Magnitude of wavelet coefficients measures the correlation between the image data and the wavelet functions. For first-derivative-based wavelet, the magnitude

Figure 6.10: Example of thresholding functions, assuming that the input data was normalized to the range of [-1, 1]. (a) Hard thresholding, (b) soft thresholding, and (c) affine thresholding. The threshold level was set to T = 0.5.

Figure 6.10: Example of thresholding functions, assuming that the input data was normalized to the range of [-1, 1]. (a) Hard thresholding, (b) soft thresholding, and (c) affine thresholding. The threshold level was set to T = 0.5.

therefore reflects the "strength" of signal variation. For second-derivative-based wavelets, the magnitude is related to the local contrast around a signal variation. In both cases, large wavelet coefficient magnitude occurs around strong edges. To enhance weak edges or subtle objects buried in the background, an enhancement function should be designed such that wavelet coefficients within certain magnitude range are amplified.

General guidelines for designing a nonlinear enhancement function E(x) are [35]:

1. An area of low contrast should be enhanced more than an area of high contrast. This is equivalent to saying that smaller values of wavelet coefficients should be assigned larger gains.

2. A sharp edge should not be blurred.

In addition, an enhancement function may be further subjected to the following constraints [36]:

1. Monotonically increasing: Monoticity ensures the preservation of the relative strength of signal variations and avoids changing location of local extrema or creating new extrema.

2. Antisymmetry: (E(-x) = -E(x)): This property preserves the phase polarity for "edge crispening."

A simple piecewise linear function [37] that satisfies these conditions is plotted in Fig. 6.11(a):

Figure 6.11: Example of enhancement functions, assuming that the input data was normalized to the range of [-1, 1]. (a) Piecewise linear function, T = 0.2, K = 20. (b) Sigmoid enhancement function, b = 0.35, c = 20. Notice the different scales of the y-axis for the two plots.

Such enhancement is simple to implement, and was used successfully for contrast enhancement on mammograms [19, 38, 39].

From the analysis in the previous subsection, wavelet coefficients with small-magnitude were also related to noise. A simple amplification of small-magnitude coefficients as performed in Eq. (6.39) will certainly also amplify noise components. This enhancement operator is therefore limited to contrast enhancement of data with very low noise level, such as mammograms or CT images. Such a problem can be alleviated by combining the enhancement with a denoising operator presented in the previous subsection [35].

A more careful design can provide more reliable enhancement procedures with a control of noise suppression. For example, a sigmoid function [37], plotted in Fig. 6.11 (b), can be used:

E(x) = a[sigm(c(x — b)) — sigm(—c(x + b))], (6.40)

where a =-, 0 < b < 1, sigm(c(1 — b)) — sigm(—c(1 + b))

andsigm( y) is defined as sigm( y) =-1—.The parameters b and c respectively

1 + e—y control the threshold and rate of enhancement. It can be easily shown that E(x) in Eq. (6.40) is continuous and monotonically increasing within the interval

[ — 1 , 1]. Furthermore, any order of derivatives of E(x) exists and is continuous. This property avoids creating any new discontinuities after enhancement.

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