Barsi, Arpad
the whole block then changes for the next block. The block size
depends on the dimensionality of the input vector from 1000 to
10 000 pixels. The classification of the original image by this
block method took 57.7 s, while the pixel-wise algorithm ran
255 minutes (265 times faster)!
2.3. Principal component analysis
Because of the close imaging bands satellite images contain
more or less redundancy. This redundancy can be detect and
reduce by the principal component analysis (PCA). The
analysis is mathematically a problem, where the eigenvalues
and eigenvactors of a quadratic matrix are calculated. The
mentioned matrix — in the image processing practice — is the
covariance or the correlation matrix of the image. In the project
the covariance matrix, then its eigenvalues and eigenvectors
were calculated. The calculated eigenvalues are normalized and
sorted (Figure 3).
Figure 3. The sorted normalized eigenvalues
of the original image expressed in %
The cumulated eigenvalues have the meaning of the increasing
information content, they're very important. In order to be able
to decrease the data amount, a special linear image
transformation is to be executed (so-called Karhunen-Loeve
transformation). The transformation matrix is built from the
corresponding eigenvectors. By the transformation the data
amount can be reduced by keeping a controlled part of the
original information content. For example essential 95 96
information content requires the first three transformed bands
(71.3 * 16.2 * 82 = 95.7 %). Satisfying with 85 %, only 2
bands are needed (87.5 %).
Considering not only the pixels but also their neighborhoods the
described PCA and the connecting transformation could reduce
the data amount to be processed. One important point of view in
the project was testing this hypothesis.
2.4. Considering neighborhood information
The neighborhood can be defined classically in two ways: 4-
and 8-neighborhood. The 4-neighborhood means only the direct
neighbors of a pixel, while after 8-neighborhood all direct and
indirect connecting pixels count (gray fields in Figure 4).
2 6/2 7
3; 1 | 4 3 | 1 | 4
5 S |5 | 8
Figure 4. The numbering order of 4- and 8- neighborhood
Speaking about a LANDSAT TM image it has 7 bands. The
intensity values are collected into a vector, which has therefore
a length of 7. If the neighborhood is to be handled the
dimension of the intensity vector is getting higher. Following
the notion of Figure 4 the intensity vector has 5 or 9 blocks with
every times 7 values. The dimension of an image handling also
the neighborhood is 35 or 63. The importance of the hypothesis
of the previous chapter is very clear!!
Testing the hypothesis, neural networks were designed without
and with principal component analysis and -transformation.
The feed-forward neural networks require training data for the
parameter definition. The original training pixel set was
updated: their intensity vector was dimensionally enlarged. The
number of inputs is 5 x 7 = 35 in 4-neighborhood and 9 x 7 =
63 in 8-neighborhood. The trained neural network should
produce a single output (class membership).
The preparation of the training and test data set was executed in
two ways. The first version was the usual indexed solution: as
in Figure 4 is shown, the original intensity vector was created .
(1), then the intensities of the above neighbor in all bands are
appended (2), then the same with the right (3), left (4) and
below (5) standing neighbors. The indices were (4j), (i-1j),
(ij-1), (ij+1), (i+1,)). This solution was rather slow in Matlab,
so a better algorithm is searched for. The second (and the
successful) algorithm doesn’t apply indices, instead of them the
original image was masked, then the whole image band was
moved one pixel up, left, right and down. The essential
intensities were collected by the mask. Comparing the speed of
the two algorithms: 755.1 s with indexing and 1.8 s with
elementary image movements!
The movements of 8-neighborhood are the combinations of
ones in 4-neighborhood. In the 8-neighborhood the second
algorithm was implemented. The training and test data
preparation took 3.6 s.
2.5. Compression and neighborhood
The PCA and the coupled transformation are very efficient
ways to decrease the data amount while the information content
is kept. The training and test sets are dramatically enlarged with
the neighborhood information, the application of PCA and the
transformation is advised.
As describing the PCA yet mentioned, with 95 % information
content only 3 bands are necessary. The combination of PCA
and neighborhood could be realized easily with extending the
former algorithms. The original image bands are transformed
with Karhunen-Loeve transformation. The results of the
transformation are similar intensity bands like the original ones
were, but they have no correlation. The first three derived bands
will give the necessary information. The neighborhood
operation of the previous chapter is executed on these input
channels. This has the main advantage that both the
neighborhood could be taken into consideration and the data
dimensionality isn't increased so drastically. The 4-
neighborhood data set contains 5 x 3 - 15 bands, the 8-
neighborhood version 9 x 3 — 27. Important data reduction! The
training set and test set of this combinations were applied to
design and qualify new neural networks.
The implemented training set preparation is just one possible
way for reducing the enormous data amount. Also a further
possibility arises: the first step is preparing the whole data sets
with neighborhood, then the intensity vectors are to be analyzed
and transformed. This second solution will decrease the amount
much effectively. In the case of 4 neighbors, the 95 %
information content needs only the first four bands — the
cumulative eigenvalues are 96.9 %! 8-neighborhood is very
similar, 4 essential transformed bands have 96.4 % information.
142 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B7. Amsterdam 2000.
