based on the results of physically spectral analysis, so that we 
can get a small number of salient features, reduce the dimen- 
sionality of hyperspectral images and keep the accuracy of clas- 
sification results. In our experiment, an AVIRIS data set is used 
to test the performance of the proposed HHT-based methods. 
Finally, the results are also compared with wavelet-based fea- 
ture extraction methods. 
2. HILBERT-HUANG TRANSFORM 
Hilbert-Huang transform (HHT), first proposed by Huang ef al. 
(1998), is a valid time-frequency analysis tool for nonlinear and 
nonstationary data. The HHT which consists of empirical mode 
decomposition (EMD) and Hilbert spectral analysis (HSA) will 
be described briefly in this section. 
2.1 Empirical mode decomposition 
Empirical mode decomposition (EMD) can decompose time- 
series data into a series of intrinsic mode functions (IMFs) 
adaptively. These IMFs include different regions of frequency, 
and each IMF has two properties (Huang, 2005): 
1. The number of extrema and the number of zero-crossing 
of an IMF must equal or differ at most by one. 
2. All the local maxima and minima of an IMF are symmet- 
ric with respect to zero. 
The EMD consists of the following steps: 
1. First, identify all the local maxima and connect them by 
cubic spline function as the upper envelope for a signal, 
x(r) . Repeat the procedure for the local minima to gener- 
ate the lower envelope. 
2. Compute the mean », of the upper and lower envelopes, 
and let x(;) minus », . We will get first proto-IMF (PIMF) 
component, 7, : 
x(f)- m, h, (1) 
The procedure which obtain IMF components is called sifting 
process. 
3. Proto-IMF, 4, , may not satisfy the definitions of IMF. 
Repeat the sifting process k times until the IMF meet the 
stoppage criteria. 
hun za =p, (2) 
4. As soon as the IMF component satisfy the criteria, we will 
get first IMF, c, , and separate c, from x(/) . 
x(t)-c, 7r, 
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5. Since the residue, 7, still contains information with long 
periods, it is treated as the data and repeat the sifting pro- 
cess. The result is 
rim C= 
(4) 
P me omm 
Finally, by Summing up equation (3) and (4), we obtain 
x(t) = Y e E. (5) 
The EMD separates variations from the mean, and each IMF 
has its own physical meaning. 
2.2 Hilbert Spectral Analysis 
Having obtained the IMF components, we can apply the Hilbert 
transform to each IMF component and compute the instantane- 
ous frequency. Then we can find the complex conjugate, & (/) , 
of an IMF, c, (#) , and have an analytic signal: 
z,) 7 e, (0) * jé (0) =a (re (6) 
a(t) is the function of instantaneous amplitude, and 6 (:) is 
the function of phase angle. As a consequence, we can express 
the original data as the real part, RP, in the following form: 
Jf dr 
X= RT 0,00" = P Y e, (7) 
Therefore, the Hilbert spectrum can be defined as: 
Hlm:4) = an Y ane T (8) 
We can also define the marginal Hilbert spectrum as: 
hy s [46.04 (9) 
0 
In summary, the HHT, consisting of EMD and HSA, can de- 
compose data adaptively and compute instantaneous frequency 
by differentiation rather than convolution. HHT is a superior 
data analysis tool for nonlinear and nonstationary data.