Research Article A New QIM-Based Watermarking Method...

Research ArticleA New QIM-Based Watermarking Method Robust to Gain Attack

Yevhen Zolotavkin and Martti Juhola

Research Center for Information and Systems School of Information Sciences University of Tampere Kanslerinrinne 133014 Tampere Finland

Correspondence should be addressed to Yevhen Zolotavkin zhzolotsuremailus

Received 26 May 2014 Accepted 22 August 2014 Published 11 September 2014

Academic Editor Harald Kosch

Copyright copy 2014 Y Zolotavkin and M JuholaThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We propose a new watermarking method based on quantization index modulation A concept of initial data loss is introduced inorder to increase capacity of thewatermarking channel under high intensity additivewhiteGaussian noise According to the conceptsome samples in predefined positions are ignored even though this produces errors in the initial stage of watermark embeddingThe proposed method also exploits a new form of distribution of quantized samples where samples that interpret ldquo0rdquo and ldquo1rdquo havedifferently shaped probability density functions Compared to well-known watermarking schemes this provides an increase ofcapacity under noise attack and introduces a distinctive feature Two criteria are proposed that express the feature numericallyThecriteria are utilized by a procedure for estimation of a gain factor after possible gain attack Several state-of-the-art quantization-based watermarking methods were used for comparison on a set of natural grayscale images The superiority of the proposedmethod has been confirmed for different types of popular attacks

1 Introduction

Digitalmedia have a great impact onmany aspects ofmodernsociety Some aspects assume that we deal with audio-visualdata that relates to a person or an organization Informationabout the relation quite often should be preserved Water-marking approach is to insert the information in the mediaitself [1] However in that case the watermark might beintentional or not altered by the third party In order toavoid an alteration the watermark needs to be robust [2]Other characteristics except robustness may also be impor-tant Watermark invisibility and payload are among themInvisibility is important to assure that the quality of themediadoes not degrade significantly as a result of watermarking [3]High data payload might be needed in some applications inorder to define many aspects of ownership

In the field of digital image watermarking (DIW) digitalimages are used as amedia (or host) DIW incorporatesmanydifferent techniques and one of the most popular amongthem is quantization index modulation (QIM) Methodsthat belong to QIM are widely used in blind watermarkingwhere neither original media nor watermark is known tothe receiver [4] For the purpose of evaluating robustnessthe watermarked image is being attacked and additive white

Gaussian noise (AWGN) is the most popular conditionfor that Theoretical limit of the channel capacity which isachievable by QIM under AWGN was first derived in [5]

In most cases quantization is implemented to somecoefficients rather than to signal samples In order to obtaincoefficients a transform is applied to a host signal It hasbeen shown that some transforms provide coefficients thatare more robust to popular image processing algorithms likeJPEG geometric modifications and so forth [6 7]

It is assumed that during quantization each of the originalcoefficient values belongs to one of equally spaced intervalsFurther inside each interval coefficients to interpret ldquo0rdquo andldquo1rdquo are selected The task of quantization is to separate coeffi-cients that represent different bits inside each interval Theseparation efficiency influences robustness and invisibilityThe result of the separation can be characterized by the sizeof original interval distribution of separated samples and thedistortion incurred by the separation

However all the known implementations of QIM are farfrom achieving the capacity limit under AWGNThe simplestscalar realization of QIM is to replace all the coefficient valuesfrom a certain interval by a single value equal either to theleft or right endpoint depending on a bit of a watermark [8]

Hindawi Publishing CorporationInternational Journal of Digital Multimedia BroadcastingVolume 2014 Article ID 910808 14 pageshttpdxdoiorg1011552014910808

2 International Journal of Digital Multimedia Broadcasting

Hence the distribution of quantized samples that representboth ldquo0rdquo and ldquo1rdquo is degenerate (or Dirac) Nevertheless thecapacity of the simplest QIM (further referred to as QIMwithout ldquosimplestrdquo) is less than 10 of the limit value forthe condition when noise and watermark energies are equalMore advanced realization of DC-QIM is to replace eachcoefficient value from an original interval by a correspondingvalue taken from one out of two disjoint intervals that aresubsets of the original one [9] A parameter 05(1 minus 120572) is tocontrol the size of these intervals relatively to the originalThedistribution for ldquo0rdquo and ldquo1rdquo in that case is uniform Parameter120572 is being adjusted depending on noise level in order tomaximize capacity Method DC-QIM is widely used andprovides the highest capacity under AWGN among knownpractical realizations However considering AWGN attackonly the most evident gap under high noise intensity iscaused by low capacity in comparison with the theoreticallimit

Some other modifications of QIM have emerged overthe past years Forbidden zone data hiding (FZDH) modifiesonly a fraction (controlled by 120572) of coefficient values ineach interval of original values [10] Despite the fact thatFZDH performs slightly worse than DC-QIM the paperrepresents a promising idea on how to reduce embeddingdistortions Another idea was proposed by the authors ofThresholded Constellation Modulation (TCM) that use twodifferent quantization rules to modify coefficients inside theoriginal interval [11] Each rule is applied only to samplesfrom a particular subinterval and 120573 defines their endpointsThe value of a shift is different for any value from a subintervalaccording to the first quantization rule The second ruleis to provide an equal shift to all the values from anothersubinterval There are two shift directions in order to embedldquo0rdquo and ldquo1rdquo

The main advantage of the techniques based on QIMwith different kinds of compensation [9ndash11] is a consid-erable robustness against AWGN The limitation is thatsynchronization is required to extract a watermark Evenminor distortion of a different kind can make embeddedinformation unreadableThe simplest realization of such kindof distortion is a gain attack (GA) which performs a constantscaling of the whole sequence of watermarked coefficientsThe scaling factor might be close to 1 and cause very littlevisual distortion but it is unknown to the receiver whichcauses asynchronous extraction Usually GA is followed byAWGN that complicates retrieval of the watermark [12]Vulnerability to GA causes one of the most critical gapsfor practical implementation of QIM-based methods withcompensation

Many different approaches have been developed toimprove robustness against GA of QIM related methods [13]Most approaches can be classified into two groups where themain idea of the first group is to estimate the unknown factorwhile the idea of second is to quantize coefficients that areinvariant to scaling of original signal

Estimation of the scaling factor requires modelling Somefeature that is unique for the watermarked and attackedsignal might be described by a model [14] The scalingfactor may be included in the model and to be a subject

to optimization An obvious complication is that a processof feature selection is not straightforward In some casesthe feature is created instead of being selected and somepermanent data agreed upon between the sender and thereceiver is a suitable example However such compulsoryagreement limits practical implementation of watermarkingmethod Other possible limitations are low model accuracyor computationally heavy optimization

For instance a kind of GA and a constant offset attack fol-lowed by AWGN are assumed in [12] The solution proposedthere is to embed a pilot signal and to use Fourier analysis toestimate the gain factor and the offset However an obviousdisadvantage of the solution is that the precision of estimatedparameters is low even for quite long pilot sequence

Themethod of recovery after GA and AWGN is proposedin [15] It uses information about dither sequence and appliesmaximum likelihood (ML) procedure to estimate the scalingfactor The estimation is based on a model of watermarkedand attacked signal Information about statistical charac-teristics of original host signal should be either known orguessed in order to define the model Another limitationof the approach is that it requires exact information aboutembedding parameter 120572 Computational complexity of theapproach is high

As an opposite for estimation invariance to GA ingeneral requires more complex transform of original signal(eg nonlinear) to obtain coefficients It is necessary tomodify coefficients to embed a watermark However a modelto estimate distortions of the host is more complex in thatcase Distortions should be controlled which limits the choiceof the kind ofQIM to one that adds less complexity to amodelof distortion This for example might result in reducing thenumber of adjustable parameters of QIM This is one of thereasons why invariant to GA approaches are more vulnerableto AWGN compared to DC-QIM

Rational dither modulation (RDM) is one of the mostpopular watermarking methods invariant to GA [16] For aparticular coefficient a ratio that depends on a norm of othercoefficients is being quantized instead of a coefficient itselfThe simplest QIM scheme is utilized in order to quantize theratio and the performance of RDM under AWGN (withoutGA) is close to the simplest QIM Other recent blindwatermarkingmethods robust toGA are proposed in [17ndash23]Nevertheless for GA invariant methods the gap is caused bythe reduced capacity under AWGN

In this paper we propose our own scalar QIM-basedwatermarking approach that is beneficial in several aspectsThe approach addresses the mentioned gaps in the literatureit both delivers higher capacity under AWGN and recoversafter GA In order to do this host signal coefficients are sepa-rated in a way that the resulting distributions for coefficientsthat interpret ldquo0rdquo and ldquo1rdquo are differentThis distinctive featureis used by a simple yet efficient procedure for estimation of ascaling factor underGA A concept of initial data loss (IDL) isintroduced in order to increase watermark channel capacityunder low watermark to noise ratios (WNRs) Accordingto IDL some fraction of wrong watermark bits is acceptedduring embedding procedure

International Journal of Digital Multimedia Broadcasting 3

The rest of the paper is organized as follows In Section 2we describe our quantization model using formal logicapproach and derive some constraints on the parametersof the model In Section 3 some important watermarkingcharacteristics of the model are evaluated analytically whilethe following Section 4 contains description for the proce-dure of recovery after GA as well as experimental resultsobtained under popular attacks In Section 5 we discuss indetail experimental conditions and compare the performanceof the proposedmethodwith the performance of well-knownmethods Section 6 concludes the paper and outlines possibledirections for improvement

2 Quantization Model

In this section we define a new model of quantization Firstit is necessary to show that according to our model theseparation of original coefficients is possible and we canembed information Formal logic approach is used to definedependencies between several conditions that are importantfor the separation of original coefficients Separation argu-ment (SA) represents the model in a compact form yet hasa clear structure which is sufficient to reason the intuitionbehind the dependencies Second it is necessary to assureconditions when SA is sound

21 Formalization of SA Symbol Σ will be used to denotea random variable whose domain is the space of originalcoefficients of a host A particular realization of Σ will bedenoted by 120589 We will further describe our model for values120589 that are in some interval of size Δ More specifically we willrefer to an interval with integer index 119896 whose left endpointis 119897119896

Δ Such an interval is referred to further as embedding

interval For any 120589 isin [119897119896

Δ 119897119896

Δ+ Δ] we define 119909 = 120589 minus 119897

119896

Δand

119883 will be used to denote a random variable which represents119909 The length Δ is selected in a way that an appropriatedocument to watermark ratio (DWR) is guaranteed afterthe separation We also assume that Δ is small enough toderive that the distribution of 119883 is uniform A randomvariable that represents separated coefficients inside 119896thinterval is denoted by 119883

1015840 and its realization is denoted by 1199091015840

Correspondingly a randomvariable that represents separatedcoefficients on the whole real number line is denoted by Σ

1015840

and its realization is denoted by 1205891015840 Each pair of an original

119909 and corresponding quantized 1199091015840 belong to the same 119896th

embedding interval so that an absolute shift is never largerthan Δ

Let us denote a watermark bit by 119887 Truncated pdfs 1198910(1199091015840)

and 1198911(1199091015840) are used to describe the distribution of 1198831015840 and

should be defined prior to quantization Parameters 1205781 and 1205990

represent fractions of IDL for 119887 = 1 and 119887 = 0 respectivelyParameters1205931 and 1205740 represent fractions of the samples whereoriginal values 119909 are to be modified by a quantizer for 119887 = 1

and 119887 = 0 respectively It is therefore assumed that thefraction of zeros in a watermark data is 1205740 +1205990 and fraction ofones is 1205931 + 1205781 Condition 1205740 + 1205990 + 1205931 + 1205781 = 1 always holdsThe result of the separation in the 119896th embedding intervaldepends on 119887 1205781 1205990 1205931 1205740 1198910(119909

1015840) and 1198911(119909

1015840) In other

words 1199091015840 is defined by quantizer119876119896Δ[sdot] that has thementioned

parameters

1199091015840= 119876119896

Δ[119909 119887 1205781 1205990 1205931 1205740 1198910 1198911] (1)

We will use SA to describe the quantizer 119876119896Δ[sdot] Each of

logical atoms 119901 119902 119903 119904 119905 119906 and V represents some conditionwhich is either true or false

119901 | 119909 leΔ1205740

1205740 + 1205990

(2)

119902 | 119909 geΔ1205781

1205931 + 1205781

(3)

119903 | 1199091015840= 119909 (4)

119904 | 1199091015840lt 119909 (5)

119905 | 1199091015840gt 119909 (6)

119906 | 1199091205740 + 1205990

Δ= 1205740 int

1199091015840

0

1198910 (1199091015840) 1198891199091015840 (7)

V | (Δ minus 119909)1205931 + 1205781

Δ= minus1205931int

1199091015840

Δ

1198911 (1199091015840) 1198891199091015840 (8)

For example (sim 119887amp119901) is true if and only if ldquo119887 = 0rdquo and 119909 isnot classified for IDL We formalize SA in the following way

((sim 119887amp119901) sup (119906amp (119904 or 119903)))

((119887amp119902) sup (Vamp (119905 or 119903)))

(((sim 119887amp sim 119901) or (119887amp sim 119902)) sup 119903)

⊨ ((119906amp (119904 or 119903)) or (Vamp (119905 or 119903)) or 119903)

(9)

It can be seen that SA is valid The conclusion of SA statesthat the separation of coefficient values inside 119896th embeddinginterval is possible which means that the proposed modelis suitable for information embedding Furthermore eachpremise represents an important dependency between inputand output of the quantizer 119876119896

Δ[sdot] and we require that each

premise is indeed true Hence it is necessary to enforcesoundness for SA

The intuition behind SA can be explained in the followingway Initially samples with labels ldquo119887 = 0rdquo and ldquo119887 = 1rdquo arenot separated in the dimension of 119909 inside the mentioned 119896thembedding interval In order to separate them we shift thosewith ldquo119887 = 0rdquo to the left and those with ldquo119887 = 1rdquo to the right Ifso shift to the right for ldquo119887 = 0rdquo or shift to the left for ldquo119887 = 1rdquois not acceptable because it would introduce distortion andon the other hand worsen separation between ldquo0rdquo and ldquo1rdquoTherefore for sim 119887 formula (119904 or 119903) is true and for 119887 formula(119905 or 119903) is true

Another consideration is that for any two 119909119894 le 119909119895 withthe same bit value we infer that quantization in a way that1199091015840

119894le 1199091015840

119895implies less distortion than if 1199091015840

119894gt 1199091015840

119895 Saving the

order we preserve cumulative distribution in respect to theorder Quantized samples 1199091015840 that interpret ldquo0rdquo are distributedaccording to pdf 1198910(119909

1015840) samples 119909

1015840 that interpret ldquo1rdquo aredistributed according to pdf 1198911(119909

1015840) Therefore 119906 or V is true

if (sim 119887amp119901) or (119887amp119902) is true respectively


1205740

1205781

1205740f0(x998400) 1205931f1(x998400)

1205931

1205990

1205740 + 1205990

1205931 + 1205781

ldquo0rdquo ldquo1rdquo

lk

lk

120589998400

120589

lk +

lk +

(u and ( )) ( and ( ))

(b and q)(simb and p)

Δ

ΔΔΔ

Δ Δ

IDL(0)

IDL(0)

IDL(1)

IDL(1)

t or rs or r

Figure 1 Illustration of the process of separation

And lastly the condition for IDL is ((sim 119887amp sim 119901) or (119887amp sim

119902)) and it is the case when 119909 is not modified and therefore 119903An illustration of an example where SA is sound is given

in Figure 1 Two positions of original values are shown onthe lower part of Figure 1 Condition (sim 119887amp119901) is satisfiedfor the first original value and condition (119887amp119902) is satisfiedfor the second Two positions of the modified values areshown on the upper part of Figure 1 After the separation themodified values satisfy conditions (119906amp(119904or119903)) and (Vamp(119905or119903))respectivelyThe areas of green segments on the lower and theupper parts of Figure 1 are equal The areas of blue segmentsare also equal As it can be seen on the upper part of Figure 1the distribution of separated coefficients in 119896th embeddinginterval depends on Δ 1205781 1205990 1205931 1205740 1198910(119909

1015840) and 1198911(119909

1015840)

Parameters of the pdfs 1198910(1199091015840) and 1198911(119909

1015840) need to be

specified in order to prove soundness for the whole range of119909 in the 119896th interval In addition formulas (7) and (8) need tobe rearranged in order to express 1199091015840 in a suitable way for thequantization form

We propose such 1198910(1199091015840) and 1198911(119909

1015840) that in general there

is no line of symmetry which can separate them insideembedding interval This feature will provide easier recoveryafter GA It is necessary to emphasize that the proposedfunctions 1198910(119909

1015840) and 1198911(119909

1015840) only describe distributions for

fractions 1205740 and 1205931 respectively (eg without taking intoaccount fractions of IDL)We introduce parameters 120572 120573 and120591 to define both pdfs 1198910(119909

