Probabilistic Visual Cryptography Schemes - Visual Cryptography and Secret Image Sharing - page 127

making mistakes in reconstructing the secret image. In some cases it is pos-
sible to sacrifice some pixel expansion in order to improve the probabilistic
reconstruction of the secret image or vice versa. Yang's model has been gener-
alized in Cimato et al. [11] showing how it is possible to trade pixel expansion
for the probability of a good reconstruction. Such a model can be seen as a
generalization of both the classical deterministic model and the probabilistic
model introduced by Yang [22]. Moreover, there exists a one-to-one mapping
between probabilistic schemes with no expansion and deterministic schemes;
such a mapping trades the contrast of the deterministic scheme with the prob-
ability factor of the probabilistic scheme. Other proposals in literature have
been introduced to deal with non-OR-based vcs and to extend the approach
to color and grayscale images.
5.2 Visual Cryptography Schemes
A formal definition of the probabilistic model has been given in [11], gen-
eralizing Yang's approach and extending the traditional denition of VCS.
In the next subsection we review the notions related to traditional VCS, be-
fore introducing the definition of probabilistic visual cryptography schemes in
subsection 5.2.2
5.2.1 The Deterministic Model
The secret image consists of black and white, where usually white color is
interpreted as transparent, so that the superposition of white pixels, let the
color of the pixel contained in the other shares pass. In order to share each
pixel of the secret image the owner of the secret, usually called the dealer,
provides each participant with a share, which is an enlarged version of the
secret pixel consisting of a certain number m of subpixels. So the shared
version of the original secret pixel will consists of m pixels, which are called
subpixels because all together they represent the original secret pixel.
The shares can be conveniently represented with n m matrices where
each row represents one share, i.e., m subpixels, and each element is either
0, for a white subpixel, or 1 for a black subpixel. A matrix representing the
shares is called the distribution matrix. Physically, the shares are given out in
the form of printed transparencies. Given a distribution matrix M and a set
Q of participants, the notation M Q refers to the submatrix of M consisting
of only the rows corresponding to participants in Q.
To reconstruct the secret image a group of participants stacks together the
shares. Since each secret pixel is represented by m pixels in the shares, the
reconstructed image will be bigger than the original (depending on m and on
the actual positions of the pixels, the image can also be distorted; a perfect