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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