Visual Cryptography and Random Grids - Visual Cryptography and Secret Image Sharing - page 183

Regarding a random grid R, the number of transparent pixels is probabilis-
tically the same as that of opaque ones so that the average light transmission
of R is also

1

2 , denoted as

1
2 :

T (R) = (7.3)

7.2.2 Superimposition of Random Grids
Let denote the generalized "or" operation, which describes the relation of
the superimposition of two random pixels or grids pixel by pixel. It is obvious
that rr (RR) is entirely the same as r (R), that is

1
2

1
2

t (rr) =

or T (RR) = (7.4)

for each pixel r in R.
Let R 1 and R 2 be two independent random grids with the same size. When
R 1 and R 2 are superimposed pixel by pixel, each pixel (either transparent or
opaque) in R 1 has an equal possibility to be stacked by a transparent pixel
or an opaque pixel in R 2 .We call r 1 = R 1 [i;j] the corresponding pixel of
r 2 = R 2 [i 0 ;j 0 ] if and only if i = i 0 and j = j 0 (the positions of r 1 at R 1 and r 2
at R 2 are the same). It is easy to see that the order of the two random grids
does not affect the superimposed result, i.e.,

R 1 R 2 = R 2 R 1 : (7.5)

Indeed, is a commutative operation.

TABLE 7.1
Results of the superimposition of two random
pixels.
r 1 r 2 r 1 r 2

0 0 0

0 1 1

1 0 1

1 1 1

Table 7.1 shows the superimposed results of the corresponding random
pixels r 1 and r 2 . There is only one outcome among the four possible combi-
nations of r 1 r 2 showing transparency. Since the four possible combinations
occur with an equal probability, the probability for r 1 r 2 to be transparent
is

1

4 . That is, the average light transmission of the superimposition of R 1 and
R 2 (r 1 and r 2 ) is

1

4 .

We summarize the aforementioned properties in Lemma 1.