1015840) and 1198911(119909

1015840) where 0 le 120572 le 120573 le 1

as shown in Figure 2(a) As can be seen the density is zero inthe subinterval (Δ(120573 minus 120572) Δ120573) which separates ldquo0rdquo from ldquo1rdquoIn Figure 2(b) we can see the distribution of the quantizedcoefficients outside 119896th embedding interval as well

Namely the proposed truncated pdfs are a linear functionand a constant

1198910 (1199091015840) =

1198881199091015840+ 120591 if 1199091015840 isin [0 Δ (120573 minus 120572)]

0 otherwise(10)

1198911 (1199091015840) =

119892 if 1199091015840 isin [Δ120573 Δ]

0 otherwise(11)

The samples that belong to IDL fraction are distributedaccording to pdfs IDL0(119909

1015840) and IDL1(119909

1015840)

IDL0 (1199091015840) =

1205740 + 1205990

Δ1205990

if 1199091015840 isin [Δ1205740

1205740 + 1205990

Δ]

0 otherwise

IDL1 (1199091015840) =

1205931 + 1205781

Δ1205781

if 1199091015840 isin [0Δ1205781

1205931 + 1205781

]

0 otherwise

(12)

22 Soundness Conditions for SA The soundness of SAis guaranteed if it is possible to satisfy (119906amp(119904 or 119903)) or(Vamp(119905 or 119903)) when (sim119887amp119901) or (119887amp119902) is true respectively Therequirement to satisfy (119906amp(119904 or 119903)) or (Vamp(119905 or 119903)) imposessome constraints on 120572 120573 119888 119892 120591 1205740 1205931 1205781 1205990 and Δ Let usfind those constraints

We start from defining parameters of 1198910(1199091015840) and 1198911(119909

1015840)

using property of pdf

int

(120573minus120572)Δ

0

1198910 (1199091015840) 1198891199091015840= 119888

(120573 minus 120572)2Δ2

2+ 120591Δ (120573 minus 120572) = 1 (13)

int

Δ

120573Δ

1198911 (1199091015840) 1198891199091015840= 119892Δ (1 minus 120573) = 1 (14)

It is easy to derive from (14) that

119892 =1

Δ (1 minus 120573) (15)

According to (4) (5) and (7) condition (119906amp(119904or119903)) is satisfiedif and only if for all 1199091015840

1199091015840 1205740 + 1205990

Δle 1205740 int

1199091015840

0

1198910 (1199091015840) 1198891199091015840 (16)

Using (10) and the fact 1199091015840 ge 0 we can derive

120591 ge1205740 + 1205990

Δ1205740

minus 1198881199091015840

2 (17)

The latter inequality should be true for all 1199091015840 isin [0 Δ(120573 minus 120572)]

which means

120591 ge max1199091015840isin[0Δ(120573minus120572)]

(1205740 + 1205990

Δ1205740

minus 1198881199091015840

2) (18)

For our particular application we chose 119888 ge 0 therefore

120591 ge1205740 + 1205990

Δ1205740

(19)

and we are using the value 120591 = (1205740+1205990)(Δ1205740) in our methodUsing (13) we can conclude that

119888 = 21205740 minus (1205740 + 1205990) (120573 minus 120572)

1205740(120573 minus 120572)2Δ2

(20)


f0(x998400)f1(x998400)

120572

120573

x998400

lk + 120589998400lk

120591ldquo0rdquo ldquo1rdquoΔ

ΔΔ Δ

ΔΔ

(a)

120589998400

k minus 1 k k + 1 k + 2 k + 3

ldquo0rdquoldquo1rdquo ldquo0rdquo ldquo0rdquoldquo1rdquo ldquo1rdquo

(b)

Figure 2 Distribution of the quantized coefficients (a) inside 119896th embedding interval (b) in five consecutive intervals

Functions 1198910(1199091015840) and 1198911(119909

1015840) can be fully defined now Let

us find dependencies that connect 120572 and 120573with 1205740 1205931 1205781 and1205990 Taking into account that in our realization 119888 ge 0 we canderive from (20) that

120573 minus 120572 le1205740

1205740 + 1205990

(21)

According to (4) (6) and (8) condition (Vamp(119905 or 119903)) issatisfied if and only if

1205931 + 1205781

Δle 1198921205931 (22)

Using (15) we find that

120573 ge1205781

1205931 + 1205781

(23)

In the experiment section of the paper the goal is to findthe highest capacity for a given WNR Different values of theparameters need to be checked for that purpose Preserving(15) and (19)ndash(21) (23) would guarantee soundness of SAand avoidance of using parametersrsquo combinations that arenot efficient for watermarking This can reduce requiredcomputations

23 Embedding Equations For the proposed pdfswe can nowdefine 119909

1015840 as a function of 119909 which is the main task of thequantizer 119876119896

Δ[sdot] Let us consider conditions (sim 119887amp119901) (119887amp119902)

separately as it is never the case when both conditions aretrue We will denote 119909

1015840 in case of (sim 119887amp119901) by 1199091015840 but in caseof (119887amp119902) the notation 1199091015840 will be used

From (7) (10) and 120591 = (1205740 + 1205990)(Δ1205740) it is clear that

05119888 11990910158402

+ 120591 1199091015840 = 120591119909 (24)

Taking into account that 1199091015840 ge 0 we derive

1199091015840 =

radic1205912 + 2119888120591119909

119888minus

120591

119888 (25)

From (8) (11) and (15) we can find that

1199091015840 = 119861119909 + Δ (1 minus 119861) 119861 =(1 minus 120573) (1205931 + 1205781)

1205931

(26)

According to (26) the values of quantized coefficientsare linearly dependent on original values while according

to (25) the dependency is nonlinear Different character ofdependency between quantized and original values for ldquo0rdquoand ldquo1rdquo is one of the key features of our approach Thisdifferentiates the proposed watermarking method from themethods previously described in the literature [10ndash12]

3 Characteristics of Quantization Model

The model was proposed in the previous section It wasshown that it is suitable for coefficient separation and theconditions necessary for soundness of SA were definedIn this section we focus on efficiency of separation Themain characteristic that can be estimated analytically is thewatermark channel capacity under AWGN It is required tocalculate such characteristic for different WNRs First weexpress WNR in terms of parameters of the quantizationscheme Second we express error rates in terms of parametersof the quantization scheme This makes it possible to includeWNR in the expression for error rates (and capacity)

31 Estimation of Quantization Distortions The variance 1205902

119899

is the only parameter of AWGN attack and WNR is definedas

WNR = 10 log10

(119863

1205902119899

) (27)

where 119863 is a watermark energy Alternatively 119863 can be seenas a distortion of a host signal induced by the quantizationLet us define119863

For the matter of convenience of the experiment it isbetter to use a single parameter (control parameter) thatcan be adjusted in order to provide the desired value of 119863While defining 119863 we choose Δ to be the control parameterand collect it in the expression for 119863 The total distortion 119863

is a sum of distortions 1198630 and 1198631 caused by two types ofshifts that are 119909 rarr 1199091015840 and 119909 rarr 1199091015840 respectively The firstdistortion component1198630 is defined as

1198630 = 1205740 int

Δ(120573minus120572)

0

1198910 (1199091015840)(1199091015840minus

1

120591int

1199091015840

0

1198910 (1199091015840) 1198891199091015840)

2

1198891199091015840

(28)

Proceeding further and using (10) we can derive that

1198630 = 1205740 int

Δ(120573minus120572)

0

(1198881199091015840+ 120591)

119888211990910158404

412059121198891199091015840 (29)


However it is clear from (19)-(20) that both parameters 119888

and 120591 depend on Δ In order to collect Δ we introduce twoindependent ofΔ parameters 119888 = 119888Δ

2 and 120591 = 120591ΔThis bringsus to

1198630 = Δ21198760

1198760 = 1205740 (1198883

241205912(120573 minus 120572)

6+

1198882

20120591(120573 minus 120572)

5)

(30)

The second distortion component1198631 is defined as

1198631 = 1205931

times int

Δ

120573Δ

1198911 (1199091015840)

times (1199091015840minus (

1205931Δ

1205781 + 1205931

int

1199091015840

120573Δ

1198911 (1199091015840) 1198891199091015840

+1205781Δ

1205781 + 1205931

))

2

1198891199091015840

(31)

Using (11) (15) and integrating in (31) we obtain

1198631 = Δ21198761

1198761 = 1205931

((1205781 + 1205931) (1 minus 120573) minus 1205931)2

3(1205781 + 1205931)2

(32)

The total quantization distortion 119863 can be expressed interms of Δ1198760 and 1198761

119863 = Δ2(1198760 + 1198761) (33)

For any combination of 1205902119899WNR 120572 120573 1205781 1205990 1205740 and 1205931

the required value of Δ is defined using (27) and (33) as

Δ = radic1205902

1198991001lowastWNR

1198760 + 1198761

(34)

32 Estimation of Error Rates Bit error rate (BER) andchannel capacity can be calculated without simulation ofwatermark embedding procedure It is important that thekind of threshold used to distinguish between ldquo0rdquo and ldquo1rdquo issuitable for analytic estimations Further we assume that theposition of the threshold remains permanent after watermarkis embedded and does not depend on attack parameters InFigure 2(b) the position of the threshold is Th for intervalsnumbered 119896 + 2119898 119898 isin Z For the intervals numbered119896 + 2119898 + 1 the position of the threshold is Δ minusTh

The absolute value of quantized sample in any interval is1205891015840 We use 120589

1015840

119899for a sample that is distorted by noise Hence

1205891015840

119899interprets ldquo0rdquo or ldquo1rdquo depending on belonging to Z or O

respectively

Z =

infin

⋃

119898=minusinfin

[2Δ119898 + 119897119896

ΔminusTh 2Δ119898 + 119897

119896

Δ+Th) (35)

O =

infin

⋃

119898=minusinfin

[2Δ119898 + 119897119896

Δ+Th 2Δ(119898 + 1) + 119897

119896

ΔminusTh) (36)

There are two cases when errors occur in non-IDLsamples An error in ldquo0rdquo is incurred by a noise if and onlyif the both following conditions are true

(1205891015840isin Z) (120589

1015840

119899isin O) (37)

An error in ldquo1rdquo occurs if and only if the following is true

(1205891015840isin O) (120589

1015840

119899isin Z) (38)

Two cases when errors occur in IDL samples can bepresented with the following conditions for ldquo0rdquo and ldquo1rdquorespectively

(1205891015840isin O) (120589

1015840

119899isin O) (39)

(1205891015840isin Z) (120589

1015840

119899isin Z) (40)

The pdf of AWGN with variance 1205902

119899can be represented

in terms of 1205891015840 and 1205891015840

119899as 119891N[120589

1015840

119899minus 1205891015840 0 120590119899] In general we can

estimate error rates for an interval with any integer index119896 + 119898 For that purpose we use generalized notations 1198910(120589

1015840)

1198911(1205891015840) IDL0(120589

1015840) and IDL1(120589

1015840) for pdfs of quantized samples

in any interval For example for even 119898 pdf 1198910(1205891015840) = 1198910[120589

1015840minus

(119897119896

Δ+Δ119898)] for odd119898 pdf 1198910(120589

1015840) = 1198910[119897

119896

Δ+ Δ(119898 + 1) minus 120589

1015840

] Wedenote 119896+119898 interval by 119868119896+119898 = [119897

119896

Δ+Δ119898 119897

119896

Δ+Δ(119898+1)]Then

the error rates for quantized samples in 119868119896+119898 can be definedas

BER0 =1205740

1205740 + 1205990

int

Oint

119868119896+119898

1198910 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

+1205990

1205740 + 1205990

int

Oint

119868119896+119898

IDL0 (1205891015840)

times 119891N [1205891015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

BER1 =1205931

1205931 + 1205781

int

Zint

119868119896+119898

1198911 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

+1205781

1205931 + 1205781

int

Zint

119868119896+119898

IDL1 (1205891015840)

times 119891N [1205891015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

(41)

Now we can show that BER0 and BER1 can be calculatedaccording to (41) for any chosen interval For that purpose itis enough to demonstrate that any component in (41) remainsthe same for every interval For example we state that

int

Oint

119868119896+119898

1198910 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

= int

Oint

Δ

0

1198910 (1199091015840) 119891N [120589

1015840

119899minus (119897119896

Δ+ 1199091015840) 0 120590119899] 119889119909

10158401198891205891015840

119899

(42)

for any119898


Let us first assume 119898 = 2119899 119899 isin Z Then 1205891015840 = 1199091015840+ 119897119896

Δ+

2Δ119899 1198910(1205891015840) = 1198910(119909

1015840) However it is also clear from (36) that

O + 2Δ119899 = O Hence

int

Oint

119868119896+2119899

1198910 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

= int

O+2119899Δint

Δ

0

1198910 (1199091015840)

times 119891N [(1205891015840

119899minus 2119899Δ)

minus (119897119896

Δ+ 1199091015840) 0 120590119899] 119889119909

1015840119889 1205891015840

119899minus 2119899Δ

(43)

and we prove the statementNow let us assume 119898 = 2119899 + 1 119899 isin Z Then 1205891015840 =

(1199091015840minus Δ) + 119897

119896

Δ+ 2Δ(119899 + 1) 1198910(120589

1015840) = 1198910(Δ minus 119909

1015840) For the matter

of convenience we accept that 119897119896Δ+ 119895Δ = 0 for some 119895 isin Z

Therefore 119891N[1205891015840

119899minus 1205891015840 0 120590119899] = 119891N[(120589

1015840

119899minus 2Δ(119899 + 1minus 119895)) minus (minus119897

119896

Δ+

(1199091015840minusΔ)) 0 120590119899] Also minus(O+ 2Δ(119899 + 1 minus 119895)) = O The property

of pdf of AWGN provides that 119891N[119910 0 120590119899] = 119891N[minus119910 0 120590119899]

and consequently

119891N [ (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))

minus (minus 119897119896

Δ+ (1199091015840minus Δ)) 0 120590119899]

= 119891N [ minus (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))

minus (119897119896

Δ+ (Δ minus 119909

1015840)) 0 120590119899]

(44)

Using the latest equation we derive that

int

Oint

119868119896+2119899+1

1198910 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

= int

minus(O+2Δ(119899+1minus119895))int

Δ

0

1198910 (Δ minus 1199091015840)

times 119891N [ minus (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))

minus (119897119896

Δ+ (Δ minus 119909

1015840)) 0 120590119899]

times 119889 Δ minus 1199091015840

times 119889 minus (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))

(45)

and we prove the statement

4 Experimental Results

In this section we describe conditions procedure and resultsof two different kinds of experiments based on analyticestimation of capacity as well as simulations The preferredindex of attack severity is WNR (indexes 120590119899 and qualityof JPEG compression are also used) For a given set ofembedding parameters the error rates and capacity are

minus12 minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 10 12

CTLNS-QIM-IDLNS-QIM

DC-QIMQIM

WNR (dB)

100

10minus2

10minus4

10minus6

C (b

itsy

mbo

l)Figure 3 Analytic-based estimation of capacity under AWGN

estimated differently using different models suitable for eachkind of experiment However for both kinds of experimentthe maximum capacity for a given level of attack severity isfound by using brute force search in the space of all adjustableparameters

41 Analytic Estimation of Watermarking Performance underAWGN In this subsection of our experiment we use 120590119899 = 1Parameters 120572 120573 1205781 1205740 1205990 and 1205931 are subjects to constraints(21) (23) 1205781 +1205740 = 05 and 1205990 +1205931 = 05 and the simulationsare repeated for each new value of WNR Then the length ofembedding interval Δ is calculated according to (34) Errorrates are calculated according to (41)

We use two variants of the proposed quantization schemewith adjustable parameters nonsymmetric QIM (NS-QIM)and nonsymmetric QIM with IDL (NS-QIM-IDL) Such adecision can be explained by a consideration that IDL isacceptable for some application but other applications mayrequire all the watermark data to be embedded correctly

In Figure 3 the plots for channel capacity towardWNRareshown for two variants of the proposedmethod aswell asDC-QIM and QIM [9] The permanent thresholding Th = Δ(120573 minus

05120572) is applied toNS-QIMandNS-QIM-IDL As a referenceCosta theoretical limit (CTL) [5] is plotted in Figure 3

CTL =1

2log2(1 + 10

01lowastWNR) (46)

Capacity is calculated analytically according to thedescription provided in the literature for DC-QIM and QIM


During the estimation the subsets Z sub Z and O sub O wereused instead of Z andO

Z =

100

⋃

119898=minus100

[2Δ119898 + 119897119896

ΔminusTh 2Δ119898 + 119897

119896

Δ+Th)

O =

100

⋃

119898=minus100

[2Δ119898 + 119897119896

Δ+Th 2Δ (119898 + 1)

+ 119897119896

ΔminusTh)

(47)

Therefore for such estimation we assume that quantizedcoefficients from the 119896th interval after AWGN are distributedonly inside [minus200Δ+119897

