International Archives of Photogrammetry and Remote Sensing, Vol. 32, Part 7-4-3 W6, Valladolid, Spain. 3-A June. 1999 
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2.3 Non-Parametric Estimation: Bit-Planes Methods 
Bit-plane methods rely on the binary representation of the dig 
ital data. A grey-scale image of word length L is partitioned into 
L bit planes, i.e. planes containing the binary digits 0 and 1. 
Each plane is statistically analysed to find out spatial correlations 
to decide whether it conveys information to a user, or not. 
2.3.1 Run-Length Method 
The analysis of each bit plane is carried out by checking 
whether its binary digits may be modelled or not as a sequence of 
discrete random variables X stemming from a Bernoulli’s pro 
cess, i.e. a random experiment with two possible outcomes, 
called success and failure, with probability p and q = 1 — p, 
respectively. The attribute of success is alternatively assigned to 
zero and one digits, in order to evaluate the behaviour of the entire 
bit plane. By considering, e.g., the ones as successes, we want to 
verify whether the random variable X obeys to a Bernoulli prob 
ability density function {pdf) with mean 
P a- = P- (7) 
and variance 
a 2 x = pq. (8) 
with p is the probability of one. The bit plane may be rejected 
as being without information to a user, if the random variable A r 
comes from a Bernoulli process, that is, the results of the exper 
iment are independent. This can be verified considering a new 
random variable Y made up by the number of failures between 
two consecutive successes, i.e. the length r of the runs of failures, 
in our case the runs of zeroes. If the succession of successes and 
failures is random, then Y exhibits a geometrical pdf with mean 
Py = 1/p (9) 
and variance 
cry- = q/p 2 (10) 
As a consequence, an estimate of E[r], namely r, is calculated as 
the average length of the runs of zeroes occurring in the bit plane. 
For a large number of experiments (i.e. a large number of run 
lengths N r i) the true value of r can be determined by considering 
that 1/r is a Gaussian random variable with mean p and variance 
qp 2 /N r i. Analogously, an estimate of p, namely p, is calculated 
as the number of occurrences of ones in the bit plane. For a large 
number of pixels N p , p is normally distributed with mean p and 
variance pq/N p . 
By enclosing the estimated values of r and p into their 99.75% 
confidence intervals, the following relationships must be verified: 
Pm in — 3 • pm in • y/(l — PMiN)/N r i < 1/r < 
Pm ax + 3 • pm ax ■ y/ (1 — pm ax )/N r i (H) 
in which 
Pm in = p — 3 • \Jp{ 1 — p)/N p 
Pm ax = p + 3 • y/p( 1 -p)/N p (12) 
Notice that in the special case of p — q — 0.5, we obtain E[r] = 
2. 
The analysis reported above has to be repeated for zeroes. Im 
age areas comprising saturated pixels, in which the bits in a sam 
ple are all zeroes or all ones, must be avoided. Successively, ver 
tical runs will be analysed, e.g. by rotating the image by 7r/2. 
2.3.2 Differences Method 
The algorithm (Barducci, 1997) may be summarized by the 
following steps: 
• Bit planes are considered. Least Significant Bit (LSB) first. 
• For each pixel in the bit plane, the absolute differences with 
each of its 4-neighbours is calculated together with their pdf. 
• The distribution of differences is analysed in order to estab 
lish whether the bit plane is completely random, or belongs to a 
textured, i.e. correlated, intensity field. In the former case, the 
bit-plane is recognized to be dominated by noise. 
• The previous steps are repeated until a non-random bit plane is 
encountered. Hence, the SNR is calculated. 
Let 
• N: overall number of differences in the bit plane; 
• rv. cumulative value of differences in the bit plane; 
• p\ probability of a difference being equal to one; 
• q: probability of a difference being equal to zero (p + q = 1); 
• A: user-specified threshold defining the confidence interval. 
Calculate the following random variable: 
It is verified that E[x] — 1, if the bit-plane is random, provided 
thatp = q = 0.5, which happens in practice. Thus, the bit-plane 
is noisy ifl — A<x<l + A, within the confidence interval. 
2.4 SNR Assessment 
The main drawback of the bit-planes methods is that, notwith 
standing the presence of the smallest amounts of information can 
be detected, it is only possible to set lower and upper bounds to 
the noise level of an image. In fact, let Ls < L be the num 
ber of bit planes recognized to be significant. For the unknown 
noise standard deviation, it holds that 2 L ~ L “~ l < a u < 2 L ~ L * 
and thus it may be either upper bounded as d^ = 2 L ~ Ls , or 
lower bounded as du ' = 2 L ~ Ls ~ 1 . Conversely, the relationship 
between the otherwise estimated d u and the number of signif 
icant bits is Ls = min{L, L - log 2 cr u }. Notice that if, say, 
0.5 < d u < 1, then one extra bit, i.e. a word length L 4- 1 bits, 
would be recommended instead. By assuming a uniform signal 
distribution between the levels 0 and 2 L — 1, the SNR will be 
estimated as 
(2 l - l) 2 
SNR {dB) = 10 x log 10 V (14) 
1L ' G xx 
and thus it will vary at steps of 6 dB, between a minimum and a 
maximum SNR, depending on whether the a u is over- or under 
estimated from the number of significant bit-planes Ls- When 
the d u is no longer constrained to be a power of two, as it hap 
pens with the parametric methods based on scatter-plots and his 
tograms, the SNR estimation is finer, but not necessarily more 
accurate. As a matter of fact, bit-plane methods are suitable for 
measuring small amounts of additive white noise from images 
of 10 to 12 bit word length. The results are accurate, notwith 
standing the ±3 dB approximation. Larger amounts of noise 
are generally measured better by parametric methods (scatter-plot 
and histogram) together with all the cases in which the noise is 
signal-dependent and/or spatially correlated.