Достаточное условие оптимальности в (и,р)дифференцируемых задачах

К. Петросян

Государственный Университет Бухареста,

Бухарест, Румыния

В статье приведены определения функциий типа (к,р)-( Р, а, р, <Л)

I, которые использованы для получения достаточных условий оптимальности в дифференцируемых многообъектных задачах, использующих обобщенные алгебраические операции Бен-Тала. В статье приведено одно условие достаточности оптимальности Каруш-КунТаккера, которое является основой для еще нескольких результатов достаточности оптимальности и для теорем дуальностей решений в для парето слабых решений.

Ключевые Слова. Обобщенные алгебраические операции Бен-Тала, функции (Н,ф)-(Р, а, р, <Л) типа I, условие достаточности КарушКун-Такера, слабые решения парето.

ВВЕДЕНИЕ

В литературе [1-2] определены некоторые обобщенные алгебраические операции сложения и умножения скалярных и векторных величин. С помощю этих операторов приведено новое обобщение выпуклости функциий, называемых (к,ф) выпуклостью [1]. Там же дано понятие (к,ф) дифференцируемости. Хю и Лю [3] получили необходимые условия Кун-Таккера для оптимальности в задачах многообъектной оптимизации, где все функции (И,ф) дифференцируемы.

Пусть п-мерное эвклидово пространство, а Я и Я+ , соответственно, множество всех реальных и множество неотрицательных реальных чисел.

Приведем определения обобщенных операций сложения и умножения Бен-Тала [3]:

Пусть к векторная непрерывная функция, определенная на Яп и со значениями из Яп. Пусть, существует обратная функция к—. Определим сложение векторов х, у е Яп относительно функции к следующим образом:

х Ф у = И 1 (к ( х) + И (у))

и умножение вектора х є Яп скалярным значением а є Я относительно функции И следующим образом:

а ® х = И(ак (х))

Пусть, р реальная функция, определенная на Я, для которой существует обратная функция р 1. Тогда сложение двух реальных чисел а є Я и в є Я относительно функции р определим как:

а [+]в = Р 1 р(а) + Р(в))

а умножение скалярных значений а є Я и в є Я относительно функции р как:

вНа = р (вР(а))

Обозначим:

т

фхг = х1 Ф х2 Ф…Ф хт, х є Яп, і = 1,2,…,т,

і=і

подпись: іподпись: а= а [+]а [+]…[+]ат ає Я , і = 1,2,…, т.

г =1

а[-]в = а[+]((1)[*]в), )веЯ.

Выше приведенны обобщенные алгебраические операции Бен-Тала. Отметим, что в[*]а может не совпадать с а[*]в для любых а, ве Я. Легко убедиться, что 1 ® х = х для Ух е Я” и 1 [•] а = а для У а е Я .

Пусть, а, ве Я и х е Я” . Тогда можно доказать следующие уравнения: р( а [• ]в) = ар(в), И (а ® х) = аИ ( х) .

а [-]в = Р(р(а )-Р(в)).

Определение 1.1. Пусть, X непустое подмножество пространства Я”. Функционал Р : X х X х Я” ^ Я будем называть (И, р) сублинейным, если

для любых х, х е X,

Р ( х, х, а1 Ф а2 )^Р ( х, х, а1)[+] Р ( х, х, а2), Уа1, а2 е Я”,

Р ( х, х, а Ф а )^а [•]Р(х,х,а) УаеЯ”, а^0,УаеЯ”.

Из вышеприведенного определения легко доказать, что если Р (к, р) суб-

 

(1)

 

Некоторые свойства обобщенных алгебраических операций Бен-Тала при­ведены в [3].

Лемма 1.1. Пусть, / — реальная функция, определенная на Я" и (к,р) —

дифференцируемая в точке х е Я" . Тогда удовлетворены следующие условия:

(a)  Пусть х1 е Я", Д е Я, 1 = 1,2,…, т Тогда

т                                 С т                    т                        С т               

ф(Д в X ) = к-‘ I £4* (х1) , фх1 = к-‘ I £ к (X )

1=1                                V 1=1                 У 1=1                     V 1 =’

(b)  Пусть, /л1, а; е Я, 1 = 1,2,…, т . Тогда

 

i rn

(«[•]a )=р_1| Zw(a )

 

Z

 

(с) Для любого а е Я функция а [•] / (к,р) дифференцируема в точке х

и V(а[•]/(X)) = а®У/(X).

