International Archives of Photogrammetry and Remote Sensing, Vol. 32, Part 7-4-3 W6, Valladolid, Spain, 3-4 June, 1999 
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2. NOISE ESTIMATION 
This section will focus on estimation of noise variance from 
true images. Unlike coherent or systematic disturbances, the 
noise is assumed to be due to a completely stochastic process 
and, thus, it is not predictable by any deterministic model. Gen 
eral solutions for the case of parametric signal-dependent noise 
models will be specialized to the particular case of the noise in 
troduced by optical imaging sensors. In this latter case, also non- 
parametric models based on bit planes will be considered. 
2.1 Signal-Dependent Noise Modelling 
A general form of signal-dependent noise may be stated as 
g(m,n) = f(m,n)-\-f(m,ny-u(m,n) (1) 
in which g(m,n) denotes the recorded noisy image value at 
pixel position (m,n), f(m,n) the noise-free image value and 
u(m, n) a random process, independent of /(m, n), stationary 
and uncorrelated, with zero mean and variance a 2 . The term 
/(m, n) 7 • u(m, n) represents the signal-dependent noise contri 
bution (Kuan, 1985). 
This model is capable of describing many physical phenomena 
originating from different noise models in digital images, through 
its exponent 7; usually, 0 < 7 < 1. Multiplicative, or speckle, 
noise (Goodman, 1976) is a limit case of signal-dependent noise, 
in which the amplitude of the noise term is proportional to the 
value of the noise-free signal having nonzero mean. It can be 
achieved with 7=1. Film-grain noise is a signal-dependent 
noise in which the amplitude of the noise term is generally taken 
to be proportional either to the square root (7 = 1/2) or to the 
cubic root (7 = 1/3) of the optical density (Naderi, 1978). Even 
tually, for 7 = 0, the model (1) reduces to plain additive signal- 
independent noise. 
2.2 Parametric Noise Estimation 
2.2.1 7 and a u both unknown: Log-scatter-plot Method 
Unless known from “a priori” information about the imaging 
system and/or process, like for speckle in radar and ultrasound 
systems, the exponent 7 need be estimated together with the noise 
standard deviation a u . To this end, let us calculate the variance 
of (1): 
crg(m,n) « cr 2 (m,n)(l+7 2 cr 2 )+/x 27 (ra,n) ■ a 2 u (2) 
in which pf(m,n) denotes the space-varying mean of / and 
equals that of g: pf(m, n) = p g (m, n), from (1). The approxi 
mation stems from the variance of / 7 which is replaced by 7 s cr/. 
Out of the two terms on the right-hand of (2), the former is domi 
nant in textured areas and null in homogeneous areas; conversely, 
the latter is negligible in high-signal areas. By considering these 
two effects separately and by taking the logarithms of both mem 
bers, one obtains: 
log a g (m,n) « 7 • log p g (m,n) + logcr u (3) 
log cr g (m, n) « logo7(m,n) + ^log(l + 7 2 -cr 2 ) (4) 
Equation (3) yields an estimate of 7, namely 7, as the regression 
coefficient of the scatter-plot of log a g versus log p g , calculated 
in homogeneous areas. The value in which the regression straight 
line crosses the ordinate axis corresponds to the term log <r u , from 
which an estimate of a u , o u , may be straightforwardly obtained. 
In order to overcome the drawback of manually identifying ho 
mogeneous areas, an automatic procedure was developed, based 
on considerations that such areas tend to produce clusters of 
points on the log-scatter-plot, which are aligned along the regres 
sion straight line. The log-scatter-plot plane is partitioned into a 
number of rectangular blocks, say, 100 x 100. Such blocks are 
sorted and labeled for decreasing number of points. Thus, the 
first blocks are those which are the most populated. A succes 
sion of regression lines, corresponding to a succession of 7s and 
one of <t u s, is calculated on the points lying inside an increas 
ing number of blocks in the log-scatter-plot. Thus, the first term 
of the succession is calculated on the most dense block, the sec 
ond on the two most dense blocks, and so on. The succession 
attains the true values after a number of terms, which depends on 
the actual percentage of homogeneous points. A stop criterion 
was devised based on that the 7 is almost always underestimated 
throughout the succession. Thus, the maximum of the 7, or better 
the median of three or five of the maxima, is chosen in order to 
reject possible outliers. The first terms of the succession, say 1%, 
are always discarded because of statistical fluctuations due to the 
small sample. 
2.2.2 Estimation of o u with known 7: Scatter-plot Method 
Whenever the 7 is otherwise known, e.g. from a model of the 
imaging process, the standard deviation of the modelled signal- 
dependent noise (2) may be stated in homogeneous areas as 
(T g (m,n) = cr u • p] (m, n) (5) 
Thus, (5) yields an estimate of a u , namely a u , either as the 
slope, or as the origin ordinate of the regression line drawn on 
the scatter-plot of <r g versus /x 7 and calculated in homogeneous 
areas. The former case holds when 7 / 0 and the line passes 
through the origin. In the latter case, 7 = 0 and the slope is zero 
as well. The block strategy can be still used to estimate the a u . 
2.2.3 Estimation of a u with known 7: Histogram Method 
If the 7 is known, both members of (2) may be divided by 
/x 27 (m, n) to yield: 
oj(m,n) _ o-/ 7 (m,n)(l +7 2 crg) 2 
pi 1 (m, n) pp(m,n) 
Again, for homogeneous areas, in which 07(771,71) = 0, (6) 
yields the noise variance a 2 . In practice, the ensemble statistics 
in (6) are replaced with the local spatial statistics and 07 (to, n) f 
0. It can be noticed, however, that the left-hand term of (6) always 
exhibits a unimodal distribution whose mode is roughly indepen 
dent of image texture. Therefore, the real valued mode, which 
can be extrapolated from a smoothed version of the histogram, 
will yield an estimate, a 2 , of the value of cr 2 . 
In the case of additive noise independent of the signal, which is 
mostly encountered in such imagers as air-borne and space-borne 
multi-spectral scanners, both scatter-plot and histogram methods 
are available to estimate the unique parameter o u . Also non- 
parametric methods can be devised, especially to improve the 
confidence of results found otherwise.