Two-Decoding-Option Image Sharing Method - Visual Cryptography and Secret Image Sharing - page 462

For simplicity, we first describe the proposed TiOISSS, which reconstructs a
lossless secret image. The compressible version is just an easy extension of the
lossless TiOISSS, and will be discussed in Section 17.4.2.

17.4.1 Design Concept
Our design concept adopts the gray subpixel into the VCS, and this grayscale
values simultaneously represents the output of the (k 1)-degree polynomial
in PISSS. In VCS, it is evident that when a subpixel is stacked by the white
subpixel, its intensity is kept unchanged. While stacking two gray subpixels,
we get a grayer color (a dark version of the color). Therefore, if we replace black
subpixels with gray subpixels in the shadow, we still can use the whiteness in
every m subpixel to distinguish the black color from the white color in the
reconstructed image. Here, we adopt the widely accepted definition of color
superimposition in [10] to dene the color mixing function C() when stacking
two subpixels with the grayscale values between 0 and 255.
The grey level of the resultant pixel by stacking the two pixels can be
expressed (approximately) as follows, in which each mixed color is produced
by a color mixing function C().

g 3 = C (g 1 ;g 2 ) = Int ((g 1 g 2 )/255) ;

where Int() function maps a real number to the nearest integer. The values
of g 1 , g 2 , g 3 are any grayscale values between 0 and 255, and "0" (respectively
"255") is a black (respectively, white) color.
It is easy to verify g 3 < gi 1 and g 3 < gi 2 , and this implies that stacking
two gray pixels of g 1 and g 2 results in a grayer pixel of g 3 . For example,
g 1 = C (g 1 ; 255) shows the grayscale value unchanged when stacking with the
white pixel and 255 = C (255; 255) shows that the stacked result is a white
color when stacking two white pixels.
Example 3. Consider Example 1, and randomly use gray subpixels gi i 2
[0; 255] instead of black subpixels in B 1 and B 0 and do not change the white
subpixel.
As a replacement in B 1 and B 0 , we have B 0 1 =

g i 0
0g j

g i 0
g j 0

and B 0 0 =

,
respectively. In the reconstructed image, it is observed that the stacked result
in the black area (using B 0 1 ) is gi i 1g j and the stacked result in the white area
(using B 0 0 ) is gi k 1W or 1W1g k , where g k < gi i and g k < gi j . Through the
whiteness, we can still visually reveal the secret. Each shadow contains 1g i 1W
(or 1W1g i ) which is a gray-and-white and noise-like image, so that one cannot
see anything from any shadow. In Figure 17.2(a) and Figure 17.2(b) are two
noise-like shadows (Shadow 1, Shadow 2) and Figure 17.2(c) shows the recon-
structed image (Shadow 1 + Shadow 2). It is observed that the secret VCS
is also revealed but is a little blurred when compared with Figure 17.1( d ).

2
We call this VCS with the matrices B 0 1 and B 0 0 the gray-subpixel based
VCS (GVCS). Matrices B 0 1 and B 0 0 are the same as B 1 and B 0 except that the