Cryptography Reference
In-Depth Information
C
B
(S
0
) that consists of all zeroes are exactly s(s 1). Thus, p
0
wjb
=
s(s1)
m(m1)
and consequently p
0
bjb
= 1
s(s1)
m(m1)
.
To compute p
0
bjw
, consider a qualified set Q. By property PW, the matrices
of the collection C
W
(S
0
) that consists of all zeroes are exactly (m`)(m`1).
Hence, p
0
wjw
=
(m`)(m`1)
m(m1)
, which gives p
0
bjw
= 1
(m`)(m`1)
m(m1)
.
Hence, for the (2;n)-threshold scheme obtained by choosing `
0
= 0 the
probabilistic factor is:
(m`)(m` 1) s(s 1)
m(m 1)
0
=
:
Case `
0
= 1;h
0
= 2.
To compute p
0
bjb
, consider that in this case a reconstructed pixel is black if
both subpixels are black (and white otherwise). If a qualified set Q is fixed,
by property PB, the matrices of the collection C
B
(S
0
) that have at least a 1
in each column are exactly (ms)(ms 1). Thus, p
0
bjb
=
(ms)(ms1)
m(m1)
.
To compute p
0
bjw
, fix any qualified set Q and consider that by property PW,
the matrices of the collection C
B
(S
0
) that have at least a 1 in each column are
exactly `(` 1). Thus, p
0
bjw
=
`(`1)
m(m1)
.
Hence, for the (2;n)-threshold scheme obtained by choosing `
0
= 1 the
probabilistic factor is:
(ms)(ms 1) `(` 1)
m(m 1)
1
=
:
Compare now the value of for the above two cases trying to gure out
whether one case is better than the other, i.e., the probabilistic factor is big-
ger and a better quality image is reconstructed. It is possible to express the
difference
1
0
as
1
0
=
m
(s)
m
(`)
m(m 1)
where
m
(x) = (mx)(mx1) +x(x1): The rst and second derivatives
of with respect to x are
@
m
@x
= 4x2m and
@
2
m
@x
2
= 4, respectively. Hence,
function
m
(x) is a convex [ function of x, with a minimum at x = m=2. A
simple algebra shows that, for n even, s =
4
n2
n1
and ` =
2
and that for
n odd, s =
4
n+
n
and ` =
2
n
n
. Since in both cases s < ` m=2, it is
possible to conclude that
m
(s) >
m
(`) and then
1
>
0
. A simple analysis
shows that the limit of
1
as n approaches innity is 5=16 ' 0:31, while the
limit of
0
as n approaches innity is 3=16 ' 0:18.
In the case of any m
0
Equations (5.2){(5.5) can be used to compute the
gives the resulting values of the probabilistic factor of S
0
over all possible
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