Лемма 1.2. Пусть, 1 = 1,2,…, т. Имеют место следующие утверждения:

(a)  Для а,Р,уе Я справедливо а[• ](в[^]х) = вН(а[• ]х) = (ав)]/.

(b)  Для в, а1 е Я справедливо

 

).

 

(c)  Для a,ß,YG R справедливо /[•](a[—]ß) = (^[•]a)[-](fHß)

(d)  Для ai, ßi g R справедливо

 

Z

. i =1

m

z

 

Z

 

ß„

 

Z

 

ß,.

 

a

 

Лемма 1.3. Пусть функция р обобщенный алгебрических операций Бен — Тала строго монотонна ир(0) = 0 . Тогда следующие условия удовлетворены:

 

 

Если у^.0, у, а, в е Я, и а<в, тогда у [•] а<у [•] в .

Если у^.0, у, а, в е Я, и а < в, тогда у[*а</[^]в.

Если у> 0, у,а,в е Я и а < в, тогда. у[»]а < х[^]в.

Если у < 0, у,а,в е Я и а^в, тогда у[•]а<к[•]в.

Если а^,вi еЯ, i еМ = {1,2,…,да} и аг.<Ддля любой i еМ , тогда

подпись: (2)подпись: а <I

. *=1

Если ai <Д для любой i е М и существует хотя бы один к е М , для которого ак < вк, то

I

 

(3)

 

 

Лемма 1.4. Пусть, р непрерывная, монотонная функция и р(0) = 0. Пусть, а, в е Я. Тогда

а < в ^ а[—]в< 0,        (4)

а<в ^ а [—] в<0 ,        (5)

а [+]в< 0 ^ а < (—1)|>]в,        (6)

а[+]в<0 ^ а<(—1)[^]в .          (7)

Далее, в этой статье мы будем предполагать, что функция И непрерывна, имеет обратную функцию и И(0) = 0. Функцию р будем предполагать непрерывной, строго монотонной с р(0) = 0 . При этих условиях нетрудно доказать, что 0 [•]а = а[•] 0 для любой а е Я .

Пусть, X непустое подмножество пространства Яп’, ¥ : X х X х Яп ^ Я(И, р) — сублинейная функция,        а

/ = ((Ъ-….-./т): Х ^ Ят и ё = (gl,8Р) : Х ^ ЯР (И,р) дифференцированные на множестве X относительно одной и той же пары (И, р) . Пусть,

Р = (Р, Р2), где р = (р, р^,…, р1т ) е Ят, р2 =(р2, р22,…, рр ) е Яр. Пусть а = (, а2), где а1: X х X ^ Я+ {0}, а2: X х X ^ Я+ {0} и ё (•,•): X х X ^ Я .

Рассмотрим следуюшую многообъектную задачу:

(MOP) min f ( x) = ( (( /2 ( x)fm ( x)), x e X ^ R так, что g (x)^0.

Обозначим через F множество всех допустимых решений задачи (MOP )„

F = {x e X : g(x)^0j .

Допустим, что F непустое множество и обозначим: M = {l, 2,…,m},

P = {l,2,…, p}

Для допустимого решения x e F обозначим J(x) = {j e P : gj (x) = o} .

Будем искать слабые решения парето для проблемы (MOP)h ^ . Понятие

слабого решения парето широко используется в методах оптимального управления и определено в [12].

Определение 2.2. Точка x называется слабым решением парето или слабым минимумом для проблемы (MOP) , если x e Fsif (x) < f (x) для всех xe F .

Определение 2.3. Пусть i e M. Скажем, что (fi, g) является

(,p)-(F, a, p, d ) — типа I в точке x e X , если для всех x e F выполняются условия:

f (x)[-f (x)^F((,a(x)0V*f(x))[+](p[*К(x,x)), ieM, (8)

(-1) [* gi (x)^F (x, x, a2 ( x) 0 V*gj (x)) [+ ) [*]d2 (x, x)), j e P. (9)

Если в вышеуказанном определении x ^ x и неравенство (8) строгое, то (fi, g) называется полустрогим (И,^)-(F, a, p, d) — типа I в точке x.

