and therefore

filters /m(ffl) and ¿m(ffl) completely cover the frequency domain:

As an example, let as consider an extension of the class of filters proposed by Mallat and Zhong [20] and precisely show how these filters can cover the space-frequency plane to provide a complete analysis:

and sin (f) is the frequency response of the discrete Laplacian operator of impulse response {1, —2,1}.

0 m,q(®) with even exponential q is an approximate Gaussian function, while the frequency responses of the channels 0 < m < N are approximately Laplacian of Gaussian. Figure 3 shows the frequency response of each distinct channel while Fig. 4 compares 0 24(ffl) and 0 2g(ffl) with related Gaussians.

2.2 Two Possible Filters

In this framework, the possible choices of filters are constrained by Eq. (1). For the class of filters defined by Eq. (2), we can derive

Under the constraint that both g (ffl) and k (ffl) are finite impulse response (FIR) filters, there are two possible design choices which are distinguished by the order of zeros in their frequency responses.

1. Laplacian filter. In this case, g (ffl) = —4 sin (2) or g(l) = {1, —2, 1}, which defines a discrete Laplacian operator, such that (g * s)(Z) = s(Z + 1) — 2s(Z)

+s(Z— 1). Accordingly, we can chose both filters g(ffl)

0 m,4« + 2p(f) — 0 m + 1,4« + 2p(f) 0 < m <(N — 1)

Both forward and inverse filters, 0 < m < N — 1, can be derived by

A A 1 2" — 1 i i m(2) = — 0 m,2«M 4 [cos(2m — 2 = ? mM-4 1 = 0

FIGURE 3 Channel frequency responses for N = 6, n = 1, and (a) p = 0 and (b) p = 1.

FIGURE 3 Channel frequency responses for N = 6, n = 1, and (a) p = 0 and (b) p = 1.

Note that the forward filters fm (rn), 0 < m < N — 1, can be interpreted as two cascaded operations, a Gaussian averaging of — 0 m2n + 2(a>) and the a Laplacian —4sin2(y), while the set of inverse filters i m (rn) are low-pass filters. For an input signal s(l), the wavelet coefficients at the points "E" (as shown in Figs 1 and 2) may be written as

Wm(ffl) = A(s * Xm)(l), where A is the discrete Laplacian operator, and Àm(l) is approximately a Gaussian filter. This means that each wavelet coefficient wm(l) is dependent on the local contrast of the original signal at each position l.

2. Gradient filter. In this case, g (m) = 2ie^t~ sin(y), or g(0) = 1, and g(1) = -1, such that (g * s) '(l) = s(l) — s(l — 1). Thus, we select the filters

FIGURE 4 (a) 024(m) compared with the Gaussian function e 2 8m2. (b) 026(m) compared with the Gaussian function e

We then derive the forward filters

L(œ) = g (œ)2m 6 m>2„ + 2(ffl) = g (œ) 1 m(rn)

and inverse filters

1 m(œ) = -etœ g (œ) y m(œ), where y m(œ) = 2m 6 m,2„ + 2 (œ) l-jr [cos(2m - 'œ)]21 4 l = o is a low-pass filter.

In this case, the associated wavelet coefficients may be written as

This formula shows that the discrete Laplacian operator can be implemented by substracting from the value of a central point its average neighborhood. Thus, an extended formula [8] can be written as

where h(i, j) is a discrete averaging filter, and * denotes convolution. Loo, Doi, and Metz [8] used an equal-weighted averaging mask:

0, otherwise.

Another way to extend the prototype formula [9] came from the idea of a Laplacian-of-Gaussian filter, which expands Eq. (7) into su(x, y) = s(x, y) - kA(s*g)(x, y) = s(x, y) - k(s*Ag)(x, y)

where g(x, y) is a Gaussian function, and Ag(x, y) is a where V is a discrete gradient operator characterized by Laplacian-of-Gaussian filter.

We mention for future reference that both extensions shown in Eqs. (8) and (9) are limited to a single scale.

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