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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004
11. If any components of Q y) in I? are negative, then move
corresponding indices from P to R9, Go to step 5.
: k ; .
12. Form another new matrix TuS by deleting all columns of
* NN ? specified by P
k)
13. Calculate aU «ys auo p (9 and go to step 3.
3.2 UFCLS Method
The above FCLS method requires a complete knowledge of
the endmember signature matrix M. For the situation of no a
priori information, it needs an unsupervised process to
generate the desired endmember information, based on which
the UFCLS method was proposed.
The least squares error (LSE) is a criterion to judge the fit
between data measurements and estimated values stand or fall.
In the inversion of linear spectral mixture model, we expect to
minimize the LSE, i.e., estimated values are extremely close to
the data measurements. The UFCLS method makes the
endmember signature values to be obtained directly from the
remote sensing image iteratively. In every iteration, we judge a
pixel to be an endmember pixel or not by the LSE, and decide
whether to regard its digital values as endmember signature
values. The idea can be described as follows.
Initially, we can select any arbitrary pixel vector as an initial
desired endmember m,. However, a good choice may be the
pixel vector with the maximum length d ( d —
b, denotes the digital values of the i-th band, / denotes the
numbers of bands). We then assume that all pixel vectors in an
image scene are pure pixels made up of my with100%
abundance. Of course, this is generally not true. So, we find a
pixel vector which has the largest LSE between itself and my,
and select it as the second endmember my. Now, we form an
endmember signature matrix M* fm, my]. Because the LSE
between my, and m, is the largest, m, is most distinct from
m,.The FCLS algorithm is then used to estimate the abundance
.
fractions of my and m, denoted by a D.) and
(2) D . vn S UA
a, ( 5) for each pixel vector Dy respectively. The
superscript indicates the number of the iteration currently being
executed. Here Dy is included in the estimated abundance
fractions to emphasize that they are functions of Dy. According
to the principle of the linear spectral mixture model, we are able
to ‘use "A (Dy) and a’ (Dy) to calculate the
^
oum Juin
estimated values Dn =o (D) mo vy, u. m,. We
^
then calculate the LSE between Dy and Dy for all image
pixel vectors Dy using the following equation.
LSE® (Dy) =
k-l T k-l
K) k
Dy-|Za (Dÿm;|} | Di-| La Dom ||
i=0 i=
The pixel vector that yields the largest LSE will be selected to
be the third endmember m;. The same procedure of using the
FCLS algorithm with M* (mj m, m;] is repeated until the
resulting LSE is small enough and below a given error
threshold.
4. EXPERIMENTAL RESULTS
4.1 Experiment 1
The data used in this section are ETM* * data, which were
obtained on 2 January 2000 over Shenzhen city and the vicinity
of it, China. The ETM* *image consists of eight bands, and we
only selected six bands in the visible and infrared spectral
region, referred to band 1 to band 5 and band 7. The spatial
resolution of the six bands is 30 m. The digital value for every
image pixel is constrained from 0 to 255. Because the ETM* *
data which we obtained here were corrected radiometrically and
geometrically initially, they were only registered. It is a
subscene of 51x51 pixels with few manmade objects for the
convenience of analysis extracted from the low right corner of
the scene, shown in Fig. 1.
Figure 1. The experiment images — Figure 2.The LSE image
of the 51x51 pixels of ms
We know from the linear spectral mixture model(1) that
solving Q actually becomes to solve p unknown parameters
from L linear equations if M and Dy are prior known, which
requires L larger than p .However, we only selected six bands,
so we can solve six unknown parameters at best. For easing the
problem of insufficient bands in multispectral imagery, and no
prior knowledge on the numbers of endmembers, we expanded
six bands to eighteen bands so that there were sufficient bands
to calculate iteratively a set of LSE,,,, the maximum of LSE.
The expanded bands were generated by taking the square root
of the cross-correlation between bandl and band4, bandl and
band5, band! and band6, band2 and band3, band2 and band 4,
band2 and band 5, band2 and band6, band3 and band4, band3
and band 5, band3 and band 6, band4 and band6, and band5 and
band6. The square root function was applied to keep the
magnitudes of the resulting image pixels similar to the original
data.
After 10 iterations using UFCLS method, a set of LSE,,, was
obtained, shown in Table 1. From the table we see that the
LSE,,, decreased constantly with the increase of iterative times,
first decreased at a high speed, but the more later the iteration
came to the more slowly it decreased. There is a distinct
borderline at ms iteration since after that the LSE,,,, decreased
very slowly, which shows that the extracted pixel vector after it
is not endmember but noise. In Fig.2 the LSE image of ms also
accounts for it. So, in this section the first six extracted pixel
vector were selected as endmembers.
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