WATERMARKING IMAGES USING 2D REPRESENTATIONS

OF SELF-INVERTING PERMUTATIONS

Maria Chroni, Angelos Fylakis and Stavros D. Nikolopoulos

Department of Computer Science, University of Ioannina, GR-45110 Ioannina, Greece

Keywords:

Watermarking Techniques, Image Watermarking Systems, Intellectual Property Rights, Color Images,

Self-inverting Permutations, 2D-representations of Permutations, Encoding, Decoding, Algorithms.

Abstract:

In this work we propose an efﬁcient and easily implemented codec system, which we named WaterIMAGE,

for watermarking images that are intended for uploading on the web and making them public online. An

important fact of our system is that it suggests a way in which an integer number can be represented in a two

dimensional grid and, thus, since images are two dimensional objects that representation can be efﬁciently

marked on them. In particular, our system uses an efﬁcient technique which is based on a 2D representation of

self-inverting permutations and mainly consists of two components: the ﬁrst component contains an encoding

algorithm which encodes an integer w into a self-inverting permutation π

∗

and a decoding algorithm which

extracts the integer w from π

∗

, while the second component contains codec algorithms which are responsible

for embedding a watermark into an image I, resulting the image I

, and extract it from I

. Our system

incorporates important properties which allow us to successfully extract the watermark w from the image I

even if the input image has been compressed with a lossy method and/or rotated. All the system’s algorithms

have been developed and tested in JAVA programming environment.

1 INTRODUCTION

Internet technology, becomes day by day an indis-

pensable tool for everyday life since most people use

it on a regular basis and do many daily activities on-

line (Garﬁnkel, 2001). As a consequence, transferring

digital informationvia the Internet, such as audio, pic-

tures, video, or software, has also become very popu-

lar during the last years.This frequent use of internet

means that measures taken for internet security are

indispensable since the web is not risk-free. One of

those risks is the fact that the web is an environment

where intellectual property is under threat.And that’s

where watermarks come to place.

Watermarking. Watermarks are symbols which are

placed into physical objects such as documents, pho-

tos and bank notes and their purpose is to carry in-

formation about an object’s authenticity. In our case

the watermarks have digital form and they are embed-

ded into digital objects, this technique is called digital

watermarking.

The watermarking problem can be described as

the problem of embedding a watermark w into an ob-

ject I and, thus, producing a new object I

, such that

w can be reliably located and extracted from I

even

after I

has been subjected to transformations (Coll-

berg and Nagra, 2010); for example, compression in

case the object is an image.

Motivation. We believe that protecting intellectual

material on the web is one of the major issues con-

cerning the proper use of the internet. Digital images

are a very characteristic part of this material found

online and our target is to make people feel free to

upload their images without hesitating because of the

fear of their work being unauthorized used.

Watermarking is the ideal solution for protecting

your property of the images and keeping them visi-

ble to the public as well, so research towards imper-

ceptible secure and robust image watermarking tech-

niques is vital. As mentioned, there are already var-

ious methods that can achieve that, but every single

method requires attention and that’s because we can

not discriminate a speciﬁc method as the best. Every

case has its ideal solution and the same idea applies

for image watermarking.

Contribution. In this work we present an efﬁcient

and easily implemented codec system, which we

named WaterIMAGE, for watermarking images that

we are interested in uploading in the web and mak-

380

Chroni M., Fylakis A. and D. Nikolopoulos S..

WATERMARKING IMAGES USING 2D REPRESENTATIONS OF SELF-INVERTING PERMUTATIONS.

DOI: 10.5220/0003936003800385

In Proceedings of the 8th International Conference on Web Information Systems and Technologies (WEBIST-2012), pages 380-385

ISBN: 978-989-8565-08-2

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

ing them public online; this way web users are now

enabled to consider how to protect their own images.

2 BASIC TOOLS

In this section we present basic tools which are used

by our image watermarking system. In particular,

we ﬁrst describe discrete structures, namely, permuta-

tions, self-inverting permutations, and bitonic permu-

tations, and then brieﬂy outline a codec system (en-

coding/decoding algorithms) which encodes an inte-

ger w into a self-inverting permutation π

∗

and extracts

it from π

∗

. Finally, we propose a 2D representation of

permutations and give a 3D representation of color

images.