119896

ΔminusTh 202Δ+119897

119896

ΔminusTh)The assumption

is a compromise between computational complexity and thefidelity of the result

As can be seen from Figure 3 both variants of theproposed method perform better than DC-QIM for WNRvalues less than minus2 dB and obviously much higher capacityprovided by DC-QIM-IDL is compared to the other methodsin that range Taking into account that DC-QIM providesthe highest capacity under AWGN compared to the otherknown in the literature methods [12 19] newly proposedmethodDC-QIM-IDL fills an important gap Reasonably thedemonstrated superiority is mostly due to IDL

42 Watermarking Performance in Simulation Based Exper-iments without GA The advantage of analytic estimation oferror rates according to (41) is that the stage of watermarkembedding can be omitted and host signal is not requiredThe practical limitation of the approach is that Z and O arejust subsets of Z andO respectively Other disadvantages arethat estimation might become even more complex in casethe threshold position is optimized depending on the levelof noise only rates for AWGN can be estimated but thereare other kinds of popular attacks [24] Therefore in thissubsection we will also simulate watermarking experimentsusing real host signals

421 Conditions for Watermark Embedding and ExtractionIn case of experiments with real signals the parameters ofthe proposed watermarking scheme must satisfy some otherconstraints instead of (34) However constraints (21) (23)1205781 + 1205740 = 05 and 1205990 + 1205931 = 05 remain the same as in theanalytic based experiment

Some lower limit of DWR has to be satisfied for water-marked host which assures acceptable visual quality DWRis calculated according to

DWR = 10 log10

(1205902

119867

119863) (48)

where 1205902119867is the variance of the host

Therefore using (33) the equation for Δ in that case is

Δ =120590119867

radic(1198760 + 1198761) 1001DWR

(49)

In contrast to analytic based experiment 120590119899 should beadjusted for different severity of the attack and is defined as

1205902

119899=

1205902

119867

1001(DWR+WNR)

(50)

After watermark is embedded and AWGN with 1205902

119899is

introduced we perform extraction and calculate channelcapacity

A variant NSC-QIM with constant (nonadjustable)parameters is also used in some experiments The intentionto adjust the parameters in order to maximize capacity isnatural However maximization requires information aboutWNR to be known before watermark embedding and trans-mission In some application areas level of noise (or severityof an attack) might change over time or remain unknownTherefore watermark should be embedded with some con-stant set of parameters depending on expected WNR

Different positions of the threshold can be used to extractawatermarkAn optimal position of the threshold is not obvi-ous Placing the threshold in the middle of the interval mightbe inefficient because the distribution of quantized samplesinside embedding interval is nonsymmetric Two kinds ofthresholding are proposed permanent and nonpermanentThe permanent position is Th = Δ(120573 minus 05120572) for the intervalswith numbers 119896 + 2119898 119898 isin Z The name ldquopermanentrdquo isbecause Th cannot be changed after embedding Its positiondepends only on 120572 120573 and Δ and does not depend on theparameters of attack

The nonpermanent position of Th is the median of thedistribution inside each interval Nonpermanent positionmay depend on the type and severity of a noiseThe advantageof nonpermanentTh is that extraction of a watermark can bedone without information about 120572 and 120573

422 Watermarking Performance for AWGN and JPEGAttacks without GA The performance of the proposedmethod was evaluated using real host signals For that pur-pose we used 87 natural grayscale images with resolution 512times 512 Each bit of a watermark was embedded by quantizingthe first singular value of SVD of 4 times 4 block This kindof transform is quite popular in digital image watermarkingand the chosen block size provides a good tradeoff betweenwatermark data payload and robustness [7 25] The value ofDWR was 28 dB An attack of AWGN was then applied toeach watermarked imageThe resulting capacity toward noisevariance is plotted for different methods in Figure 4

It can be seen that the resulting capacity after AWGNattack is the highest for NS-QIM The other two methodswhose performance is quite close to NS-QIM are DC-QIMand FZDH Compared to DC-QIM the advantage is moreobvious for higher variance However for moderate variancethe advantage is more obvious compared to FZDH

Methods QIM and RDMdo not have parameters that canbe adjusted to different variance Under some circumstancesadjustment is not feasible for NS-QIM as well We havechosen constant parameters 120572 = 005 and 120573 = 035 for NSC-QIM in order to provide a fair comparison with QIM andRDM The plots for NSC-QIM QIM and RDM are marked


0 40 80 120 160 200

100

10minus1

10minus2

10minus3

10minus4

10minus5

Var

DC-QIMNS-QIMFZDHTCM

QIMNSC-QIMRDM

C (b

itsy

mbo

l)

Figure 4 Capacity under AWGN for natural grayscale images

by squares triangles and crosses respectively in Figure 4As can be seen NSC-QIM performs considerably better thanQIM and RDM and the advantage is especially noticeable forhigher noise variance

Other image processing techniques except additive noiseare able to destroy a watermark and one of them is JPEGcompression which is quite popular The capacity of theproposed watermarking method was also compared withothermethods and the procedure of embeddingwas the sameas in AWGNcase However this time JPEG compressionwithdifferent levels of quality was considered as an attack Theresults are plotted in Figure 5

According to the plots in Figure 5 the performance ofNS-QIM in general is very close to that of DC-QIM butis slightly worse for low 119876 factor The methods FZDH andTCMprovide lower capacity thanNS-QIM andDC-QIM butin general are quite close to them The worst performanceis demonstrated by QIM and RDM and the disadvantage isespecially noticeable for low 119876 For NSC-QIM with 120572 = 005

and120573 = 035 the performance is considerably better than thatforQIMandRDMunder lowQbut isworse for higher qualityof JPEG compression

43 Procedure forGARecovery It has been demonstrated thatfor some popular types of attack the performance of NS-QIMis comparable or better than that of DC-QIMThementionedDC-QIM is considered to be one of the best quantizationmethods for watermarking but it is extremely vulnerable toGA On the other hand the performance of RDM is not asgood under AWGN and JPEG attacks and is comparable tothat of QIM In this subsection we propose a procedure forGA recovery in order to fill an important gap in the literatureand introduce a watermarking method that provides highefficiency under AWGN as well as GAThe procedure utilizes

DC-QIMNS-QIMFZDHTCM

QIMNSC-QIMRDM

100

10minus1

10minus2

20 30 40 50 60 70 80 90 100

Q of JPEG ()

C (b

itsy

mbo

l)Figure 5 Capacity under JPEG for natural grayscale images

features that are unique for the proposed approach and havenot been discussed in the field of watermarking before

We are proposing several criteria that will be used by theprocedure to provide robustness againstGA forNS-QIMThecriteria exploit nonsymmetric distribution inside embeddinginterval and help to recover a watermarked signal after theattack It is presumed that a constant gain factor is appliedto the watermarked signal (followed by AWGN) and the taskis either to estimate the factor or the resulting length ofembedding interval

Let us denote the actual gain factor by 120582 and our guessabout it by 120582

1015840 The length of the embedding interval (whichis optimal for watermark extraction) is modified as a result ofGA and is denoted by Δ = 120582Δ Our guess about Δ is Δ1015840 = 120582

1015840Δ

The core of the procedure of recovery after GA is the fol-lowing For each particular value Δ1015840 noisy quantized samples1205891015840

119899are being projected on a single embedding interval

1199091015840

119899=

1205891015840

119899mod Δ

1015840 if

[[[

[

1205891015840

119899minus 119897119896

Δ

Δ1015840

]]]

]

mod 2 = 0

Δ1015840minus (1205891015840

119899mod Δ

1015840) otherwise

(51)

One of the following criteria is being applied to therandom variable1198831015840

119899isin [0 Δ

1015840]

1198621 (Δ1015840) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

median (1198831015840

119899)

Δ1015840minus 05

10038161003816100381610038161003816100381610038161003816100381610038161003816

1198622 (Δ1015840) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

119864 ([1198831015840

119899]119908

)

[Δ1015840]119908

10038161003816100381610038161003816100381610038161003816100381610038161003816

119908 = 2119898 + 1 119898 isin N

(52)


0035

003

0025

002

0015

001

0005

09 95 10 105 11

998400

Crite

rion

1

Δ

(a)

0

1

2

3

4

5

6

7

8

9 95 10 105 11

Crite

rion

2

times10minus4

998400Δ

(b)

Figure 6 Plots of criteria values toward guessed length of embedding interval (a) criterion 1198621 (b) criterion 1198622

The value of Δ10158401015840 that maximizes one of the proposed

criteria should be chosen as the best estimate of Δ

Δ10158401015840= argmax

Δ101584011986212 (Δ

1015840) (53)

The intuition behind the proposed procedure of recoveryfrom GA is the following The variance of the coefficients ofthe host signal is much larger than the length of embeddinginterval Embedding intervals are placed next to each otherwithout gaps and even small error in estimation of Δ results inconsiderable mismatch between positions of samples insidecorresponding embedding intervals In other words wrongassumption about Δ makes distribution of 1198831015840

119899very close to

uniform However in case Δ1015840 is close to Δ the distribution

of 1198831015840119899demonstrates asymmetry because the distribution of

quantized samples inside embedding interval (before GA isintroduced) is indeed asymmetric Hence criteria 1198621 and 1198622

are just measures of asymmetry The main advantage of theprocedure is simplicity and low computational demand

Experimental results demonstrate high level of accuracyof the proposed procedure of recovery after GA Grayscaleimage Lenatif with dimension 512 times 512 was used as a hostsignal for that purpose A random watermark sequence wasembedded into the largest singular values of SVD of 4 times

4 blocks using NS-QIM with 120572 = 005 and 120573 = 035The AWGN attack was applied after the embedding so thatWNR = minus5 dB The length of embedding interval was 10However we use notation Δ = 10 because the value is notknown to the receiver and during watermark extraction theproposed recovery procedure was usedThe interval of initialguess was Δ plusmn 10 so that Δ1015840 isin [9 11] Such an initial guessreflects real needs for recovery after GA because a gain factorthat is outside the range 09sim11 causes considerable visualdistortions in most cases The initial guess interval was splitby equally spaced 1000 steps and for each step the recoveryprocedure was applied The plots for values of 1198621 and 1198622

119908 = 5 toward guessed values of Δ are shown in Figures 6(a)and 6(b) respectively

Despite the fact that for the sameΔ the difference betweenvalues of1198621 and1198622 is huge the shapes of the plots are similarThe criteria reach their maximum at 10042 and 9998 for 1198621and 1198622 respectively which are quite precise estimates of theactual Δ used during watermark embedding

44 Performance for AWGN and JPEG Attacks with GA Theembedding constraints for the current experiment are thesame as described in Section 421 Among the quantizationmethods used for comparison the only method robust to GAis RDMTherefore only RDMwas used as a reference to NS-QIM andNSC-QIMunder GA followed by AWGNand JPEGattacks respectively The exact information about Δ was notused for extraction in NS-QIM and NSC-QIM cases which isequivalent to GA with unknown scaling factor

The watermark embedding domain was the same asin previous tests first singular values of SVD of 4 times 4blocks from 512 times 512 grayscale images were quantizedDWR = 28 dB In case of RDM the quantized value of aparticular coefficient is based on the information about thelast 100 previous coefficients For NSC-QIM the parametersof embedding were 120572 = 005 and 120573 = 035 For both AWGNand JPEG attacks the same as previously ranges of parameterswere used

However during watermark extraction no informationexcept initial guess interval Δ plusmn 10 was used in NS-QIMandNSC-QIMcases Criterion1198621was used for the estimationof actual Δ Nonpermanent thresholding was applied to bothmodifications of the proposed watermarking method Incontrast to that RDM does use the exact information aboutquantization step The resulting capacity toward AWGNvariance is plotted for each method in Figure 7

It can be seen from Figure 7 that both NS-QIM andNSC-QIM outperform RDMThe advantage of the proposedmethod is more evident for larger variance of the noise


0 40 80 120 160 200

100

10minus1

10minus2

10minus3

10minus4

10minus5

Var

NS-QIMNSC-QIMRDM

C (b

itsy

mbo

l)

Figure 7 Capacity under GA followed by AWGN

100

10minus1

10minus2

20 40 60 80 100

Q of JPEG ()

NS-QIMNSC-QIMRDM

C (b

itsy

mbo

l)

Figure 8 Capacity under GA followed by JPEG compression

The capacity plots for NS-QIM NSC-QIM and RDM incase of JPEG attack are shown in Figure 8

FromFigure 8we can conclude that bothmodifications ofthe proposed watermarking method supply higher capacitythan RDM when 119876 lt 50 However only NS-QIMoutperforms RDM in case119876 gt 50 and NSC-QIM performsworse than RDM for that range

5 Discussion

In the experiment section we have estimated the capacityof the proposed method in both analytical and empirical

ways Following both ways we can witness that the proposedmethod provides higher capacity compared to the otherreference methods In this section we are to discuss in moredetail measures of watermarking efficiency conditions of theexperiments and the reasons of superiority of NS-QIM-IDL

Channel capacity 119862 is one of the most important mea-sures for watermarking as it indicates the maximum amountof the information that can be transmitted by a singleembedded symbol [1 12] However some authors in theiroriginal papers refer to error rates instead [13 16 19ndash21] It canbe demonstrated that calculations of 119862 using error rates arestraightforward [26] Capacity can be calculated according tothe following expression

119862 = max119901em(sim119887)

[119901 (sim 119887 119887) log2(

119901 (sim 119887 119887)

119901em (sim 119887) 119901ex (119887))

+ 119901 (119887 sim 119887) log2(

119901 (119887 sim 119887)

119901em (119887) 119901ex (sim 119887))

+ 119901 (sim 119887 sim 119887) log2(

119901 (sim 119887 sim 119887)

119901em (sim 119887) 119901ex (sim 119887))

+ 119901 (119887 119887) log2(

119901 (119887 119887)

119901em (119887) 119901ex (119887))]

(54)

where for instance 119901(sim119887 119887) denotes joint probability ofembedding symbol sim119887 and extracting symbol 119887 119901em(119887) and119901ex(119887) denote probabilities of embedding and extracting ofsymbol 119887 Probabilities of extracting a particular symbol canbe calculated using joint probabilities

119901ex (119887) = 119901 (sim 119887 119887) + 119901 (119887 119887)

119901ex (sim 119887) = 119901 (119887 sim 119887) + 119901 (sim 119887 sim 119887)

(55)

Joint probabilities can be expressed using 119901em(sdot) and errorrates

119901 (sim 119887 119887) = 119901em (sim 119887)BERsim119887

119901 (119887 sim 119887) = 119901em (119887)BER119887

119901 (sim 119887 sim 119887) = 119901em (sim 119887) (1 minus BERsim119887)

119901 (119887 119887) = 119901em (119887) (1 minus BER119887)

(56)

Embedding probabilities for the methods proposed in thispaper are

119901em (sim 119887) = 1205740 + 1205990

119901em (119887) = 1205781 + 1205931

(57)

As a contrast to the watermarking approach proposed inthis paper the QIM-based methods known in the literatureassume equal embedding probabilities and provide equalerror rates for ldquo0rdquo and ldquo1rdquo [12 19] For all the mentionedin the experimental section methods (QIM DC-QIM RDAFZDH TCM and the proposed methods) the results werecollected under equal conditions of each kind of attack In


order to compare efficiency of the proposed methods withsome other state-of-the-art papers in watermarking [13 21]their channel capacity can be calculated based on the dataprovided in those papers From (54)ndash(56) we derive thatQIM-based watermarking which has been presented in theliterature capacity is

119862 = 1 + BERlog2(BER) + (1 minus BER) log

2(1 minus BER) (58)

The largest singular values of SVD of 4 times 4 blockswere used by all the methods for watermark embedding inthe empirical estimations of capacity Such a domain is anatural choice formanywatermarking applications because itprovides a good tradeoff between robustness invisibility anddata payload [7 27 28] Commonly the largest singular val-ues are being quantized [25] The robustness of a watermarkembedded in the domain can be explained by a considerationthat the largest singular values have a great importance Forexample compared to a set of the coefficients of discretecosine transform (DCT) the set of singular values has morecompact representation for the same size of a segment of animage [29] At the same time the block size of 4 times 4 is enoughto avoid some visible artefacts and this guarantees invisibilityunder DWR = 28 dB The data payload of 1 bit per 16 pixelsis sufficient for inclusion of important copyright informationand for image size 512 times 512 provides capacity of 2 kB

Among the reference (and state of the art) methods usedfor comparison no one performs better than the proposedwatermarking methods simultaneously under both AWGNand GA Hence the proposed methods fill the gap existingin watermarking literature This is thanks to several newadvancements used for embedding and extraction of a water-mark

