CONTRAST-ENHANCED VISUAL CRYPTOGRAPHY...
Transcript of CONTRAST-ENHANCED VISUAL CRYPTOGRAPHY...
Chapter 3
CONTRAST-ENHANCED VISUAL CRYPTOGRAPHY SCHEMES
3.1 Introduction
Contrast is one of the most important parameters of visual
cryptography schemes as a performance measure. Usually, the
reconstructed secret image will be darker than the original secret image.
This chapter presents different methods to improve the contrast of visual
cryptography schemes. In the first method, Additional Basis Matrix
(ABM) is used to improve the contrast of the reconstructed secret image.
By using ABM for the white pixels, the contrast of the reconstructed
secret image can be improved significantly from the contrast levels in the
Noar and Shamir schemes. The second method presents a visual
cryptography scheme based on Perfect Reconstruction of White Pixels
(PRWP). The third method combines the above two techniques to
enhance the contrast of VCS. Experiments were also conducted based on
these different VCS to test and prove their efficiency.
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3.2 Contrast-Enhanced Visual Cryptography Schemes with
ABM1,2
3.2.1 The Model
In this method, one additional basis matrix is used to represent
new pixel patterns which can be represented as AS0. The basis matrix AS0
is used to share white pixels in the secret data. The AS0 can be defined by
an n x m Boolean matrix, AS0 = [asij], where
asij = 1 ⇔ the jth subpixel in the ith share is black.
asij = 0 ⇔ the jth subpixel in the ith share is white.
Formula 3.1 (Additional Relative Difference)
Let [sij0] be an n x m basis matrix and [asij
0] be an additional basis
matrix of the same order. Then,
α* = (α1 + α )/2
1 Thomas Monoth and Babu Anto P, Contrast-Enhanced Visual Cryptography Schemes Based on Additional
Pixel Patterns, Proc. of the IEEE International Conference on Cyber Worlds (CW 2010), NTU, Singapore,
pp. 171-178, 2010. (IEEE Computer Society). 2 Thomas Monoth and Babu Anto P, Achieving Optimal Contrast in Visual Cryptography Schemes Without
Pixel Expansion, International Journal of Recent Trends in Engineering (IJRTE), Vol.1, No.1, pp. 468-
471, 2009. (Academy Publisher, Finland ).
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where
α1 = (ωH(S1) – ωH(AS0 ))/ m
α = (ωH(S1) – ωH(S0 ))/ m
ωH (S1) is the hamming weight (the number of ones) of the m-
vector V of any k of the n rows in S1 and ωH (AS0 ) is the hamming
weight of the m-vector V of any k of the n rows in the additional basis
matrix AS0.
Formula 3.2 (Additional Contrast)
Let α* be the additional relative difference and m be the pixel
expansion. The formula to compute contrast in different VCS with ABM is
β* = α*.m , β* ≥ 1
3.2.2 The Construction of 2-out-of-2 VCS with ABM
The visual cryptography scheme based on ABM is explained based
on a 2-out-of-2 VCS with 4-subpixel layout. The new pixel patterns for
the method are shown in Table 3.1. By increasing the number of pixel
patterns for white pixels, the contrast of the reconstructed image can be
improved without adding any computational complexity.
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Table 3.1 The pixel patterns for 2-out-of-2 VCS with ABM
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The AS0 for 2-out-of-2 VCS can be designed according to the new
pixel layout as
AS0 = ⎥
⎦
⎤⎢⎣
⎡00010001
Therefore the collection of matrices C0 is obtained by permuting
the columns of matrix S0 plus permuting the columns of matrix AS0 and
the collection of matrices C1 is obtained by permuting the columns of
matrix S1. The matrices C0 and C1 can be designed as:
C0 = { π ⎥⎦
⎤⎢⎣
⎡01010101
, π ⎥⎦
⎤⎢⎣
⎡00010001
}
C1 = { π ⎥⎦
⎤⎢⎣
⎡01101001
}
The α* and β* of the reconstructed secret data in the VCS with
ABM method are computed as:
α* = 5/8
β* = 2.5
We have noted that the additional basis matrix (AS0) does not
satisfy the security conditions of the Noar and Shamir VCS. But this does
not affect the final security of the VCS with ABM, because it employs
random basis column pixel expansion for sharing black and white pixels.
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3.2.3 The Construction of 2-out-of-n VCS with ABM
The AS0 for 2-out-of-n visual cryptography scheme can be
designed by the following definition.
Definition 3.1 (Additional Basis Matrix for 2-out-of-n VCS):
Let AS0 be an n x m Boolean matrix for 2-out-of-n visual
cryptography scheme with ABM . Then AS0 is said to be an additional
basis matrix, if
(1) AS0 should have n - 2 columns of weight n and remaining
columns of weight zero.
The AS0 designed for 2-out-of-n VCS is illustrated by 2-out-of-3
VCS using the definition (3.1) as:
AS0 =
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
001001001
The basis matrix AS0 is used to share white pixels in the 2-out-of-3
VCS.
The matrices C0 and C1 can be considered as:
C0 = { π ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
011011011
, π ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
001001001
}
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C1 = { π
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
011101110
}
The α* and β* of the reconstructed secret data in this method are
computed as
α* = 1/2
β* = 1.5
3.2.4 The Construction of n-out-of-n VCS with ABM
The construction of AS0 for n-out-of-n visual cryptography scheme
is based on the following definition.