2. ДОСТАТОЧНОЕ УСЛОВИЕ ОПТИМАЛЬНОСТИ

Теорема 2.1. Допустим, существует допустимое решение x e F и A = (, A2,…, Am)e Rm, A> 0, ni>0, j e J(x) так, что:

подпись: f
v i=i
подпись: 0a,- ®v* f (х))ф @x ®v* g; (x))
= v ’) i &(x) v ’’
X f      

= 0, (10)

Если г еМ(£,(()), функция (й,р)-(¥,а,р,й) — типа I в точке х и

а'(Х)-1 [] Е (Р)[‘К(•,Х))|[+]

_ /=1 _ )

)

А

‘(, Х )-1 [

 

(11)

 

[+]

 

>0.

 

где g() = ) , ^()-, то х — слабый минимум для задачи (МОР)й ^ .

Доказательство: Так как (10) выполняется, то из (Ь,ф) — подлинейности функции ¥ , для всех х е X получаем:

(                                                 ч (

 

¥

 

(12)

 

Проведем доказательство методом от противного. Допустим х не являет­ся слабым минимумом для задачи (МОР)й ^. Тогда существует допустимое

решение х задачи (МОР) ^, так что £ (х) < /, (х), г е М .

Так как X > 0 , значит существует хотя бы одно к так, что Хк > 0, X/ >0,

V/ е м , г ф к.

Из леммы 1.3 (а)-(с) получаем

Хк []/к (х )<Х Н/ (х) Х Щ (х )^Х Щ (х) , VІ е М , г Ф к .

Из леммы 1.3 (е) и (4) получаем:

 

Е

г=1

п

Е

 

Е

 

(13).

 

 

подпись: для выше-Из предположения (к,ф)-(¥, а, р, й) — типа I для (£,

указанного х получим

£ (х )Н£ (х )>¥ (х, x, а (х х (х ))[+](р [*]й 2 (х х)), г е М

= (-1) [•] ё] (х)(х, х а2 (х х) ® V*g] (х)) [+] (р2 [• ]й2 (x, х)), ] е 3(х).

Из (к, <р) -подлинейности Р получим:

/г ( )[“]./; (х )^’ ( х )МР ( x, x, У*/г ( х ))[ + ](Р [К ( ^ х))

0^а2 (х х)[*]Р(х, x, V*g] (х))[+](р [• ]й2 (х, х)), 7 е 3(х)

Так как , Лг^0, г е М , |лi ^0, ] е 3 (х) , из леммы 1.3 (а), имеем:

Л []( ( )[~]/г (х ))^Л [](а’ ( ^ х )МР ( х, X, У*/г ( х ))[ + ](Р [•]й 2 (, х) г е м

°^; [•](а 2 (х, х )[]р (х (, V* gJ (х ))[+](р2 КИ 2 (x, х))), 7 е 3 (х)

Из леммы 1.2 (а)-(с) имеем:

(Л]/ (х ))[-](Л (х ))г

^а1 (х, х)•](Лг []Р(х,х, V*/ (х)))[+](Лр [•]2 (х,х)) , г е М 0^2 (хх)[*](^; []р(хx, V*g; (х)))[+)р)[*К (x, х)), ] е 3(х)

Из (к, <р) -подлинейности Р имеем:

(ЛгН/ (х ))[-](Лг []/г (х ))г

^а1 (х, х)[]Р(х, х,Лг (х))[+]((Лр )[]й2 (х, х)) , г е М 0^2 ( х)[.]р( x, !л} (х))[+]((^р)[*К (Я х)) , 7 е 3(х)

Из леммы 1.3 (е) получаем:

подпись: z
i =1
подпись: я.[•]/, (x ))[-]( [•]/, (x )))>

подпись: i
i=1
подпись: >a1 (x x ) [ •] F x, Xi ® Vf ( x ) j [+] f (Яр1 j [ • ] d2 ( *, x

V

f

 

jеJ (х)

 

 

Из (h, p) -подлинейности функции F и (1), используя (12), получаем:

0 = F

 

jeJ (x

 

V V i=1

 

 

подпись: zподпись: <F(x,(, Ai ®V*f (x)[+] z F(x,X 0V*g, (x))< 0,

jeJ(x)

которое является противоречием. Следовательно, слабый минимум для проблемы (mop )h ^.