2.1 Self-inverting Permutations

Informally, a permutation of a set of objects S is an

arrangement of those objects into a particular order,

while in a formal (mathematical) way a permutation

of a set of objects S is deﬁned as a bijection from S

to itself (i.e., a map S → S for which every element of

S occurs exactly once as image value)(Sedgewick and

Flajolet, 1996; Golumbic, 1980).

Hereafter, we shall say that π

∗

is a permutation

over the set N

Deﬁnition 2.1.1. Let π = (π

,π

,...,π

) be a permu-

tation over the set N

, where n > 1. The inverse of the

permutation π is the permutation τ = (q

,...,q

)

with q

= π

= i. A self-inverting permutation (or,

for short, SIP) is a permutation that is its own inverse:

= i.

By deﬁnition, every permutation has a unique in-

verse, and the inverse of the inverse is the original

permutation. Clearly, a permutation is a SIP (self-

inverting permutation) if and only if all its cycles are

of length 1 or 2; hereafter, we shall denote a 2-cycle as

c = (x,y) and an 1-cycle as c = (x), or, equivalently,

c = (x, x).

The permutation π

∗

= (5,6, 9, 8,1,2,7, 4, 3) is a

SIP with cycles: (1,5), (2,6), (3,9), (4,8), and (7,7).

2.2 Encoding Numbers as SIPs

Next, we present an algorithm for encoding an inte-

ger w into a self-inverting permutation π

∗

and an al-

gorithm for extracting w from π

∗

; both algorithms run

in O(n) time, where n is the length of the binary rep-

resentation of the integer w (author’s algorithms). The

encoding process uses the notion of Bitonic Permuta-

tions which we brieﬂy describe below.

Bitonic Permutations. The key-object in our algo-

rithm for encoding integers as self-inverting permu-

tations is the bitonic permutation: a permutation π =

(π

,π

,...,π

) over the set N

is called bitonic if ei-

ther monotonically increases and then monotonically

decreases, or else monotonically decreases and then

monotonically increases. For example, the permuta-

tions π

= (1,4,6, 7, 5,3,2) and π

= (6,4,3, 1, 2,5,7)

are both bitonic.

Our encoding algorithm uses only bitonic permu-

tations that monotonically increase and then mono-

tonically decrease. Let π be such a bitonic permuta-

tion over the set N

and let π

, π

i+1

be the two con-

secutive elements of π such that π

> π

i+1

. Then,

the sequence X = (π

,π

,...,π

) is called ﬁrst in-

creasing subsequence of π and the sequence Y =

(π

i+1

,π

i+2

,...,π

) is called ﬁrst decreasing subse-

quence of π.

We next give some notations and terminology

we shall use throughout the encoding and decod-

ing process. Let w be an integer number. We de-

note by B = b

···b

the binary representation of

w. If B

= b

···b

and B

= d

···d

be two

binary numbers, then the number B

||B

is the bi-

nary number b

···b

···d

. The binary se-

quence of the number B = b

···b

is the sequence

∗

= (b

,...,b

) of length n.

Let B = b

···bn be a binary number. Then,

flip(B) = b

′

···b

′

is the binary number such that

′

= 0 (1 resp.) if and only if b

= 1 (0 resp.),

1 ≤ i ≤ n.

Algorithm W-to-SIP: We brieﬂy outline an algo-

rithm for encoding an integer as self-inverting permu-

tation. Our algorithm, which we call Encode W-to-

SIP, takes as input an integer w, computes the binary

representation b

···b

of w, and then produces a

self-inverting permutation π

∗

in O(n) time (Chroni

and Nikolopoulos, 2010).

Algorithm SIP-to-W: Having presented the encod-

ing algorithm Encode W-to-SIP, let us now present

an extraction algorithm, that is, an algorithm for de-

coding a self-inverting permutation. More precisely,

our extraction algorithm, which we call Decode SIP-

to-W, takes as input a self-inverting permutation π

∗

produced by the algorithm Encode W-to-SIP and re-

turns its corresponding integer w. The time complex-

ity of the decode algorithm is also O(n), where n is the

length of the permutation π

∗

(Chroni and Nikolopou-

los, 2010).

2.3 2DM Representations

Given a permutation π over the set N

= {1,2,...,n},

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381

3 4

Figure 1: A 2DM representation of the self-inverting per-

mutation π = (5,6,9,8,1,2,7,4,3).

we ﬁrst deﬁne a two-dimensional representation (2D-

representation) of the permutation π that is useful for

studying properties which help us to deﬁne, later, a

more suitable representation of π for efﬁcient use in

our watermarking system.