Definition 3.2 (Additional Basis Matrix for n-out-of-n VCS):
Let AS0 be an n x m Boolean matrix for n-out-of-n VCS with
ABM. Then AS0 is said to be additional basis matrix for n ≥3, if
1. asij0 = 0 ⇔ 1 ≤ i ≤ n and j < n.
2. asij0 = 1 ⇔ 1 ≤ i ≤ n and n≤ j ≤ 2n-1.
The AS0 for n-out-of-n VCS is expressed by 3-out-of-3 VCS using
the definition (3.2) as:
AS0 =
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
001100110011
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The collection of basis matrices C0 and C1 are
C0 = { π
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
110001101010
, π ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
001100110011
}
C1 = { π
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
110010101001
}
The relative difference and contrast for 3-out-of-3 VCS with ABM
is calculated as
α* = 3/8
β* = 1.5
3.2.5 The Construction of k-out-of-n VCS with ABM The k-out-of-n VCS with ABM, the AS0 can be designed according
to the following definition.
Definition 3.3 (Additional Basis Matrix for k-out-of-n VCS):
Let AS0 be an n x m Boolean matrix. Then AS0 is said to be an
additional basis matrix for k-out-of-n VCS, if
(1) asij0 = 1 ⇔ 1 ≤ i ≤ n and j ≤ ((m/2)+1)
(2) asij0 = 0 ⇔ 1 ≤ i ≤ n and ((m/2)+1) < j ≤ l*2k-1 .
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The k-out-of-n VCS with ABM can be best described by considering a 3-out-of-6 VCS case. The matrix AS0 for 3-out-of-6 VCS can be designed as:
AS0 =
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
000001111111000001111111000001111111000001111111000001111111000001111111
The matrices C0 and C1 can be designed as:
C0={ π
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
010100110110001101010110010101100011011001010011001101100101011000110101
, π
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
000000111111000000111111000000111111000000111111000000111111000000111111
}
C1 = { π
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
100110101100101010011100100111001010110010011010101011001001110010101001
}
The α* and β* for the scheme are
α* = 1/6
β* = 2
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3.2.6 The Experimental Results of VCS with ABM
3.2.6.1 The 2-out-of-2 VCS
The Figure 3.1 and 3.2 represents 2-out-of-2 VCS using ABM for
two different images. Consider the first Secret Image (SI1).
(a) (b)
(c) (d)
Figure 3.1 The 2-out-of-2 VCS with ABM of SI1: (a)SI1, (b) S1,
(c) S2, (d) S1+S2
For the second Secret Image (SI2),
(a) (b)
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(c ) (d)
Figure 3.2 The 2-out-of-2 VCS with ABM of SI2: (a) SI2, (b)
S1, (c) S2, and (d) S1+S2
Table 3.2 The details of the pixels in SI1 for the 2-out-of-2
VCS with ABM
Image No. of Columns
No. of Rows
No. of Black Pixels
No. of White Pixels
Total Pixels
Secret image 250 100 7510 17490 25000
Share1 + Share 2 250 100 13773 11227 25000
Table 3.3 The details of the pixels in SI2 for the 2-out-of-2
VCS with ABM
Image No. of
Columns No. of Rows
No. of Black Pixels
No. of White Pixels
Total Pixels
Secret image 200 200 16665 23335 40000
Share1 + Share 2 200 200 25186 14814 40000
3.2.6.2 The 2-out-of-n VCS
The figures 3.3 and 3.4 are shows the 2-out-of-3 VCS with ABM:
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(a) (b)
(c) (d)
(e) (f)
(g)
Figure 3.3 The 2-out-of-3 VCS with ABM of SI1: (a) SI1 (b) S1, (c)
S2, (d) S3, (e) S1+S2, (e) S1+S3, and (f) S2+S3
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(a) (b)
(c) (d)
(e) (f)
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(g)
Figure 3.4 The 2-out-of-3 VCS with ABM of SI2: (a) SI2 (b) S1, (c)
S2, (d) S3, (e) S1+S2, (e) S1+S3, and (f) S2+S3
Table 3.4 The details of the pixels in SI1 for the 2-out-of-3 VCS
with ABM
Image No. of Columns
No. of Rows
No. of Black Pixels
No. of White Pixels
Total Pixels
Secret image 250 100 7510 17490 25000
Share2 + Share 3 250 100 16541 8459 25000
Table 3.5 The details of the pixels in SI2 for the 2-out-of-3 VCS
with ABM
Image No. of Columns
No. of Rows
No. of Black Pixels
No. of White Pixels
Total Pixels
Secret image 200 200 16665 23335 40000
Share2 + Share 3 200 200 28460 11540 40000
3.2.6.3 The n-out-of-n VCS
The 3-out-of-3 VCS with ABM applied to two different images are
shown below.
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(a) (b)
(c) (d)
(e) (f)
(g) (h)
Figure 3.5 The 3-out-of-3 VCS with ABM of SI1: (a) SI1 (b) S1, (c) S2,
(d) S3, (e) S1+S2, (f) S1+S3, (g) S2+S3, and (h) S1+S2+S3
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(a) (b)
(c) (d)
(e) (f)
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(g) (h)
Figure 3.6 The 3-out-of-3 VCS with ABM of SI2: (a) SI2 (b) S1, (c)
S2,(d)S3,(e)S1+S2, (f) S1+S3,(g) S2+S3, and (h) S1+S2+S3
Table 3.6 The details of the pixels in SI1 for the 3-out-of-3 VCS
with ABM
Image No. of
Columns No. of Rows
No. of Black Pixels
No. of White Pixels
Total Pixels
Secret image 250 100 7510 17490 25000
Share1+ Share 2+
Share3 250 100 18085 6915 25000
Table 3.7 The details of the pixels in SI2 for the 3-out-of-3 VCS
with ABM
Image No. of
Columns No. of Rows
No. of Black Pixels
No. of White Pixels
Total Pixels
Secret image 200 200 16665 23335 40000
Share1+ Share 2+
Share3 200 200 30676 9324 40000
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3.2.6.4 The k-out-of-n VCS
The Figures represent the 3-out-of-6 VCS with ABM.
