuzzy Methods in Nonlinear Signal Processing: Part II—Applications - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 94

(a) (b)

FIGURE 3.18
Tw o 512 × 512 8-bit/pixel gray scale images: (a) Lenna and (b) Albert.

performance. This similarity in performance is a result of the optimization
that yields a membership spread parameter of 01, which is very small

compared to the range of signal values, [0, 255]. Thus, in the impulsive noise
case, the optimal weighted median filter utilizes crisp order statistics as the
output. This is an intuitive result because in the impulsive case, samples are
either unaltered or bear no information. Thus, if the filter is centered on a
sample that is not corrupted, it is best to simply perform the identity opera-
tion and weighted averaging has no advantage. On the other hand, if the filter
is centered on a corrupted sample, then this sample contains no information
about the original signal, and the best output selection is a crisp order statis-
tics that can preserve local structure. Thus, there is no advantage to averaging
samples and the optimization procedure correctly yields a crisp filter. Similar
results are obtained in the median case.
Consider next the performance of the filters operating in a contaminated
Gaussian noise environment. In this case, the observation image is corrupted
by

λ = 0 .

indicates mixing pro-
portion of two Gaussian processes with variance 10 and 100. The MAE perfor-
mance of each of the filters operating in the contaminated Gaussian environ-
ment is shown in Figure 3.19b as a function of the contamination parameter,
for

( 10 , 100 , ) contaminated Gaussian noise, where

in the range 0 to 0.45. In this case there is a clear separation in the per-
formance of the crisp and fuzzy filters. Selected weighted averaging among
similarly valued samples is advantageous in this case because there is always
background Gaussian noise, regardless of the mixture parameter value. The
figure shows a fairly consistent performance advantage for the fuzzy filters
across the entire range of contamination. The performance gain is primarily
due to two factors: the fuzzy versions of the filters (1) smooth the background