In this representation, the elements of the permu-

tation π = (π

,π

,...,π

) are mapped in speciﬁc cells

of an n× n matrix A as follows:

• integer i −→ entry A(π

−1

,i)

or, equivalently,

• the cell at row i and column π

is labeled by the

number π

, for each i = 1,2,...,n.

Based on the previous 2D representation of a per-

mutation, we next propose a two-dimensional marked

representation (2DM representation) of a permutation

which is an efﬁcient tool for watermarking images.

In our 2DM representation, a permutation π over

the set N

= {1,2,...,n} is represented by an n × n

matrix A

∗

as follows:

• the cell at row i and column π

is marked by a

speciﬁc symbol, for each i = 1,2,...,n.

Figure 1 shows the 2DM representation of the permu-

tation π. Note that, as in the 2D representation, there

is also one symbol in each row and in each column of

the matrix A

∗

We next present an algorithm which extracts the

permutation π from its 2DM representation matrix.

More precisely, let π be a permutation over N

and

let A

∗

be the 2DM representation matrix of π (see,

Figure 1); given the matrix A

∗

, we can easily extract

π from A

∗

in linear time (in the size of matrix A

∗

) by

the following algorithm:

Algorithm Extract π from 2DM

Input: the 2DM representation matrix A

∗

of π;

Output: the permutation π;

1. For each row i of matrix A

∗

, 1 ≤ i ≤ n, do:

ﬁnd the marked cell and let j be its column;

set π

← j;

2. Return the permutation π;

Remark 2.3.1. It is easy to see that the resulting per-

mutation π, after the execution of Step 1, can be taken

by reading the matrix A

∗

from top row to bottom

row and write down the positions of its marked cells.

Since the permutation π is a self-inverting permuta-

tion, its 2D matrix A has the following property:

• A(i, j) = j if π

= j, and

• A(i, j) = 0 otherwise, 1 ≤ i, j ≤ n.

Thus, the corresponding matrix A

∗

is symmetric:

• A

∗

(i, j) = A

∗

( j,i) = “mark” if π

= j, and

• A

∗

(i, j) = A

∗

( j,i) = 0 otherwise, 1 ≤ i, j ≤ n.

Based on this property, it is also easy to see that the

resulting permutation π can be also taken by reading

the matrix A

∗

from left column to right column and

write down the positions of its marked cells.

2.4 Color Images

A digital image is a numeric representation of a 2-

dimensional image; it has a ﬁnite set of values, called

picture elements or pixels, that represent the bright-

ness of a given color at any speciﬁc point in the im-

age.

In our system we use the RGB model, where the

name comes from the initials of the three additive col-

ors Red, Green and Blue. The range of colors can be

represented on the Cartesian 3-dimensional system.

The axes x, y and z are used for the red green and

blue color respectively

In our system, since a color is a triple of integers

(x,y,z), a digital image I of resolution N × M (i.e., it

contains N rows and M columns of pixels) is stored in

a three-dimensional matrix Img of size N × M × 3 as

follows:

if the pixel I(i, j) of the image I has (x,y,z)

color, then Img(i, j,1) = x, Img(i, j,2) = y, and

Img(i, j,3) = z.

3 OUR IMAGE WATERMARKING

SYSTEM

Having proposed an efﬁcient method for encoding in-

tegers as self-inverting permutations using the bitonic

property of a permutation, and the 2DM representa-

tion of self-inverting permutations, we next describe

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382

the two main algorithms of our image watermark-

ing system; the encoding algorithm Encode SIP-to-

IMAGE which encodes a self-inverting permutation

∗

, corresponding to watermark w, into an image I

resulting the watermarked image I

and the decoding

algorithm Decode IMAGE-to-SIP which extracts the

permutation π

∗

from the image I

3.1 Embed Watermark into Image

We next describe the algorithm Encode SIP-to-

IMAGE of our codec system which embeds a self-

inverting permutation (SIP) π

∗

into an image I; recall

that, in our system we use a SIP π

∗

over the set N

∗

for

encoding the watermark w, where n

∗

= 2n+1 and n is

the length of the binary representation of the integer

w (author’s technique); see, Subsection 2.2.