(a) (b)
(c) (d)
(e) (f)
(g) (h)
(i) (j)
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(k)
Figure 3.7 The 3-out-of-6 VCS with ABM of SI1: (a) SI1 (b) S1,(c)
S2, (d) S3, (e) S4, (f) S5, (g) S6, (h) S3+S4, (i) S1+S2+S3,
(j) S1+S2+S4, (k) S2+S3+S4
(a) (b)
(c) (d)
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(e) (f)
(g) (h)
(i) (j)
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(k)
Figure 3.8 The 3-out-of-6 VCS with ABM of SI2: (a) SI2 (b) S1,(c)
S2, (d) S3, (e) S4, (f) S5, (g) S6,(h) S3+S4, (i) S1+S2+S3,
(j) S1+S2+S4, (k) S2+S3+S4
Table 3.8 The details of the pixels in SI1 for the 3-out-of-6 VCS
with ABM
Image No. of Columns
No. of Rows
No. of Black Pixels
No. of White Pixels
Total Pixels
Secret image 100 250 7510 17490 25000
Share1+ Share2 +
Share3 100 250 14669 10331 25000
Table 3.9 The details of the pixels in SI2 for the 3-out-of-6 VCS
with ABM
Image No. of Columns
No. of Rows
No. of Black Pixels
No. of White Pixels
Total Pixels
Secret image 200 200 16665 23335 40000
Share1 + Share2 +
Share3 200 200 25197 14803 40000
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3.2.7 Analysis of Experimental Results in VCS with ABM
3.2.7.1 The Comparison of Relative Difference and Contrast
Comparison of the relative difference (α) and contrast (β) of Noar
and Shamir scheme with VCS using ABM are shown in the table 3.10.
Table 3.10 The contrast and relative difference of VCS with ABM
VCS Noar & Shamir VCS VCS with ABM α β α* β*
2-out-of-2t ½ 2 5/8 2.5
2-out-of-3 1/3 1 1/2 1.5
3-out-of-3f-3 1/4 1 3/8 1.5
3-out-of-6-ouf-4 1/12 1 1/6 2
The results in Table 3.10 show that the relative difference and
contrast of the VCS with ABM method are better compared to those of
the Noar & Shamir VCS.
3.2.7.2 The Graphical Representation of Pixels
Next, the contrast of the VCS with ABM based on pixel by pixel is
analysed with the help of graphs. Consider the pixels in the secret image
and reconstructed image for different VCS:
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Figure 3.9 The graphical representation of 2-out-of-2 VCS
with ABM using the image SI1
Figure 3.10 The graphical representation of 2-out-of-2 VCS
with ABM using the image SI2
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Figure 3.11 The graphical representation of 2-out-of-3 VCS
with ABM using the image SI1
Figure 3.12 The graphical representation of 2-out-of-3 VCS
with ABM using the image SI2
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Figure 3.13 The graphical representation of 3-out-of-3 VCS
with ABM using the image SI1
Figure 3.14 The graphical representation of 3-out-of-3 VCS
with ABM using the image SI2
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Figure 3.15 The graphical representation of 3-out-of-6 VCS
with ABM using the image SI1
Figure 3.16 The graphical representation of 3-out-of-6 VCS
with ABM using the image SI2
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By analyzing the graphs, one can see that the number of white
pixels in the secret image is greater than the number of black pixels in
both the secret images. But the number of black pixels in the
reconstructed secret images is greater than that of white pixels in Noar &
Shamir Scheme. From this it is very clear that increasing the black pixels
in the reconstructed images will reduce the contrast of the reconstructed
images. Therefore to enhance the contrast, number of white pixel should
be increased. This is guaranteed in the VCS with ABM method.
3.2.7.3 The Percentage of Increase in White Pixels
The comparisons between increasing and decreasing of white and
black pixels in Noar & Shamir VCS and VCS with ABM for two different
images are shown in Table 3.11.
Table 3.11 Percentage of decrease and increase in black and white
pixels
VCS
SI1 SI2
Noar & Shamir VCS VCS with ABM Noar & Shamir
VCS VCS with ABM
% of White Pixels
% of Black Pixels
% of White Pixels
% of Black Pixels
% of White Pixels
% of Black Pixels
% of White Pixels
% of Black Pixels
2-out-of-2t- 34.81 65.19 44.91 54.93 29.02 70.98 37.03 62.97
2-out-of-3 23.22 76.78 33.84 66.16 19.40 80.60 28.85 71.15
3-out-of-3f- 17.31 82.69 27.66 72.34 14.56 85.44 23.31 76.69
3-out-of-6- 29.76 70.24 41.32 58.68 27.43 72.57 37.01 62.99
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Analysing the data in Table 3.11, we see that the number of white
pixels in the VCS with ABM is increased by approximately 10%
compared to Noar & Shamir VCS in both the images. Increasing white
pixels in the reconstructed image will in turn increase the contrast. From
this it can be seen that VCS with ABM method provides better contrast in
both the images.