ЛИТЕРАТУРА

Avriel M. Nonlinear Programming: Analysis and Method, Prentice-Hall, Englewood Cliffs, New Jersey (1976).

Ben-Tal A. On Generalized Means and Generalized Convex Functions, J. Optim. Theory Appl. 21 (1977). PP. 1-13.

Xu Y. H., Liu S.Y. Kuhn-Tucker Necessary Conditions for (h, )-multiobjective Optimization Problems, Journal of Systems and Complexity 17 (2004). PP. 472-484.

Aghezzaf B., Hachimi M. Sufficiency and Duality in Multiobjective Programming, involving Generalized ■ £’) — convexity. J. Ath. Anal. 258 (2001). PP. 617-628.

Hanson M.A., Mond B. Further Generalizations of Convexity in Mathematical Programming, J. Inform. Optim. Sci. 3 (1986). PP. 25-32.

Hanson M.A., Mond B. Necessary and Sufficient Conditions in Constrained Optimization, Math. Programming 37 (1987). PP. 51-58.

Kaul R.N., Suneja S.K., Srivastava M.K. Optimality Criteria and Duality in Multiple Objective Optimization Involving Generalized Invexity, J. Optim. Theory Appl. 80 (1994). PP. 465482.

Liang Z.A., Huang H.X., Pardalos P.M. Optimality Conditions and Duality for A Class of Nonlinear Factional Programming Problems, Optim. Theory Appl. 110 (2001). PP. 611-619.

Preda V. On Efficiency and Duality for Multiobjective Programs, J. Math. Anal. Appl. 166 (1992). PP. 365-377.

Rue da N.G., Hanson M.A. Optimality Criteria in Mathematical Programming Involving Generalized Invexity / Math. Anal. Appl. 130 (1998). PP. 375-385.

Rueda N.G., Hanson M.A., Singh C. Optimality and Duality with Generalized Convexity, J. Optim. Theory Appl. 86 (1995). PP. 491-500.

Sawaragi Y., Nakayama H., Tanino T. Theory of Multiobjective Optimization, Academic Press, Orlando (1985).

Aghezzaf B., Hachimi M. Generalized Invexity and Duality in Multiobjective Programming Problems, J. Global Optim 18 (2000). PP. 91-100.

OPTIMIZATION SUFFICIANCY CONDITION IN THE (h,q>)DIFFERENTIABLE PROBLEMS K. Petrosyan

In this article we define a class of functions ( h,ç)-( F, a, p, d ) type I,

which are used in obtained sufficient conditions of optimality in differentiable multiobjective problems, using generalized algebric operations of Ben-Tal. We presented one sufficient condition of optimality of Karush-Kuhn-Tucker, which is used in obtaining some other optimality conditions and theorems of duality for weak pareto solutions. Keywords: Generalized algebric operations of Ben-Tal.,

(h,ç)-(F,a, p,d) type I functions, sufficient conditions of KarushKuhn-Tucker, weak pareto solutions.

Y^K 004.056.53

A NOVEL THRESHOLD-BASED IMAGE WATERMARKING ALGORITHM USING DWT2

M. Khalili, D. Asatryan

Institute for Informatics and Automation Problems of NAS Armenia, Russian-Armenian (Slavonic) University

In this paper, a novel threshold-based image watermarking is proposed in which a scrambled binary image by ATM, after encryption, is embedded into sub-images of the first channel wavelet decomposition of CIELab color space using block processing technique. The experimental results show that the proposed method improves security and imperceptibility compared the similar works and has stronger robustness against JPEG compression, different noises and motion filter attacks.

Keywords. Threshold-based watermarking, block processing technique, DWT2, ATM, CIELab color space.

