Multiscale Analysis of Hyperspectral Data
Using Wavelets for Spectral Feature Extraction
3. Wavelet-Based Feature Extraction
3.1 Wavelet Transform
The continuous wavelet transform (CWT) which decomposes signals over dilated and translated wavelets was first introduced by Grossmann and Morlet (1984). The wavelet transform of a function
¦Î L
2(
R) is defined by
where
yu,s(x) is obtained by introducing a scale factor s and a translation factor u to the mother wavelet function
y(x) :
The wavelet transform
W¦(u,s) is a function of the scale and the spatial position x. It measures the variation in
¦ in the neighborhood of u, whose size is proportional to s. When the scale s varies from its maximum to zero, the decay of the wavelet coefficients characterizes the regularity of
¦ in the neighborhood of u. This is the essential idea in detecting the absorption band position from the reflectance spectra.
3.2 Modulus Maxima of Wavelet Transform
A wavelet
y with n vanishing moments can be written as the n
th order derivative of a function
q , that is
y=(-1)
n q(n) , thus the resulting wavelet transform is a multiscale differential operator:
Suppose the convolution
averages f(x) over a domain proportional to s. Let
y1=-
q' and
y2=
q'' be two wavelets, thus the wavelet transforms,
W1¦(u,s) and
W2¦(u,s), are respective to the first and second derivative of
¦(x) smoothed by
Figure 1. The positions of the local maxima of and the zero-crossing of
. For a fixed scale, the local maxima of
W1¦(u,s) and the zero-crossings of
W¦(u,s) will correspond to the inflection points of
(figure 1). For all scales, the local maxima points of
W1¦(u,s) can be connected as a set of maximal lines in the scale-space plane (u, s). Similarly, the zero-crossings of
W2¦(u,s) define a set of smooth curves that often look like fingerprints. By detecting the positions of the local maxima or zero-crossings from a coarse to fine scale, we can obtain the positions of the singularities of a signal. These two methods are very similar, but the local maxima approach has several important advantages (Mallat and Hwang, 1992). The smoothing function
q can be viewed as the impulse response of a low-pass filter. An important example often used in signal processing is the Gaussian function. In this study, the wavelet is defined as the first derivative of the Gaussian function.
3.3 The Estimation of Lipschitz Regularity
To characterize the singular structure of a signal, it is necessary to quantify local regularities precisely. Lipschitz exponents provide not only uniform regularity measurements over time intervals, but also pointwise Lipschitz regularity at any point v of a signal. The relationship between the decay of the wavelet transform amplitude across scales and the pointwise Lipschitz regularity of the signal was described by Jaffard (1991). He proved a necessary and sufficient condition in the wavelet transform for estimating the Lipschitz regularity of
¦ at a point v. Assume that the wavelet
y has n vanishing moments and n derivatives with a fast decay. If
¦Î L
2(
R) is Lipschitz
a£n at v, then there exists A>0 such that
"(u,s)
Î Rx
R+ ,
In order to simplify the above condition, we assume that
y has a compact support equal to [-C,C] . The cone of influence of v in the scale-space plane is the set of points (u, s) such that v is included in the support of
(Mallat, 1997). Since the support of
is equal to [u-Cs,u+Cs], the cone of influence of v is defined by
½u-v
½£Cs . This is illustrated in Figure 2. Since
½u-v
½£Cs, the conditions (4) and (5) can be written as
which is equivalent to the uniform Lipschitz condition given by Mallat (1997). In this study, we assumed that all modulus maxima converging to v are included in a cone. The potential singularity at v is isolated. Function
¦ is uniformly Lipschitz
a in the neighborhood of v if and only if there exist A>0 such that each modulus maximum (u, s) in the cone satisfies (6). In order to estimate the Lipschitz exponent we rewrite (5) as
.The Lipschitz regularity at v is thus the maximum slope of log
2½W¦(u,s)
½ as a function of log
2s along the maxima lines converging to v.
4. Experiments
4.1 Test data
The test data are a set of hyperspectral data delivered from AVIRIS. The data has 220 spectral bands from 0.4um to 2.5um with about 10 nm spectral resolution. The spectral curves of three different materials are shown in Figure3.
Figure 3. Three kinds of hyperspectral curves from AVIRIS images.
(a) grass (b) soybean (c) corn
Figure 4. W¦(u,s), wavelet transform of the spectrums,
(a) grass (b) soybean (c) corn
Figure 5. The modulus maxima of W¦(u,s)
(a) grass (b) soybean (c) corn
Figure 6. 3D view ofW¦(u,s)
(a) grass (b) soybean (c) corn
Figure 7. The spectral features extracted by modulus maxima of
W¦(u,s)