3.2.7.4 The Reconstructed Images
Finally compare and analyse the clarity of the reconstructed secret
images of Noar & Shamir visual cryptography scheme with that in the
VCS with ABM as shown in table 3.11.
Table 3.12 The reconstructed images in Noar & Shamir VCS
and VCS with ABM
VCS VCS with Naor & Shamir
Method VCS with ABM Method
2-out-of-2
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2-out-of-3
3-out-of-3
3-out-of-6
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From the table 3.12, one can see that the VCS with ABM method
produces improved clarity images than Noar and Shamir VCS.
3.3 Visual Cryptography Schemes with PRWP
The existing pixel patterns for the visual cryptography scheme are based
on the perfect reconstruction of black pixels (PRBP). Mathematically in
PRBP the white pixels are represented by 0 and the black pixel by 1. In the
usual binary image, the number of white pixels is much larger than the
number of black pixels. Therefore, the perfect reconstructions of black pixels
in visual cryptography schemes can decrease the contrast. Here, a visual
cryptography scheme which is focused on the perfect reconstruction of white
pixels (PRWP) and hence can provide better clarity is presented. As in the
case of all existing binary image file formats, PRWP represents white pixel
by 1 and black pixel by 0.
3.3.1 The Model
Let P = {1, . . . , n} be a set of elements called participants, and let
2P denote all the subsets of P. Let ΓQual 2P and ΓForb 2P, where ΓQual
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∩ ΓForb = Ø. Here the members of ΓQual are referred to as qualified sets
and the members of ΓForb are called forbidden sets. The pair (ΓQual, ΓForb)
is called the access structure of the scheme.
Define Γ0 to consist of all the minimal qualified sets:
Γ0 = {A ΓQual : B∉ ΓQual for all B A, B A }
The secret image consists of a collection of black and white pixels.
Each pixel appears in n versions called shares. Each share is a collection
of m black and white subpixels. The resulting structure can be described
by an n × m Boolean matrix S = [sij ] where
(sij) = 0 ⇔ the jth subpixel in the ith share is black.
(sij) = 1 ⇔ the jth subpixel in the ith share is white.
Therefore the gray level of the combined share, obtained by
stacking the transparencies i1, . . . , is, is proportional to the hamming
weight ωH (V) of the m-vector V = OR(ri1, . . . , ris), where ri1 , . . . , ris are
the rows of S associated with the transparencies stacked. This gray level
is interpreted by the visual system of the users as black or white in
according with some rule of contrast.
Definition 3.4 Let (ΓQual , ΓForb) be an access structure on a set of n
participants. Two collections of n × m Boolean matrices C0 and C1
constitute a visual cryptography scheme (ΓQual, ΓForb) VCS with PRWP if
there exists values β(m) and threshold 1 ≤ tX ≤ m satisfying:
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1. Any qualified set X = {i1, i2, . . . , ip} ΓQual can recover the
shared image by stacking their transparencies.
Formally, for any M C0, the “or” V of rows i1, i2, . . . , ip satisfies
ωH (V) ≥ tX – β(m); whereas, for any M C1, it results that ωH (V)
≤ tX.
2. Any nonqualified set X = {i1, i2, . . . , ip} ΓForb has no
information on the shared image.
Formally, the two collections of p×m matrices Dt, with t {0, 1},
obtained by restricting each n × m matrix in Ct to rows i1, i2, . . . ,
ip are indistinguishable in the sense that they contain the same
matrices with the same frequencies.
The first condition is related to the contrast of the image. The
number β(m) is referred to as the contrast of the image. The second
condition is security, which implies that by inspecting the shares of a
nonqualified subset of participants one cannot gain any advantage in
deciding whether the shared pixel was white or black.
3.3.2 The Construction of Basis Matrices
Let (ΓQual , ΓForb) be an access structure on a set of n participants.
A (ΓQual , ΓForb) VCS with PRWP with relative difference α(m), contrast
β(m) and threshold 1 ≤ tX ≤ m is realized using the n × m basis matrices
MS0 and MS1 if the following two conditions hold:
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1. If X = {i1, i2, . . . , ip} ΓQual is a qualified set, then the “or” V of
rows {i1, i2, . . . , ip} of MS0 satisfies ωH (V ) ≥ tX – β(m); whereas,
for MS1 it results that ωH (V ) ≤ tX.
2. If X = {i1, i2, . . . , ip} ΓForb is not a qualified set then the two
p×m matrices obtained by restricting MS0 and MS1 to rows {i1, i2,
. . . , ip} are equal up to a column permutation.
The collections C0 and C1 are obtained by permuting the columns
of the corresponding matrix (MS0 for C0 and MS1 for C1) in all possible
ways.
Formula 3.3 (Relative Difference):
Let ωH(MS0) and ωH (MS1) be the hamming weight corresponding
to the basis matrices MS0 and MS1. Then relative difference α(m) is
defined as:
α(m) = (ωH (MS0) – ωH (MS1))/ m
Formula 3.4(Contrast):
Let α(m) be the relative difference and m be the pixel expansion.
The formula to compute contrast in different VCS with PRWP is:
β(m) = α(m).m , β(m) ≥ 1
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3.3.3 The Construction of 2-out-of-2 VCS with PRWP
The basic idea of visual cryptography scheme with PRWP can be
explained by 2-out-of-2 VCS. The pixel layouts for the scheme are as
shown in table 3.13.