In the case when AWGN is applied at the absence ofGA the benefit is caused mostly by IDL and the kind ofthresholding during watermark extraction From Figure 3it can be noticed that even without IDL variant NS-QIMdelivers slightly higher capacity under low WNRs comparedto DC-QIM However the capacity rises dramatically for lowWNRs if we switch to NS-QIM-IDL It is remarkable that theform of capacity plot in the latter case does not inherit thesteepness demonstrated by the other methods Instead theplot shape is similar to CTL but is placed at a lower positionThe explanation of such phenomena is in the quantizationprocess According to IDL we refuse to modify sampleswhose quantization brings the highest embedding distortionIn case these samples are quantized they are placed closerto the threshold which separates ldquo0rdquo and ldquo1rdquo Therefore theinformation interpreted by these samples is the most likely tobe lost under low WNRs Predicting the loss of informationwe might accept that fact and introduce IDL instead It is akind of ldquoaccumulationrdquo of embedding distortion which canbe ldquospentrdquo on making the rest of embedded informationmore robust Another unique feature is the proposed way ofnonpermanent thresholding In contrast to the permanentthresholding the information about 120572 120573 is not requiredfor watermark extraction Hence during embedding theseparameters can be adjusted to deliver higher capacity even incase there is no way to communicate new parameters to thereceiver

The proposed method is in advantageous position com-pared to RDM in the case when GA is used to attackthe watermarked image As one of its stages GA assumesAWGN and this explains superiority of NS-QIM over RDMin general The success of recovery is due to easy and efficientprocedure that utilizes a unique feature introduced by theproposedmethodsThe feature is created during quantizationand is a result of different quantization rules for ldquo0rdquo and ldquo1rdquo

The proposed estimation of scaling factor in this paperhas some advantages compared to other known retrievingprocedures For instance a model of a host is used in [15]to estimate the scaling factor In contrast to that we exploitthe unique asymmetric feature of the proposed quantizationapproach and this feature is not dependent on a hostThe onlyimportant assumption about the host is that its variance ismuch larger than the size of embedding interval As soon asthis holds the estimation is not dependent on themodel of thehost which is a contrast to [15] Also our recovery proceduredoes not use any additional information except interval guessfor Δ which can be given roughlyThese improvements implymore efficient retrieval after GA which in addition requiresfewer samples

The nonpermanent thresholding was proposed with theaim to avoid transmitting any additional information to thereceiver For example different size of embedding interval Δand different parameters 120572 120573 can be used to watermark dif-ferent images Nevertheless a watermark can be extracted incase the recovery procedure and nonpermanent thresholdingare used Such featuremight be beneficial in adaptation to theconditions that change

In the paper we do not consider a constant offset attackIn some other papers like [12 14 19] it is assumed to beapplied in conjunction with GA Further modifications of theproposed recovery procedure are needed to copewith it Alsoanother criterion that exploits different features compared1198621

and 1198622 might be useful for that task Apart from this goalwe would like to experiment with other concepts of IDL Forexample it might be reasonable to allow for those samplesto be shifted during quantization procedure Such shifts mayincrease chances for those samples to be interpreted correctlyafter an attack is applied

6 Conclusions

Thenewwatermarkingmethodbased on scalarQIMhas beenproposed It provides higher capacity under different kindsof attacks compared to other existing methodsThe proposedNS-QIM-IDLmethod is themost beneficial in case ofGAandAWGN The advantages of the method are due to its uniqueapproach towatermark embedding aswell as a newprocedureof recovery and extraction

The main features of the unique approach to watermarkembedding are a new kind of distribution of quantizedsamples and IDL In general there is no line of symmetryinside embedding interval for the new distribution of quan-tized samples This feature is used to recover a watermarkafter GA The feature of IDL can reduce distortions intro-duced to a host signal which are caused by watermarkingThis is done by letting some watermark bits to be interpreted


incorrectly at the initial phase of embedding and before anyattack occurs The proposed IDL is extremely beneficial forlowWNRs under AWGN attack

The new procedure of recovery after GA exploits thenonsymmetric distribution of quantized samples One outof two different criteria might be chosen to serve as agoal function for the procedure The criteria behave in asimilar way despite the differences in realization It has beendemonstrated experimentally that the proposed recoveryprocedure estimates the original length of embedding inter-val with deviation of 002 even in case when WNR is quitelow Nonpermanent thresholding was proposed in order toavoid transmitting additional information to the site wherewatermark extraction is done The technique is simple andestablishes the threshold in the position of the median of thedistribution inside embedding interval

The mentioned advancements implied considerable per-formance improvement Under conditions of AWGN andJPEG attacks (at the absence of GA) the capacity of theproposed method is at the same or higher level comparedto DC-QIM The most advantageous application of NS-QIM-IDL is under AWGN for WNRs around minus12 dB whereit performs up to 104 times better than DC-QIM Underthe condition of GA followed by high level of AWGN theperformance of the proposedmethod is up to 103 times higherthan that of RDM For the case when GA is followed by JPEGwith119876 = 25 the capacity of the proposedmethod is up to 10times higher than that of RDM Superiority of the proposedmethods under AWGN as well as GA allows narrowingthe gap between watermarking performances achievable intheory and in practice

Conflict of Interests

The authors declare that there is no conflict of interestsregarding to the publication of this paper

References

[1] I Cox M Miller J Bloom J Fridrich and T Kalker DigitalWatermarking and Steganography Morgan Kaufmann SanFrancisco Calif USA 2nd edition 2007

[2] M Barni F Bartolini V Cappellini and A Piva ldquoRobustwatermarking of still images for copyright protectionrdquo inProceedings of the 13th International Conference onDigital SignalProcessing (DSP rsquo97) vol 2 pp 499ndash502 Santorini Greece July1997

[3] H R Sheikh and A C Bovik ldquoImage information and visualqualityrdquo IEEE Transactions on Image Processing vol 15 no 2pp 430ndash444 2006

[4] T Chen ldquoA framework for optimal blind watermark detectionrdquoinProceedings of the 2001Workshop onMultimedia and SecurityNew Challenges pp 11ndash14 Ottawa Canada 2001

[5] M H M Costa ldquoWriting on dirty paperrdquo IEEE Transactions onInformation Theory vol 29 no 3 pp 439ndash441 1983

[6] E Ganic and A M Eskicioglu ldquoRobust DWT-SVD domainimage watermarking embedding data in all frequenciesrdquo inProceedings of the Multimedia and Security Workshop (MM ampSec rsquo04) pp 166ndash174 September 2004

[7] K Loukhaoukha ldquoImage watermarking algorithm based onmultiobjective ant colony optimization and singular valuedecomposition inwavelet domainrdquo Journal of Optimization vol2013 Article ID 921270 10 pages 2013

[8] B Chen andGWornell ldquoDithermodulation a new approach todigital watermarking and information embeddingrdquo in SecurityandWatermarking ofMultimedia Contents vol 3657 of Proceed-ings of SPIE pp 342ndash353 April 1999

[9] B Chen and G W Wornell ldquoQuantization index modulationa class of provably good methods for digital watermarkingand information embeddingrdquo IEEETransactions on InformationTheory vol 47 no 4 pp 1423ndash1443 2001

[10] E Esen and A Alatan ldquoForbidden zone data hidingrdquo inProceedings of the IEEE International Conference on ImageProcessing pp 1393ndash1396 October 2006

[11] M Ramkumar and A N Akansu ldquoSignalling methods for mul-timedia steganographyrdquo IEEE Transactions on Signal Processingvol 52 no 4 pp 1100ndash1111 2004

[12] J J Eggers R Bauml R Tzschoppe and B Girod ldquoScalarCosta scheme for information embeddingrdquo IEEE Transactionson Signal Processing vol 51 no 4 pp 1003ndash1019 2003

[13] J Oostveen T Kalker and M Staring ldquoAdaptive quantizationwatermarkingrdquo in Security Steganography andWatermarking ofMultimedia Proceedings of SPIE pp 296ndash303 San Jose CalifUSA January 2004

[14] X Kang J Huang and W Zeng ldquoImproving robustness ofquantization-based image watermarking via adaptive receiverrdquoIEEE Transactions on Multimedia vol 10 no 6 pp 953ndash9592008

[15] I D Shterev and R L Lagendijk ldquoAmplitude scale estimationfor quantization-based watermarkingrdquo IEEE Transactions onSignal Processing vol 54 no 11 pp 4146ndash4155 2006

[16] F Perez-Gonzalez C Mosquera M Barni and A AbrardoldquoRational dither modulation a high-rate data-hiding methodinvariant to gain attacksrdquo IEEE Transactions on Signal Process-ing vol 53 no 10 pp 3960ndash3975 2005

[17] F Ourique V Licks R Jordan and F Perez-Gonzalez ldquoAngleQIM a novel watermark embedding scheme robust againstamplitude scaling distortionsrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo05) pp 797ndash800 March 2005

[18] M Zareian and H Tohidypour ldquoRobust quantisation indexmodulation-based approach for image watermarkingrdquo IETImage Processing vol 7 no 5 pp 432ndash441 2013

[19] X Zhu and J Ding ldquoPerformance analysis and improvementof dither modulation under the composite attacksrdquo EurasipJournal on Advances in Signal Processing vol 2012 no 1 article53 2012

[20] M A Akhaee S M E Sahraeian and C Jin ldquoBlind imagewatermarking using a sample projection approachrdquo IEEETrans-actions on Information Forensics and Security vol 6 no 3 pp883ndash893 2011

[21] N K Kalantari and S M Ahadi ldquoA logarithmic quantizationindex modulation for perceptually better data hidingrdquo IEEETransactions on Image Processing vol 19 no 6 pp 1504ndash15172010

[22] E Nezhadarya J Wang and R K Ward ldquoA new data hidingmethod using angle quantization index modulation in gradientdomainrdquo in Proceedings of the IEEE International Conference onAcoustics Speech and Signal Processing (ICASSP 11) pp 2440ndash2443 Prague Czech Republic May 2011


[23] M Zareian and A Daneshkhah ldquoAdaptive angle quantizationindex modulation for robust image watermarkingrdquo in Proceed-ings of the IEEE Global Communications Conference (GLOBE-COM rsquo12) pp 881ndash884 Anaheim Calif USA December 2012

[24] C Song S Sudirman M Merabti and D Llewellyn-JonesldquoAnalysis of digital image watermark attacksrdquo in Proceedingof the 7th IEEE Consumer Communications and NetworkingConference (CCNC rsquo10) pp 1ndash5 Las Vegas Nev USA January2010

[25] V Gorodetski L Popyack V Samoilov and V Skormin ldquoSVD-based approach to transparent embedding data into digitalimagesrdquo in Proceedings of the International Workshop on Infor-mation Assurance in Computer Networks Methods Models andArchitectures for Network Security (MMM-ACNS rsquo01) pp 263ndash274 2001

[26] R Gallager Information Theory and Reliable CommunicationJohn Wiley amp Sons New York NY USA 1968

[27] Y Zolotavkin and M Juhola ldquoA new blind adaptive water-marking method based on singular value decompositionrdquo inProceedings of the International Conference on Sensor NetworkSecurity Technology and Privacy Communication System (SNSand PCS rsquo13) pp 184ndash192 Nangang China March 2013

[28] Y Zolotavkin and M Juhola ldquoSVD-based digital image water-marking on approximated orthogonal matrixrdquo in Proceedings ofthe 10th International Conference on Security and Cryptography(SECRYPT 13) pp 321ndash330 July 2013

[29] X Jun and W Ying ldquoToward a better understanding of DCTcoefficients in watermarkingrdquo in Proceedings of The Pacific-Asia Workshop on Computational Intelligence and IndustrialApplication (PACIIA rsquo08) vol 2 pp 206ndash209 Wuhan ChinaDecember 2008

International Journal of

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RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Hence the distribution of quantized samples that representboth ldquo0rdquo and ldquo1rdquo is degenerate (or Dirac) Nevertheless thecapacity of the simplest QIM (further referred to as QIMwithout ldquosimplestrdquo) is less than 10 of the limit value forthe condition when noise and watermark energies are equalMore advanced realization of DC-QIM is to replace eachcoefficient value from an original interval by a correspondingvalue taken from one out of two disjoint intervals that aresubsets of the original one [9] A parameter 05(1 minus 120572) is tocontrol the size of these intervals relatively to the originalThedistribution for ldquo0rdquo and ldquo1rdquo in that case is uniform Parameter120572 is being adjusted depending on noise level in order tomaximize capacity Method DC-QIM is widely used andprovides the highest capacity under AWGN among knownpractical realizations However considering AWGN attackonly the most evident gap under high noise intensity iscaused by low capacity in comparison with the theoreticallimit

Some other modifications of QIM have emerged overthe past years Forbidden zone data hiding (FZDH) modifiesonly a fraction (controlled by 120572) of coefficient values ineach interval of original values [10] Despite the fact thatFZDH performs slightly worse than DC-QIM the paperrepresents a promising idea on how to reduce embeddingdistortions Another idea was proposed by the authors ofThresholded Constellation Modulation (TCM) that use twodifferent quantization rules to modify coefficients inside theoriginal interval [11] Each rule is applied only to samplesfrom a particular subinterval and 120573 defines their endpointsThe value of a shift is different for any value from a subintervalaccording to the first quantization rule The second ruleis to provide an equal shift to all the values from anothersubinterval There are two shift directions in order to embedldquo0rdquo and ldquo1rdquo

The main advantage of the techniques based on QIMwith different kinds of compensation [9ndash11] is a consid-erable robustness against AWGN The limitation is thatsynchronization is required to extract a watermark Evenminor distortion of a different kind can make embeddedinformation unreadableThe simplest realization of such kindof distortion is a gain attack (GA) which performs a constantscaling of the whole sequence of watermarked coefficientsThe scaling factor might be close to 1 and cause very littlevisual distortion but it is unknown to the receiver whichcauses asynchronous extraction Usually GA is followed byAWGN that complicates retrieval of the watermark [12]Vulnerability to GA causes one of the most critical gapsfor practical implementation of QIM-based methods withcompensation

Many different approaches have been developed toimprove robustness against GA of QIM related methods [13]Most approaches can be classified into two groups where themain idea of the first group is to estimate the unknown factorwhile the idea of second is to quantize coefficients that areinvariant to scaling of original signal

Estimation of the scaling factor requires modelling Somefeature that is unique for the watermarked and attackedsignal might be described by a model [14] The scalingfactor may be included in the model and to be a subject

to optimization An obvious complication is that a processof feature selection is not straightforward In some casesthe feature is created instead of being selected and somepermanent data agreed upon between the sender and thereceiver is a suitable example However such compulsoryagreement limits practical implementation of watermarkingmethod Other possible limitations are low model accuracyor computationally heavy optimization

For instance a kind of GA and a constant offset attack fol-lowed by AWGN are assumed in [12] The solution proposedthere is to embed a pilot signal and to use Fourier analysis toestimate the gain factor and the offset However an obviousdisadvantage of the solution is that the precision of estimatedparameters is low even for quite long pilot sequence

Themethod of recovery after GA and AWGN is proposedin [15] It uses information about dither sequence and appliesmaximum likelihood (ML) procedure to estimate the scalingfactor The estimation is based on a model of watermarkedand attacked signal Information about statistical charac-teristics of original host signal should be either known orguessed in order to define the model Another limitationof the approach is that it requires exact information aboutembedding parameter 120572 Computational complexity of theapproach is high

As an opposite for estimation invariance to GA ingeneral requires more complex transform of original signal(eg nonlinear) to obtain coefficients It is necessary tomodify coefficients to embed a watermark However a modelto estimate distortions of the host is more complex in thatcase Distortions should be controlled which limits the choiceof the kind ofQIM to one that adds less complexity to amodelof distortion This for example might result in reducing thenumber of adjustable parameters of QIM This is one of thereasons why invariant to GA approaches are more vulnerableto AWGN compared to DC-QIM

Rational dither modulation (RDM) is one of the mostpopular watermarking methods invariant to GA [16] For aparticular coefficient a ratio that depends on a norm of othercoefficients is being quantized instead of a coefficient itselfThe simplest QIM scheme is utilized in order to quantize theratio and the performance of RDM under AWGN (withoutGA) is close to the simplest QIM Other recent blindwatermarkingmethods robust toGA are proposed in [17ndash23]Nevertheless for GA invariant methods the gap is caused bythe reduced capacity under AWGN

In this paper we propose our own scalar QIM-basedwatermarking approach that is beneficial in several aspectsThe approach addresses the mentioned gaps in the literatureit both delivers higher capacity under AWGN and recoversafter GA In order to do this host signal coefficients are sepa-rated in a way that the resulting distributions for coefficientsthat interpret ldquo0rdquo and ldquo1rdquo are differentThis distinctive featureis used by a simple yet efficient procedure for estimation of ascaling factor underGA A concept of initial data loss (IDL) isintroduced in order to increase watermark channel capacityunder low watermark to noise ratios (WNRs) Accordingto IDL some fraction of wrong watermark bits is acceptedduring embedding procedure








Δ 119897119896


119896

Δand




1015840









1015840) and 1198911(119909

1015840) In other


parameters

1199091015840= 119876119896

Δ[119909 119887 1205781 1205990 1205931 1205740 1198910 1198911] (1)



119901 | 119909 leΔ1205740

1205740 + 1205990

(2)

119902 | 119909 geΔ1205781

1205931 + 1205781

(3)