Introduction

With the development of network and multimedia technologies, multimedia copyright protection and content authentication have become serious problems that need to be solved urgently. Digital watermarking technology provides a strong solution for it [1]. Most of the digital watermarking algorithms uses transform domain techniques to embed the watermark information in a robust and imperceptible way into the host image. The most commonly used transform domain techniques are DCT, DWT, and SVD [2]. Digital watermarking based on wavelet analysis can effectively use the characteristics of the human visual system and can be compatible with international compression standard. In this domain, the embedded watermark signal energy can be distributed to all the pixels on the space [3]. For watermarking in DWT domain, certainly the embedding strategy affects robustness of watermarks, while the selection of embedding frequency bands is another important issue. As we all know that schemes in low frequency bands will bring about stronger robustness, but human’s eyes are just like low filters and sensitive to even slight modifications in low frequency coefficients, so improper methods can easily spoil the image quality. While watermark hidden in the middle or lower frequency bands is more liable to be suppressed by compression [4]. In this paper, we propose a new technique in low frequency based on B-spline wavelet filter and improve the obtained results in [7]. Cause of use of B-spline function wavelet is that, B-spline functions, do not have compact support, and are demonstrated more appropriate orthogonal wavelet base and have better smoothness property for digital image wavelet decomposition [5, 6]. For more security of the proposed scheme, before watermark embedding process, the binary watermark image after scrambling by Transform Map Arnold (ATM) method is reshaped to a sequence and then a random binary sequence R of size n is adopted to encrypt the watermark, where n is the size of the watermark. This adopting process uses a pseudo-random number generator to determine the pixel to be used on a given key. On the other hand, the RGB channels of the host image are converted to CIELab channels and then the first channel is pre-filtered to enhance embedding process. After that, low frequency subband of wavelet decomposition of its first channel, is quantized and divided to different sub-blocks with the certain sub-block size to embed the encrypted watermark.

Arnold Transform Map

Usually scrambling transform is used in the pretreatment stage of the watermark as a way of encryption. Generally, a meaningful watermark image becomes meaningless and disorderly (chaotic) after scrambling transform. Without the scrambling algorithm and the key, the attacker will not recover the watermark at all even if it has been extracted from the watermarked image. So the scrambling transform gives a secondary security for the digital products. In addition, after scrambling transform, the spatial relationships of the pixels of an image has been destroyed completely, which makes it evenly distributed in all the space, so the robustness of the algorithm was improved in this way [8]. ATM or Arnold Transform Map [9] is a kind of image scrambling methods, called as Arnold’s cat mapping. The discrete Arnold transformation is defined as follows [10]:

Xn+1

a b

Xn

= A

Xn

Yn+1

c d

Yn

Yn

mod N

подпись: (1)where, a, b, c and d are positive integers, and |A| = ad bc.1, so only three among the four parameters of a, b, c and d are independent. Xn+1, Yn+1, Xn and Yn are integers in {0,1,2,…,N-1}. In this paper the extended Arnold transform in [11] is used to scramble watermarking of copyright protection.

CIELab Color Space

In 1976, the International Commission on Illumination (CIE) recommended the CIE L*a*b*, or CIELab color space for color quality estimation [12]. The color space CIELab is a perceptually uniform color space created by nonlinear transformations of tristimulus XYZ values to overcome the non-uniformity of color spaces which had been discussed by Macadam [13]. The main intention was to provide a standard and approximate uniform color space which can be used to compare the color values easily. In this color space the differences between points plotted in the color space correspond to visual differences between the colors plotted. The CIE recommended to use XYZ coordinate system to transform RGB to L*a*b*. The equations in [14, 15] are used to transfer RGB to CIELab and CIELab to

Watermarking Procedure

A digital watermarking system usually consists of embedding framework and extraction framework. The block diagram of the proposed watermarking approach is shown in Figure 1.

Figure 1. Block diagrams of the proposed watermarking approach; (a) Embedding procedure, (b) Extracting procedure.

The Embedding Framework

The steps involved in the embedding of watermark image into the in LL3 coefficients of the host image are described as follows:

Step 1: Convert RGB channels of a host image H into CIELab channels.

Step 2: pre-filtering the first channel.

Step 3: Decompose the first channel into three levels with ten DWT subbands, F(Y). The subband LL3 is taken as the target subband for embedding watermarks.

Step 4: Creating a sign matrix to save the signs of selected target subband coefficients.

Step 5: Quantization of the selected embedding coefficients.

Step 6: Divide of the Target subband into the different sub-blocks. In this paper each sub-block size is equal to 16.

Step 7: Determining the maximum message size of each sub-blocks.

