For linear enhancement, the selection of filters g m(ffl) (and thus k m(ffl)) makes little difference. However, this is not true for the nonlinear case. For the nonlinear approach described later, we show that a Laplacian filter should be favored. By selecting a Laplacian filter, we can be assured that positions of extrema will be unchanged and that no new extrema will be created within each channel. This is possible for the following reasons:

1. Laplacian filters are zero-phase. No spatial shifting in the transform space will occur as a result of frequency response.

2. A monotonically increasing function E(x) will not produce new extrema. (At some point E\f (xo)] is an extremum if and only if f (xo) was an extreme singularity.)

4.3 A Nonlinear Enhancement Function

Designing a nonlinear enhancement scheme is difficult for two reasons: (1) the problem of defining a criterion of optimality for contrast enhancement, and (2) the complexity of analyzing nonlinear systems. We adopted the following guidelines in designing nonlinear enhancement functions:

1. An area of low contrast should be enhanced more than an area of high contrast. This is equivalent to saying that small values of wavelet coefficients w at some level m, denoted by wm, should have larger gains.

2. A sharp edge should not be blurred.

Experimentally, we found the following simple function to be advantageous:

x — (K — 1)T, if x< - T E(x) = Kx, if |x| < T < x + S(x) (13)

where K > 1. This expression for enhancement may also be reformulated as a simple change in value (Eq. (13) right) where

— (K — 1)T, if x < — T S(x)= (K — 1)x, if |x| < T. (K — 1)T, if x>T

At each level m, the enhancement operator 5m has two free parameters: threshold Tm and gain Km. In our experimental studies, Km = Ko, 0 < m < N — 1, and Tm = t ^ max{wm\n]}, where 0<T < 1 was user specified. For t = 1.0, the wavelet coefficients at levels 0 < m < N — 1 were multiplied by a gain of K0, shown previously to be mathematically equivalent to unsharp masking. Thus this nonlinear algorithm includes unsharp masking as a subset. Figure 6 shows a numerical example, comparing linear and nonlinear enhancement. Note the lack of enhancement for the leftmost edge in the case of the linear operator.

Specifically, an enhanced signal Se(l) can be written as

0 0

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