Steganography in Halftone Images - Visual Cryptography and Secret Image Sharing - page 349

tained by referencing both Y 1 and W. Let W b be the collection of the locations
of all the black pixels in W, and W w the collection of the white pixel locations.
In constructing Y 2 , we will force the pixel value at all locations belonging to
W w to be identical to Y 1 . In other words, values at these locations are merely
copied from Y 1 to Y 2 . That is,

y 2 (i;j) = y 1 (i;j) 8(i;j) 2 W w (13.10)

For the remaining pixels in W b , Y 2 needs to look natural and thus DHSED
applies error diffusion with the same error kernel. However, DHSED seeks to
make Y 2 different from Y 1 statistically so that when they are overlaid, pixels
in W b would tend to be darker. To achieve this, DHSED first morphologically
dilate, W b with a structuring element D consisting of a (2L + 1) (2L + 1)
matrix. We denote the dilated W b as C.

C = W b D (13.11)

which can be interpreted as a L-pixel expansion of W b in all directions.
For the pixels outside C, DHSED copies Y 2 from Y 1 using (13.10) but
forces the error for Y 2 to be zero, i.e. i.e. 2 (i;j) = 0 for (i;j) 62 C. Note that the
error for Y 1 are nonzero in general for the same locations.
Let E = CW b = C\W w be the "border" of the secret pattern, obtained
by removing W b from the expanded region C. For the pixels in E, (13.1) and
(13.2) are still applied while (13.3) and (13.4) are not. (13.10) will be used to
replace (13.3) since E W w and Y 2 pixels in W w are copied from Y 1 . We will
use (13.12) to replace (13.4).

e 2 (i;j) = maxfminfu 2 (i;j) y 2 (i;j); 127g;127g (13.12)
which is basically (13.4) with a limiter. As Y 2 pixels in E are copied from Y 1
with artificial zero error outside C, there are chances that u2(i, 2 (i;j)y 2 (i;j) is
outside 127. The limiter would then help to make the e 2 (i;j) reasonable.
For the pixels in W b , DHSED uses regular error diffusion to generate Y 2
so that region W b in Y 2 still has the same characteristic texture as regular
error diusion. But the "phase" of the texture will be dierent compared to
the corresponding region in Y 1 since the error outside the region C is different
in Y 1 and Y 2 .
The overlaying operation is equivalent to applying the logical AND oper-
ation between images Y 1 and Y 2 . Since the pixels in region W w of Y 1 and Y 2
have been forced to be identical, the overlaid pixel values are simply the reg-
ular error diffused pixel values. However, in region W b , although the texture
in Y 1 and Y 2 maintains the same characteristic, there is an artificially intro-
duced phase shift such that collocated Y 1 and Y 2 pixels tend to be statistically
independent. As a result, the overlaying operation tends to give darker local
intensity thus revealing the secret pattern W. A more detailed analysis of this
will be given in Section 13.4.
Using Lena as the test image and Figure 13.5 as the secret binary pattern