Step 8: For more security of watermarks, first, the watermark W is scrambled for key times with presented ATM algorithm in [14] and then reshaped to a sequence; after that, a random binary sequence R of size n is adopted to encrypt the watermark, where n is the size of the watermark image. This adopting process uses a pseudorandom number generator to determine the pixel to be used on a given key.

Step 9: Embedding the watermark using the correlation properties of additive pseudo-random noise patterns according to equation shown in below:

f / + k *W if W = 0

Iwx,y (uv) ” {I * W otherwise            (2)

k. x,y 1

where k denotes a gain factor for completely controlling the imperceptibility of watermarked images and the robustness of watermarks and also IW is the resulting watermarked image.

Step 10: Apply the sign matraix.

Step 11: Perform Inverse DWT on new first channel with all changed and unchanged DWT coefficients.

Step 12: Reconvert CIELab channels of the changed host image into RGB channels. Step 13: Save key times of ATM, random binary sequence R and index of the embedded subband as the authenticated key.

The Extraction Framework

The proposed algorithm is a blind watermarking algorithm and thus the original host image is not required to extract the watermark. Execration algorithm is the same as embedding one and pre-filtering is used before applying DWT transform to better separate watermark information from host image. The watermark extraction procedure is described in details in the following steps:

Step 1: Convert RGB channels of a watermarked image H into CIELab channels.

Step 2: pre-filtering the first channel.

Step 3: Decompose the first channel into three levels with ten DWT subbands. The subband LL3 is taken as the target subband for extraction watermarks.

Step 4: Quantization of the selected embedding coefficients.

Step 5: Divide of the Target subband into the different sub-blocks.

Step 6: Determining the maximum message size of each sub-blocks.

Step 7: Computation of threshold T as follows:

T Correlation(HL) + Correlation(LH) (3)

~ 2

Step 8: Computation of the threshold T and each embedded coefficient correlation in sub-blocks, separately.

Step 9: The sequence encrypted watermark is extracted as follows:

W = 0 if Ct )T (4)

[W = 1 Othetwise

Step 10: The encrypted image watermark is produced by reconverting the extracted sequence watermark.

Step 11: Scramble the encrypted image watermark with the same ATM algorithm with the same key times.

EXPERIMENTAL RESULTS

The proposed perceptual watermarking framework was implemented for evaluating security, imperceptibility and robustness. Three 512^512 famous images: Lenna, Peppers and Baboon, shown in figure 2(a-c) were taken as the host images to embed a 27×27 binary watermark image, shown in figure 2(d). For gain factor k, value 1.0 was taken entire implementation of the proposed scheme. Also for the entire test results in this paper, MATLAB R2007a software is used. Also for computing the wavelet transforms, 9-7 biorthogonal spline (B-spline) wavelet filter are used.

Table 1. Watermark imperceptibility results

Image

Lenna

Peppers

I

iaboon

Extracted

Watermark

M.

KH

M#

KH

M.

KH

PSNR (dB)

86.34

85.94

87.28

Corr

0.9996

0.9997

0.9993

NC

1.00

1.00

1.00

Error Bit%

0

0

0

Table 1 shows the watermark imperceptibility results. It can be seen that in addition of enhancement the security, the proposed watermarking approach yields satisfactory results in watermark imperceptibility and robustness and improves the earlier works such as [7]. The PSNRs of the watermarked images produced by the proposed approach are all greater than 85 dB, NCs between original watermark images and extracted watermark images are all equal 1, and correlations between host images and watermarked images are all greater than 0.999, which are perceptually imperceptible.

After able to achieve the desired watermark imperceptibility the watermarked images were tested under JPEG compression, Gaussian and ““Salt & Pepper”” noises and motion attacks to evaluate the robustness of the proposed scheme.

P%.

(a)        (b)

M*

KH

<J)

Figure 2. The host images for watermarking;

(a-c) Lenna, Peppers and Baboon, (d) Watermark image

Robustness to JPEG Compression

In this experiment, the watermarked images compressed with different jpeg qualities Qs: 5, 10, 15, 20, 25, 50, 70 and 90 and then the watermark images extracted from the compressed watermarked images. The obtained results from compression experiment and the extracted watermarks from compressed watermarked images are shown in tables 2 and 3, respectively.

