Cryptography Reference
In-Depth Information
k
2m(m−1)
,
α =
0≤α≤1,
(8.1)
where k is the total length of the black-and-white border in the image and
2m(m−1) is the maximum possible border length obtained from an mm
checkerboard pattern. Eq. (8.1) is defined globally, i.e., α is calculated over
the whole image area. It gives us the global complexity of a binary image.
However, we can also use α for a local image complexity (e.g., an 88 pixel-
size area). We will use such α as our local complexity measure in this chapter.
8.1.2 Analysis of Informative and Noise-Like Regions
Informative images are simple, while noise-like images are complex. However,
this is only true in cases where such binary images are part of a natural image.
In this section we will discuss how many image patterns are informative and
how many patterns are noise-like. We will begin by introducing a conjuga-
tion operation of a binary image.
Conjugation of a Binary Image
Let P be a 2
N
2
N
size black-and-white image with black as the foreground
area and white as the background area. W and B denote all-white and all-
black patterns, respectively. We introduce two checkerboard patterns W
c
and
B
c
, where W
c
has a white pixel at the upper-left position, and B
c
is its
complement, i.e., the upper-left pixel is black (See Fig. 8.1). We regard black
and white pixels as having a logical value of 1 and 0, respectively.
P
∗
P
WB
B
c
c
Fig. 8.1.
Illustration of each binary pattern (N =4).
P is interpreted as follows. Pixels in the foreground area have the B pat-
tern, while pixels in the background area have the W pattern. Now we define
P
∗
as the conjugate of P which satisfies:
(1) The foreground area shape is the same as P .
(2) The foreground area has the B
c
pattern.
(3) The background area has the W
c
pattern.
Correspondence between P and P
∗
is one-to-one, onto. The following prop-
erties hold true and are easily proved for such conjugation operation.⊕des-
ignates the exclusive OR operation,