Table 3.13 The pixel layout for 2-out-of-2 VCS with PRWP Original Pixel Pixel Value Share1 Share2 Share1+ Share2
1
1
0
0
The basis matrices, MS0 and MS1 are:
MS0= ⎥⎦
⎤⎢⎣
⎡1001
MS1 = ⎥⎦
⎤⎢⎣
⎡0101
The relative difference α(m) and contrast β(m) can be computed as:
α(m) = ½
β(m) = 1
The matrices C0 and C1 are :
C0 = { ⎥⎦
⎤⎢⎣
⎡1001
, ⎥⎦
⎤⎢⎣
⎡0110
} and C1 = { ⎥⎦
⎤⎢⎣
⎡0101
, ⎥⎦
⎤⎢⎣
⎡1010
}
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While observing the basis matrices MS0 and MS1, S0 of VCS
becomes MS1 of PRWP scheme and S1 becomes MS0. Therefore, in VCS
with PRWP scheme, to share a white pixel, the dealer randomly selects
one of the matrices in C1, and to share a black pixel, the dealer randomly
selects one of the matrices in C0 of Noar & Shamir scheme. From the
results, both the relative difference and contrast of VCS with PRWP are
equal to that of Noar & Shamir scheme.
3.3.4 The Experimental Results of VCS with PRWP
For assessing the feasibility, some experiments were conducted
using 2-out-of-2, 2-out-of-3, 3-out-of-3 and 3-out-of-6 VCS with PRWP.
3.3.4 .1 The 2-out-of-2 VCS
The 2-out-of-2 VCS with PRWP applied to two different images is
shown below.
(a) (b)
(c) (d)
Figure 3.17 The 2-out-of-2 VCS with PRWP of SI1: (a) SI1, (b) S1, (c) S2 and (d) S1+S2
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(a) (b)
(c) (d)
Figure 3.18 The 2-out-of-2 VCS with PRWP of SI2: (a) SI2, (b) S1,
(c) S2 and (d) S1+S2
Table 3.14 The details of the pixels in SI1 for the 2-out-of-2 VCS
with PRWP
Image No. of
Columns No. of Rows
No. of Black Pixels
No. of White Pixels
Total Pixels
Secret image 250 100 7510 17490 25000
Share1 + Share 2 250 100 3719 21281 25000
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Table 3.15 The details of the pixels in SI2 for the 2-out-of-2 VCS
with PRWP
Image No. of
Columns No. of Rows
No. of Black Pixels
No. of White Pixels
Total Pixels
Secret image 200 200 16665 23335 40000 Share1 + Share 2 200 200 8365 31635 40000
3.3.4.2 The 2-out-of-n VCS
The Figures 3.19 and 3.20 are obtained by using 2-out-of-3 VCS
with PRWP.
(a) (b)
(c) (d)
(e) (f)
(g)
Figure 3.19 The 2-out-of-3 VCS with PRWP of SI1: (a) SI1 (b) S1, (c) S2, (d) S3, (e) S1+S2, (f) S1+S3, and (g) S2+S3
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(a) (b)
(c) (d)
(e) (f)
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(g)
Figure 3.20 The 2-out-of-3 VCS with PRWP of SI2: (a) SI2 (b) S1,
(c) S2, (d) S3, (e) S1+S2, (f) S1+S3, and (g) S2+S3
Table 3.16 The details of the pixels in SI1 for the 2-out-of-3 VCS
with PRWP
Image No. of
Columns No. of Rows
No. of Black Pixels
No. of White Pixels
Total Pixels
Secret image 250 100 7510 17490 25000
Share1+ Share 2 250 100 2437 22563 25000
Share 1 + Share 3 250 100 2437 22563 25000
Share2 + Share 3 250 100 2437 22563 25000
Table 3.17 The details of the pixels in SI2 for the 2-out-of-3 VCS
with PRWP
Image No. of Columns
No. of Rows
No. of Black Pixels
No. of White Pixels
Total Pixels
Secret image 200 200 16665 23335 40000
Share1+ Share 2 200 200 5449 34551 40000
Share 1+ Share 3 200 200 5449 34551 40000
Share2 + Share 3 200 200 5449 34551 40000
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3.3.4.3 The n-out-of-n VCS
The 3-out-of-3 VCS with PRWP using two different images are
given below:
(a) (b)
(c) (d)
(e) (f)
(g) (h)
Figure 3.21 The 3-out-of-3 VCS with PRWP of SI1: (a) SI1 (b) S1, (c)
S2, (d) S3, (e) S1+S2, (f) S1+ S3, (g) S2+S3, and (h)
S1+S2+S3
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(a) (b)
(c) (d)
(e) (f)
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(g) (h)
Figure 3.22 The 3-out-of-3 VCS with PRWP of SI2: (a) SI2 (b) S1,
(c) S2, (d) S3, (e) S1+S2, (f) S1 +S3, (g) S2+S3, and (h)
S1+S2+ S3
Table 3.18 The details of the pixels in SI1 for the 3-out-of-3 VCS
with PRWP
Image No. of
Columns No. of Rows
No. of Black Pixels
No. of White Pixels
Total Pixels
Secret image 250 100 7510 17490 25000
Share1 + Share
2+ Share3 250 100 1886 23114 25000
Table 3.19 The details of the pixels in SI2 for the 3-out-of-3 VCS
with PRWP
Image No. of
Columns No. of Rows
No. of Black Pixels
No. of White Pixels
Total Pixels
Secret image 200 200 16665 23335 40000
Share1 + Share 2 +
Share3 200 200 4210 35790 40000
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3.3.4.4 The k-out-of-n VCS
The Figures 3.23 and 3.24 depict 3-out-of-6 VCS applied to two
different images.