The algorithm takes as input a SIP π

∗

and an

image I, in which the user wants to embed the

watermark w = π

∗

, and produces the watermarked

image I

; it works as follows:

Step 1: The algorithm ﬁrst computes the 2DM

representation of the permutation π

∗

, that is, it

computes the n

∗

× n

∗

array A (see, Subsection 2.3);

the entry (i, π

∗

) of the array A contains the symbol

“*”, 1 ≤ i ≤ n

∗

Step 2: Next, the algorithm computes the size N × M

of the input image I and do the following: if N is an

even number it removes the pixels from the bottom

row of I and reduces N by 1, while if M is an even

number it removes the pixels from the right column

of I and reduces M by 1. The resulting image has size

∗

× M

∗

, where N

∗

and M

∗

are both odd numbers.

Step 3: Let n

∗

be the size of the SIP π

∗

and let N

∗

≤

∗

. Now the algorithm takes the input image I and

places on it an imaginary grid G , which covers almost

the whole image I, having

∗

× n

∗

grid-cells C

(I)

each C

(I) of size

⌊N

∗

⌋ × ⌊N

∗

⌋

where, 1 ≤ i, j ≤ n

∗

It places the imaginary grid G on I as follows:

it ﬁrst locates the central pixel P

cent

of the image I,

which is at position (⌊N

∗

/2⌋ + 1, ⌊M

∗

/2⌋ + 1), then

locates the central pixel p

of the central grid-cell

(I), where i = ⌊n

∗

/2⌋ + 1, and places the grid G

on image I such that both P

cent

and p

have the same

position in I.

Step 4: Then it scans the image and goes to each grid-

cell C

(I) (there are always n

∗

× n

∗

grid-cells in any

image) and locates the central pixel p

of the grid-

cell C

(I) and also the four pixels p

, p

, and

around it, 1 ≤ i, j ≤ n

∗

; hereafter, we shall call

these four pixels cross pixels.

Then, it computes the difference between the

brightness of the central pixel p

and the average

brightness of the twelve pixels around it, that is, the

pixels p

ℓ1

, p

ℓ2

, and p

ℓ3

(ℓ = 1, 2,3,4), and stores this

value in the variable dif(p

) (see, Figure 2).

Finally, it computes the maximum absolute value

of all n

∗

× n

∗

differences dif(p

), 1 ≤ i, j ≤ n

∗

, and

stores it in the variable Maxdif(I).

Step 5: The algorithm goes again to each central pixel

of each grid-cellC

and if the corresponding entry

A(i, j) contains the symbol “*”, then it increases

• the brightness k

of the central pixel p

, and

• the brightness k

, k

, and k

of its cross pix-

els.

Actually, it ﬁrst increases the central pixel p

by the

value e

so that it surpasses the image’s maximum

difference Maxdif(I) by a constant c; that is,

• k

+ e

= Maxdif(I) + c

and, then, it sets the brightness of the four cross pixels

, p

, and p

equal to k

In our system we use c = 5, and thus the bright-

ness k

of the central pixel of each grid-cell C

increased by e

, where

= Maxdif(I) − k

+ 5 (1)

where, 1 ≤ i, j ≤ n

∗

Let I

be the resulting image after increasing the

brightness of the n

∗

central and the corresponding

cross pixels, with respect to π, of the image I. Here-

after, we call the n

∗

central pixels of I as 2DM-pixels;

recall that, p

is a 2DM-pixel if A(i,π

) = “*”, or,

equivalently, the cell (i,π

) of the matrix A is marked.

Step 6: The algorithm returns the watermarked image

3.2 Extract Watermark from Image

Next we describe our decoding algorithm which is

responsible for extracting the watermark w = π

∗

form

image I

. In particular, the algorithm, which we call

Decode IMAGE-to-SIP, takes as input a watermarked

image I

and returns the SIP π

∗

which corresponds

to integer watermark w; the steps of the algorithm are

the following:

WATERMARKINGIMAGESUSING2DREPRESENTATIONSOFSELF-INVERTINGPERMUTATIONS

383

grid-cell

Figure 2: The brightness k

ℓ

of the central and cross pixels

ℓ

of the grid-cell C

(I), 0 ≤ ℓ ≤ 4, and the brightness k

ℓm

of the cycle-cross pixels p

ℓm

, 1 ≤ ℓ ≤ 4 and m = 1,2,3.

