Visual Cryptography for Color Images - Visual Cryptography and Secret Image Sharing - page 44

2.4.1 The VV Schemes
The model considered in [10], which we will call the VV model, requires a
special property: if we superpose pixels with different colors then the resulting
pixel is black. As we have explained earlier, this property is not natural.
When we superpose two pixels with different colors, we get a third color that
depends on the colors of the two superposed pixels. In some particular cases
the resulting color is actually black, but it is not black in most cases.
Verheul and van Tilborg propose a trick that "implements" such a prop-
erty. The trick works as follows: each pixel is divided into c subpixels, where
c is the number of colors in the secret image, subpixels i gets color number i,
while all other subpixels get painted with black, as shown in Figure 2.9.
FIGURE 2.9
(See color insert.) The VV trick for the case of 4 colors. Subpixels with different
colors are never superposed.
This trick implements the required property and makes the VV model
equivalent to the SC model because in the resulting scheme subpixels with
different colors are never superposed. However, to implement the trick, we
have to pay an extra pixel expansion factor of c and a considerable fraction
of the original pixel gets annihilated in the reconstruction.
The schemes of [10] are constructed using finite fields that satisfy certain
conditions. We refer the reader to the original paper for a detailed description
of the construction.
Assuming that c > 2 is a prime power, the construction produces
(k;k)-threshold schemes with c colors for any k;
(k;c 1)-threshold schemes with c colors for k < c;
(k;c)-threshold schemes with c colors for k < c, if k 1 and c 1 are not
relatively prime.
The pixel expansion of the schemes is m = c k ; this includes the pixel
expansion m = c k1 due to the construction of the scheme and the extra c
factor due to implementation of the special property of the VV model.
The reconstruction guarantees that there is at least one pixel of the original
color and no pixels with other colors, that is h = 1 and ` = 0. The contrast
property property considered is the weak one. However, we note that when ` =