Two-Decoding-Option Image Sharing Method - Visual Cryptography and Secret Image Sharing - page 470

and Lin's scheme and our scheme are 3072 3072 pixels, 2548 2548 pixels
and 512 512 pixels, respectively. Finally the pixel expansions are m JIN =
(3072 3072)=(512 512) = 36, m (L)

LIN = (2548 2548)=(512 512) = 24.76

and m (L)

PRO = (512 512)=(512 512) = 1.

2
Example 6. Consider Lin and Lin's scheme and our scheme in Example
5 but use the compressible versions with a compression ratio R = 6:5.
By J(), we compress the original Lena into the compressed image I C with
PSNR = 38.88 dB. Now, Lin and Lin's (2, 4)-TiOISSS has enough space in
I 0 (jI 0 j = jIj) to embed the information of I C . By using I C as the original gray-
level secret image in our TiOISSS, we only need the 201201 halftone secret
images for sharing information bits. For the compression version, the pixel
expansions of (2, 4)-TiOISSS are m (C)

LIN = m = 4 and m (C)

PRO = m=(kgR) =

1=R = 1=6:5 = 0:154.

2
When comparing the pixel expansions among these three TiOISSSs [6, 9,
20], we consider the lossless version. This is a fair approach since all three
schemes can reconstruct the same quality of the reconstructed image. From
Equations (17.4), (17.7), and (17.8), it is obvious that our pixel expansion
m (L)

PRO < m JIN (since k 2 and g 1, so m=(k g) < m=2 < 9m). When
comparing m (L)

LIN and m (L)

PRO , one can verify that our TiOISSS has the lesser
pixel expansion than Lin and Lin's TiOISSS for most values of k, n, and m.
In the following theorem, we theoretically prove that m (L)

PRO < m (L) LIN when

both TiOISSSs use Naor{Shamir optimal (n, n)-VCS, Naor{Shamir optimal
(3, n)-VCS, and Naor{Shamir (2, n)-VCS [11].
Theorem: The pixel expansion m (L)

PRO = m=(kg) of the proposed (k, n)-
TiOISSS is lesser than the pixel expansion m (L)

m
w

LIN = (n=k)m8=log 2

of Lin and Lin's TiOISSS when using Naor{Shamir optimal (n, n)-VCS, Naor{
Shamir optimal (3, n)-VCS, and Naor{Shamir (2, n)-VCS.
Proof: When using Naor{Shamir optimal (n;n)-VCS (note: k = n) with
m = 2 (n1) , w = g = m=2, we have

m
w

m
w

m (L)

LIN = (n=k) m 8=log 2

= m 8=log 2

m(m1)(m=2+1)
(m=2)(m=21)1

m!
(m=2)!(m=2)!

= 8m=log 2 = 8m=log 2

> 8m=log 2 m m=2 = 8m= ((m=2) log 2 (m)) = 16= log 2 2 (n1)
= 2 4 =(n 1) > 2=n = m=(kg) = m (L)

PRO (since g = m=2 and k = n):
(17.10)
When using Naor{Shamir (3, n)-VCS with m = 2n2, w = g = m=2, we