Visual Cryptography for Color Images - Visual Cryptography and Secret Image Sharing - page 43

The ND (No Darkening) Model.
The ND model is as the SC model but it considers the darkening problem.
Thus, again we cannot superpose pixels with different colors, but if we su-
perpose several pixels with the same color we get a darker version of that
color.
The General Model.
In the General model, there are no restrictions about the superposition of pix-
els and the superposition operation satisfies the real properties of color super-
position. This means the darkening problem is considered. Very few schemes
have been defined for this model.
2.3.4 Base Matrices
Given a matrix B, we denote by C(B) the set of matrices obtained by permut-
ing in all possible ways the columns of B. In most schemes, the c collections
C i are obtained by fixing c matrices Bi, i , i = 1; 2;:::;c, and letting C i = C(B i ).
The matrices B i are called the "base matrices." Base matrices constitute an
ecient representation of a scheme. Indeed, the dealer has to store only the
base matrices and in order to randomly choose a matrix from C(Bi) i ) it has to
randomly choose a permutation of the columns of the base matrix Bi. i .
Notice that the security property for a base matrices scheme is equivalent
to: Given a forbidden set X, the matrices B i jX, for i = 1; 2;:::;c are the
same up to a permutation of the columns.
2.4 Schemes for the SC Model
In this section, we review the known schemes for the SC model. Verheul and
van Tilborg [10] were the first to consider visual cryptography schemes for
color images. Their model is equivalent to the SC one; as we will see shortly
their model requires a special property, which can be easily implemented us-
ing the SC model. The schemes of [10] were improved first by [2] and then
by [7, 11]. Paper [3] provides a lower bound on the pixel expansion and also
the construction of (n;n)-threshold schemes that achieve the lower bound.
It turns out that the (n;n)-threshold schemes of [7, 11] also have optimal
pixel expansion, which means that the schemes of [3] and those of [7, 11] are
equivalent.