Visual Cryptography from Halftone Error Diffusion - Visual Cryptography and Secret Image Sharing - page 11

As another example, consider a 2-out-of-3 halftone VSS scheme where a
secret image pixel is encoded into a halftone cell of size q = 12. The number
of SIPs is = 6. By Eqn. 1.4, the optimal number of ABPs is x = 4. Thus,
the configuration matrix T of the 6 non-SIPs is given by:

2 3

1 1 1 1 M M

4 5 :

T = 1 1 M M 1 1 (1.6)

M M 1 1 1 1

From the content of T, it is concluded that 1=3 of the pixels on a share will
be ABPs.

1.3.2 Distribution of SIPs and ABPs
The locations of the SIPs do not depend on the share images or the secret im-
age, but only on the HVC expansion q and the underlying VSS scheme. Thus,
the distribution of SIPs can be generated prior to the generation of halftone
shares. For security purposes, the SIPs should be randomly distributed. To
achieve good image quality, it is also desirable to distribute the SIPs homo-
geneously so that one SIP is maximally separated from its neighboring SIPs.
Since the SIPs are maximally separated, the quantization error caused by an
SIP will be diffused away before the next SIP is encountered leading to visu-
ally pleasing halftone shares. Similarly, the distribution of ABPs can also be
determined a priori. As SIPs, ABPs should be distributed as homogenously
as possible and maximally separated from each other. Since there is a strong
correlation between the distribution of SIPs and the distribution of ABPs, the
distributions of SIPs and ABPs should be optimized jointly to avoid low fre-
quency spectral interference among them [4]. The SIPs and ABPs should also
be maximally separated from each other. The jointly optimized distributions
of SIPs and ABPs are generated based on a method of blue noise multitoning
as follows [4].
We first construct a constant grayscale image with gray level g =

P i=0 g i z i , where g i is a tone arbitrarily chosen between 0 and 1 and gi i 6= g j
for i 6= j. zi i is the pixel density for the pixels with tone gi. i . The value of zi i ,
together with w, depends on q, and the structure of the conguration matrix
T. By using the blue noise multitone error diffusion, an output image with
w+1 tones is produced. The distribution of pixels with tone gi i indicates a pixel
distribution denoted by Z i . Let z 0 = =q, then Z 0 indicates the distribution
of SIPs. The distribution of ABPs in a share is a subset of fZ i g;i = 1;:::;w.
An example following the previous 2-out-of-3 halftone VSS example is used
to illustrate how to set zi, i , g i , and w, where a secret image pixel is encoded
into a halftone cell of size q = 12. Among the q pixels, = 6 SIPs are
characterized by basis matrix S 0 or S 1 ; q = 6 non-SIPs are characterized
by the configuration matrix T. Thus, the matrix Ri, i , i = 0; 1, is constructed