119903 | 1199091015840= 119909 (4)

119904 | 1199091015840lt 119909 (5)

119905 | 1199091015840gt 119909 (6)

119906 | 1199091205740 + 1205990

Δ= 1205740 int

1199091015840

0

1198910 (1199091015840) 1198891199091015840 (7)

V | (Δ minus 119909)1205931 + 1205781

Δ= minus1205931int

1199091015840

Δ

1198911 (1199091015840) 1198891199091015840 (8)


((sim 119887amp119901) sup (119906amp (119904 or 119903)))

((119887amp119902) sup (Vamp (119905 or 119903)))



(9)






119894le 1199091015840


119894gt 1199091015840

119895 Saving the


1015840) samples 119909





1205740

1205781

1205740f0(x998400) 1205931f1(x998400)

1205931

1205990

1205740 + 1205990

1205931 + 1205781


lk

lk

120589998400

120589

lk +

lk +

(u and ( )) ( and ( ))


Δ

ΔΔΔ

Δ Δ

IDL(0)

IDL(0)

IDL(1)

IDL(1)

t or rs or r





1015840) and 1198911(119909

1015840)


1015840) need to be





1015840) and 1198911(119909



1015840) and 1198911(119909

1015840) where 0 le 120572 le 120573 le 1



1198910 (1199091015840) =

1198881199091015840+ 120591 if 1199091015840 isin [0 Δ (120573 minus 120572)]

0 otherwise(10)

1198911 (1199091015840) =

119892 if 1199091015840 isin [Δ120573 Δ]

0 otherwise(11)


1015840) and IDL1(119909

1015840)

IDL0 (1199091015840) =

1205740 + 1205990

Δ1205990

if 1199091015840 isin [Δ1205740

1205740 + 1205990

Δ]

0 otherwise

IDL1 (1199091015840) =

1205931 + 1205781

Δ1205781

if 1199091015840 isin [0Δ1205781

1205931 + 1205781

]

0 otherwise

(12)



1015840)


int

(120573minus120572)Δ

0

1198910 (1199091015840) 1198891199091015840= 119888

(120573 minus 120572)2Δ2

2+ 120591Δ (120573 minus 120572) = 1 (13)

int

Δ

120573Δ

1198911 (1199091015840) 1198891199091015840= 119892Δ (1 minus 120573) = 1 (14)


119892 =1

Δ (1 minus 120573) (15)


1199091015840 1205740 + 1205990

Δle 1205740 int

1199091015840

0

1198910 (1199091015840) 1198891199091015840 (16)


120591 ge1205740 + 1205990

Δ1205740

minus 1198881199091015840

2 (17)


which means


(1205740 + 1205990

Δ1205740

minus 1198881199091015840

2) (18)


120591 ge1205740 + 1205990

Δ1205740

(19)


119888 = 21205740 minus (1205740 + 1205990) (120573 minus 120572)

1205740(120573 minus 120572)2Δ2

(20)


f0(x998400)f1(x998400)

120572

120573

x998400

lk + 120589998400lk


ΔΔ Δ

ΔΔ

(a)

120589998400

k minus 1 k k + 1 k + 2 k + 3


(b)


Functions 1198910(1199091015840) and 1198911(119909



120573 minus 120572 le1205740

1205740 + 1205990

(21)


1205931 + 1205781

Δle 1198921205931 (22)


120573 ge1205781

1205931 + 1205781

(23)








05119888 11990910158402

+ 120591 1199091015840 = 120591119909 (24)


1199091015840 =

radic1205912 + 2119888120591119909

119888minus

120591

119888 (25)


1199091015840 = 119861119909 + Δ (1 minus 119861) 119861 =(1 minus 120573) (1205931 + 1205781)

1205931

(26)






119899


WNR = 10 log10

(119863

1205902119899

) (27)




1198630 = 1205740 int

Δ(120573minus120572)

0

1198910 (1199091015840)(1199091015840minus

1

120591int

1199091015840

0

1198910 (1199091015840) 1198891199091015840)

2

1198891199091015840

(28)


1198630 = 1205740 int

Δ(120573minus120572)

0

(1198881199091015840+ 120591)

119888211990910158404

412059121198891199091015840 (29)





1198630 = Δ21198760

1198760 = 1205740 (1198883

241205912(120573 minus 120572)

6+

1198882

20120591(120573 minus 120572)

5)

(30)


1198631 = 1205931

times int

Δ

120573Δ

1198911 (1199091015840)

times (1199091015840minus (

1205931Δ

1205781 + 1205931

int

1199091015840

120573Δ

1198911 (1199091015840) 1198891199091015840

+1205781Δ

1205781 + 1205931

))

2

1198891199091015840

(31)


1198631 = Δ21198761

1198761 = 1205931

((1205781 + 1205931) (1 minus 120573) minus 1205931)2

3(1205781 + 1205931)2

(32)


119863 = Δ2(1198760 + 1198761) (33)



Δ = radic1205902

1198991001lowastWNR

1198760 + 1198761

(34)



1015840


1205891015840


respectively

Z =

infin

⋃

119898=minusinfin

[2Δ119898 + 119897119896

ΔminusTh 2Δ119898 + 119897

119896

Δ+Th) (35)

O =

infin

⋃

119898=minusinfin

[2Δ119898 + 119897119896

Δ+Th 2Δ(119898 + 1) + 119897

119896

ΔminusTh) (36)


(1205891015840isin Z) (120589

1015840

119899isin O) (37)


(1205891015840isin O) (120589

1015840

119899isin Z) (38)


(1205891015840isin O) (120589

1015840

119899isin O) (39)

(1205891015840isin Z) (120589

1015840

119899isin Z) (40)



in terms of 1205891015840 and 1205891015840

119899as 119891N[120589

1015840



1015840)

1198911(1205891015840) IDL0(120589

1015840) and IDL1(120589



1015840minus

(119897119896

Δ+Δ119898)] for odd119898 pdf 1198910(120589

1015840) = 1198910[119897

119896

Δ+ Δ(119898 + 1) minus 120589

1015840


119896

Δ+Δ119898 119897

119896

Δ+Δ(119898+1)]Then


BER0 =1205740

1205740 + 1205990

int

Oint

119868119896+119898

1198910 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

+1205990

1205740 + 1205990

int

Oint

119868119896+119898

IDL0 (1205891015840)

times 119891N [1205891015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

BER1 =1205931

1205931 + 1205781

int

Zint

119868119896+119898

1198911 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

+1205781

1205931 + 1205781

int

Zint

119868119896+119898

IDL1 (1205891015840)

times 119891N [1205891015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

(41)


int

Oint

119868119896+119898

1198910 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

= int

Oint

Δ

0

1198910 (1199091015840) 119891N [120589

1015840

119899minus (119897119896

Δ+ 1199091015840) 0 120590119899] 119889119909

10158401198891205891015840

119899

(42)

for any119898



Δ+

2Δ119899 1198910(1205891015840) = 1198910(119909


O + 2Δ119899 = O Hence

int

Oint

119868119896+2119899

1198910 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

= int

O+2119899Δint

Δ

0

1198910 (1199091015840)

times 119891N [(1205891015840

119899minus 2119899Δ)

minus (119897119896

Δ+ 1199091015840) 0 120590119899] 119889119909

1015840119889 1205891015840

119899minus 2119899Δ

(43)


(1199091015840minus Δ) + 119897

119896

Δ+ 2Δ(119899 + 1) 1198910(120589

1015840) = 1198910(Δ minus 119909



Therefore 119891N[1205891015840

119899minus 1205891015840 0 120590119899] = 119891N[(120589

1015840


119896

Δ+



and consequently

119891N [ (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))


Δ+ (1199091015840minus Δ)) 0 120590119899]

= 119891N [ minus (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))

minus (119897119896

Δ+ (Δ minus 119909

1015840)) 0 120590119899]

(44)


int

Oint

119868119896+2119899+1

1198910 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

= int


Δ

0

1198910 (Δ minus 1199091015840)

times 119891N [ minus (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))

minus (119897119896

Δ+ (Δ minus 119909

1015840)) 0 120590119899]

times 119889 Δ minus 1199091015840

times 119889 minus (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))

(45)





CTLNS-QIM-IDLNS-QIM

DC-QIMQIM

WNR (dB)

100

10minus2

10minus4

10minus6

C (b

itsy

mbo







CTL =1

2log2(1 + 10

01lowastWNR) (46)




Z =

100

⋃

119898=minus100

[2Δ119898 + 119897119896

ΔminusTh 2Δ119898 + 119897

119896

Δ+Th)

O =

100

⋃

119898=minus100

[2Δ119898 + 119897119896

Δ+Th 2Δ (119898 + 1)

+ 119897119896

ΔminusTh)

(47)


119896


119896







DWR = 10 log10

(1205902

119867

119863) (48)



Δ =120590119867

radic(1198760 + 1198761) 1001DWR

(49)


1205902

119899=

1205902

119867

1001(DWR+WNR)

(50)


119899is









0 40 80 120 160 200

100

10minus1

10minus2

10minus3

10minus4

10minus5

Var

DC-QIMNS-QIMFZDHTCM

QIMNSC-QIMRDM

C (b

itsy

mbo

l)







DC-QIMNS-QIMFZDHTCM

QIMNSC-QIMRDM

100

10minus1

10minus2

20 30 40 50 60 70 80 90 100

Q of JPEG ()

C (b

itsy

mbo






1015840Δ



1199091015840

119899=

1205891015840

119899mod Δ

1015840 if

[[[

[

1205891015840

119899minus 119897119896

Δ

Δ1015840

]]]

]

mod 2 = 0

Δ1015840minus (1205891015840

119899mod Δ

1015840) otherwise

(51)


119899isin [0 Δ

1015840]

1198621 (Δ1015840) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

median (1198831015840

119899)

Δ1015840minus 05

10038161003816100381610038161003816100381610038161003816100381610038161003816

1198622 (Δ1015840) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

119864 ([1198831015840

119899]119908

)

[Δ1015840]119908

10038161003816100381610038161003816100381610038161003816100381610038161003816

119908 = 2119898 + 1 119898 isin N

(52)


0035

003

0025

002

0015

001

0005

09 95 10 105 11

998400

Crite

rion

1

Δ

(a)

0

1

2

3

4

5

6

7

8

9 95 10 105 11

Crite

rion

2

times10minus4

998400Δ

(b)




Δ10158401015840= argmax

Δ101584011986212 (Δ

1015840) (53)


119899very close to














0 40 80 120 160 200

100

10minus1

10minus2

10minus3

10minus4

10minus5

Var

NS-QIMNSC-QIMRDM

C (b

itsy

mbo

l)


100

10minus1

10minus2

20 40 60 80 100

Q of JPEG ()

NS-QIMNSC-QIMRDM

C (b

itsy

mbo

l)




5 Discussion




119862 = max119901em(sim119887)

[119901 (sim 119887 119887) log2(

119901 (sim 119887 119887)

119901em (sim 119887) 119901ex (119887))

+ 119901 (119887 sim 119887) log2(

119901 (119887 sim 119887)

119901em (119887) 119901ex (sim 119887))

+ 119901 (sim 119887 sim 119887) log2(

119901 (sim 119887 sim 119887)

119901em (sim 119887) 119901ex (sim 119887))

+ 119901 (119887 119887) log2(

119901 (119887 119887)

119901em (119887) 119901ex (119887))]

(54)


119901ex (119887) = 119901 (sim 119887 119887) + 119901 (119887 119887)

119901ex (sim 119887) = 119901 (119887 sim 119887) + 119901 (sim 119887 sim 119887)

(55)


119901 (sim 119887 119887) = 119901em (sim 119887)BERsim119887

119901 (119887 sim 119887) = 119901em (119887)BER119887


119901 (119887 119887) = 119901em (119887) (1 minus BER119887)

(56)


119901em (sim 119887) = 1205740 + 1205990

119901em (119887) = 1205781 + 1205931

(57)





2(1 minus BER) (58)









6 Conclusions









References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















Δ 119897119896


119896

Δand




1015840









1015840) and 1198911(119909

1015840) In other


parameters

1199091015840= 119876119896

Δ[119909 119887 1205781 1205990 1205931 1205740 1198910 1198911] (1)



119901 | 119909 leΔ1205740

1205740 + 1205990

(2)

119902 | 119909 geΔ1205781

1205931 + 1205781

(3)

119903 | 1199091015840= 119909 (4)

119904 | 1199091015840lt 119909 (5)

119905 | 1199091015840gt 119909 (6)

119906 | 1199091205740 + 1205990

Δ= 1205740 int

1199091015840

0

1198910 (1199091015840) 1198891199091015840 (7)

V | (Δ minus 119909)1205931 + 1205781

Δ= minus1205931int

1199091015840

Δ

1198911 (1199091015840) 1198891199091015840 (8)


((sim 119887amp119901) sup (119906amp (119904 or 119903)))

((119887amp119902) sup (Vamp (119905 or 119903)))



(9)






119894le 1199091015840


119894gt 1199091015840

119895 Saving the


1015840) samples 119909





1205740

1205781

1205740f0(x998400) 1205931f1(x998400)

1205931

1205990

1205740 + 1205990

1205931 + 1205781


lk

lk

120589998400

120589

lk +

lk +

(u and ( )) ( and ( ))


Δ

ΔΔΔ

Δ Δ

IDL(0)

IDL(0)

IDL(1)

IDL(1)

t or rs or r





1015840) and 1198911(119909

1015840)


1015840) need to be





1015840) and 1198911(119909



1015840) and 1198911(119909

1015840) where 0 le 120572 le 120573 le 1



1198910 (1199091015840) =

1198881199091015840+ 120591 if 1199091015840 isin [0 Δ (120573 minus 120572)]

0 otherwise(10)

1198911 (1199091015840) =

119892 if 1199091015840 isin [Δ120573 Δ]

0 otherwise(11)


1015840) and IDL1(119909

1015840)

IDL0 (1199091015840) =

1205740 + 1205990

Δ1205990

if 1199091015840 isin [Δ1205740

1205740 + 1205990

Δ]

0 otherwise

IDL1 (1199091015840) =

1205931 + 1205781

Δ1205781

if 1199091015840 isin [0Δ1205781

1205931 + 1205781

]

0 otherwise

(12)



1015840)


int

(120573minus120572)Δ

0

1198910 (1199091015840) 1198891199091015840= 119888

(120573 minus 120572)2Δ2

2+ 120591Δ (120573 minus 120572) = 1 (13)

int

Δ

120573Δ

1198911 (1199091015840) 1198891199091015840= 119892Δ (1 minus 120573) = 1 (14)


119892 =1

Δ (1 minus 120573) (15)


1199091015840 1205740 + 1205990

Δle 1205740 int

1199091015840

0

1198910 (1199091015840) 1198891199091015840 (16)


120591 ge1205740 + 1205990

Δ1205740

minus 1198881199091015840

2 (17)


which means


(1205740 + 1205990

Δ1205740

minus 1198881199091015840

2) (18)


120591 ge1205740 + 1205990

Δ1205740

(19)


119888 = 21205740 minus (1205740 + 1205990) (120573 minus 120572)

1205740(120573 minus 120572)2Δ2

(20)


f0(x998400)f1(x998400)

120572

120573

x998400

lk + 120589998400lk


ΔΔ Δ

ΔΔ

(a)

120589998400

k minus 1 k k + 1 k + 2 k + 3


(b)


Functions 1198910(1199091015840) and 1198911(119909



120573 minus 120572 le1205740

1205740 + 1205990

(21)


1205931 + 1205781

Δle 1198921205931 (22)


120573 ge1205781

1205931 + 1205781

(23)








05119888 11990910158402

+ 120591 1199091015840 = 120591119909 (24)


1199091015840 =

radic1205912 + 2119888120591119909

119888minus

120591

119888 (25)


1199091015840 = 119861119909 + Δ (1 minus 119861) 119861 =(1 minus 120573) (1205931 + 1205781)

1205931

(26)






119899


WNR = 10 log10

(119863

1205902119899

) (27)




1198630 = 1205740 int

Δ(120573minus120572)

0

1198910 (1199091015840)(1199091015840minus

1

120591int

1199091015840

0

1198910 (1199091015840) 1198891199091015840)

2

1198891199091015840

(28)


1198630 = 1205740 int

Δ(120573minus120572)

0

(1198881199091015840+ 120591)

119888211990910158404

412059121198891199091015840 (29)





1198630 = Δ21198760

1198760 = 1205740 (1198883

241205912(120573 minus 120572)

6+

1198882

20120591(120573 minus 120572)

5)

(30)


1198631 = 1205931

times int

Δ

120573Δ

1198911 (1199091015840)

times (1199091015840minus (

1205931Δ

1205781 + 1205931

int

1199091015840

120573Δ

1198911 (1199091015840) 1198891199091015840

+1205781Δ

1205781 + 1205931

))

2

1198891199091015840

(31)


