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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.