1

An Improved Detector for Spread-Spectrum basedWatermarking using Independent Component

Analysis: Performance AnalysisHafiz Malik, Ashfaq Khokhar, and Rashid Ansari

Department of Electrical and Computer Engineering University of Illinois at Chicago, Illinois, USA{hmalik, ashfaq, ansari}@ece.uic.edu

Abstract

This paper presents a novel blind watermark detection/decoding scheme for spread spectrum (SS) based watermarkingexploiting the fact that the embedded watermark and the host signal are mutually independent. The proposed detector assumesthat the host signal and the watermark obey non-Gaussian distributions. The proposed scheme employs the theory of blind sourceseparation (BSS) using independent component analysis (ICA) to cancel the host-signal interference at the detector/decoder. Thepaper presents analytical results showing that the proposed detector performs significantly better than the existing blind detectorsused for SS-based watermarking. The paper also shows that the detection performance approaches to that of an informed detectoras interference due to the host signal in the estimated watermark using BSS approaches to zero.

I. INTRODUCTION

Technological and economical factors such as the growth of the Internet, the proliferation of low-cost and reliable storagedevices, the deployment of seamless broadband networks, the availability of state-of-the-art digital media production andediting technologies, and the development of efficient multimedia data compression schemes have made digital forgeries andunauthorized sharing of digital media a wide spread reality. This form of piracy has subjected the entertainment industryto enormous annual revenue losses. For example, music industry alone claims multi-million illegal music downloads on theInternet every week. It is therefore imperative to have robust technologies to protect copyrighted digital media from illegalsharing and tampering. Traditional digital data protection techniques, such as encryption and scrambling, alone cannot provideadequate protection of copyrighted contents, because these technologies are unable to protect digital content once they aredecrypted or unscrambled. Digital watermarking technology complements cryptography for protecting digital content even afterit is deciphered [1].

Digital watermarking is a process of embedding imperceptible content protection and/or content authentication information(watermark) into the digital content to be protected (the host media). Although the performance expected from a givenwatermarking system depends on the target application area [1], [2] robust embedding scheme and efficient detection procedureare inherently desired.

The growing concern over copyright protection of digital contents is a major driving force behind the ongoing research inthe area of digital watermarking. In general, the existing watermark detectors are classified into two categories: (a) informeddetectors, which assume that the host signal is available at the detector during watermark detection process, and (b) blinddetectors, which assume that the host signal is not available at the detector for watermark detection. Similarly, watermarkembedding schemes can be classified into two major groups: (a) blind embedding implies that the watermark embedderdoes not exploit the host signal information during the watermark embedding process (spread spectrum based watermarkingschemes [?], [1]–[3], [7]–[14] fall in this category), and (b) informed embedding implies that the watermark embedder exploitsthe properties of the host signal during the watermark embedding process (quantization index modulation based watermarkingschemes [?], [1]–[3], [15], [17] belong to this category).

In order to meet the fidelity requirement of the watermarked signal, the power of the embedded watermark (watermarkstrength) is generally kept much lower than the host signal power. The informed watermark detection schemes for SS-basedwatermarking employ statistical characterization of the host signal to develop an optimal or near-optimal watermark detector(in the maximum likelihood (ML) sense) [17], [18]. The salient features of such schemes include increased robustness againstinterference (i.e. antijamming capability), simplicity, and low computational complexity. Blind watermark detectors for SS-basedwatermarking perform poorly, particularly in terms of decoding error probability due to the presence of host-signal interferenceat the decoder. The nonzero decoding error probability at the watermark decoder even in the absence of attack-channel distortionis one of the limitations of the existing blind watermark detectors.

The main motivation of this paper is to design a blind detector for SS-based watermarking schemes capable of canceling thehost-signal interference at the detector, hence improving decoding as well as detection performance. Towards this end, we usethe theory of independent component analysis (ICA) by posing watermark detection as a blind source separation problem. Theproposed detector assumes that the received watermarked signal is a linear mixture and the underlying independent components

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(the host signal and the watermark) obey non-Gaussian distributions. The proposed detector falls in the category of blinddetectors. We show that the watermark estimation problem in SS-based watermarking systems is equivalent to blind sourceseparation (BSS) problem. Therefore, ICA framework could be used to estimate the watermark in SS-based watermarkedsignals. The proposed ICA-based detector first estimates hidden independent components (i.e., the watermark and the hostsignal) from the received watermarked signal using ICA framework; then these estimated components are used to detect theembedded watermark. Theoretical analysis shows that the proposed ICA-based detector performs significantly better than theexisting watermark detectors that operate without reducing the host-signal interference at the watermark detector for watermarkdetection [17], [18]. The proposed ICA-based watermark detector is applicable to SS-based watermarking of all media types,i.e. audio, video and images. However, in this paper digital audio is used as the host media for watermark embedding, detection,and performance analysis of the proposed ICA-based detector.

II. BASICS OF SS-BASED WATERMARKING

The spread-spectrum (SS) based watermarking system can be modeled using a classical secure communication model [1],as shown in Fig. 1.

