Two-Decoding-Option Image Sharing Method - Visual Cryptography and Secret Image Sharing - page 465

/* Phase 1: does not need computation; visually decode the secret I 0 by
stacking k shadows;
Phase 2: needs computation; decode the secret I by using Lagrange inter-
polation */
Phase 1 (preview phase):
2-1) G -1 (G i 1 ;G i 2 ;:::;G i k );
/* stack k shadows to visually preview the halftone secret image */
Phase 2 (perfect-reconstruction phase):
2-2) Obtain (Pi1 i 1 ;P i 2 ;:::;P i k ) from (Gi1 i 1 ;G i 2 ;:::;G i k ) by discarding the
white subpixels.
2-3) I = P -1 (P i 1 ;P i 2 ;:::;P i k ).
/* use Lagrange interpolation to reconstruct the gray-level secret image

*/

Our TiOISSS contains two decoding phases: the preview phase and the
perfect-reconstruction phase. A halftone secret image can be visually pre-
viewed by simply stacking shadows in Phase 1. The preview phase may be
used when a computer is temporarily not available, or in a scenario verifying
whether the shadows are correct or not in a distributed multimedia system,
as mentioned in the introduction. On the other hand, by extracting the gray
subpixels from shadows we may perfectly reconstruct the gray-level secret
image when the computer finally is available (or after successful verification
in a distributed multimedia system). We call the second decoding phase the
perfect-reconstruction phase because we can gain a lossless secret image.
Considering security, the proposed (k;n)-TiOISSS is a combination of two
(k;n)-threshold schemes: the GVCS and the PISSS. So our scheme still retains
the threshold property. An attacker could not stack less than k shadows to
retrieve the black-and-white secret. Also, he cannot use the gray values of less
than k shadows to reconstruct the (k1)-degree polynomial. Thus, combining
these two ISSSs together assures the secrecy of the threshold scheme.
Example 4 shows the proposed (2, 2)-TiOISSSs using different w and m
to demonstrate different shadow sizes and the resolutions of the reconstructed
images in the preview phase.
Example 4. Construct two (2, 2)-TiOISSSs using base matrices: (1) B 1 =

10
01

10
10

110
101

110
110

, respectively.
The secret image is 512 512 Lena from the USC-SIPI image database.
A 512 512 gray-level Lena image I is shown in Figure 17.3(a). By (2,
2)-PISSS, we obtain two 512 256 gray-level noise-like shadows P 1 and P 2 ,
as shown in Figure 17.3(b) and Figure 17.3(c).

and B 0 = (2) B 1 = and B 0 =

10
01

10
10

Case (1) B 1 = , B 0 = :

Since m = 2 and g = 1, we then resize and halftone I to get a halftone
image I 0 by H(), and the size is jI 0 j = jIj=(k g) = jIj=2. This 512 256
halftone Lena is shown in Figure 17.4(a). Output two shadows G i =G(I 0 ),
i =1, 2, by using (2, 2, 2, 1)-GVCS, and the values of gray subpixels are chosen