Cryptography Reference
In-Depth Information
∗
P
= P⊕W
c
;
(8.2)
∗
∗
(P
)
= P ;
(8.3)
∗
= P.
P
(8.4)
Let α(P ) be the complexity of a given image P , then we have
∗
α(P
)=1−α(P ).
(8.5)
It is evident that the combination of each local conjugation (e.g., 88area)
makes an overall conjugation (e.g., 512512 area). Eq. (8.5) says that every
binary image pattern P has its counterpart P
∗
is always symmetrical against P regarding α =0.5. For example, if P has a
complexity of 0.7, then P
∗
. The complexity value of P
∗
has a complexity of 0.3.
Criterion to Segment a Bit-Plane into Informative and Noise-Like
Regions
We are interested in how many binary image patterns are informative and
how many patterns are noise-like with regard to the complexity measure α.
Firstly, as we think 88 is a good size for local area, we want to know the
total number of 88 binary patterns in relation to α value. This means we
must check all 2
64
different 88 patterns. However, 2
64
is too huge to make
an exhaustive check by any means. Our practical approach is as follows. We
first generate as many random 88 binary patterns as possible, where each
pixel value is set random, but has equal black-and-white probability. Then we
make a histogram of all generated patterns in terms of α. This simulates the
distribution of 2
64
binary patterns. Fig. 8.2 shows the histogram for 4,096,000
88 patterns generated by our computer. This histogram shape almost exactly
fits the normal distribution function as shown in the figure. We would expect
this by application of the central limit theorem. The average value of the
complexity α was exactly 0.5. The standard deviation was 0.047 in α.We
denote this deviation by σ (Sigma in Fig. 8.2).
Fig. 8.2.
Histogram of randomly generated 88 binary patterns.
