Resolving the Alignment Problem in Visual Cryptography - Visual Cryptography and Secret Image Sharing - page 294

1. (Contrast) For the collection C 0 and a share matrix s 2 C 0 , by
v a vector resulting from the OR of any k out of the n rows of s. If
w(v) denotes the average of the Hamming weights of v, over all the
share matrices in C 0 , then w(v) l.
2. (Contrast) For the collection C 1 , the value of w(v) satises

w(v) h.
3. (Security) For any i 1 < i 2 < < i t in f1; 2; ;ng with
t < k, the two collections of tm 0 matrices D j , j = 0; 1, obtained by
restricting each nm 0 matrix in C j , j = 0; 1, to rows i 1 ;i 2 ;i t , are
indistinguishable in the sense that they contain the same matrices
with the same frequencies.

The denition of PVCS in [14] only considers the case with n 1 share
matrices, we extend their denition to the nm 0 case. And the definition of
PVCS in [4] used the factor to reect the contrast, we use the values l and
h (darkness grey-levels) to reect the contrast. The same point of the three
definitions of PVCS is that, for a particular pixel in the original secret image,
the qualified participants can only correctly represent it in the recovered secret
image with a certain probability. Because the human eyes always average the
high frequency black and white dots into gray areas, so the average value of
the Hamming weight of the black dots in the area reflects the grayness of the
area. The PVCS does not require the satisfaction of the difference in grayness
for each pixel in the recovered secret image as the DVCS does. It only reflects
the difference in grayness in the overall view.
The contrast of the DVCS is fulfilled for each pixel (consisting of m sub-
pixels) in the recovered secret image, however, this is quite different in the
PVCS. The application of the average contrast, denoted by , rst appeared
in [3]. This term is often used in the PVCS, see [4, 14, 11, 7], where the
traditional contrast of the PVCS does not exist. Here we define the average
contrast to be the average value of the overall contrast of the recovered secret
image, i.e., the mean value of the contrast of all the pixels in the recovered
secret image. According to our denition of the contrast = h m , the average
contrast can be calculated by the formula =

m 0 , where h and l are the
mean values of w(v) for the black and white pixels in the overall recovered
secret image respectively, and m 0 is the pixel expansion of the PVCS. Because
the number of pixels is large in the recovered secret image, the values h and
l are equivalent to the mean values of the w(v) in the collections C 1 and C 0 ,
respectively. Note that, the DVCS also has the average contrast, and many
proposed DVCS's in the literature have = , see examples in [10, 5, 1],
etc. When comparing, the DVCS that has = then, in the overall view,
the clearness of the recovered secret image of the PVCS is the same as the
clearness of the recovered secret image of a DVCS. However, because of the
probabilistic nature, a PVCS is disadvantaged in displaying the details of the
original secret image, for example, a thin line in the original secret image is
likely to be displayed as a dotted line in a PVCS.

hl