(a) (b)
(c) (d)
(e) (f)
(g) (h)
(i) (j)
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(k)
Figure 3.23 The 3-out-of-6 VCS with PRWP of SI1: (a) SI1 (b)
S1,(c) S2, (d) S3, (e) S4, (f) S5, (g) S6, (h) S3+S4, (i)
S1+S2+ S3, (j) S1+S2+ S4, (k) S2+S3+S4
(a) (b)
(c) (d)
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(e) (f)
(g) (h)
(i) (j)
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(k)
Figure 3.24 The 3-out-of-6 VCS with PRWP of SI2: (a) SI2 (b)
S1,(c)S2,(d)S3,(e) S4, (f)S5,(g)S6,(h)S3+S4,(i) S1+S2+S3,
(j) S1+S2 +S4, (k)S2+S3+S4
Table 3.20 The details of the pixels in SI1 for the 3-out-of-6 VCS
with PRWP
Image No. of Columns
No. of Rows
No. of Black Pixels
No. of White Pixels
Total Pixels
Secret image 100 250 7510 17490 25000
Share1 + Share2 +
Share3 100 250 5428 19572 25000
Share1 + Share2 +
Share4 100 250 5318 19682 25000
Share2 + Share3 +
Share4 100 250 5381 19619 25000
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Table 3.21 The details of the pixels in SI2 for the 3-out-of-6 VCS
with PRWP
Image No. of
Columns No. of Rows
No. of Black Pixels
No. of White Pixels
Total Pixels
Secret image 200 200 16665 23335 40000
Share1 + Share2 +
Share3 200 200 9729 30271 40000
Share1 + Share2 +
Share4 200 200 9768 30232 40000
Share2 + Share3 +
Share4 200 200 9808 30166 40000
3.3.5 Analysis of Experimental Results in VCS with PRWP
This section focuses on comparing the number of pixels in the
reconstructed images of VCS with PRWP and Noar & Shamir scheme
with the help of graphs.
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Figure 3.25 The graphical representation of 2-out-of-2 VCS
with PRWP using the image SI1
Figure 3.26 The graphical representation of 2-out-of-2 VCS
with PRWP using the image SI2
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Figure 3.27 The graphical representation of 2-out-of-3 VCS
with PRWP using the image SI1
Figure 3.28 The graphical representation of 2-out-of-3 VCS
with PRWP using the image SI2
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Figure 3.29 The graphical representation of 3-out-of-3 VCS
with PRWP using the image SI1
Figure 3.30 The graphical representation of 3-out-of-3 VCS
with PRWP using the image SI2
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Figure 3.31 The graphical representation of 3-out-of-6 VCS
with PRWP using the image SI1
Figure 3.32 The graphical representation of 3-out-of-6 VCS
with PRWP using the image SI2
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The graphs (Figure 3.25 to 3.32) show that the number of white
pixels in the secret image is greater than the number of black pixels in
both the secret images (SI1 and SI2). In the reconstructed secret images by
using Noar & Shamir scheme, the number of black pixels is larger than
the number of white pixels, which in turn reduces the contrast. In order to
retain the contrast, the change of number of white (black) pixels in the
original image to the reconstructed secret image should be as small as
possible. In VCS with PRWP method, the rate of this change can reduced
to a considerable extent. From this one can reach the conclusion that the
VCS with PRWP method gives a clearer image than Noar & Shamir
scheme.
3.3.5.1 The Comparison of the Reconstructed Images
Finally the clarity of the reconstructed images in Noar & Shamir
scheme and VCS based on PRWP is compared by considering 2-out-of-2,
2-out-of-3, 3-out-of-3 and 3-out-of-6 VCS.
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Table 3.22 The Comparison of reconstructed images between Noar
& Shamir VCS and VCS with PRWP
VCS VCS with Naor & Shamir Method VCS with PRWP Method
2-out-of-2
2-out-of-3
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3-out-of-3
3-out-of-6
From the table 3.22, we find that VCS with PRWP method
achieves more clear images than Noar and Shamir VCS.
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3.4 Contrast-Enhanced VCS based on PRWP with ABM
3.4.1 The Model
The visual cryptography scheme with PRWP can improve the
clarity of reconstructed images. But by analysing the experimental results,
we know that the number of black pixels in the reconstructed image is
very less compared to the original image. Therefore increasing the black
pixels in the reconstructed image can improve the contrast. In order to
increase the black pixels use additional basis matrix (ABM) to represent
the new pixel pattern. The ABM for VCS with PRWP is represented by
AS1. The matrix AS1 is used to share black pixels in the secret image. The
AS1 can be defined by an n x m Boolean matrix, AS1 = [asij1], where
asij1 = 1 ⇔ the jth subpixel in the ith share is white.
asij1 = 0 ⇔ the jth subpixel in the ith share is black.
Formula 3.5(Additional Relative Difference for PRWP with ABM )
Let [sij1] be an n x m basis matrix and [asij
1] be an additional basis
matrix of the same order. Then,
α (m)* = (α2 + α(m) )/2
where
α2 = (ωH(MS0) – ωH (AS1 ))/ m
α(m) = (ωH (MS0) – ωH (MS1))/ m
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where ωH(MS0) is the hamming weight (the number of ones) of the
m-vector V of any k of the n rows in MS0 and ωH (AS1) is the hamming
weight of the m-vector V of any k of the n rows in the additional basis
matrix AS1.
