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(1) 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004 
d pl Y gln-2pha[n]- a, * [2p] Q) 
N= 
where A[n]- h[-n] and gn] = g[-n]. a, is the approximation 
coefficients at scale 2/, and q i and d ,, are respectively the 
/ J 
approximation and detail components at scale 2/*'. There are 
some necessary and sufficient conditions associated with the 
conjugate mirror filters h and g, so that the perfect 
reconstruction of signal x can be achieved without losing 
information. Figure 2 shows the diagram of a fast wavelet 
decomposition calculated with a cascade of filtering with 4A and 
g followed by a factor 2 sub-sampling. Assume that the length 
of 4. is N, one may notice that the sub-sampling procedure in 
/ 
the wavelet decomposition shown in Figure 2 which reduces the 
length of 4, | to N/2 achieves the dimensionality reduction of 
1+ 
a, In practice, the original signal x in Figure 2 is always 
J 
expressed as a sequence of coefficients 0, A multilevel 
orthogonal wavelet decomposition of 4, is composed of 
wavelet coefficients of signal x at scales2^ « 2/ « 2^ plus the 
remaining approximation at the largest scale 2" : 
[(2 3, 1:0] (3) 
It is calculated from a, by iterating formula (1) and (2). 
9-4 
  
Figure 2. Fast orthogonal wavelet decomposition 
2.) Linear Wavelet Feature Extraction 
The sub-sampling shown in Figure 2 motives us to reduce the 
dimensionality of hyperspectral data by wavelet decomposition. 
Firstly, the wavelet decompositions of (1) and (2) were 
implemented on the hyperspectral data, and then only the 
M z 2" first scaling and wavelet coefficients at scales 2/ » 2' 
are selected as features. One may prove that the selected 
features [{d;};,,a,] are useful for data representation. 
FAIRS) 2 
Because the linear wavelet transformation of x from large scale 
wavelet coefficients are equivalent to the finite element 
approximation over uniform grids, we call this method Linear 
Wavelet Feature Extraction (Linear WFE). 
In this method, the large amplitude wavelet coefficients at small 
scales would not be selected as features. However, the wavelet 
coefficients with large amplitudes are generated by the 
singularities of the spectral curve which may involve important 
information for representation or classification. Hsu (2003) 
suggested that the approximation and detail components at each 
scale of linear WFE should be combined together to extract 
885 
better features of hyperspectral images for classification. This 
can be done by non-linear wavelet feature extraction. 
2.3 Non-Linear Wavelet Feature Extraction 
Linear WFE method which selects the M wavelet coefficients 
independently of original spectrum x at larger scales can be 
improved by choosing the M wavelet coefficients depending on 
the x. This can be done by sorting the coefficients 
Hd] calculated by the multilevel orthogonal wavelet 
decomposition in decreasing order. Then the M largest 
amplitude wavelet coefficients are selected as the important 
features of x for classification. The non-linear approximation 
calculated from the M largest amplitude wavelet coefficients 
including the approximation and detail information can be 
interpreted as an adaptive grid approximation, where the 
approximation scale is refined in the neighborhood of 
singularities (Mallat, 1999). Thus this feature extraction method 
based on the non-linear approximation is called Non-Linear 
Wavelet Feature Extraction (Non-linear WFE). 
2.4 Best Basis Feature Extraction 
2.4.1 Wavelet Packets: Wavelet packets were introduced 
by Coifman et al. (1992) by generalizing the link between 
multiresolution approximations and wavelets. In the orthogonal 
wavelet decomposition algorithm described in Section 2.1, only 
the approximation coefficients are split iteratively into a vector 
of approximation coefficients and a vector of detail coefficients 
at a coarser scale. In the wavelet packet situation, each detail 
coefficients vector is also decomposed into two parts using the 
same approach as in approximation vector splitting. This 
recursive splitting shown in Figure 3 defines a complete binary 
tree of wavelet packet spaces where each parent node is divided 
in two orthogonal subspaces. The nodes of the binary tree 
represent the subspaces of a signal with different time- 
frequency localization characteristics. Any node in the binary 
tree can be labelled by ( j, p), where 2/ is the scale and p is the 
number of nodes that are on its left at the same scale. Suppose 
that we have already constructed a wavelet packet space Ww? 
and its orthogonal basis q^ SW (-2 at node ( j, p). 
The two successor wavelet packet orthogonal bases at the 
children nodes are defined by the splitting relations (Coifman et 
al.; 1992; Mallat, 1999): 
Ss 
y = Y nn? 0 - 2/2) (4) 
yit = enw! (t-2"n) (5) 
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