Visual Cryptography for Color Images - Visual Cryptography and Secret Image Sharing - page 41

columns in the distribution matrices). If D is the matrix selected for the dis-
tribution of the shares then the 5 subpixels in the first share will have colors
11 M 1, while those in the second share will have colors R 112.
Given a distribution matrix M and a set of participants X, we denote
with MjX the submatrix of M obtained by considering only the rows of M
corresponding to the participants in X.
As for the black and white case, the definition of a scheme must satisfy
the security and the contrast properties:
Security property: Given a forbidden set X, jXj < k, the c collections of
jXjm matrices, D i , i = 1; 2;:::;c, consisting of MjX for each M 2C i , contain
the same matrices with the same frequencies. This property guarantees that
a forbidden set of participants has no information on the secret image.
Contrast property: The contrast property has to guarantee that the secret
image will be visible for a qualified set of participants. For B&W-VC this
property uses two thresholds ` and h, with ` < h, and requires that when the
secret pixel is white, the number of black subpixels in the reconstruction is at
most ` and when the secret pixels is black, the number of black subpixels is
at least h. Many papers that deal with color images generalize this definition
requiring that in the reconstructed pixel there are at least h subpixels of color
i, where i is the color of the secret pixel, and for any other color j 6= i there
are at most ` subpixels with color j. Notice that this definition can be used
only if the shares palette is equal to the secret palette. Moreover, it allows
the possibility that the reconstructed pixel is made up of an overwhelming
majority of subpixels with a wrong color. For example if h = 4, l = 3, and
c = 10 it is possible to have in the reconstructed pixel only 4 subpixels with the
right color while other 27 = 3 9 have (mixed) wrong colors. The annihilator
color can be present without any restriction.
Probably a better definition of the contrast property should require that
in the reconstructed image there be at least h subpixels with the right color
and at most ` subpixels with wrong colors. That is, the number of subpixels
with the right color should be greater than the number of subpixels with a
wrong color (counting all the subpixels with wrong colors).
We will refer to the first property as the weak contrast property and to the
second one as the strong contrast property.
Next, we provide a formalization of such properties. Define the add (M) for
a distribution matrix M to be the vector whose j th component is the add of
column j in M and define w i (v) for a vector v to be the number of elements
equal to color i, for i = 1; 2;:::;c, that is for any color in the secret palette.
Moreover we dene w i (v) to be the number of elements in v different from
color i and from the annihilator color.
Weak 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 j ( add (MjX)) ` for any j in the shares palette
and j 6= i. Note that the annihilator color is not considered, that is, it is
allowed that many pixels be .