Visual Cryptography from Halftone Error Diffusion - Visual Cryptography and Secret Image Sharing - page 5

to generate mutually exclusive sets of pixels. To this end, the method of error
diffusion is modified so as to produce multitone output pixels where the pixels
of each tone are assigned to a pixel set. Multitone error diffusion is obtained
by simply replacing the thresholding block by a multilevel quantization block
in halftone error diffusion. The number of output levels of the quantization
block is the same as the number of tones of the multitone image [15]. Multitone
error diffusion can generate multitone images where the pixels of each tone are
homogeneously distributed. The multitone error diffusion algorithm proposed
in [4] is used here for the generation of mutually exclusive pixel sets. This
algorithm jointly optimizes the distribution of multitone pixels by locating the
pixels of different tones in a correlated fashion so that the mutual interference
between different tones is minimized and multitone pixels are well separated
from each other. Refer to [4] for details.
1.2 Visual Secret Sharing
We provide a brief description on visual cryptography where the key concepts
will be referenced in subsequent sections. Please refer to [19, 2] for more details
on VSS.
1.2.1 Notion and Formal Definitions
To illustrate the principles of VSS, consider a simple 2-out-of-2 VSS scheme
shown in Figure 1.3. Each pixel p taken from a secret binary image is encoded
into a pair of black and white subpixels in each of the two shares. If p is
white/black, one of the first/last two columns tabulated under the white/black
pixel in Figure 1.3 is selected. The selection is random such that each column
is selected with a 50% probability. Then, the first two subpixels in that column
are assigned to share 1 and the following two subpixels are assigned to share
2. Independent of whether p is black or white, p is encoded into two subpixels
of black-white or white-black with equal probabilities. Thus, an individual
share gives no clue as to whether p is black or white [26, 25]. Now consider
the superposition of the two shares as shown in the last row of Figure 1.3.
If the pixel p is black, the superposition of the two shares outputs two black
subpixels corresponding to a gray level 1. If p is white, it results in one white
and one black subpixel, corresponding to a gray level 1/2. Then by stacking
two shares together, we can obtain the full information of the secret image.
Figure 3.24 shows an example of the application of the 2-out-of-2 VSS
scheme. Figure 1.4(a) shows a secret binary image SI to be encoded. Ac-
cording to the encoding rule shown in Figure 1.3, each pixel p of SI is split
into two subpixels in each of the two shares, as shown in Figure 1.4(b) and
Figure 1.4(c). Superimposing the two shares leads to the output secret im-