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.

Chapter-3

Department of Information Technology, Kannur University 52

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 ).

Contrast-Enhanced Visual Cryptography Schemes


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.

Chapter-3


Table 3.1 The pixel patterns for 2-out-of-2 VCS with ABM



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.

Chapter-3


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

}



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

Chapter-3


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 .



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

Chapter-3


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)



(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:

Chapter-3


(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



(a) (b)

(c) (d)

(e) (f)

Chapter-3


(g)


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


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


with ABM


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.



(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

Chapter-3


(a) (b)

(c) (d)

(e) (f)



(g) (h)


S2,(d)S3,(e)S1+S2, (f) S1+S3,(g) S2+S3, and (h) S1+S2+S3


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


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

Chapter-3


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)



(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)

Chapter-3


(e) (f)

(g) (h)

(i) (j)



(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


with ABM


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


with ABM


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

Chapter-3


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:



Figure 3.9 The graphical representation of 2-out-of-2 VCS

with ABM using the image SI1



Chapter-3








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

Chapter-3


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



2-out-of-3

3-out-of-3

3-out-of-6

Chapter-3


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



∩ 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:

Chapter-3


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:



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

Chapter-3


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

}



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

Chapter-3


(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


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




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

Chapter-3


(a) (b)

(c) (d)

(e) (f)



(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


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


with PRWP


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

Chapter-3


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



(a) (b)

(c) (d)

(e) (f)

Chapter-3


(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


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


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



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)

Chapter-3


(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)



(e) (f)

(g) (h)

(i) (j)

Chapter-3


(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


with PRWP


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




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.

Chapter-3



with PRWP using the image SI1



Chapter-3






Chapter-3


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.



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

Chapter-3


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.



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

Chapter-3


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.



Table 3.23 The 4-subpixel layouts for 2-out-of-2 VCS based on PRWP with ABM

Chapter-3


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.



(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

Chapter-3



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




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)



(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)

Chapter-3


(e) (f)

(g)


(a) SI2 (b) S1, (c) S2,(d)S3, (e) S1+S2, (f) S1+S3, and (g)

S2+S3



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





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)

Chapter-3


(g) (h)


(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)



(e) (f)

( g) (h)


(a) SI2 (b) S1, (c) S2, (d) S3, (e)S1+S2, (f) S1+S3,(g)

S2+S3, and (h) S1+S2+S3




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

Chapter-3




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)



(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)

Chapter-3


(c) (d)

(e) (f)

(g) (h)



(i) (j)

(k)


(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




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

Chapter-3




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.



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

Chapter-3


Figure 3.42 The graphical representation of 2-out-of-2 VCS based




Chapter-3










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.

Chapter-3


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



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.

Chapter-3


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.

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