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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004
where R is the autocorrelation matrix of the image. Now we can
compute the output matrix y. The value of each pixel is ideally
related to the abundance of the target material. Theoretically,
the value of 1 means that the pixel has a spectrum like the target
and 0 means the absence of target spectrum in the pixel.
3. THRESHOLDING
For decision making to separate target from non target pixels, a
threshold is necessary. One of most reliable way to find a
threshold is using Receiver Operating Characteristic (ROC)
Curves. It has been used with the Neyman-Pearson method in
signal detection theory (Bradley 1997). It can be used to
visualize a classifier performance in order to select the proper
decision threshold. The ROC Curves compare a series of
similarity image classification results for different threshold
values with ground truth information. A probability of detection
(Pd) versus a probability of false alarm (Pfa) curve and a Pd
versus a threshold curve are reported for each selected class
(rule band).
For calculating of ROC curves, Confusion Matrix is needed. A
confusion matrix is a form of contingency table showing the
differences between the ground true data and classified images
and it is computed by cross tabulation technique. In case of a
single class classification or target detection we obtain a
confusion matrix such as given on Table |.
Confusion Matrix Classified Classes
> 0 ] sum
True 0 Tn Fp Cn
Classes 1 Fn Tp Cp
sum Rn Rp N
Table 1. A Confusion Matrix For Target Detection Case
The elements of this matrix are defined as:
Cn=Tn+Fp
Cp=Fn+Tn
Rn = Tn + Fn
Rp=Fp+Tp
Cn+Cp=Rn+Rp=N
(10)
Tn (true negative) is the number of non target pixels which are
correctly classified as non target. P(Tn) is its probability or rate
as calculated using : P(Tn)=Tn/Cn.
Tp (true positive) is the number of target pixels which are
correctly classified as target and P(Tp) is its rate as obtained
using: P(Tp)=Tp/Cp. It is also called probability of detection:
Pd.
Fp (false positive) is the number of non target pixels which are
incorrectly classified as target and P(Fp) is its probability as
calculated by: P(Fp)=Fp/Cn. It is also called probability of
false alarm: Pfa.
Fn (false negative) is the number of target pixels which are
incorrectly classified as non target and P(Fn) is its probability
as calculated by: P(Fn)=Fn/Cp.
This matrix and its elements must be calculated for a set of
thresholds. In practice we fix a number of thresholds between
the minimum and maximum values of rule data. Then, for each
threshold, a Pd and Pfa could be calculated. With each triple of
(thr, Pd, Pfa) we can plot two curves: A ROC that contains the
Pd against the Pfa and another curve that contains the Pd
against the threshold. An example of ROC curves are presented
on Figure 2.
51
ROC Curve ROC Thresh
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Figure 2. (a) Curves of the Pd versus the Pfa and (b) the Pd
versus the thresholds.
With these curves we can easily find a convenient threshold by
defining a level of false alarm or probability of false positive.
4. IMAGERY
We have applied the above techniques to CASI (Compact
Airborne Spectrographic Imager) hyperspectral images. CASI is
an airborne push-broom sensor that covers a range of
electromagnetic waves from 0.41pm to 0.95um. CASI has a
flexible spectral resolution capability. It means that the image
data may have different numbers of bands, maximum to 288.
Spatial resolution of CASI is a function of its IFOV and altitude
of airborne platform. It can vary from | to 10 meters. Dynamic
range of sensor is another parameter which produces the image
data with 12 bits or 4096 grey levels. CASI also is equipped
with a GPS and an INS for In/Off fly rectification and geo-
referencing of images.
The data for this experiment consists of two images on the same
scene. The first image was acquired at the altitude of 1293m;
the spatial resolution of the image is then 2m. The number of
bands for this image was fixed to 32 channels. The second
image was taken at 2540m, with 4m in spatial resolution and 48
spectral bands. Both images were acquired over the city of
Toulouse in the South of France on March 2001.
5. EXPEREMENTS
To perform tests with the proposed measures, we have selected
an area containing man-made objects like roads, buildings and
green spaces: two windows of the CASI images above this area
were selected. The first part has a size of 64x64 pixels with 48
bands and a spatial resolution of 4 meters (Figure 3a), and the
second is 128x128 pixels with 32 bands and 2m for spatial
resolution (Figure 3b). To compare and evaluate the results, we
extracted a true data map by visual interpretation of the building
materials of the scene for both images (Figures 3c and 3d). A
target spectrum of building. materials has been extracted by
collecting and averaging the spectra of manually selected pixels
for both sample data (Figures 3e and 3f).
We have applied the three mapping methods corresponding to
the three spectral similarity measures and matching operator. As
they are explained above: Modified Spectral Angle Similarity
(MSAS), Spectral Value Similarity, (SSV) and output of
Constrain Energy Minimizing operator or simply CEM. As
mentioned, since the values of SSV are in [0,/2], we have
stretched them linearly to [0,1]. Due to the noises, the values of
CEM output are not exactly in [0,1], then we have stretched
them to [0 1]. But as the most similarity between the target and
an unknown vector should be zero, the stretching is the
following:
CEM -1-(CEM - Min, ) (Max, - Min) (1)