Formula 3.6 (Additional Contrast for PRWP with ABM)
Let α(m)* be the additional relative difference for PRWP and m be
the pixel expansion. The formula to compute contrast in different VCS
with PRWP and ABM is
β(m)* = α(m)* .m , β(m)* ≥ 1
3.4.2 The Construction of 2-out-of-2 VCS based on PRWP with ABM
The visual cryptography scheme based on PRWP with ABM is
illustrated by a 2-out-of-2 VCS with 4-subpixel layout. The new pixel
patterns for the method are shown in Table 3.23. By increasing the
number of pixel patterns for black pixels, the contrast of the reconstructed
image can be improved without adding any computational complexity.
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Table 3.23 The 4-subpixel layouts for 2-out-of-2 VCS based on PRWP with ABM
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The AS1 for 2-out-of-2 VCS can be designed according to new
pixel layout as
AS1 = ⎥
⎦
⎤⎢⎣
⎡00010001
The matrices C0 and C1 can be designed as:
C0 = { π ⎥⎦
⎤⎢⎣
⎡01101001
}
C1 = { π ⎥⎦
⎤⎢⎣
⎡01010101
, π ⎥⎦
⎤⎢⎣
⎡00010001
}
The α(m)* and β(m)* are calculated as
α(m)* = 5/8
β(m)* = 2.5
3.4.3 The Experimental Results
For evaluating the feasibility of the scheme, some experiments
were conducted using 2-out-of-2, 2-out-of-3, 3-out-of-3 and 3-out-of-6
VCS.
3.4 .3.1 The 2-out-of-2 VCS based on PRWP with ABM
The figures represent 2-out-of-2 VCS of two different secret
images using PRWP with ABM.
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(a) (b)
(c) (d)
Figure 3.33 The 2-out-of-2 VCS based on PRWP with ABM of SI1:
(a) SI1, (b) S1, (c) S2 and (d) S1+ S2
(a) (b)
(c) (d)
Figure 3.34 The 2-out-of-2 VCS based on PRWP with ABM of SI2: (a) SI2, (b) S1, (c) S2 and (d) S1+S2
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Table 3.24 The details of the pixels in SI1 for the 2-out-of-2 VCS
based on PRWP with ABM
Image No. of
Columns No. of Rows
No. of Black Pixels
No. of White Pixels
Total Pixels
Secret image 250 100 7510 17490 25000
Share1 + Share 2 250 100 4785 20215 25000
Table 3.25 The details of the pixels in SI2 for the 2-out-of-2 VCS
based on PRWP with ABM
Image No. of Columns
No. of Rows
No. of Black Pixels
No. of White Pixels
Total Pixels
Secret image 200 200 16665 23335 40000
Share1 + Share 2 200 200 10654 29346 40000
3.4.3.2 The 2-out-of-n VCS based on PRWP with ABM
The 2-out-of-3 VCS based on PRWP with ABM using two
different images are:
(a) (b)
(c) (d)
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(e) (f)
(g)
Figure 3.35 The 2-out-of-3 VCS based on PRWP with ABM of SI1: (a)
SI1 (b) S1, (c) S2, (d) S3, (e) S1+S2, (f) S1+S3, and (g) S2+S3
(a) (b)
(c) (d)
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(e) (f)
(g)
Figure 3.36 The 2-out-of-3 VCS based on PRWP with ABM of SI2:
(a) SI2 (b) S1, (c) S2,(d)S3, (e) S1+S2, (f) S1+S3, and (g)
S2+S3
Table 3.26 The details of the pixels in SI1 for the 2-out-of-3 VCS
based on PRWP with ABM
Image No. of
Columns No. of Rows
No. of Black Pixels
No. of White Pixels
Total Pixels
Secret image 250 100 7510 17490 25000
Share1+ Share 2 250 100 3695 21305 25000
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Table 3.27 The details of the pixels in SI2 for the 2-out-of-3 VCS
based on PRWP with ABM
Image No. of
Columns No. of Rows
No. of Black Pixels
No. of White Pixels
Total Pixels
Secret image 200 200 16665 23335 40000
Share1+ Share 2 200 200 8176 31824 40000
3.4.3.3 The n-out-of-n VCS based on PRWP with ABM
Figure 3.37 and 3.38 show a 3-out-of-3 VCS applied to two
different images using PRWP with ABM:
(a) (b)
(d) (c)
(e) (f)
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(g) (h)
Figure 3.37 The 3-out-of-3 VCS based on PRWP with ABM of SI1:
(a) SI1 (b) S1, (c) S2, (d) S3, (e) S1+S2, (f) S1+S3, (g)
S2+S3, and (h) S1+S2+S3
(a) (b)
(c) (d)
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(e) (f)
( g) (h)
Figure 3.38 The 3-out-of-3 VCS based on PRWP with ABM of SI2:
(a) SI2 (b) S1, (c) S2, (d) S3, (e)S1+S2, (f) S1+S3,(g)
S2+S3, and (h) S1+S2+S3
Table 3.28 The details of the pixels in SI1 for the 3-out-of-3 VCS
based on PRWP with ABM
Image No. of Columns
No. of Rows
No. of Black Pixels
No. of White Pixels
Total Pixels
Secret image 250 100 7510 17490 25000
Share1 + Share
2+ Share3 250 100 2943 22057 25000
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Table 3.29 The details of the pixels in SI2 for the 3-out-of-3 VCS
based on PRWP with ABM
Image No. of
Columns No. of Rows
No. of Black Pixels
No. of White Pixels
Total Pixels
Secret image 200 200 16665 23335 40000
Share1 + Share 2
+ Share3 200 200 6672 33328 40000
3.4.3.4 The k-out-of-n VCS based on PRWP with ABM
The Figures 3.39 and 3.40 depict 3-out-of-6 VCS based on PRWP
with ABM:
