Visual Cryptography for Color Images - Visual Cryptography and Secret Image Sharing - page 42

Strong contrast property: There must exist h and `, integers 0 ` < h m,
such that given a qualied set X, jXj = k, for any M 2 C i , it holds that
w i ( add (MjX)) h and w i ( add (MjX)) `. Also in this case the annihilator
color can be present without restriction.
In the black and white case, the thresholds ` and h, together with the pixel
expansion m have been used to define several variants of the contrast metric,
such as = h`, = (h`)=m, and = (h`)=(h+`). Similar measures have
been used for color schemes and we will specify the definition of the contrast
when presenting the schemes. However, for Color-VC schemes we need to
account for the presence of the annihilator color in the reconstructed image
and this makes the contrast less important. We will evaluate the annihilator
presence that we can dene as = b=m, where b is the number of pixels that
get annihilated in the reconstruction process.

2.3.3 The SC, ND, and General Models
The schemes that we will review in the rest of the chapter can be classified,
based on the formal model that they use, into three classes. In the next para-
graph we define three formal models for Color-VC.

The SC (Same Color) model.

The SC model does not allow the superposition of pixels with different colors,
with the exception of the identity () and the annihilator () colors. Hence,
the shares have to be constructed in such a way that each column in the
distribution matrices have elements taken from the set fi;;g, for some color
i. Thus, when we superpose several transparencies, we never have a pixel of
color i superposed with a pixel of color j.
Moreover the darkening problem is ignored. That is, it is assumed that
superposing several pixels with color i, the resulting color is still i.
An example of a distribution matrix for such kinds of schemes is the fol-
lowing (we have used three colors, denoted with the numbers 1; 2, and 3):

2
4

3
5

3 1 1 12 2 23 3312
3 1 11 2 2233312
3 11 1 22 2 33312
31 1 12 2 2 33 312

D =

As can be noted, in each column, we either have colors ;, or pixels with
a color = 1; 2, or 3. We never have a column that mixes two dierent colors
in the set f1; 2; 3g.
This restriction and the fact that the darkening problem is ignored avoids
the complications that derive from color superposition.