1198631 = Δ21198761

1198761 = 1205931

((1205781 + 1205931) (1 minus 120573) minus 1205931)2

3(1205781 + 1205931)2

(32)


119863 = Δ2(1198760 + 1198761) (33)



Δ = radic1205902

1198991001lowastWNR

1198760 + 1198761

(34)



1015840


1205891015840


respectively

Z =

infin

⋃

119898=minusinfin

[2Δ119898 + 119897119896

ΔminusTh 2Δ119898 + 119897

119896

Δ+Th) (35)

O =

infin

⋃

119898=minusinfin

[2Δ119898 + 119897119896

Δ+Th 2Δ(119898 + 1) + 119897

119896

ΔminusTh) (36)


(1205891015840isin Z) (120589

1015840

119899isin O) (37)


(1205891015840isin O) (120589

1015840

119899isin Z) (38)


(1205891015840isin O) (120589

1015840

119899isin O) (39)

(1205891015840isin Z) (120589

1015840

119899isin Z) (40)



in terms of 1205891015840 and 1205891015840

119899as 119891N[120589

1015840



1015840)

1198911(1205891015840) IDL0(120589

1015840) and IDL1(120589



1015840minus

(119897119896

Δ+Δ119898)] for odd119898 pdf 1198910(120589

1015840) = 1198910[119897

119896

Δ+ Δ(119898 + 1) minus 120589

1015840


119896

Δ+Δ119898 119897

119896

Δ+Δ(119898+1)]Then


BER0 =1205740

1205740 + 1205990

int

Oint

119868119896+119898

1198910 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

+1205990

1205740 + 1205990

int

Oint

119868119896+119898

IDL0 (1205891015840)

times 119891N [1205891015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

BER1 =1205931

1205931 + 1205781

int

Zint

119868119896+119898

1198911 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

+1205781

1205931 + 1205781

int

Zint

119868119896+119898

IDL1 (1205891015840)

times 119891N [1205891015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

(41)


int

Oint

119868119896+119898

1198910 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

= int

Oint

Δ

0

1198910 (1199091015840) 119891N [120589

1015840

119899minus (119897119896

Δ+ 1199091015840) 0 120590119899] 119889119909

10158401198891205891015840

119899

(42)

for any119898



Δ+

2Δ119899 1198910(1205891015840) = 1198910(119909


O + 2Δ119899 = O Hence

int

Oint

119868119896+2119899

1198910 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

= int

O+2119899Δint

Δ

0

1198910 (1199091015840)

times 119891N [(1205891015840

119899minus 2119899Δ)

minus (119897119896

Δ+ 1199091015840) 0 120590119899] 119889119909

1015840119889 1205891015840

119899minus 2119899Δ

(43)


(1199091015840minus Δ) + 119897

119896

Δ+ 2Δ(119899 + 1) 1198910(120589

1015840) = 1198910(Δ minus 119909



Therefore 119891N[1205891015840

119899minus 1205891015840 0 120590119899] = 119891N[(120589

1015840


119896

Δ+



and consequently

119891N [ (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))


Δ+ (1199091015840minus Δ)) 0 120590119899]

= 119891N [ minus (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))

minus (119897119896

Δ+ (Δ minus 119909

1015840)) 0 120590119899]

(44)


int

Oint

119868119896+2119899+1

1198910 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

= int


Δ

0

1198910 (Δ minus 1199091015840)

times 119891N [ minus (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))

minus (119897119896

Δ+ (Δ minus 119909

1015840)) 0 120590119899]

times 119889 Δ minus 1199091015840

times 119889 minus (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))

(45)





CTLNS-QIM-IDLNS-QIM

DC-QIMQIM

WNR (dB)

100

10minus2

10minus4

10minus6

C (b

itsy

mbo







CTL =1

2log2(1 + 10

01lowastWNR) (46)




Z =

100

⋃

119898=minus100

[2Δ119898 + 119897119896

ΔminusTh 2Δ119898 + 119897

119896

Δ+Th)

O =

100

⋃

119898=minus100

[2Δ119898 + 119897119896

Δ+Th 2Δ (119898 + 1)

+ 119897119896

ΔminusTh)

(47)


119896


119896







DWR = 10 log10

(1205902

119867

119863) (48)



Δ =120590119867

radic(1198760 + 1198761) 1001DWR

(49)


1205902

119899=

1205902

119867

1001(DWR+WNR)

(50)


119899is









0 40 80 120 160 200

100

10minus1

10minus2

10minus3

10minus4

10minus5

Var

DC-QIMNS-QIMFZDHTCM

QIMNSC-QIMRDM

C (b

itsy

mbo

l)







DC-QIMNS-QIMFZDHTCM

QIMNSC-QIMRDM

100

10minus1

10minus2

20 30 40 50 60 70 80 90 100

Q of JPEG ()

C (b

itsy

mbo






1015840Δ



1199091015840

119899=

1205891015840

119899mod Δ

1015840 if

[[[

[

1205891015840

119899minus 119897119896

Δ

Δ1015840

]]]

]

mod 2 = 0

Δ1015840minus (1205891015840

119899mod Δ

1015840) otherwise

(51)


119899isin [0 Δ

1015840]

1198621 (Δ1015840) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

median (1198831015840

119899)

Δ1015840minus 05

10038161003816100381610038161003816100381610038161003816100381610038161003816

1198622 (Δ1015840) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

119864 ([1198831015840

119899]119908

)

[Δ1015840]119908

10038161003816100381610038161003816100381610038161003816100381610038161003816

119908 = 2119898 + 1 119898 isin N

(52)


0035

003

0025

002

0015

001

0005

09 95 10 105 11

998400

Crite

rion

1

Δ

(a)

0

1

2

3

4

5

6

7

8

9 95 10 105 11

Crite

rion

2

times10minus4

998400Δ

(b)




Δ10158401015840= argmax

Δ101584011986212 (Δ

1015840) (53)


119899very close to














0 40 80 120 160 200

100

10minus1

10minus2

10minus3

10minus4

10minus5

Var

NS-QIMNSC-QIMRDM

C (b

itsy

mbo

l)


100

10minus1

10minus2

20 40 60 80 100

Q of JPEG ()

NS-QIMNSC-QIMRDM

C (b

itsy

mbo

l)




5 Discussion




119862 = max119901em(sim119887)

[119901 (sim 119887 119887) log2(

119901 (sim 119887 119887)

119901em (sim 119887) 119901ex (119887))

+ 119901 (119887 sim 119887) log2(

119901 (119887 sim 119887)

119901em (119887) 119901ex (sim 119887))

+ 119901 (sim 119887 sim 119887) log2(

119901 (sim 119887 sim 119887)

119901em (sim 119887) 119901ex (sim 119887))

+ 119901 (119887 119887) log2(

119901 (119887 119887)

119901em (119887) 119901ex (119887))]

(54)


119901ex (119887) = 119901 (sim 119887 119887) + 119901 (119887 119887)

119901ex (sim 119887) = 119901 (119887 sim 119887) + 119901 (sim 119887 sim 119887)

(55)


119901 (sim 119887 119887) = 119901em (sim 119887)BERsim119887

119901 (119887 sim 119887) = 119901em (119887)BER119887


119901 (119887 119887) = 119901em (119887) (1 minus BER119887)

(56)


119901em (sim 119887) = 1205740 + 1205990

119901em (119887) = 1205781 + 1205931

(57)





2(1 minus BER) (58)









6 Conclusions









References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1205740

1205781

1205740f0(x998400) 1205931f1(x998400)

1205931

1205990

1205740 + 1205990

1205931 + 1205781


lk

lk

120589998400

120589

lk +

lk +

(u and ( )) ( and ( ))


Δ

ΔΔΔ

Δ Δ

IDL(0)

IDL(0)

IDL(1)

IDL(1)

t or rs or r





1015840) and 1198911(119909

1015840)


1015840) need to be





1015840) and 1198911(119909



1015840) and 1198911(119909

1015840) where 0 le 120572 le 120573 le 1



1198910 (1199091015840) =

1198881199091015840+ 120591 if 1199091015840 isin [0 Δ (120573 minus 120572)]

0 otherwise(10)

1198911 (1199091015840) =

119892 if 1199091015840 isin [Δ120573 Δ]

0 otherwise(11)


1015840) and IDL1(119909

1015840)

IDL0 (1199091015840) =

1205740 + 1205990

Δ1205990

if 1199091015840 isin [Δ1205740

1205740 + 1205990

Δ]

0 otherwise

IDL1 (1199091015840) =

1205931 + 1205781

Δ1205781

if 1199091015840 isin [0Δ1205781

1205931 + 1205781

]

0 otherwise

(12)



1015840)


int

(120573minus120572)Δ

0

1198910 (1199091015840) 1198891199091015840= 119888

(120573 minus 120572)2Δ2

2+ 120591Δ (120573 minus 120572) = 1 (13)

int

Δ

120573Δ

1198911 (1199091015840) 1198891199091015840= 119892Δ (1 minus 120573) = 1 (14)


119892 =1

Δ (1 minus 120573) (15)


1199091015840 1205740 + 1205990

Δle 1205740 int

1199091015840

0

1198910 (1199091015840) 1198891199091015840 (16)


120591 ge1205740 + 1205990

Δ1205740

minus 1198881199091015840

2 (17)


which means


(1205740 + 1205990

Δ1205740

minus 1198881199091015840

2) (18)


120591 ge1205740 + 1205990

Δ1205740

(19)


119888 = 21205740 minus (1205740 + 1205990) (120573 minus 120572)

1205740(120573 minus 120572)2Δ2

(20)


f0(x998400)f1(x998400)

120572

120573

x998400

lk + 120589998400lk


ΔΔ Δ

ΔΔ

(a)

120589998400

k minus 1 k k + 1 k + 2 k + 3


(b)


Functions 1198910(1199091015840) and 1198911(119909



120573 minus 120572 le1205740

1205740 + 1205990

(21)


1205931 + 1205781

Δle 1198921205931 (22)


120573 ge1205781

1205931 + 1205781

(23)








05119888 11990910158402

+ 120591 1199091015840 = 120591119909 (24)


1199091015840 =

radic1205912 + 2119888120591119909

119888minus

120591

119888 (25)


1199091015840 = 119861119909 + Δ (1 minus 119861) 119861 =(1 minus 120573) (1205931 + 1205781)

1205931

(26)






119899


WNR = 10 log10

(119863

1205902119899

) (27)




1198630 = 1205740 int

Δ(120573minus120572)

0

1198910 (1199091015840)(1199091015840minus

1

120591int

1199091015840

0

1198910 (1199091015840) 1198891199091015840)

2

1198891199091015840

(28)


1198630 = 1205740 int

Δ(120573minus120572)

0

(1198881199091015840+ 120591)

119888211990910158404

412059121198891199091015840 (29)





1198630 = Δ21198760

1198760 = 1205740 (1198883

241205912(120573 minus 120572)

6+

1198882

20120591(120573 minus 120572)

5)

(30)


1198631 = 1205931

times int

Δ

120573Δ

1198911 (1199091015840)

times (1199091015840minus (

1205931Δ

1205781 + 1205931

int

1199091015840

120573Δ

1198911 (1199091015840) 1198891199091015840

+1205781Δ

1205781 + 1205931

))

2

1198891199091015840

(31)


1198631 = Δ21198761

1198761 = 1205931

((1205781 + 1205931) (1 minus 120573) minus 1205931)2

3(1205781 + 1205931)2

(32)


119863 = Δ2(1198760 + 1198761) (33)



Δ = radic1205902

1198991001lowastWNR

1198760 + 1198761

(34)



1015840


1205891015840


respectively

Z =

infin

⋃

119898=minusinfin

[2Δ119898 + 119897119896

ΔminusTh 2Δ119898 + 119897

119896

Δ+Th) (35)

O =

infin

⋃

119898=minusinfin

[2Δ119898 + 119897119896

Δ+Th 2Δ(119898 + 1) + 119897

119896

ΔminusTh) (36)


(1205891015840isin Z) (120589

1015840

119899isin O) (37)


(1205891015840isin O) (120589

1015840

119899isin Z) (38)


(1205891015840isin O) (120589

1015840

119899isin O) (39)

(1205891015840isin Z) (120589

1015840

119899isin Z) (40)



in terms of 1205891015840 and 1205891015840

119899as 119891N[120589

1015840



1015840)

1198911(1205891015840) IDL0(120589

1015840) and IDL1(120589



1015840minus

(119897119896

Δ+Δ119898)] for odd119898 pdf 1198910(120589

1015840) = 1198910[119897

119896

Δ+ Δ(119898 + 1) minus 120589

1015840


119896

Δ+Δ119898 119897

119896

Δ+Δ(119898+1)]Then


BER0 =1205740

1205740 + 1205990

int

Oint

119868119896+119898

1198910 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

+1205990

1205740 + 1205990

int

Oint

119868119896+119898

IDL0 (1205891015840)

times 119891N [1205891015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

BER1 =1205931

1205931 + 1205781

int

Zint

119868119896+119898

1198911 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

+1205781

1205931 + 1205781

int

Zint

119868119896+119898

IDL1 (1205891015840)

times 119891N [1205891015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

(41)


int

Oint

119868119896+119898

1198910 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

= int

Oint

Δ

0

1198910 (1199091015840) 119891N [120589

1015840

119899minus (119897119896

Δ+ 1199091015840) 0 120590119899] 119889119909

10158401198891205891015840

119899

(42)

for any119898



Δ+

2Δ119899 1198910(1205891015840) = 1198910(119909


O + 2Δ119899 = O Hence

int

Oint

119868119896+2119899

1198910 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

= int

O+2119899Δint

Δ

0

1198910 (1199091015840)

times 119891N [(1205891015840

119899minus 2119899Δ)

minus (119897119896

Δ+ 1199091015840) 0 120590119899] 119889119909

1015840119889 1205891015840

119899minus 2119899Δ

(43)


(1199091015840minus Δ) + 119897

119896

Δ+ 2Δ(119899 + 1) 1198910(120589

1015840) = 1198910(Δ minus 119909



Therefore 119891N[1205891015840

119899minus 1205891015840 0 120590119899] = 119891N[(120589

1015840


119896

Δ+



and consequently

119891N [ (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))


Δ+ (1199091015840minus Δ)) 0 120590119899]

= 119891N [ minus (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))

minus (119897119896

Δ+ (Δ minus 119909

1015840)) 0 120590119899]

(44)


int

Oint

119868119896+2119899+1

1198910 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

= int


Δ

0

1198910 (Δ minus 1199091015840)

times 119891N [ minus (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))

minus (119897119896

Δ+ (Δ minus 119909

1015840)) 0 120590119899]

times 119889 Δ minus 1199091015840

times 119889 minus (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))

(45)





CTLNS-QIM-IDLNS-QIM

DC-QIMQIM

WNR (dB)

100

10minus2

10minus4

10minus6

C (b

itsy

mbo







CTL =1

2log2(1 + 10

01lowastWNR) (46)




Z =

100

⋃

119898=minus100

[2Δ119898 + 119897119896

ΔminusTh 2Δ119898 + 119897

119896

Δ+Th)

O =

100

⋃

119898=minus100

[2Δ119898 + 119897119896

Δ+Th 2Δ (119898 + 1)

+ 119897119896

ΔminusTh)

(47)


119896


119896







DWR = 10 log10

(1205902

119867

119863) (48)



Δ =120590119867

radic(1198760 + 1198761) 1001DWR

(49)


1205902

119899=

1205902

119867

1001(DWR+WNR)

(50)


119899is









0 40 80 120 160 200

100

10minus1

10minus2

10minus3

10minus4

10minus5

Var

DC-QIMNS-QIMFZDHTCM

QIMNSC-QIMRDM

C (b

itsy

mbo

l)







DC-QIMNS-QIMFZDHTCM

QIMNSC-QIMRDM

100

10minus1

10minus2

20 30 40 50 60 70 80 90 100

Q of JPEG ()

C (b

itsy

mbo






1015840Δ



1199091015840

119899=

1205891015840

119899mod Δ

1015840 if

[[[

[

1205891015840

119899minus 119897119896

Δ

Δ1015840

]]]

]

mod 2 = 0

Δ1015840minus (1205891015840

119899mod Δ

1015840) otherwise

(51)


119899isin [0 Δ

1015840]

1198621 (Δ1015840) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

median (1198831015840

119899)

Δ1015840minus 05

10038161003816100381610038161003816100381610038161003816100381610038161003816

1198622 (Δ1015840) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

119864 ([1198831015840

119899]119908

)

[Δ1015840]119908

10038161003816100381610038161003816100381610038161003816100381610038161003816

119908 = 2119898 + 1 119898 isin N

(52)


0035

003

0025

002

0015

001

0005

09 95 10 105 11

998400

Crite

rion

1

Δ

(a)

0

1

2

3

4

5

6

7

8

9 95 10 105 11

Crite

rion

2

times10minus4

998400Δ

(b)




Δ10158401015840= argmax

Δ101584011986212 (Δ

1015840) (53)