(a) (b)
(c) (d)
(e) (f)
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(g) (h)
(i) (j)
(k)
Figure 3.39 The 3-out-of-6 VCS based on PRWP with ABM of SI1: (a)
SI1 (b) S1,(c) S2, (d) S3, (e) S4, (f) S5 , (g) S6 ,(h) S3+S4,
(i)S1+S2+S3, (j) S1+S2+S4, (k) S2+S3+S4
(a) (b)
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(c) (d)
(e) (f)
(g) (h)
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(i) (j)
(k)
Figure 3.40 The 3-out-of-6 VCS based on PRWP with ABM of SI2:
(a) SI2 (b) S1,(c)S2, (d)S3, (e)S4,(f) S5, (g) S6, (h)
S3+S4, (i) S1+S2 +S3, (j) S1+S2+S4, (k) S2+S3+S4
Table 3.30 The details of the pixels in SI1 for the 3-out-of-6 VCS
based on PRWP with ABM
Image No. of Columns
No. of Rows
No. of Black Pixels
No. of White Pixels
Total Pixels
Secret image 100 250 7510 17490 25000
Share1 + Share2 +
Share3 100 250 6709 18291 25000
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Table 3.31 The details of the pixels in SI2 for the 3-out-of-6 VCS
based on PRWP with ABM
Image No. of
Columns No. of Rows
No. of Black Pixels
No. of White Pixels
Total Pixels
Secret image 200 200 16665 23335 40000
Share1 + Share2 +
Share3 200 200 12432 27568 40000
3.4.4 Analysis of Experimental Results
For the analysis of experimental results, first compare the relative
difference (α(m)) and contrast (β(m)) of VCS based on PRWP and VCS
based on PRWP with ABM.
Table 3.32 The contrast and relative difference of VCS with PRWP
and VCS based on PRWP with ABM
VCS VCS with PRWP
VCS with PRWP and ABM
α(m) β(m) α(m)* β (m)* 2-out-of-2t-of-2 ½ 2 5/8 2.5
2-out-of-3 1/3 1 ½ 1.5
3-out-of-3f-3 1/4 1 3/8 1.5
3-out-of-6-ouf-4 1/12 1 1/6 2
The results in the Table 3.32 shows that the relative difference and
contrast of the VCS with PRWP with ABM method are better compared
to those of the VCS with PRWP.
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3.4.4.1 The Graphical Representation of the Pixels
Consider the VCS based on PRWP and VCS based on PRWP with
ABM scheme based on pixel by pixel approach with the help of graphs.
Figure 3.41The graphical representation of 2-out-of-2 VCS based
on PRWP with ABM using the image SI1
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Figure 3.42 The graphical representation of 2-out-of-2 VCS based
on PRWP with ABM using the image SI2
Figure 3.43 The graphical representation of 2-out-of-3 VCS based
on PRWP with ABM using the image SI1
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Figure 3.44 The graphical representation of 2-out-of-3 VCS based
on PRWP with ABM using the image SI2
Figure 3.45 The graphical representation of 3-out-of-3 VCS based
on PRWP with ABM using the image SI1
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Figure 3.46 The graphical representation of 3-out-of-3 VCS based
on PRWP with ABM using the image SI2
Figure 3.47 The graphical representation of 3-out-of-6 VCS based
on PRWP with ABM using the image SI1
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Figure 3.48 The graphical representation of 3-out-of-6 VCS based
on PRWP with ABM using the image SI2
By analyzing the graphs (Figure 3.41 to 3.48), one can see that the number of white pixels is greater than that of black pixels in the secret images. But the black pixels are reduced and white pixels are increased significantly in the reconstructed secret images by using VCS with PRWP. Therefore, to enhance the contrast of reconstructed secret images requires increasing the black pixels and decreasing the white pixels. This is achieved in VCS based on PRWP with ABM.
3.4.4.2 The Reconstructed Images
Finally investigate the clarity of the reconstructed images in VCS based on PRWP and PRWP with ABM using 2-out-of-2, 2-out-of-3, 3-out-of-3 and 3-out-of-6 VCS.
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Table 3.33 The comparison of reconstructed images between VCS
with PRWP and VCS based on PRWP with ABM
VCS VCS with PRWP VCS with PRWP and ABM
2-out-of-2
2-out-of-3
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3-out-of-3
3-out-of-6
From the table 3.33, it is clear that the VCS based on PRWP with
ABM scheme achieves contrast-enhanced images than VCS with PRWP
scheme.
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3.5 Conclusion
This chapter presents two new methods for contrast-enhanced
visual cryptography schemes. These methods are explained and
implemented with examples. The contrast of the presented visual
cryptography schemes and traditional VCS are compared here. Using
these methods, some experiments were also conducted based on different
VCS. These results are analysed by using tables and graphs and are also
compared with the features of existing visual cryptography schemes.