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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004 
In image classification for a pixel, viewed as a statistical 
variable C, the uncertainty in class C; is defined as: 
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for i = L.,n, where X denotes the available data; the 
uncertainty is measured in bits. Generally, the true class of the 
pixel is not known and, as a consequence, the amount of 
information required revealing the pixel's class is unknown. 
The entropy of the pixel is therefore defined as the expected 
information content of a piece of information that would reveal 
its true class. To this end, the entropy measure combines the 
uncertainties in the various classes of the pixel by weighting 
them by their probabilities: 
n P(C=C,/X) 
-2 2 (5) 
iz] 
P(C - C, / X)* Log 
As another measure of weighted uncertainty, the quadratic 
score (Glasziou and Hilden, 1989) is briefly discussed here. The 
quadratic score is built on the notion of confirmation. The 
uncertainty in a single class for a pixel is the amount of 
probability required to establish this class with complete 
accuracy. The uncertainty in class Ci isdefined as 1-P(C=C;/X), 
where X once more denotes the available data. The quadratic 
score of the pixel is then: 
QS 0-P(CCxNp Pcr) 
i=l 
(6) 
This measure exhibits the same behavior in its minimum and 
maximum values as does the entropy measure. The two 
measures differ, however, in their slopes as is shown in 
Figure(1). The slope of the entropy measure is steeper than the 
slope of the quadratic score. As a result, the entropy measure 
for example more strongly weighs small deviations from 
probabilities equal to zero or one than the quadratic score. 
Uncertainty 
A Y 
[-————— 
Entropy 
   
  
   
  
Quadratic Score 
  
  
Probability 
Figure 1. Relation between quadratic score and entropy 
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As can be seen in Figure(1) we can argue that when entropy is 
increased we have a lot of chaos information (many objects are 
in a given pixel) and we are nd$pyertain about the labeling 
(class) of the pixel. This means we have considerable 
radiometric overlap between classes and vice versa. When 
uncertainty is decreased we have less chaos and we are more 
certain about labeling of desired pixels. Therefore radiometric 
overlap between classes is low; as a result they are separated 
from each other. 
We can use this concept to design a relation between 
uncertainty and accuracy of classified images. Thus as will be 
shown (section 3.2) the amount of uncertainty is a good 
indicator to investigate the accuracy of a map. Traditional 
approaches for accuracy assessment of thematic maps use 
ground truth. However, usually this ground truth is usually 
inherently unreliable. Hence, it is not a good idea to compare 
the extracted information (with a specific level of uncertainty) 
with a reference data set that is uncertain itself. 
3round truth could be non-representative (i.e. only partly 
covering the general characteristics of a particular land cover 
class), insufficient, incomplete (overlooked classes) or even 
outdated and thus lay an unstable foundation for accuracy 
assessment. Additionally the collection of this data is often a 
time-consuming and money-swallowing activity which in order 
to get rid of which, it is simply replaced by a visual inspection 
of some cartographic document or the image itself. 
3. TESTS 
Regarding the mentioned questions in the section 2, we have 
investigated the inverse relation between uncertainty and 
accuracy. To this end we have produced some synthetic images 
and (using some well known ground truth) and have classified 
them. Finally some accuracy and uncertainty related measures 
(URMs) have been calculated. Relation between these 
parameters is the major theme of the experiment. 
3.1 Generation of the Synthetic Images 
In this case study some synthetic images are used generated by 
a simple algorithm. For each image 3 spectral bands have been 
generated. Firstly in order to simulate the imaging process and 
generation of these bands in each case, we generate a ground 
truth map. This is used to generate the spectral bands of the 
synthetic images and in addition to evaluate the actual accuracy 
of the classification results. The general ground truth map has 
10 spectral classes with the various radiometric overlaps 
between them. This ground truth map can be generated 
automatically or manually. In this case study this map has been 
generated manually and regarding the real world it was tried to 
include various shapes of the possible objects [Figure 2.A ]. 
It was assumed that the statistical distribution of the image data 
(pixel values) is a multi dimensional normal distribution. This 
assumption doesn't affect the final results and just simplify the 
band generation and avoid the wrong assumption of the 
distribution of the data that is used in the maximum likelihood 
(MLH) classification. For generation of the images we have to 
consider some values for mean and variance vectors. Therefore 
we have a mean and variance value for each class per band 
(totally 30 values for means and 30 values for the variances). 
Covariances between all of the bands were assumed to be zero 
for the sake of simplicity and the little effect of them.