119899very close to














0 40 80 120 160 200

100

10minus1

10minus2

10minus3

10minus4

10minus5

Var

NS-QIMNSC-QIMRDM

C (b

itsy

mbo

l)


100

10minus1

10minus2

20 40 60 80 100

Q of JPEG ()

NS-QIMNSC-QIMRDM

C (b

itsy

mbo

l)




5 Discussion




119862 = max119901em(sim119887)

[119901 (sim 119887 119887) log2(

119901 (sim 119887 119887)

119901em (sim 119887) 119901ex (119887))

+ 119901 (119887 sim 119887) log2(

119901 (119887 sim 119887)

119901em (119887) 119901ex (sim 119887))

+ 119901 (sim 119887 sim 119887) log2(

119901 (sim 119887 sim 119887)

119901em (sim 119887) 119901ex (sim 119887))

+ 119901 (119887 119887) log2(

119901 (119887 119887)

119901em (119887) 119901ex (119887))]

(54)


119901ex (119887) = 119901 (sim 119887 119887) + 119901 (119887 119887)

119901ex (sim 119887) = 119901 (119887 sim 119887) + 119901 (sim 119887 sim 119887)

(55)


119901 (sim 119887 119887) = 119901em (sim 119887)BERsim119887

119901 (119887 sim 119887) = 119901em (119887)BER119887


119901 (119887 119887) = 119901em (119887) (1 minus BER119887)

(56)


119901em (sim 119887) = 1205740 + 1205990

119901em (119887) = 1205781 + 1205931

(57)





2(1 minus BER) (58)









6 Conclusions









References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










f0(x998400)f1(x998400)

120572

120573

x998400

lk + 120589998400lk


ΔΔ Δ

ΔΔ

(a)

120589998400

k minus 1 k k + 1 k + 2 k + 3


(b)


Functions 1198910(1199091015840) and 1198911(119909



120573 minus 120572 le1205740

1205740 + 1205990

(21)


1205931 + 1205781

Δle 1198921205931 (22)


120573 ge1205781

1205931 + 1205781

(23)








05119888 11990910158402

+ 120591 1199091015840 = 120591119909 (24)


1199091015840 =

radic1205912 + 2119888120591119909

119888minus

120591

119888 (25)


1199091015840 = 119861119909 + Δ (1 minus 119861) 119861 =(1 minus 120573) (1205931 + 1205781)

1205931

(26)






119899


WNR = 10 log10

(119863

1205902119899

) (27)




1198630 = 1205740 int

Δ(120573minus120572)

0

1198910 (1199091015840)(1199091015840minus

1

120591int

1199091015840

0

1198910 (1199091015840) 1198891199091015840)

2

1198891199091015840

(28)


1198630 = 1205740 int

Δ(120573minus120572)

0

(1198881199091015840+ 120591)

119888211990910158404

412059121198891199091015840 (29)





1198630 = Δ21198760

1198760 = 1205740 (1198883

241205912(120573 minus 120572)

6+

1198882

20120591(120573 minus 120572)

5)

(30)


1198631 = 1205931

times int

Δ

120573Δ

1198911 (1199091015840)

times (1199091015840minus (

1205931Δ

1205781 + 1205931

int

1199091015840

120573Δ

1198911 (1199091015840) 1198891199091015840

+1205781Δ

1205781 + 1205931

))

2

1198891199091015840

(31)


1198631 = Δ21198761

1198761 = 1205931

((1205781 + 1205931) (1 minus 120573) minus 1205931)2

3(1205781 + 1205931)2

(32)


119863 = Δ2(1198760 + 1198761) (33)



Δ = radic1205902

1198991001lowastWNR

1198760 + 1198761

(34)



1015840


1205891015840


respectively

Z =

infin

⋃

119898=minusinfin

[2Δ119898 + 119897119896

ΔminusTh 2Δ119898 + 119897

119896

Δ+Th) (35)

O =

infin

⋃

119898=minusinfin

[2Δ119898 + 119897119896

Δ+Th 2Δ(119898 + 1) + 119897

119896

ΔminusTh) (36)


(1205891015840isin Z) (120589

1015840

119899isin O) (37)


(1205891015840isin O) (120589

1015840

119899isin Z) (38)


(1205891015840isin O) (120589

1015840

119899isin O) (39)

(1205891015840isin Z) (120589

1015840

119899isin Z) (40)



in terms of 1205891015840 and 1205891015840

119899as 119891N[120589

1015840



1015840)

1198911(1205891015840) IDL0(120589

1015840) and IDL1(120589



1015840minus

(119897119896

Δ+Δ119898)] for odd119898 pdf 1198910(120589

1015840) = 1198910[119897

119896

Δ+ Δ(119898 + 1) minus 120589

1015840


119896

Δ+Δ119898 119897

119896

Δ+Δ(119898+1)]Then


BER0 =1205740

1205740 + 1205990

int

Oint

119868119896+119898

1198910 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

+1205990

1205740 + 1205990

int

Oint

119868119896+119898

IDL0 (1205891015840)

times 119891N [1205891015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

BER1 =1205931

1205931 + 1205781

int

Zint

119868119896+119898

1198911 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

+1205781

1205931 + 1205781

int

Zint

119868119896+119898

IDL1 (1205891015840)

times 119891N [1205891015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

(41)


int

Oint

119868119896+119898

1198910 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

= int

Oint

Δ

0

1198910 (1199091015840) 119891N [120589

1015840

119899minus (119897119896

Δ+ 1199091015840) 0 120590119899] 119889119909

10158401198891205891015840

119899

(42)

for any119898



Δ+

2Δ119899 1198910(1205891015840) = 1198910(119909


O + 2Δ119899 = O Hence

int

Oint

119868119896+2119899

1198910 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

= int

O+2119899Δint

Δ

0

1198910 (1199091015840)

times 119891N [(1205891015840

119899minus 2119899Δ)

minus (119897119896

Δ+ 1199091015840) 0 120590119899] 119889119909

1015840119889 1205891015840

119899minus 2119899Δ

(43)


(1199091015840minus Δ) + 119897

119896

Δ+ 2Δ(119899 + 1) 1198910(120589

1015840) = 1198910(Δ minus 119909



Therefore 119891N[1205891015840

119899minus 1205891015840 0 120590119899] = 119891N[(120589

1015840


119896

Δ+



and consequently

119891N [ (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))


Δ+ (1199091015840minus Δ)) 0 120590119899]

= 119891N [ minus (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))

minus (119897119896

Δ+ (Δ minus 119909

1015840)) 0 120590119899]

(44)


int

Oint

119868119896+2119899+1

1198910 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

= int


Δ

0

1198910 (Δ minus 1199091015840)

times 119891N [ minus (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))

minus (119897119896

Δ+ (Δ minus 119909

1015840)) 0 120590119899]

times 119889 Δ minus 1199091015840

times 119889 minus (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))

(45)





CTLNS-QIM-IDLNS-QIM

DC-QIMQIM

WNR (dB)

100

10minus2

10minus4

10minus6

C (b

itsy

mbo







CTL =1

2log2(1 + 10

01lowastWNR) (46)




Z =

100

⋃

119898=minus100

[2Δ119898 + 119897119896

ΔminusTh 2Δ119898 + 119897

119896

Δ+Th)

O =

100

⋃

119898=minus100

[2Δ119898 + 119897119896

Δ+Th 2Δ (119898 + 1)

+ 119897119896

ΔminusTh)

(47)


119896


119896







DWR = 10 log10

(1205902

119867

119863) (48)



Δ =120590119867

radic(1198760 + 1198761) 1001DWR

(49)


1205902

119899=

1205902

119867

1001(DWR+WNR)

(50)


119899is









0 40 80 120 160 200

100

10minus1

10minus2

10minus3

10minus4

10minus5

Var

DC-QIMNS-QIMFZDHTCM

QIMNSC-QIMRDM

C (b

itsy

mbo

l)







DC-QIMNS-QIMFZDHTCM

QIMNSC-QIMRDM

100

10minus1

10minus2

20 30 40 50 60 70 80 90 100

Q of JPEG ()

C (b

itsy

mbo






1015840Δ



1199091015840

119899=

1205891015840

119899mod Δ

1015840 if

[[[

[

1205891015840

119899minus 119897119896

Δ

Δ1015840

]]]

]

mod 2 = 0

Δ1015840minus (1205891015840

119899mod Δ

1015840) otherwise

(51)


119899isin [0 Δ

1015840]

1198621 (Δ1015840) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

median (1198831015840

119899)

Δ1015840minus 05

10038161003816100381610038161003816100381610038161003816100381610038161003816

1198622 (Δ1015840) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

119864 ([1198831015840

119899]119908

)

[Δ1015840]119908

10038161003816100381610038161003816100381610038161003816100381610038161003816

119908 = 2119898 + 1 119898 isin N

(52)


0035

003

0025

002

0015

001

0005

09 95 10 105 11

998400

Crite

rion

1

Δ

(a)

0

1

2

3

4

5

6

7

8

9 95 10 105 11

Crite

rion

2

times10minus4

998400Δ

(b)




Δ10158401015840= argmax

Δ101584011986212 (Δ

1015840) (53)


119899very close to














0 40 80 120 160 200

100

10minus1

10minus2

10minus3

10minus4

10minus5

Var

NS-QIMNSC-QIMRDM

C (b

itsy

mbo

l)


100

10minus1

10minus2

20 40 60 80 100

Q of JPEG ()

NS-QIMNSC-QIMRDM

C (b

itsy

mbo

l)




5 Discussion




119862 = max119901em(sim119887)

[119901 (sim 119887 119887) log2(

119901 (sim 119887 119887)

119901em (sim 119887) 119901ex (119887))

+ 119901 (119887 sim 119887) log2(

119901 (119887 sim 119887)

119901em (119887) 119901ex (sim 119887))

+ 119901 (sim 119887 sim 119887) log2(

119901 (sim 119887 sim 119887)

119901em (sim 119887) 119901ex (sim 119887))

+ 119901 (119887 119887) log2(

119901 (119887 119887)

119901em (119887) 119901ex (119887))]

(54)


119901ex (119887) = 119901 (sim 119887 119887) + 119901 (119887 119887)

119901ex (sim 119887) = 119901 (119887 sim 119887) + 119901 (sim 119887 sim 119887)

(55)


119901 (sim 119887 119887) = 119901em (sim 119887)BERsim119887

119901 (119887 sim 119887) = 119901em (119887)BER119887


119901 (119887 119887) = 119901em (119887) (1 minus BER119887)

(56)


119901em (sim 119887) = 1205740 + 1205990

119901em (119887) = 1205781 + 1205931

(57)





2(1 minus BER) (58)









6 Conclusions









References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













1198630 = Δ21198760

1198760 = 1205740 (1198883

241205912(120573 minus 120572)

6+

1198882

20120591(120573 minus 120572)

5)

(30)


1198631 = 1205931

times int

Δ

120573Δ

1198911 (1199091015840)

times (1199091015840minus (

1205931Δ

1205781 + 1205931

int

1199091015840

120573Δ

1198911 (1199091015840) 1198891199091015840

+1205781Δ

1205781 + 1205931

))

2

1198891199091015840

(31)


1198631 = Δ21198761

1198761 = 1205931

((1205781 + 1205931) (1 minus 120573) minus 1205931)2

3(1205781 + 1205931)2

(32)


119863 = Δ2(1198760 + 1198761) (33)



Δ = radic1205902

1198991001lowastWNR

1198760 + 1198761

(34)



1015840


1205891015840


respectively

Z =

infin

⋃

119898=minusinfin

[2Δ119898 + 119897119896

ΔminusTh 2Δ119898 + 119897

119896

Δ+Th) (35)

O =

infin

⋃

119898=minusinfin

[2Δ119898 + 119897119896

Δ+Th 2Δ(119898 + 1) + 119897

119896

ΔminusTh) (36)


(1205891015840isin Z) (120589

1015840

119899isin O) (37)


(1205891015840isin O) (120589

1015840

119899isin Z) (38)


(1205891015840isin O) (120589

1015840

119899isin O) (39)

(1205891015840isin Z) (120589

1015840

119899isin Z) (40)



in terms of 1205891015840 and 1205891015840

119899as 119891N[120589

1015840



1015840)

1198911(1205891015840) IDL0(120589

1015840) and IDL1(120589



1015840minus

(119897119896

Δ+Δ119898)] for odd119898 pdf 1198910(120589

1015840) = 1198910[119897

119896

Δ+ Δ(119898 + 1) minus 120589

1015840


119896

Δ+Δ119898 119897

119896

Δ+Δ(119898+1)]Then


BER0 =1205740

1205740 + 1205990

int

Oint

119868119896+119898

1198910 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

+1205990

1205740 + 1205990

int

Oint

119868119896+119898

IDL0 (1205891015840)

times 119891N [1205891015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

BER1 =1205931

1205931 + 1205781

int

Zint

119868119896+119898

1198911 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

+1205781

1205931 + 1205781

int

Zint

119868119896+119898

IDL1 (1205891015840)

times 119891N [1205891015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

(41)


int

Oint

119868119896+119898

1198910 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

= int

Oint

Δ

0

1198910 (1199091015840) 119891N [120589

1015840

119899minus (119897119896

Δ+ 1199091015840) 0 120590119899] 119889119909

10158401198891205891015840

119899

(42)

for any119898



Δ+

2Δ119899 1198910(1205891015840) = 1198910(119909


O + 2Δ119899 = O Hence

int

Oint

119868119896+2119899

1198910 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

= int

O+2119899Δint

Δ

0

1198910 (1199091015840)

times 119891N [(1205891015840

119899minus 2119899Δ)

minus (119897119896

Δ+ 1199091015840) 0 120590119899] 119889119909

1015840119889 1205891015840

119899minus 2119899Δ

(43)


(1199091015840minus Δ) + 119897

119896

Δ+ 2Δ(119899 + 1) 1198910(120589

1015840) = 1198910(Δ minus 119909



Therefore 119891N[1205891015840

119899minus 1205891015840 0 120590119899] = 119891N[(120589

1015840


119896

Δ+



and consequently

119891N [ (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))


Δ+ (1199091015840minus Δ)) 0 120590119899]

= 119891N [ minus (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))

minus (119897119896

Δ+ (Δ minus 119909

1015840)) 0 120590119899]

(44)


int

Oint

119868119896+2119899+1

1198910 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

= int


Δ

0

1198910 (Δ minus 1199091015840)

times 119891N [ minus (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))

minus (119897119896

Δ+ (Δ minus 119909

1015840)) 0 120590119899]

times 119889 Δ minus 1199091015840

times 119889 minus (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))

(45)





CTLNS-QIM-IDLNS-QIM

DC-QIMQIM

WNR (dB)

100

10minus2

10minus4

10minus6

C (b

itsy

mbo







CTL =1

2log2(1 + 10

01lowastWNR) (46)




Z =

100

⋃

119898=minus100

[2Δ119898 + 119897119896

ΔminusTh 2Δ119898 + 119897

119896

Δ+Th)

O =

100

⋃

119898=minus100

[2Δ119898 + 119897119896

Δ+Th 2Δ (119898 + 1)

+ 119897119896

ΔminusTh)

(47)


119896


119896







DWR = 10 log10

(1205902

119867

119863) (48)



Δ =120590119867

radic(1198760 + 1198761) 1001DWR

(49)


1205902

119899=

1205902

119867

1001(DWR+WNR)

(50)


119899is









0 40 80 120 160 200

100

10minus1

10minus2

10minus3

10minus4

10minus5

Var

DC-QIMNS-QIMFZDHTCM

QIMNSC-QIMRDM

C (b

itsy

mbo

l)







DC-QIMNS-QIMFZDHTCM

QIMNSC-QIMRDM

100

10minus1

10minus2

20 30 40 50 60 70 80 90 100

Q of JPEG ()

C (b

itsy

mbo






1015840Δ



1199091015840

119899=

1205891015840

119899mod Δ

1015840 if

[[[

[

1205891015840

119899minus 119897119896

Δ

Δ1015840

]]]

]

mod 2 = 0

Δ1015840minus (1205891015840

119899mod Δ

1015840) otherwise

(51)


119899isin [0 Δ

1015840]

1198621 (Δ1015840) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

median (1198831015840

119899)

Δ1015840minus 05

10038161003816100381610038161003816100381610038161003816100381610038161003816

1198622 (Δ1015840) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

119864 ([1198831015840

119899]119908

)

[Δ1015840]119908

10038161003816100381610038161003816100381610038161003816100381610038161003816

119908 = 2119898 + 1 119898 isin N

(52)


0035

003

0025

002

0015

001

0005

09 95 10 105 11

998400

Crite

rion

1

Δ

(a)

0

1

2

3

4

5

6

7

8

9 95 10 105 11

Crite

rion

2

times10minus4

998400Δ

(b)




Δ10158401015840= argmax

Δ101584011986212 (Δ

1015840) (53)


119899very close to














0 40 80 120 160 200

100

10minus1

10minus2

10minus3

10minus4

10minus5

Var

NS-QIMNSC-QIMRDM

C (b

itsy

mbo

l)


