Steganography in Halftone Images - Visual Cryptography and Secret Image Sharing - page 366

and y 1 (i;j) and y 2 (i;j) are approximately independent with

P[y(i;j) = 255] = P[y 1 (i;j) \y 2 (i;j) = 255]
= P[y 1 (i;j) = 255] P[y 2 (i;j) = 255] = (A=255) 2

(13.40)

P[y(i;j) = 0] = 1 (A=255) 2 (13.41)

such that

E[y(i;j)] = A 2 =255

(13.42)

Thus, the intensity of DHSED is expected to be greater than or equal to that
of DHCED, with equality if A = 255. The difference in average intensity is

intensity = E[y(i;j)j(i;j) 2 W w ] E[y(i;j)j(i;j) 2 W b ]
= AA 2 =255 (13.43)

The contrast is

contrast = E[y(i;j)j(i;j) 2 W w ] E[y(i;j)j(i;j) 2 W b ]
E[y(i;j)j(i;j) 2 W w ]
= AA 2 =255
A

255 A
255

= (13.44)

We also simulate DHSED on Ramp to verify the equations above. The
hidden pattern is still the Column in Figure 13.15. The DHSED generated Y 1
and Y 2 are shown in Figures 13.16 and 13.23 , respectively. In Figure 13.28,
the row-wise average intensity of Y 1 and Y 2 are plotted against the average
intensity of the corresponding row in X. As expected, they are very similar
to X, which verifies (13.39).
Figure 13.24 is the overlaid image Y . Similar to DHCED, the Y of DHSED
is identical to Y 1 and Y 2 in the W w region, with intensity decreasing from top
to bottom. In the W b region, the intensity of Y is as high as X at the top
row. But as the intensity of X decreases towards the bottom of the image,
the intensity of Y decreases faster in the center than on the two sides. In
other words, the intensity of Y decreases faster in W b than in W w , making
W b visible in Y .
Similar to DHCED, to show the exact behavior of DHSED in W b in Y , we
compute the average intensity in the W b for each row of Y and plot it against
the average intensity of the corresponding row in X in Figure 13.25. Three
curves are obtained for 3 values of L, namely L = 1; 5; 10. Also shown is the
theoretical curve according to (13.42). It can be observed that the 3 empirical
curves match the theoretical curve very well. This also suggests that the choice
of L does not have much effect on the intensity of W b in Y .
We also compute the difference between the average intensity in W b and
W w regions for each row of Y , and plot this intensity against the average
intensity of the corresponding row in X in Figure 13.26. Similarly, the contrast
is computed and plotted in Figure 13.27. Also shown in the two figures are