Modified Fisher's Linear Discriminant Analysis for ... - IEEE Xplore

506 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 4, NO. 4, OCTOBER 2007
Fig. 4. AVIRIS Cuprite image scene. (a) Spectral band image. (b) Spatial
locations of five pure pixels corresponding to the following minerals: alunite
(A), buddingtonite (B), calcite (C), kaolinite (K), and muscovite (M).
Fig. 3.
ROC curves in the HYDICE for the test set.
TABLE I
(LARGEST)NUMBER OF DETECTED PIXELS (N D ) IN THE
TEST SET WHEN NO FALSE ALARM EXISTS (N F =0)
TABLE II
(SMALLEST)NUMBER OF FALSE ALARM PIXELS (N F ) WHEN
ALL PIXELS IN THE TEST SET ARE DETECTED (N D = 28)
test set were detected, whereas TCIMF using the original data
detected 19 pixels. By slightly decreasing the threshold, all the
28 pixels could be detected although the false alarm was not
zero any more. Table II lists the smallest N F when all the
28 panel pixels in the test set were still detected, corresponding
to η =0.5 for MFLDA and η =0.2 for TCIMF. In this case,
N F = 150 from MFLDA, which is much smaller than 6952
from TCIMF.
B. AVIRIS Experiment
To compare the four LDA techniques, the Airborne Visible/
Infrared Imaging Spectrometer (AVIRIS) Cuprite scene as
shown in Fig. 4 was used, which is well understood mineralogically
[5]. At least five minerals were present, namely:
1) alunite (A), 2) buddingtonite (B), 3) calcite (C), 4) kaolinite
(K), and 5) muscovite (M). The approximate spatial locations
of these minerals are marked in Fig. 4(b). However, no pixel
level ground truth is available. Due to the scene complexity, the
actual number of classes p T is much greater than five.
To compare the performance of FLDA, MFLDA, CFLDA,
and CLDA, SAM and TCIMF were applied to the original
and transformed data. To conduct FLDA and CFLDA, training
samples were generated by comparing with the five material
endmembers using SAM, and the number of training samples
for the five classes were 63, 59, 69, 72, and 63, respectively.
As shown in Fig. 5(a), with the original data, SAM could not
classify these five minerals, but TCIMF provided accurate result
as shown in Fig. 5(b). Fig. 5(c) and (d) shows the SAM and
TCIMF results using the 4-D FLDA-transformed data, respectively,
where the SAM result was slightly improved but the classification
result was still incorrect and the TCIMF result was
much worse than that in Fig. 5(b) when the original data were
used. If SAM was applied to the 4-D MFLDA-transformed
data, the classification was improved as in Fig. 5(e), and the
TCIMF result in Fig. 5(f) was as good as in Fig. 5(b) using the
189-band data. The CFLDA on the 189-band original data was
shown in Fig. 5(g), which included a lot of misclassifications
due to the use of S W that was estimated under the assumption
that only five classes were present. The CLDA on the original
data in Fig. 5(h) did as well as the TCIMF in Fig. 5(b) since
the matrices R and Σ in the operators have the same role on
background suppression.
To perform quantitative comparison, Table III lists the spatial
correlation coefficients between the result from the use of LDAtransformed
data and that from the TCIMF on the original
data, where a value closer to one is associated with better
classification. It is obvious that MFLDA outperforms FLDA
and CFLDA, and it performs comparably to CLDA but on the
data with much lower dimensionality. This means Σ is a better
term than S W when the actual number of classes and their
information are difficult or even impossible to obtain.
V. C ONCLUSION
The original FLDA is modified for hyperspectral image
dimension reduction when enough class training samples are
unavailable. This situation comes from the existence of mixed