(3) 
class membership 
lasses, pixels and 
nt (Í « m.« eo), 
d |* | is the 
a pixel X; and a 
class membership 
| membership are 
ound. 
ber of processing 
lassify input data 
ward multi-layer 
input, hidden and 
cation angle, the 
ber of bands and 
values. The units 
^ trial and error 
layer denote the 
nit in a layer is 
hese connections 
1 is multiplied by 
in the beginning) 
euron nel, in the 
(4) 
. = the weight of 
insformed by an 
that neuron. The 
ion is a sigmoid 
(5) 
a gain parameter. 
the connections is 
arning algorithms 
used. Among the 
lgorithm has been 
his algorithm, an 
of target (known) 
| iteratively. The 
nverges to some 
btained. 
(6) 
ork output vector 
propriate weights 
work is assumed 
ired accuracy, the 
adjusted weights are used to determine the outputs of the entire 
image. Sometimes, while determining the weights, a learning 
rate and a momentum factor are also adopted. The network 
outputs are called as activation levels. For fuzzy classification, 
these activation levels are scaled to range from 0 to 1 for a pixel 
to produce fuzzy outputs (Foody, 1996). 
2.5 Knowledge Based (KB) System 
KB system is used to classify remote sensing data on the basis 
of knowledge acquired from the experts in the field. The 
conventional way of representing the knowledge is to formulate 
a set of rules in the form of If-Then-Else statements. A fuzzy 
knowledge base consists of production of fuzzy rules (Tso and 
Mather, 2001). For example, a rule for crisp classification may 
be written as, 
If DN value is between 100 and 250 
Then the pixel is assigned to the class ‘vegetation’ 
Whereas, for fuzzy classification, a fuzzy rule may take the 
form as, 
If DN value in middle 
Then the pixel is assigned to ‘vegetation’ with strength w. 
where w = class membership value. 
As is clear above representations of fuzzy methods that mixed 
pixels are taken into account only in the allocation stage of 
classification. 
3. MIXED PIXELS IN TRAINING STAGE 
OF A CLASSIFICATION 
Conventionally, a supervised classification assumes that 
training pixels are pure. However, it may be difficult to define a 
training set of an appropriate size containing only pure pixels, 
and therefore the urge to include mixed pixels. To incorporate 
mixed pixels in training stage, actual class proportions of pixels 
may have to be known beforehand from known data sets. The 
training data statistics, such as mean and variance covariance, 
may be weighted by actual proportions to generate fuzzy 
statistical parameters (Wang, 1990). MLC to be implemented in 
fully fuzzy mode utilizes this concept. In supervised version of 
FCM algorithm, the mixed pixels are incorporated by default 
due to the fuzzy c-partition matrix, which depends upon class 
membership value of the mixed pixel. In order to include mixed 
pixels in LMM, the model is first run in reverse mode by 
inputting the class proportions of the mixed pixels to determine 
the end member spectra, which is then input to LMM to run in 
forward mode to determine class proportions of unclassified 
mixed pixels. The concept is to rectify the class spectral 
responses derived from a training set containing mixed pixels to 
simulate the response that would have been derived from pure 
pixels (Foody and Arora, 1996). For ANN classifier, the target 
output of the pixels are assigned the actual proportions of the 
classes instead of the code of the class, thus accounting for 
mixed pixels in training stage. 
4, MIXED PIXELS IN TESTING STAGE 
OF A CLASSIFICATION 
The accuracy of the classification is assessed in the testing 
stage. A typical strategy is to select a sample of testing pixels, 
and matching their class allocation with the actual class on 
IAPRS & SIS, Vol.34, Part 7, “Resource and Environmental Monitoring", Hyderabad, India, 2002 
reference data. The pixels of agreement and disagreement are 
summarized in an error matrix (Congalton, 1991), which is then 
used to derive a range of accuracy measures such as overall 
accuracy, producer’s and user’s accuracy, and kappa coefficient 
etc. (Arora and Ghosh, 1998). These measures may be 
appropriate when a pixel is associated with one class in the 
classification and one class in reference data (i.e., the pure 
pixels). Therefore, use of these measures to evaluate fuzzy 
classifications may degrade the accuracy. This is primarily due 
to the fact that in order to use these measures, the fuzzy 
classification has to be hardened by specifying the pixel with 
the class having the highest class membership to produce crisp 
classification. Therefore, mixed pixels may have to be used in 
the testing stage to derive alternative accuracy measures such as 
entropy, Euclidean distance, cross-entropy and correlation 
coefficient, each has its own merits and demerits. 
Entropy shows how the strength of class membership in the 
classification output is partitioned between the classes for each 
pixel (Foody, 1995). The value of entropy is maximized when 
the probability of class membership is partitioned evenly 
between all the classes and minimized when it is associated 
entirely with one class. This is, however, only appropriate for 
situations in which the output of the classification is fuzzy 
whereas the reference data are crisp. There may be ambiguity 
present in the reference data as these are also often not error- 
free, and may therefore be fuzzy. To accommodate fuzziness in 
both the classification output and the reference data, measures 
such as Euclidean and L1 distances (Foody and Arora, 1996), 
and cross-entropy (Foody, 1995), which measure the closeness 
between the two data sets may be used. A small value of these 
measures indicates that the classification is accurate. To assess 
the accuracy of individual classes of a fuzzy classification, a 
correlation coefficient obtained from the two fuzzy outputs may 
be used. The higher the correlation coefficient, higher is the 
classification accuracy of a class. Recently, the concept of 
fuzzy error matrix has also been put forth (Binaghi et al., 1999) 
to assess the accuracy of fuzzy classification but its efficacy 
needs to be explored. 
5. CONCLUSIONS 
Fuzzy classification methods are attractive for land cover 
classifications from remote sensing data. Most of the studies 
have focused on generation of fuzzy outputs and thus 
considered the mixed pixels only in the allocation stage. When 
the image contains abundance of mixed pixels (i.e. IRS Wifs), 
their incorporation in training and testing stages of a 
classification becomes mandatory in order to produce 
appropriate land cover classifications. 
REFERENCES 
Arora, M.K. and S.K. Ghosh, 1998. Classification accuracy 
indices: definitions, comparisons and a brief review, Asian 
Pacific Remote Sensing and GIS Journal, 10(2), pp. 1-9. 
Bastin, L., 1997. Comparison of fuzzy c-mean classification, 
linear mixture modelling and MLC probabilities as tools for 
unmixing coarse pixels. International Journal of Remote 
Sensing, 18, pp. 3629 — 3648. 
Bezdek, J.C., R. Ehrlich, and W. Full, 1984. FCM: the fuzzy c- 
means clustering algorithm, Computers and Geosciences, 10, 
pp. 191-203.