The algorithms were validated in two steps. First, the algorithms were applied to synthetic mixture data. The fractions were generated randomly and the spectral observations generated using an endmember matrix that closely reflected the spectral reflectances found typically in AVHRR data (again from the AVHRR Africa data set.) The errors involved in estimating the original fractions (which were, of course, known deterministically) are listed below.
The algorithms were then tested on the AVHRR data
covering a large portion of
the African continent. The size of the images is
433x487 and five spectral
bands are used. Two of the bands are in the visual
spectrum, two others
are thermal bands and the last is NDVI.
The NDVI band is actually
derived from the red and infra-red bands and reflects
the presence
or absence of green vegetation. The endmember matrix for
this image were derived from a thematic classification map that was obtained by
the maximum likelihood method.
Using the classification map, regions that contained
"pure" classes were
selected. The spectral response for each class
was averaged over its region, and the procedure was
repeated over all
the spectral bands.
The main classes present in this data set were
unvegetated land,grassland and tropical forest.
Accordingly these three
cover types were chosen as the `pure' classes.
The fractional maps generated by the algorithms were compared to known vegetation regions as well as the thematic classification map. The fraactional maps generated by the algorithms are shown below in figures 1-3. The results obtained with these algorithms are compared with those obtained using the conventional mixture unmixing algorithm used commonly by earth scientists. It may be observed from Figure.1 that the conventional algorithm fails to produce estimates in in a portion of the desert region in Africa. The algorithms illustrated here are more robust. Also, the conventional algorithm does not reflect the presence of grassland in the rift valley. Our algorithms accurately estimate the presence of grass in this region. In order to facilitate comparison, the thematic classification map is also shown (figure.).
Figure 1: Desert Estimate - Iterative
Figure 2: Grass Estimate - Iterative
Figure 3: Forest Estimate - Iterative
Figure 4: Water Estimate - Iterative
We now compare the estimates produced by two algorithms.
In the absence of accurate knowledge of the fractions
of the classes present, the estimation error is used
as a criterion of
performance. The estimation error produced by the two
algorithms has been tabulated below :
For this AVHRR data set, it appears that the non-iterative algorithm produces better estimates than the iterative one. On the other hand there are some other data sets where the iterative algorithm performs better. In all cases, the algorithms yield better estimates than some of the algorithms for mixture unmixing popular in the earth sciences community.