Step 1: The algorithm places again the same imagi-

nary n

∗

× n

∗

grid on image I

and locates the central

pixel p

of each grid-cell C

(I), 1 ≤ i, j ≤ n

∗

; there

are n

∗

× n

∗

central pixels in total. Then, it ﬁnds the

∗

central pixels p

, p

,..., p

∗

, among the n

∗

× n

∗

with the maximum brightness using a known sorting

algorithm (Cormen et al., 2001).

Step 2: In this step, the algorithm takes the n

∗

grid-

cell C

,...,C

∗

of the image I

which correspond

to n

∗

central pixels p

, p

,..., p

∗

, and compute an

∗

× n

∗

matrix A

∗

as follows:

• Initially, set A

∗

(i, j) ← 0, 1 ≤ i, j ≤ n

∗

;

• For each grid-cell C

, 1 ≤ m ≤ n

∗

, do:

if (i, j) is the position of the grid-cell C

in the

grid G then set A

∗

(i, j) ← “∗ ”;

It is easy to see that, the n

∗

× n

∗

matrix A

∗

is exactly

the 2DM representation of the self-inverting permu-

tation π

∗

embedded in image I

by the algorithm En-

code SIP-to-IMAGE.

Then, the permutation π

∗

can be extracted

from the matrix A

∗

using the algorithm Ex-

tract π from 2DM; see, Subsection 2.3.

Step 3: Finally, the algorithm returns the self-

inverting permutation π

∗

4 PERFORMANCE

Our image watermarking system mainly consists of

four algorithms, each of which is responsible for a

particular codec operation:

• Encode W-to-SIP: algorithm for encoding an in-

teger w into a self-inverting permutation π

∗

;

• Decode SIP-to-W: algorithm for extracting w

from π

∗

;

• Encode SIP-to-IMAGE: algorithm for encoding a

self-inverting permutation π

∗

into an integer I;

• Decode IMAGE-to-SIP: algorithm for extracting

∗

form the watermarked image;

The two algorithms that are considered the basic ones

for our system are those responsible for embedding a

SIP into an image and extracting the SIP from it.

We next discuss some issues concerning the per-

formance of our image watermarking system. In par-

ticular, we mainly focus on the embedding algorithm

Encode SIP-to-IMAGE and the efﬁciency of water-

marking image I

produced by this algorithm, and

also on important properties of the n

∗

× n

∗

matrix A

∗

which stores the 2DM representation of a SIP. Finally

we show the time and space performance of our sys-

tem by computing the complexity of their algorithms.

It is worth noting that our system incorporates

such properties which allow us to successfully extract

the watermark w from the image I

even if the in-

put image I

of algorithm Decode IMAGE-to-SIPhas

been compressed with a lossy method and/or rotated.

We have evaluated the embedding and extracting

algorithms by testing them on various and different in

characteristics images that were initially in JPEG for-

mat and we had positive results as the watermark was

successfully extracted at every case even if the image

was converted back into JPEG format. What is more,

the method is open to extensions as the same method

might be used with a different marking procedure part

of the Encode SIP-to-IMAGE algorithm.

All the system’s algorithms have been initially

developed and tested in MATLAB (Gonzalez et al.,

2003) and then redeveloped and also tested in JAVA.

Compression. The experimental results show that the

watermark w is “well hidden” in the image I

. We be-

lieve that it is because we mark the image by chang-

ing the difference between the brightness of the 2DM-

pixels p

of the n

∗

× n

∗

imaginary grid and its 12

neighborhood pixels around it, that is, the pixels p

ℓ1

ℓ2

, and p

ℓ3

, for ℓ = 1,2,3,4 (see, Figure 2 and also

Step 2 of the embedding algorithm Encode SIP-to-

IMAGE); recall that, we also set the brightness of the

four cross pixels of each 2DM-pixel p

, that is, the

pixels p

, p

, and p

, to be equal to the bright-

ness of the 2DM-pixel p

Note that, we change the brightness of the 2DM-

pixels by increasing them so that they surpass the im-

age’s maximum difference Maxdif(I) by a constant c,

where in our implementation c = 5. We add ﬁve be-

cause if we compress the image the values of the pix-

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384

els may slightly change, and we want our watermark

to be robust. We also believe that this technique de-

spite being simple, it is efﬁcient because the bright-

ness of each of the n

∗

marked central pixels does

not have a great difference from the brightness of the

12 neighborhood pixels and thus the modiﬁed central

pixel, along with the cross pixels, does not change

something signiﬁcantly in the image.

