Steganography in Halftone Images - Visual Cryptography and Secret Image Sharing - page 354

128 u 2 (i;j); u 2 (i;j) < 128
127 u 2 (i;j); u 2 (i;j) 128

u(i;j) = (13.16)

We further note that adding a distortion u(i;j) to u 2 (i;j) may be inter-
preted as a distortion to the original multitone image pixel x2(i, 2 (i;j). Dening
x(i;j) = u(i;j), the input to the normal quantizer u 0 2 (i;j) may be written
as

u 0 2 (i;j) = u 2 (i;j) + u(i;j)
= x 2 (i;j) + X
k;l2N

e 2 (ik;j l) h(k;l) + u(i;j)

= x 0 2 (i;j) + X
k;l2N

e 2 (ik;j l) h(k;l) (13.17)

x 0 2 (i;j) = x 2 (i;j) + u(i;j) = x 2 (i;j) + x(i;j)

(13.18)

The interpretation of (13.18) is that the output halftone image using
DHCED in fact represents X 0 2 , not X 2 , in the sense of that it can be ob-
tained from X 0 2 directly using standard error diusion. Thus, jxj can be
treated as a measure of the distortion introduced by the DHCED process. To
control this distortion, we define a threshold T that determines whether or not
the pixel should be toggled, allowing a trade-off between distortion and visual
quality of the watermarked halftone image. If jxj is less than T, the pixel
toggling will be performed, and vice versa. If T decreases, the visual quality
of the watermarked image Y 2 will improve at the price of lower contrast of the
secret pattern when the two halftone images are superimposed.
Consider any location (i;j) 2 W w . If X 1 and X 2 are the same image,
DHCED would copy y 1 (i;j) to y 2 (i;j) so that Y 2 values are effectively obtained
by applying error diffusion to X 1 . The error e 2 (i;j) will be computed as in
normal error diffusion. When Y 1 and Y 2 are overlaid, the regular error diffused
value y1(i, 1 (i;j) will be revealed. The overlaying operation will reveal a local
intensity similar to the local intensity of X 1 , which is typical for regular error
diffusion.
If X 1 and X 2 are different, DHCED would not force y2(i, 2 (i;j) to be identical
to y 1 (i;j) at (i;j) 2 W w . Instead, it merely treats y1(i, 1 (i;j) as the favored value
of y2(i, 2 (i;j). And DHCED performs the same operation as in W b .
The proposed DHCED for the case of identical X 1 and X 2 is simulated.
Using Lena as the test image and Figure 13.5 as the secret binary pattern W,
Figure 13.4 is Y 1 and Figure 13.9 is Y 2 generated using DHCED with respect
to Y 1 and W. A threshold of T = 10 is used. Note that Figure 13.9 looks like
Figure 13.4 which verifies that DHCED can give halftone images with good
visual quality. Figure 13.10 shows the image obtained by overlaying Figure
13.9 and Figure 13.4. The secret pattern W is clearly visible in Figure 13.7
verifying that DHSED is an effective visual cryptography method.