Steganography in Halftone Images - Visual Cryptography and Secret Image Sharing - page 358

Consider the case when X 1 and X 2 are the same image X. Consider a rectan-
gular region R of constant intensity A in X. Suppose that the left half of R is
in the white region W w of W and the right half in the black region W b of W.
We assume that error diffusion is effective in both the left and right halves of
R such that the average image intensity is preserved. Thus, the probability
distribution of a halftone pixel y 1 (i;j) in the region R of Y 1 is

P[y 1 (i;j) = 255] = A=255 (13.19)

P[y 1 (i;j) = 0] = (255 A)=255 (13.20)

such that the expected value is

E[y 1 (i;j)] = 0 P[y 1 (i;j) = 0] + 255 P[y 1 (i;j) = 255] = A (13.21)

for (i;j) 2 R. Typically, the percentage of white and black pixels in Y 1 in R
are A=255 100% and (255 A)=255 100% respectively, distributed evenly
in R.
The generation of Y 2 using DHCED is effectively the application of error
diffusion to the equivalent noisy multitone image. And the error diffusion
should be eective, as usual. The distortion x(i;j) introduced by DHCED
is equally likely to be positive and negative and its magnitude is bounded
by T. The average intensity in R should still be approximately A. Thus, the
probability distribution of a halftone pixel y 2 (i;j) in the region R of Y 2 is
approximately

P[y 2 (i;j) = 255] A=255 (13.22)

P[y 2 (i;j) = 0] (255 A)=255 (13.23)

such that

E[y 2 (i;j)] = 0 P[y 2 (i;j) = 0] + 255 P[y 2 (i;j) = 255] A (13.24)

for (i;j) 2 R A .

Let y(i;j) = y 1 (i;j) T y 2 (i;j) be the output pixel obtained by overlaying
Y 1 and Y 2 . The y(i;j) would be white only if both y 1 (i;j) and y 2 (i;j) are
white. For the proposed DHCED, the y2(i, 1 (i;j) and y 2 (i;j) are designed to be
dependent.
In the left half of R, which is in W w , DHCED would simply copy y1(i, 1 (i;j)
as y 2 (i;j) such that P[y 2 (i;j) = y 1 (i;j)] = 1. Then
P[y(i;j) = 255] = P[y 1 (i;j) \ y 2 (i;j) = 255] = P[y 1 (i;j) = 255] = A=255
(13.25)

such that

E[y(i;j)] = 0 P[y(i;j) = 0] + 255 P[y(i;j) = 255] = A (13.26)

In other words, y(i;j) = y 1 (i;j)) = y 2 (i;j) for the region W w and the overlay
operation does not change the halftone pixel values at all.