International Archives of Photogrammetry and Remote Sensing, Vol. 32, Part 7-4-3 W6, Valladolid, Spain, 3-4 June, 1999
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4. INFORMATION ASSESSMENT
5. EXPERIMENTAL RESULTS
The proposed compression method can be adopted to measure
the information content of multispectral data. In fact, once the
standard deviation of the observation noise, assumed to be addi
tive, white and independent of the signal, has been estimated, the
minimum net bit-rate (i.e. not including any overhead) achieved
by the compression process, and thus taking into account the en
tropy of the noise, may be utilized to yield an estimate of the
true information content of the source, i.e. of one band of the
multispectral image arranged in a causal sequence in which the
previous bands are known.
Let <J 2 u {k) denote the variance of the observation random noise
superimposed to the k-th band of a multi/hyper-spectral image.
Let also r(k) denote the average bit rate per pixel needed to loss-
lessly decompress the k-th band. Prediction errors produced by
the de-correlation algorithm may be modelled as a random pro
cess given by the sum of two independent contributions: the ran
dom prediction errors calculated from a hypothetically noise-free
image plus the observation noise. In the absence of information,
the former are identical to zero and the overall entropy of the k-
th band is equal to the entropy of its observation noise, h u (k),
which may be approximated by the differential entropy (Jayant,
1984) of the noise pdf :
h u (k) = ^ • log 2 [27re • a 2 u (k)\ (15)
assuming that the noise is Gaussian.
By considering both contributions simultaneously, the overall
bit rate of the k-th band may be written as
r(k) = ^ ■ log 2 {27re • [cr 2 (k) + al(k)]} (16)
in which a 2 (k) is the variance of the random multispectral pre
diction errors of the noise-free image, which are related to the
useful information h s (k) by
h a {k) = ^ • log 2 [27re • cr 2 (k)] (17)
Conversely, cr 2 (k) can be defined as
a 2 (k)
A
2^2-hs (k)
27re
(18)
Therefore, the entropy rate of a hypothetically noise-free version
of the k-th band of the multispectral image may be regarded as its
true information rate. By deriving a 2 (k) from (16) and replacing
it into (17), such an information may be estimated as
h s (k) = ^ • l°g 2 [2 2 r(fe) — 27re • d-l(k)}
= I •log 2 [2 2 ' r(fe) _2 2 ' l “ (fc) ] (19)
from the code rate r(k) and from the estimated noise variance
d 2 {k). Notice that no assumption is made on the Gaussian
distribution of prediction errors, which is not generally verified
(Aiazzi, 1999b), but only on the observation noise. In fact, the
term a 2 (k) in (16) is the equivalent variance that the multispec
tral residuals should have to provide an entropy h s (k), if they
were Gaussian.
Analogously, the noise variance can be easily derived, once the
entropy rate of the source is otherwise known (Roger, 1996b).
Experiments are twofold, aiming at measuring the noise vari
ance, or the SNR, of Landsat TM and of AVIRIS data, in order to
estimate their useful multi-spectral information.
5.1 Noise estimation
Results of the noise estimation methods will be presented for
the cases of simulated and true noisy data. The former are used to
calibrate the algorithms and to show the accuracy of the results.
Experiments were first carried out on images corrupted with sim
ulated noise, of known 7 and a 2 . White Gaussian noise with
7 = 0 and a 2 = 400 was superimposed to a noise-free optical
image (SPOT-P). Also a speckled version (7 = 1, a 2 = 0.069,
4-look amplitude statistics (Goodman, 1976)) was generated.
Fig. 4. Scatter-plots of logarithm of local standard deviation
vs. logarithm of local mean with block partition and re
gression straight lines superimposed: SPOT-P image cor
rupted with noise: (a) additive (7 — 0, a 2 = 400); (b)
multiplicative (7 = 1, a\ = 0.069). Regression lines are
calculated using only the highlighted clusters of points.
