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.