58 GONZALO R. ARCE AND JOSE L. PAREDES
Figure 2.16: (a) Image with background noise sharpened with (b) the LUM sharpener, (c)
the FIR sharpener, (d) the WM sharpener, and the permutation WM sharpener with (e) L = 1
and (f) L = 2.
2.6 Optimal Frequency Selection WM Filtering
We now consider the design of a robust band-pass recursive WM filter using the
LMA adaptive optimization algorithm. The performance of the optimal recursive
WM filter is compared with the performances of a linear FIR filter, a linear IIR filter,
and a nonrecursive WM filter all designed for the same task. Moreover, to show the
noise attenuation capability of the recursive WM filter and compare it with those
of the other filters, we used an impulse-noise-corrupted test signal. Examples are
shown in one-dimensional signals for illustration purposes but the extension to
two-dimensional signals is straightforward.
CHAPTER 2: IMAGE ENHANCEMENT AND ANALYSIS 59
The application at hand is the design of a 62-tap band-pass RWM filter with
passband 0.075 < r < 0.125 (normalized Nyquist frequency = 1). We used white
Gaussian noise with zero mean and variance equal to one as input training sig-
nals. The desired signal was provided by the output of a large FIR filter (122-tap
linear FIR filter) designed by MATLAB'S M-file fir| function. The 31 feedback filter
coefficients were initialized to small random numbers (on the order of 10-3). The
feed-forward filter coefficients were initialized to the values output by MATLAB'S
fir1 with 31 taps and the same passband of interest. A variable step size/~(n) was
used in both adaptive optimizations, where the step size/~(n) changes according
to/./0e -n/100 with/~0 = 10 -2.
A signal that spanned the range of frequencies of interest was used as a test
signal. Figure 2.17a depicts a linear swept-frequency signal spanning instanta-
neous frequencies from 0 to 400 Hz, with a sampling rate of 2 kHz. Figure 2.17b
shows the chirp signal filtered by the 122-tap linear FIR filter used to produce the
desired signal during the training stage. Figure 2.17c shows the output of a 62-tap
linear FIR filter used for comparison purposes.
The adaptive optimization algorithm described in Section 2.2 was used to op-
timize a 62-tap nonrecursive WM filter admitting negative weights; the filtered
signal attained is shown in Fig. 2.17d. Note that the nonrecursive WM filter tracks
the frequencies of interest but fails to attenuate completely the frequencies out of
the desired passband. MATLAB'S yulewalk function was used to design a 62-tap
linear IIR filter with passband 0.075 < co < 0.125; Fig. 2.17e depicts its output.
Finally, Fig. 2.17f shows the output of the optimal recursive WM filter determined
by the LMA training algorithm described in Sec. 2.2.2. Note that the frequency
components of the test signal that are not in the passband are attenuated com-
pletely. Moreover, the RWM filter generalizes very well on signals that were not
used during the training stage.
Comparing the different filtered signals in Fig. 2.17, we see that the recur-
sive filtering operation performs much better than its nonrecursive counterpart
having the same number of coefficients. Likewise, to achieve a specified level
of performance, a recursive WM filter generally requires considerably fewer filter
coefficients than the corresponding nonrecursive WM filter.
To test the robustness of the different filters, we next contaminated the test
signal with additive a-stable noise (Fig. 2.18a); the impulse noise was generated
using the parameter a set to 1.4. (Fig. 2.18a is truncated so that the same scale
is used in all plots.) Figures 2.18b and 2.18d show the filter outputs of the linear
FIR and IIR filters, respectively; both outputs are severely affected by the noise.
On the other hand, the nonrecursive and recursive WM filters' outputs, Fig. 2.18c
and 2.18e, remain practically unaltered. Figure 2.18 clearly depicts the robust
characteristics of median-based filters.
To better evaluate the frequency response of the various filters, we performed
a frequency domain analysis. Due to the nonlinearity inherent in the median
operation, traditional linear tools, such as transfer-function-based analysis, cannot
be applied. However, if the nonlinear filters are treated as a single-input, single-

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