Rotation. The watermarked images produced by our

embedding method have a property worth to be ref-

erenced. And this is certain characteristics noticed

at the 2DM representation of the image’s watermarks

which in our system are self-inverting permutations.

Sometimes an image might show an indeterminate de-

piction, such as a night sky or an aerial view. These

types of images might be rotated changing the coor-

dinates of the watermark’s marks making invalid the

watermark that we are about to extract. Also it is

about an indeterminate depiction which does not al-

low someone to tell which is the right angle of the

image.

Thanks to our embedding method’s properties this

problem can be overcome. It has to do with the co-

ordinates of the marks of a 2DM representation of a

self-inverting permutation found on image I

. Those

coordinates allow us to turn the image into the initial

angle and then extract the watermark successfully.

The 2DM representation of a self-inverting per-

mutation has the following properties:

(i) The main diagonal of the n

∗

× n

∗

symmetric ma-

trix A

∗

have always one and only one marked cell,

and

(ii) this marked cell are always in the entries (i,i) of

∗

, where i = ⌈

∗

⌉ + 1,⌈

∗

⌉ + 2,...,n

∗

If the main diagonal of matrix A

∗

has no marked

cell then we rotate the image by 90 degrees. Addition-

ally, if the marked cell of the main diagonal is in entry

(i,i) with i ≤ ⌈

∗

⌉, then we turn the image by 180 de-

grees and thus we end up at the initial image from

which we are able to extract the right watermark.

Time and Space Complexity. The total time perfor-

mance of our codec system, neglecting the conver-

sion of the input image I into raw raster format, is

N × n

∗

for embedding the watermark w into I and

N + M + (n

∗

× n

∗

) × log(n

∗

× n

∗

) for extracting w

from the watermarked image I

. Moreover, the extra

space needed by our codec system is linear in the size

of the input image, i.e., it uses only some extra aux-

iliary variables and an auxiliary matrix for the 2DM

representation of the self-inverting permutation.

5 CONCLUDING REMARKS

In this paper, we proposed a codec system, which we

named WaterIMAGE, for watermarking images that

are intended for online publication.

The proposed WaterIMAGE system has the fol-

lowing design and functional advantages:

• it is an efﬁcient image watermarking system; the

experimental results showed that the watermark w

is “well hidden” in the image I

• its embeddingmethod incorporates properties that

allow us to successfully extract the watermark w

for the image I

even if the image I

has been

compressed with a lossy method and/or rotated,

• it is a simple and easily implemented system, and

• ﬁnally, as far as the time and space needed for the

encoding/decodingprocess, it performs very well.

We should point out that the main feature of our Wa-

terIMAGE system is the fact that it uses a combina-

torial object to watermark an image; we show that

an integer can be efﬁciently represented by a self-

inverting permutation which, in turn, can be repre-

sented in the 2-dimensional space and, thus, this rep-

resentation forms a suitable watermarking object for

images. In our system we propose a marking method

but apart from that the investigation of alternative and

more efﬁcient methods for marking an image using a

2D representation of a permutation are an open prob-

lem for further research.

REFERENCES

Chroni, M. and Nikolopoulos, S. (2010). Encoding water-

mark integers as self-inverting permutations. In Proc.

Int'l Conference on Computer Systems and Tech-

nologies (CompSysTech'10), volume ACM ICPS 471,

pages 125 – 130.

Collberg, C. and Nagra, J. (2010). Surreptitious Software.

Addison-Wesley.

Cormen, T., Leiserson, C., Rivest, R., and Stein, C. (2001).

Introduction to Algorithms. MIT Press, 2nd edition.

Garﬁnkel, S. (2001). Web Security, Privacy and Commerce.

O’Reilly, 2nd edition.

Golumbic, M. (1980). Algorithmic Graph Theory and Per-

fect Graphs. Academic Press, Inc., New York.

Gonzalez, R. C., Woods, R. E., and Eddins, S. L. (2003).

Digital Image Processing using Matlab. Prentice-

Hall.

Sedgewick, R. and Flajolet, P. (1996). An Introduction to

the Analysis of Algorithms. Addison-Wesley.

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