Extended Visual Cryptography for Photograph Images - Visual Cryptography and Secret Image Sharing - page 109

4.4.2 Random-Dot Shadow Images
Due to the severe tradeoff between relative differences of the shadow image
and reconstructed image, some studies focused on the quality of the recon-
structed image by giving up extended capability. For instance, in 2000, Blundo
et al. examined a secret sharing scheme with an access structure that can re-
construct a gray-scale image with g gray levels and specified the upper limit of
relative differences 1 ; ; g1 as well as the lower limit of pixel expansion
m as below [4]:

1
(g 1)2 k1 ; m (g 1)2 k1 :

minf 1 ; ; g1 g

Iwamoto and Yamamoto precisely discussed an (n;n) secret sharing scheme
for a gray-scale image in 2002 [15].
Another important concern is pixel expansion. Both studies explained
above accomplish gray levels by controlling the number of white/black subpix-
els, which means pixel expansion is inevitable. The research for reducing the
pixel expansion in VSSS is mostly based on a probabilistic approach such as
Random Grids revisited by Shyu [37]. Since Random Grids have been already
explained in Section 4.2.3, here we discuss another kind of probabilistic ap-
proach. Ito et al. proposed a secret sharing scheme with m = 1 by introducing
randomness into a conventional (k;n) VSSS in 1999 [14]. Their scheme rst
determines a basis matrix according to the value of a secret pixel, black or
white. Then it randomly selects one of a column of the basis matrix, which
stands for values of corresponding subpixels in shadow images, and uses them
as those of shadow pixels. Chen et al. proposed the similar scheme in 2007 [6].
They also proposed to use histogram equalization for enhancing contrast. We
will explain this contrast enhancement technique later in Section 4.4.7.

4.4.3 Similar Shadow Images
Another approach is entirely opposite to the studies explained in the previ-
ous section. It limits the relative difference of secret image to nearly zero and
attempts to enhance the relative difference of shadow images as much as pos-
sible. Some research has succeeded in achieving a full relative difference for
shadow images, namely, S ' 1, by using very similar shadow images. The
key of this approach is the reconstructed secret image. The secret image can
be observed with the trace of a shadow image. In other words, by overlapping
very similar shadow images, one can observe obscure dark logos or text within
the shadow image.
This type of scheme was first proposed by Oka et al. in 1996 as a water-
marking technique [33]. One can conceal one's signature within a halftoned
image so that one can claim one's copyright on the image. It uses multiple sub-
pixels, i.e., density patterns, to represent a gray pixel of the original shadow
image. The second shadow image is generated by rearranging dots, i.e., black