International Archives of Photogrammetry and Remote Sensing, Vol. 32, Part 7-4-3 W6, Valladolid, Spain, 3-4 June, 1999 
172 
(b) 
Fig. 5. Scatter-plots of local standard deviation vs. local mean 
of SPOT-P: (a) corrupted with additive Gaussian noise 
of a'i — 400; (b) corrupted with multiplicative noise of 
cr'i — 0.069. Regression lines are calculated using only 
the highlighted points. 
Method 
7 7 
2 -'2 
Gu ® u 
Log-Scatter-Plot 
0 0.12 
1 0.97 
400 172.4 
0.069 0.113 
Scatter-Plot 
0 
1 
400 419.4 
0.069 0.072 
Histogram 
0 
1 
400 433.4 
0.069 0.077 
Table 1. True and estimated parameters of the noise model for 
the simulated noisy images and the three parametric 
methods. 
Table 1 summarizes the results obtained. When the 7 is known, 
the scatter-plot method is more accurate than the other two meth 
ods; thus, it has been chosen for the subsequent experiments con 
cerning true data. Figures 4-6 show the plots from which the 
noise parameters are retrieved. The size of the local windows in 
which local statistics are measured is non-critical, at least in the 
range 3 x 3 to 7 x 7, provided that the noise is white. 
1200 
(a) 
2400 
(b) 
Fig. 6. Histograms of (normalized) local variances: (a) additive 
Gaussian noise (7 = 0) with <y\ — 400; (b) speckle noise 
(7 = 1) with = 0.017. 
Concerning the bit-planes methods, since both of them lead 
to a lower and an upper estimate, oi" ’ = 2 • oi\ of the stan 
dard deviation of the noise, we have verified that the o u from the 
scatter-plot method always lie between oi l) and cr[ u; if, starting 
from the LSB, the last noisy bit-plane is retained. 
5.2 Information Assessment 
The test set comprises one Landsat TM image, with 8 bit/pel 
and 6 bands out of 7. In fact, Band 6 (thermal infrared) was omit 
ted mainly because of its poorer resolution (120m x 120m vs. 
30m x 30m) and scarce spectral correlation with the other bands. 
The test site, two bands of which are portrayed in Fig. 7 as ex 
amples of visible and infrared observations, is in Italy: part of the 
valley of river Adige, near Trento. Table 2 reports the estimated 
parameters for the six bands, the first of which is encoded in in- 
tra mode, i.e. without reference to any other previously encoded 
band. To achieve an optimum multi-spectral de-correlation, the 
different bands available were arranged to form a sequence that 
maximizes the average cross-correlation between any couple of 
consecutive bands (Wang, 1995). The optimum bidirectional se 
quence, was found to be: 1 -» 3, 1 —» 2 4- 3, 3 —» 7, 4 4— 7, 
4 —> 5 <— 7. The difference in rate between causal and non- 
causal prediction is small, since the latter provides an improve 
ment of four hundredths of bit for the optical bands and nearly 
eight hundredths for the infrared channels (Aiazzi, 1999d).