It can be seen that the extracted watermarks are excellent if the quality factors are greater than 20. PSNRs are still greater than 74 dB and the maximum error bit rates are all less than 11% even if the quality factors is less than 10. Also, it can be seen that the extracted watermarks still show satisfactory quality under the high quality factor compression 5. Comparing the obtained results in [7], it was found that the proposed scheme is more robust in JPEG compression experiment.

Robustness to Gaussian Noise

In this experiment, Gaussian noise with zero mean and different variances Vs:

1, 0.3, 0.5, 0.7, 0.9 and 1 applied on the watermarked images and then the watermark images extracted from the noisy watermarked images. Table 4 shows the obtained results from Gaussian noise experiment and table 5 shows the extracted watermarks after applying different variances. From these results it can be found that the proposed scheme satisfies robustness against Gaussian noise attacks and enhances the obtained results in [7]. The PSNRs are all greater than 47 dB. In variance of 1, the maximum error bit rate is less than 22% and the extracted watermarks are still recognizable.

Robustness to “Salt & Pepper” Noise

In this experiment, the “Salt & Pepper” noise with zero mean and different noise densities N.Ds: 0.1, 0.2, 0.3, 0.4 and 0.5 introduced in the watermarked images

and then the watermark images extracted from the noisy watermarked images. Table 6 shows the obtained results from “Salt & Pepper” noise attack. The obtained results show that the maximum error bit rate is still less than 13% and in noise density of

1the minimum PSNRs are all greater than 52 dB. The extracted watermarks from noisy watermarked images are shown in table 7. It can be seen that in noise density 0.3 the extracted watermark images are near perfect and they are still recognizable in noise density 0.5. Therefore, it can be found that the proposed scheme satisfies robustness against “Salt & Pepper” noise and improves the obtained results in [7].

Table 2. The obtained results from JPEG compression experiment

Lenna

Q

PSNR

(db)

NC

Error Bit

%

10

83.41

0.8684

6.03

20

86.60

0.9939

0.27

50

92.82

1.0000

0.00

90

111.98

1.0000

0.00

Peppers

Q

PSNR

(db)

NC

Error Bit

%

10

80.97

0.7631

10.83

20

83.69

0.9511

2.19

50

88.57

0.9847

0.68

90

97.80

1.0000

0.00

B

laboon

Q

PSNR

(db)

NC

Error Bit

%

10

77.63

0.8604

6.72

20

80.10

0.9818

0.82

50

84.45

0.9969

0.13

90

92.29

1.0000

0.00

Robustness to Motion Filter

In this experiment, the motions filter with different angles 0s: 10, 20, 30, 40 and 50 in a counterclockwise direction applied on the watermarked images and then the watermark images extracted from the filtered watermarked images. The obtain results from this experiment and the extracted watermarks are shown in tables 8 and 9, respectively. As it is obvious, all of the extracted watermarks are recognizable. The maximum error bit rate is less than 17% and PSNRs are all greater than 76 dB. Thus, the proposed scheme satisfies robustness against filter motion, thoroughly.

CONCLUSIONS

In this paper we introduced a threshold-based image watermarking scheme based on block processing, ATM, CIELab color space and DWT2 which improves the similar works such as [7]. The observations regarding the proposed watermarking scheme are summarized as follows:

For more security of the proposed scheme, before watermark embedding process, the binary watermark image after scrambling by ATM method is reshaped to a sequence and then a random binary sequence R of size n is adopted to encrypt the watermark, where n is the size of the watermark. This adopting process uses a pseudorandom number generator to determine the pixel to be used on a given key.

After converting the RGB channels of the host image to CIELab channels, the first channel is pre-filtered to enhance embedding process. After that, low frequency subband of wavelet decomposition of its first channel, is quantized and divided to different sub-blocks with the certain sub-block size to embed the encrypted watermark.

The proposed scheme satisfies security, imperceptibility and robustness against different attacks such as JPEG-compression, Gaussian and “Salt & Pepper” noises and filter motion.

The proposed scheme has no need to original image in watermark extracting process.

Table 3. The extracted watermarks in JPEG experiment

Q

Lenna

Peppers

Baboon

5

m

1 VI’J

10

№.