100

10minus1

10minus2

20 40 60 80 100

Q of JPEG ()

NS-QIMNSC-QIMRDM

C (b

itsy

mbo

l)




5 Discussion




119862 = max119901em(sim119887)

[119901 (sim 119887 119887) log2(

119901 (sim 119887 119887)

119901em (sim 119887) 119901ex (119887))

+ 119901 (119887 sim 119887) log2(

119901 (119887 sim 119887)

119901em (119887) 119901ex (sim 119887))

+ 119901 (sim 119887 sim 119887) log2(

119901 (sim 119887 sim 119887)

119901em (sim 119887) 119901ex (sim 119887))

+ 119901 (119887 119887) log2(

119901 (119887 119887)

119901em (119887) 119901ex (119887))]

(54)


119901ex (119887) = 119901 (sim 119887 119887) + 119901 (119887 119887)

119901ex (sim 119887) = 119901 (119887 sim 119887) + 119901 (sim 119887 sim 119887)

(55)


119901 (sim 119887 119887) = 119901em (sim 119887)BERsim119887

119901 (119887 sim 119887) = 119901em (119887)BER119887


119901 (119887 119887) = 119901em (119887) (1 minus BER119887)

(56)


119901em (sim 119887) = 1205740 + 1205990

119901em (119887) = 1205781 + 1205931

(57)





2(1 minus BER) (58)









6 Conclusions









References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Δ+

2Δ119899 1198910(1205891015840) = 1198910(119909


O + 2Δ119899 = O Hence

int

Oint

119868119896+2119899

1198910 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

= int

O+2119899Δint

Δ

0

1198910 (1199091015840)

times 119891N [(1205891015840

119899minus 2119899Δ)

minus (119897119896

Δ+ 1199091015840) 0 120590119899] 119889119909

1015840119889 1205891015840

119899minus 2119899Δ

(43)


(1199091015840minus Δ) + 119897

119896

Δ+ 2Δ(119899 + 1) 1198910(120589

1015840) = 1198910(Δ minus 119909



Therefore 119891N[1205891015840

119899minus 1205891015840 0 120590119899] = 119891N[(120589

1015840


119896

Δ+



and consequently

119891N [ (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))


Δ+ (1199091015840minus Δ)) 0 120590119899]

= 119891N [ minus (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))

minus (119897119896

Δ+ (Δ minus 119909

1015840)) 0 120590119899]

(44)


int

Oint

119868119896+2119899+1

1198910 (1205891015840) 119891N [120589

1015840

119899minus 1205891015840 0 120590119899] 119889120589

10158401198891205891015840

119899

= int


Δ

0

1198910 (Δ minus 1199091015840)

times 119891N [ minus (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))

minus (119897119896

Δ+ (Δ minus 119909

1015840)) 0 120590119899]

times 119889 Δ minus 1199091015840

times 119889 minus (1205891015840

119899minus 2Δ (119899 + 1 minus 119895))

(45)





CTLNS-QIM-IDLNS-QIM

DC-QIMQIM

WNR (dB)

100

10minus2

10minus4

10minus6

C (b

itsy

mbo







CTL =1

2log2(1 + 10

01lowastWNR) (46)




Z =

100

⋃

119898=minus100

[2Δ119898 + 119897119896

ΔminusTh 2Δ119898 + 119897

119896

Δ+Th)

O =

100

⋃

119898=minus100

[2Δ119898 + 119897119896

Δ+Th 2Δ (119898 + 1)

+ 119897119896

ΔminusTh)

(47)


119896


119896







DWR = 10 log10

(1205902

119867

119863) (48)



Δ =120590119867

radic(1198760 + 1198761) 1001DWR

(49)


1205902

119899=

1205902

119867

1001(DWR+WNR)

(50)


119899is









0 40 80 120 160 200

100

10minus1

10minus2

10minus3

10minus4

10minus5

Var

DC-QIMNS-QIMFZDHTCM

QIMNSC-QIMRDM

C (b

itsy

mbo

l)







DC-QIMNS-QIMFZDHTCM

QIMNSC-QIMRDM

100

10minus1

10minus2

20 30 40 50 60 70 80 90 100

Q of JPEG ()

C (b

itsy

mbo






1015840Δ



1199091015840

119899=

1205891015840

119899mod Δ

1015840 if

[[[

[

1205891015840

119899minus 119897119896

Δ

Δ1015840

]]]

]

mod 2 = 0

Δ1015840minus (1205891015840

119899mod Δ

1015840) otherwise

(51)


119899isin [0 Δ

1015840]

1198621 (Δ1015840) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

median (1198831015840

119899)

Δ1015840minus 05

10038161003816100381610038161003816100381610038161003816100381610038161003816

1198622 (Δ1015840) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

119864 ([1198831015840

119899]119908

)

[Δ1015840]119908

10038161003816100381610038161003816100381610038161003816100381610038161003816

119908 = 2119898 + 1 119898 isin N

(52)


0035

003

0025

002

0015

001

0005

09 95 10 105 11

998400

Crite

rion

1

Δ

(a)

0

1

2

3

4

5

6

7

8

9 95 10 105 11

Crite

rion

2

times10minus4

998400Δ

(b)




Δ10158401015840= argmax

Δ101584011986212 (Δ

1015840) (53)


119899very close to














0 40 80 120 160 200

100

10minus1

10minus2

10minus3

10minus4

10minus5

Var

NS-QIMNSC-QIMRDM

C (b

itsy

mbo

l)


100

10minus1

10minus2

20 40 60 80 100

Q of JPEG ()

NS-QIMNSC-QIMRDM

C (b

itsy

mbo

l)




5 Discussion




119862 = max119901em(sim119887)

[119901 (sim 119887 119887) log2(

119901 (sim 119887 119887)

119901em (sim 119887) 119901ex (119887))

+ 119901 (119887 sim 119887) log2(

119901 (119887 sim 119887)

119901em (119887) 119901ex (sim 119887))

+ 119901 (sim 119887 sim 119887) log2(

119901 (sim 119887 sim 119887)

119901em (sim 119887) 119901ex (sim 119887))

+ 119901 (119887 119887) log2(

119901 (119887 119887)

119901em (119887) 119901ex (119887))]

(54)


119901ex (119887) = 119901 (sim 119887 119887) + 119901 (119887 119887)

119901ex (sim 119887) = 119901 (119887 sim 119887) + 119901 (sim 119887 sim 119887)

(55)


119901 (sim 119887 119887) = 119901em (sim 119887)BERsim119887

119901 (119887 sim 119887) = 119901em (119887)BER119887


119901 (119887 119887) = 119901em (119887) (1 minus BER119887)

(56)


119901em (sim 119887) = 1205740 + 1205990

119901em (119887) = 1205781 + 1205931

(57)





2(1 minus BER) (58)









6 Conclusions









References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Z =

100

⋃

119898=minus100

[2Δ119898 + 119897119896

ΔminusTh 2Δ119898 + 119897

119896

Δ+Th)

O =

100

⋃

119898=minus100

[2Δ119898 + 119897119896

Δ+Th 2Δ (119898 + 1)

+ 119897119896

ΔminusTh)

(47)


119896


119896







DWR = 10 log10

(1205902

119867

119863) (48)



Δ =120590119867

radic(1198760 + 1198761) 1001DWR

(49)


1205902

119899=

1205902

119867

1001(DWR+WNR)

(50)


119899is









0 40 80 120 160 200

100

10minus1

10minus2

10minus3

10minus4

10minus5

Var

DC-QIMNS-QIMFZDHTCM

QIMNSC-QIMRDM

C (b

itsy

mbo

l)







DC-QIMNS-QIMFZDHTCM

QIMNSC-QIMRDM

100

10minus1

10minus2

20 30 40 50 60 70 80 90 100

Q of JPEG ()

C (b

itsy

mbo






1015840Δ



1199091015840

119899=

1205891015840

119899mod Δ

1015840 if

[[[

[

1205891015840

119899minus 119897119896

Δ

Δ1015840

]]]

]

mod 2 = 0

Δ1015840minus (1205891015840

119899mod Δ

1015840) otherwise

(51)


119899isin [0 Δ

1015840]

1198621 (Δ1015840) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

median (1198831015840

119899)

Δ1015840minus 05

10038161003816100381610038161003816100381610038161003816100381610038161003816

1198622 (Δ1015840) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

119864 ([1198831015840

119899]119908

)

[Δ1015840]119908

10038161003816100381610038161003816100381610038161003816100381610038161003816

119908 = 2119898 + 1 119898 isin N

(52)


0035

003

0025

002

0015

001

0005

09 95 10 105 11

998400

Crite

rion

1

Δ

(a)

0

1

2

3

4

5

6

7

8

9 95 10 105 11

Crite

rion

2

times10minus4

998400Δ

(b)




Δ10158401015840= argmax

Δ101584011986212 (Δ

1015840) (53)


119899very close to














0 40 80 120 160 200

100

10minus1

10minus2

10minus3

10minus4

10minus5

Var

NS-QIMNSC-QIMRDM

C (b

itsy

mbo

l)


100

10minus1

10minus2

20 40 60 80 100

Q of JPEG ()

NS-QIMNSC-QIMRDM

C (b

itsy

mbo

l)




5 Discussion




119862 = max119901em(sim119887)

[119901 (sim 119887 119887) log2(

119901 (sim 119887 119887)

119901em (sim 119887) 119901ex (119887))

+ 119901 (119887 sim 119887) log2(

119901 (119887 sim 119887)

119901em (119887) 119901ex (sim 119887))

+ 119901 (sim 119887 sim 119887) log2(

119901 (sim 119887 sim 119887)

119901em (sim 119887) 119901ex (sim 119887))

+ 119901 (119887 119887) log2(

119901 (119887 119887)

119901em (119887) 119901ex (119887))]

(54)


119901ex (119887) = 119901 (sim 119887 119887) + 119901 (119887 119887)

119901ex (sim 119887) = 119901 (119887 sim 119887) + 119901 (sim 119887 sim 119887)

(55)


119901 (sim 119887 119887) = 119901em (sim 119887)BERsim119887

119901 (119887 sim 119887) = 119901em (119887)BER119887


119901 (119887 119887) = 119901em (119887) (1 minus BER119887)

(56)


119901em (sim 119887) = 1205740 + 1205990

119901em (119887) = 1205781 + 1205931

(57)





2(1 minus BER) (58)









6 Conclusions









References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 40 80 120 160 200

100

10minus1

10minus2

10minus3

10minus4

10minus5

Var

DC-QIMNS-QIMFZDHTCM

QIMNSC-QIMRDM

C (b

itsy

mbo

l)







DC-QIMNS-QIMFZDHTCM

QIMNSC-QIMRDM

100

10minus1

10minus2

20 30 40 50 60 70 80 90 100

Q of JPEG ()

C (b

itsy

mbo






1015840Δ



1199091015840

119899=

1205891015840

119899mod Δ

1015840 if

[[[

[

1205891015840

119899minus 119897119896

Δ

Δ1015840

]]]

]

mod 2 = 0

Δ1015840minus (1205891015840

119899mod Δ

1015840) otherwise

(51)


119899isin [0 Δ

1015840]

1198621 (Δ1015840) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

median (1198831015840

119899)

Δ1015840minus 05

10038161003816100381610038161003816100381610038161003816100381610038161003816

1198622 (Δ1015840) =

10038161003816100381610038161003816100381610038161003816100381610038161003816

119864 ([1198831015840

119899]119908

)

[Δ1015840]119908

10038161003816100381610038161003816100381610038161003816100381610038161003816

119908 = 2119898 + 1 119898 isin N

(52)


0035

003

0025

002

0015

001

0005

09 95 10 105 11

998400

Crite

rion

1

Δ

(a)

0

1

2

3

4

5

6

7

8

9 95 10 105 11

Crite

rion

2

times10minus4

998400Δ

(b)




Δ10158401015840= argmax

Δ101584011986212 (Δ

1015840) (53)


119899very close to














0 40 80 120 160 200

100

10minus1

10minus2

10minus3

10minus4

10minus5

Var

NS-QIMNSC-QIMRDM

C (b

itsy

mbo

l)


100

10minus1

10minus2

20 40 60 80 100

Q of JPEG ()

NS-QIMNSC-QIMRDM

C (b

itsy

mbo

l)




5 Discussion




119862 = max119901em(sim119887)

[119901 (sim 119887 119887) log2(

119901 (sim 119887 119887)

119901em (sim 119887) 119901ex (119887))

+ 119901 (119887 sim 119887) log2(

119901 (119887 sim 119887)

119901em (119887) 119901ex (sim 119887))

+ 119901 (sim 119887 sim 119887) log2(

119901 (sim 119887 sim 119887)

119901em (sim 119887) 119901ex (sim 119887))

+ 119901 (119887 119887) log2(

119901 (119887 119887)

119901em (119887) 119901ex (119887))]

(54)


119901ex (119887) = 119901 (sim 119887 119887) + 119901 (119887 119887)

119901ex (sim 119887) = 119901 (119887 sim 119887) + 119901 (sim 119887 sim 119887)

(55)


119901 (sim 119887 119887) = 119901em (sim 119887)BERsim119887

119901 (119887 sim 119887) = 119901em (119887)BER119887


119901 (119887 119887) = 119901em (119887) (1 minus BER119887)

(56)


119901em (sim 119887) = 1205740 + 1205990

119901em (119887) = 1205781 + 1205931

(57)





2(1 minus BER) (58)









6 Conclusions









References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0035

003

0025

002

0015

001

0005

09 95 10 105 11

998400

Crite

rion

1

Δ

(a)

0

1

2

3

4

5

6

7

8

9 95 10 105 11

Crite

rion

2

times10minus4

998400Δ

(b)




Δ10158401015840= argmax

Δ101584011986212 (Δ

1015840) (53)


119899very close to














0 40 80 120 160 200

100

10minus1

10minus2

10minus3

10minus4

10minus5

Var

NS-QIMNSC-QIMRDM

C (b

itsy

mbo

l)


100

10minus1

10minus2

20 40 60 80 100

Q of JPEG ()

NS-QIMNSC-QIMRDM

C (b

itsy

mbo

l)




5 Discussion




119862 = max119901em(sim119887)

[119901 (sim 119887 119887) log2(

119901 (sim 119887 119887)

119901em (sim 119887) 119901ex (119887))

+ 119901 (119887 sim 119887) log2(

119901 (119887 sim 119887)

119901em (119887) 119901ex (sim 119887))

+ 119901 (sim 119887 sim 119887) log2(

119901 (sim 119887 sim 119887)

119901em (sim 119887) 119901ex (sim 119887))

+ 119901 (119887 119887) log2(

119901 (119887 119887)

119901em (119887) 119901ex (119887))]

(54)


119901ex (119887) = 119901 (sim 119887 119887) + 119901 (119887 119887)

119901ex (sim 119887) = 119901 (119887 sim 119887) + 119901 (sim 119887 sim 119887)

(55)


119901 (sim 119887 119887) = 119901em (sim 119887)BERsim119887

119901 (119887 sim 119887) = 119901em (119887)BER119887


119901 (119887 119887) = 119901em (119887) (1 minus BER119887)

(56)


119901em (sim 119887) = 1205740 + 1205990

119901em (119887) = 1205781 + 1205931

(57)





2(1 minus BER) (58)









6 Conclusions









References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 40 80 120 160 200

100

10minus1

10minus2

10minus3

10minus4

10minus5

Var

NS-QIMNSC-QIMRDM

C (b

itsy

mbo

l)


100

10minus1

10minus2

20 40 60 80 100

Q of JPEG ()

NS-QIMNSC-QIMRDM

C (b

itsy

mbo

l)




5 Discussion




119862 = max119901em(sim119887)

[119901 (sim 119887 119887) log2(

119901 (sim 119887 119887)

119901em (sim 119887) 119901ex (119887))

+ 119901 (119887 sim 119887) log2(

119901 (119887 sim 119887)

119901em (119887) 119901ex (sim 119887))

+ 119901 (sim 119887 sim 119887) log2(

119901 (sim 119887 sim 119887)

119901em (sim 119887) 119901ex (sim 119887))

+ 119901 (119887 119887) log2(

119901 (119887 119887)

119901em (119887) 119901ex (119887))]

(54)


119901ex (119887) = 119901 (sim 119887 119887) + 119901 (119887 119887)

119901ex (sim 119887) = 119901 (119887 sim 119887) + 119901 (sim 119887 sim 119887)

(55)


119901 (sim 119887 119887) = 119901em (sim 119887)BERsim119887

119901 (119887 sim 119887) = 119901em (119887)BER119887


119901 (119887 119887) = 119901em (119887) (1 minus BER119887)

(56)


119901em (sim 119887) = 1205740 + 1205990

119901em (119887) = 1205781 + 1205931

(57)





2(1 minus BER) (58)









6 Conclusions









References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












2(1 minus BER) (58)









6 Conclusions









References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















References

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article A New QIM-Based Watermarking Method...

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