Visual Cryptography for Color Images - Visual Cryptography and Secret Image Sharing - page 51

the search space for good schemes, it guarantees that the reconstructed pixels
are exactly of the same original color (and not a darker version of it).
If we consider a model that requires this special property the c-color (n;n)-
threshold schemes of [4] are optimal with respect to the pixel expansion:

Theorem 3 [4] If the shares are restricted to be such that for any superposi-
tion it is possible to have at most one colored pixel, any c-color (n;n)-threshold
scheme has pixel expansion m c2 n2 .

Other results presented in [4]:

A construction of c-color (2;n)-threshold with pixel expansion m = c(n1).

A matching lower bound m c(n 1).

2

A construction of c-color (2;n)-threshold with contrast =

cn . The contrast
is dened as = (h `)=m and the thresholds h and ` satisfy the weak
contrast property.

An upper bound on the contrast cn . This matches the construction for
k = 2.

2.6 Schemes for the General Model
In this last section we finally describe schemes for Color-VC that consider
the General model, that is we consider schemes that superimpose pixels with
dierent colors. In the rest of the section, we present several (2; 2)-threshold
schemes from [7, 1] and a construction for (2;n)-threshold schemes from [1].

2.6.1 (2; 2)-Threshold Schemes
In this section, we present schemes for the particular case of k = n = 2.

Scheme 1 [7] The secret palette is f Y ; C ; G g while the shares palette is

f Y ; C ; G ;;g. The base matrices are:

Y C
YC

C Y
CY

YC
CY

S C = S G =

S Y =

It is easy to see that for this scheme the pixel expansion is m = 4 and we
have h = 2, ` = 0. The annihilator presence is = 1=2 because 2 out of 4
pixels are annihilated.