International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004
Some proposed dimension reduction methods are based on
stochastic theory such as the Principal Component Analysis
(PCA), Discriminant Analysis Feature Extraction (DAFE) and
Decision Boundary Feature Extraction (DBFE). These
techniques are not so effective for dimension reduction of
hyperspectral data for example DAFE and DBFE need to very
large number of training samples for estimating the statistical
properties of the hyperspectral data in the original feature space.
PCA is effective at compression information in multivariate
data sets by computing orthogonal projections that maximize
the amount of data variance. It is typically performed through
the egin-decompositon of the spectral covariance matrix of an
image cube. The information can then be presented in the form
of component images, which are projections of the image cube
on to the eigenvectors, the component images corresponding to
the large eigenvalues are presumed to preserve the majority of
the information about the scene. Unfortunately information
content in hyperspectral images dose not always coincide with
such projections (Chang, 2000). This rotational transform is
also time-consuming because of its global nature (Kaewpijit ef
al, 2003). Finally, since it is a global transformation, it does not
preserve local spectral signatures and therefore might not
preserve all information useful to obtain a good classification.
For these reasons, some authors have proposed a dimension
reduction method based on wavelet decomposition.
This paper attempts to transform the spectral data from the
original feature space to a reduced feature space by using a
discrete wavelet transform. The principle of this method is to
apply a discrete wavelet transform to hyperspectral data in the
spectral domain and at each pixel location. This does not only
reduce the data, volume but it also can preserve the
characteristics of the spectral of signature. This is due to
intrinsic property of wavelet transforms of preserving of high
and low frequency during the signal decomposition, therefore
preserving peaks and valleys found in typical spectra. In
addition, some of sub bands especially the low pass filter, can
eliminate anomalies found in one of the bands.
Our experimental results for representative sets of
hyperspectral data have confirmed that the wavelet spectral
reduction as compare to PCA provides better or comparable
classification accuracy while can reduce the computational
requirement.
This paper is organized as follows. Section 2 provides an
overview of the automatic multiresoluton wavelet analysis for
dimension reduction of hyperspectral data. Section 3 discusses
the automatic selection of level of decomposition. Section 4
presents results for the automatic wavelet reduction. This is
accomplished by investigating the impact of the wavelet
reduction on classification accuracies for different conventional
classification methods and Section 5 provides our concluding
remark for this work.
2. AUTOMATIC MULTIRESOLUTION WAVELET
ANALYSIS
Wavelet transforms are the basis of many powerful tools that
are now being used in remote sensing applications, e.g.,
compression, registration, fusion, and classification (Kaewpijit
et al, 2003).Wavelet transform can provide a domain in which
both time and scale information can be studied simultaneously
62
giving a time-scale representation of the signal under
observation. A wavelet transform can be obtained by projection
the signal onto shifted and scaled version of a basic function,
This function is known as the mother wavelet, Vt )- A
“mother wavelet” must satisfy this condition (Mathur, 2002).
s (sy?
su ds < eo (1)
This condition implies that the wavelet has a zero average
[wGodx-0 Q)
And the shifted and scaled version of the mother wavelet forms
a basis of functions. These basis functions can be represented
as
1 t=b
V s) mU (3)
Joe. lw
where a represents the scaling factor and b the translation factor.
Wavelet transforms may be either discrete or continuous. In
this paper only Discrete Wavelet Transform (DWT) is
considered. For dyadic DWT the scale variables are power of 2
and the shift variables are none overlapping and discrete.
One property that most wavelet systems satisfy is the
multiresolution analysis (MRA) property. In this paper Mallat
(1989) algorithm is utilized to compute these transforms.
Following the Mallat algorithm, two filters [the lowpass filter
(L) and its corresponding highpass filter (H)] are applied to the
signal, followed by dyadic decimation removing every other
elements of the signal, thereby halving its overall length. This
is done recursively by reapplying the same procedure to the
result of the filter subbands to be an increasingly smoother
version of the original vector as shown in figure2. In this paper,
such 1-D discrete Wavelet transform will be used for reducing
hyperspectral data in the spectral domain for each pixel
individually. This transform will decompose the hyperspectral
of each pixel into a set of composite bands that are linear,
weighted combination of the original spectral bands. In order to
control the smoothness one of the simplest and most localized
Daubechies filter, called DAVB4 has been used. This filter has
only four coefficients (Kaewpijit e/ al, 2003).
High pass filter
\
Low pass Alter €
Bown sampling
ity a factor of2
fii Input hyperspectral signature
à; 7 Apprisimation cocfficients at level à
it, = Deralf coefficients ut level /
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