KB

Ilf

m

15

M.

KH

-■MU

Ktt

M.

KH

20

M.

KH

M*

KH

M.

KH

25

M.

KH

Mo

KH

M.

KH

[

50

M*

KH

M.

KH

M.

KH

75

M*

KH

M.

KH

M.

KH

90

M#

KH

M.

KH

M#

KH

Table 4. The obtained results from Gaussian noise experiment

Lenna

Vari

ance

Pi ^ z £ CM

NC

Error Bit

%

0.1

72.64

1.0000

0.00

0.3

61.52

0.9969

0.13

0.5

56.42

0.9878

0.54

0.7

54.40

0.9079

4.11

0.9

54.12

0.7135

12.75

1.0

54.11

0.5069

21.39

Peppers

Variance

Pi ^ z £ Ph

NC

Error Bit

%

0.1

70.67

1.0000

0.00

0.3

58.31

0.9969

0.13

0.5

52.71

0.9694

1.37

0.7

50.59

0.9418

2.60

0.9

50.03

0.8957

4.66

1.0

49.98

0.7342

12.07

Baboon

Variance

Pi ^ z £ Ph

NC

Error Bit

%

0.1

71.04

1.0000

0.00

0.3

58.06

1.0000

0.00

0.5

51.54

1.0000

0.00

0.7

48.65

0.9786

0.96

0.9

47.84

0.8514

6.85

1.0

47.78

0.7004

13.99

Table 5. The extracted watermarks in Gaussian noise experiment

V

Lenna

Peppers

Baboon

0.5

M.

KH

M*

KH

M.

KH

0.7

Mi

KH

M*

KH

M*

KH

0.9

ivfi

Me

KH

.Mv

m

1.0

:№

Rtf

Table 6. The obtained results from “Salt & Pepper” noise experiment

Lenna

N.D

Pi ^ Z £ ^ 3

CM

NC

Error Bit

%

0.1

65.64

1.0000

0.00

0.2

59.69

0.9909

0.41

0.3

56.14

0.9404

2.74

0.4

53.60

0.8868

5.48

0.5

51.66

0.7924

10.56

Peppers

N.D

Pi ^ N b) ^ 3

Ph

NC

Error Bit

%

0.1

66.81

0.9847

0.68

0.2

60.81

0.9427

2.60

0.3

57.23

0.8958

4.80

0.4

54.74

0.8174

8.50

0.5

52.81

0.7488

12.34

Baboon

N.D

Pi ^ N b) ^ 3

Ph

NC

Error Bit

%

0.1

66.58

0.9939

0.27

0.2

60.47

0.9819

0.82

0.3

57.01

0.9246

3.56

0.4

54.50

0.8596

6.85

0.5

52.58

0.7772

11.24

Table 7. The extracted watermarks in “Salt & Pepper” noise experiment

N.D

Lenna

Peppers

Baboon

0.1

M.

KH

M.

KH

M.

KH

0.2

M.

KH

M.

KII

M.

KH

0.3

N1.

Kti

M.

KH

Mv

ita

Table 8. The obtained results from motion filter experiment

Lenna

Degree

Pi ^ Z £ ^ 3

CM

NC

Error Bit

%

10

83.07

0.7651

10.69

20

83.27

0.7612

10.83

30

83.95

0.8611

6.17

40

84.46

0.8705

5.76

50

84.53

0.8302

7.54

Peppers

Degree

Pi ^ Z £ ^ 3

Ph

NC

Error Bit

%

10

83.43

0.7315

12.20

20

83.29

0.7289

12.34

30

83.55

0.8179

8.09

40

83.80

0.8181

8.09

50

83.63

0.7726

10.15

Baboon

Degree

Pi ^ Z £ ^ 3

Ph

NC

Error Bit

%

10

78.18

0.6567

16.18

20

77.38

0.6772

16.28

30

77.17

0.8156

8.91

40

77.10

0.8206

8.64

50

76.73

0.7665

11.52

Table 9. The extracted watermarks in motion filter experiment

Degree

Lenna

Peppers

Baboon

10

Kb

m

kH

H

20

KH

ü

30

•M.»

Kii

•fy!.»

kb

■Mi

KH

m

 

■M.

KiS

 

40

 

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