Fuzzy Methods in Nonlinear Signal Processing: Part II—Applications - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 76

FIGURE 3.4
Original Lake Charles DEM data set. (From M. Asghar and K.E. Barner, IEEE Trans.

Visualization Comput. Graphics ,7,76-93, Mar. 2001. c

( 150 × 150 )

2001 IEEE. With permission.)

preserved in the decomposition while simultaneously smoothing fine details,
as the figures show.
In determining the behavior of a decomposition method, it is instructive to
examine the detail signals, X i h , X i , and X i d . Utilizing the multiresolution struc-

ture in Figure 3.3, the level-one detail signals of the Lake Charles decompo-
sition are shown in Figure 3.7. The figure shows that the detail signals in the
linear decomposition contain significant power due to the edge smoothing
of the wavelet transform. In contrast, the edge-preserving properties of the
median filter are evident in the identically zero detail coefficients along most
edges in the median decomposition. The detail signals show that the median
affine decomposition has better edge preservation character than the linear
decomposition, while also smoothing details in a more desirable fashion than
the median.
As a final example, consider the case in which the observed data set is cor-
rupted by independent, additive noise. Specifically, let 10% of the samples in
the Lake Charles data set be corrupted by zero mean, unit variance Gaussian
noise. The additive noise can either be attributed to the sensing, transmission,
and storage of the data, or can be interpreted as small-scale details. In either
case, the noise terms should be significantly attenuated in the decomposition
approximations. Figure 3.8 shows a realization of the corrupted Lake Charles
data set and the resulting level-one approximations for the wavelet, median,
and scale invariant median affine decompositions. As the figure shows, the
wavelet decomposition spreads the noise terms across neighboring samples

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