5.1 Incorporating Wavelet Shrinkage into Contrast Enhancement

The method of denoising by wavelet shrinkage can be incorporated trivially into a nonlinear enhancement framework by simply adding an extra segment to the enhancement function E(x), defined earlier in Eq. (13):

K(x — Tn), if Tn < x < Te x +(K — l)Te — KTn, if x > Te.

However, there are two arguments that favor shrinking gradient coefficients instead of Laplacian coefficients.

First, gradient coefficients exhibit a higher signal-to-noise ratio (SNR). For any shrinkage scheme to be effective, it is essential that the magnitude of the signal's components be larger than that of existing noise. It is thus sensible to define the SNR as the maximum magnitude of a signal over the maximum magnitude of noise. For example, consider a soft edge model

Some nonlinear enhancement methods [11] do not take into account the presence of noise. In general, noise exists in both digitized and digital images, due to the detector device and quantization. As a result of nonlinear processing, noise may be amplified and may diminish the benefits of enhancement.

Denoising an image is a difficult problem for two reasons. Fundamentally, there is no absolute boundary to distinguish a feature from noise. Even if there are known characteristics of a certain type of noise, it may be theoretically impossible to completely separate the noise from features of interest. Therefore, most denoising methods may be seen as ways to suppress very high-frequency and incoherent components of an input signal.

A naive method of denoising that is equivalent to low-pass filtering is naturally included in any dyadic wavelet framework. That is, simply discard channels at the highest resolution, and enhance coefficients confined to lower space-frequency channels. The problem associated with this linear denoising approach is that edges are blurred significantly. This flaw makes linear denoising mostly unsuitable for contrast enhancement in medical images. In order to achieve edge-preserved denoising, more sophisticated methods based on wavelet analysis were proposed in the literature. Mallat and Hwang [22] connected noise behavior to singularities. Their algorithm relied on a multiscale edge representation. The algorithm traced modulus wavelet maxima to evaluate local Lipschitz exponents and deleted maxima points with negative Lipschitz exponents. Donoho [23] proposed nonlinear wavelet shrinkage. This algorithm reduced wavelet coefficients toward zero based on a level-dependent threshold.

The first and second derivatives of f (n) are f'(x) =

with the magnitude of local extrema |f'(Xo)| = A|6|/3 and lf"(Xo)l = 2Af>2j3\[3, respectively. In this simple model, we can assume that noise is characterized by a relatively small A value and large ¿6 value. Clearly, gradient coefficients have a higher SNR than those of Laplacian coefficients since ¿6 contributes less to the magnitude of the function's output. Figures 7b and 7c show first and second derivatives, respectively, for an input signal (a) with two distinct edges.

In addition, boundary contrast is not affected by shrinking gradient coefficients. As shown in Fig. 7, coefficients aligned to the boundary of an edge are local extrema in the case of a first derivative (gradient), and zero crossings in the case of a second derivative (Laplacian). For a simple pointwise shrinking operator, there is no way to distinguish the points marked "B" from the points marked "A" (Fig. 7c). As a result, regions around each "A" and "B" point are diminished, while the discontinuity in "B" (Fig. 7d) sacrifices boundary contrast.

In the previous section, we argued that nonlinear enhancement is best performed on Laplacian coefficients. Therefore, in order to incorporate denoising into our enhancement algorithm, we split the Laplacian operator into two cascaded gradient operators. Note that

FIGURE 7 (a) Signal with two edges, (b) First derivative (gradient), (c) Second derivative (Laplacian). (d) Shrunken Second derivative.

& m(ffl) = -4[sin(2m = g m>1(ffl) g m>2(ffl), where

I m,i(ffl) = sin(f), g m,2(ffl) = ^2«sin(f), if m = 0,

g mj1(ffl) = g m,2(ffl) = 2isin(2m"1 ffl), otherwise.

Denoising by wavelet shrinkage [23] can then be incorporated into this computational structure as illustrated in Fig. 8, where the shrinking operator can be written as

Note that the shrinking operator is a piecewise linear and monotonically nondecreasing function. Thus, in practice, the shrinking operator will not introduce artifacts.

The threshold Tn is a critical parameter in the shrinking operation. For a white noise model and orthogonal wavelet, Donoho [23] suggested the formula Tn = \J2log(N)a/y/N, where N is the length of an input signal and a is the standard deviation of the wavelet coefficients. However, the dyadic wavelet we used is not an orthogonal wavelet. Moreover, in our 2D applications, a shrinking operation is applied to the magnitudes of the gradient coefficients instead of the actual wavelet coefficients. Therefore, the method of threshold estimation proposed by Voorhees and Poggio [24] for edge detection may be more suitable.

In our "shrinking" operation, only the magnitudes of the gradient of a Gaussian low-passed signal are modified. As pointed out by Voorhees et al. [24], for white Gaussian noise, the probability distribution function of the magnitudes of gradient is characterized by the Rayleigh distribution:

FIGURE 8 (a) Noisy input signal (contaminated by white Gaussian noise). (b) Nonlinear enhancement without denoising, Gm = 10, N = 4, t = 0.1. (c) Nonlinear enhancement of levels 2-3, Gm = 10, t = 0.1; levels 0-1 set to zero. (d) Nonlinear enhancement with adaptive wavelet shrinkage denoising, Gm = 10, N = 4, t = 0.1.

FIGURE 8 (a) Noisy input signal (contaminated by white Gaussian noise). (b) Nonlinear enhancement without denoising, Gm = 10, N = 4, t = 0.1. (c) Nonlinear enhancement of levels 2-3, Gm = 10, t = 0.1; levels 0-1 set to zero. (d) Nonlinear enhancement with adaptive wavelet shrinkage denoising, Gm = 10, N = 4, t = 0.1.

To estimate a histogram (probability) of ||A/1| was computed, and then iterative curve fitting was applied. Under this model, the probability p of noise removal for a particular threshold x can be calculated by

Figure 8 compares the performance of different approaches. In (b), we observed that enhancement without any denoising results in distracting background noise. In (c), edges are smeared and broadened by low-pass enhancement. Only in (d), with wavelet shrinkage enabled, we are able to achieve the remarkable result of denoising and contrast enhancement simultaneously.

To demonstrate the denoising process, Figs 9a and 9b show both nonlinear enhancement of wavelet coefficients without and with denoising, respectively, for the original input signal

shown in Fig. 9a. Figure 9c shows the associated curve-fitting for threshold estimation.

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