M ESSAGE

E NCODER

M ESSAGE

D ECODER

+ +

H OST M EDIA D ATA

E MBEDDING K EY

I NPUT M ESSAGE

A DVERSARY

A TTACK

Watermark Embedder Watermark Detector

x

n s

w b

K

x X

b w

P ERCEPTUAL

M ASK E STIMATION

ˆ b

M ESSAGE

D ETECTOR 1 H

Attack Channel

0 H

Fig. 1. Depiction of a Perceptual Based Data Hiding System with Blind Receiver as a Standard Secure Communication Model

In Fig. 1, s ∈ RN is a vector containing coefficients of an appropriate transform of the host signal. It is assumed that thecoefficients, si : i = 0, 1, · · · , N − 1, are independent and identically distributed (i.i.d.) random variables (r.v.) with zero meanand variance. A watermark, w ∈ {±1}N , is generated using (a) binary information vector b used to encode input messageM and (b) a data embedding secret key K. We assume that the watermark element wi and the host signal coefficients si

are mutually independent. The watermarked signal x is obtained by adding an amplitude-modulated watermark w to the hostsignal s. In order to meet the fidelity requirement of the watermarked media x, i.e, to ensure that the embedded watermark isimperceptible, the amplitude-modulated watermark is spectrally shaped according to a perceptual mask α, estimated based onthe human auditory system (HAS) and the host signal s, i.e., α = f(x, HAS).

The watermarked signal x can be expressed as

x = s + α ·wb (1)

where [·] denotes element-wise product of the two vectors.Embedding distortion generally serves as a fidelity measure for a given watermarking scheme and can be expressed as:

de = x− s (2)

The mean-squared error embedding distortion is expressed as:

De = ‖de‖2= ‖x− s‖2 = ‖α ·w‖2

=1N

N−1∑

i=0

α2i = σ2

w (3)

where ‖ · ‖ represents the Euclidian norm.The signal distortion due to an active adversary attack can be viewed as channel noise, n, shown in Fig. 1. The resulting

watermarked signal, x, is processed for watermark detection and is expressed as:

x = x + n (4)

The SS-based watermarking schemes use probabilistic characterization of the host data to develop an optimal or near-optimalwatermark detector (in ML sense). The statistical characterizations of real-world host media are available in spatial domain as

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well as in the transform domain. For example, the probability distribution function (pdf) of the stationary speech samples intime domain can be approximated by Laplacian distribution [23].

In this paper we use watermarked audio signals to evaluate the performance of the proposed detector. The performance willbe assessed by computing the average probability of error. The embedding is performed using a frequency-selective spreadspectrum (FSSS) based audio watermarking scheme [14] that we proposed for embedding watermarks in the discrete wavelettransform (DWT) domain.

The probability density of the DWT coefficients of a sampled real-world audio signal, si, can be approximated by a Laplaciandistribution (as discussed in Appendix I), i.e.

fsi=

βi

2e−βi|si| :| si |< ∞ (5)

where βi =√

2σsi

The error probability, Pe, of a decoded bit under zero-channel distortion scenario, i.e. ni = 0, can be calculated under theassumption (a) the watermarked sample xi is obtained by adding binary amplitude-modulated watermark wi, i.e. αiwib and(b) the detection is performed using Perez-Gonzalez et al’s optimal decoder (in ML sense) [17], [18], that operates withoutcanceling the host-signal interference. In addition, for simplicity of the performance analysis we assume that only one binarydigit information b ∈ {±1} is embedded in the host media. In [18] it is shown that the optimal ML decoder estimates b = 1if xiwi > 0 and an error will occur when xiwi < 0, given that b = 1 was embedded [17]. Assuming that wi takes values ±1with probability 1

2 , the average error probability Pe is given by

Pe = Pr{xiwi < 0|b = 1}=

∫ 0

−∞fsi(τ − α) dτ (6)

Using Laplacian model, this can be shown

Pe =12e−√

2/λi (7)

where λi = σsi

σwi

From Eq. (7) it is clear that even in the absence of attack-channel distortion, zero decoding error probability is not achievable.Moreover, Pe is a function of parameter λ, which is generally referred as signal-to-watermark ratio (SWR), when expressedin dB i.e. SWR = 20 log10 λ. The value of the parameter λ offers a tradeoff between fidelity of the embedded watermark anddecoding error probability, with a larger value of λ providing better fidelity but poor Pe performance.

Similarly when the same information bit b is embedded into samples with selected indices ζ of the host audio s, then thewatermarked audio is given as:

xi = si + αiwib : i ∈ ζ (8)

Let us assume that the watermarked signal used for detection is free of attack-channel distortion, i.e. n = 0, and all hiddenmessages have same a priori probability. In this case a ML decoder minimizes the decoding error probability is given as

lnfx(x|b+)fx(x|b−)

= lnfx(x− αwb+)fx(x− αwb−)

> 0 (9)

where b+(b−) represents that binary information b = +1(b = −1) is embedded in the selected indices.The ML sufficient statistic based on Laplacian distribution for si can be written as,

r :=∑

i∈ζ

βi (|xi + αiwi| − |xi − αiwi|) (10)

If b = 1 was embedded, then r can be expressed as:

r =∑

i∈ζ

βi (|xi + 2αiwi| − |xi|) (11)

Here the ML detector is a bit-by-bit hard decoder, i.e.,

b = sgn(r) (12)

In order to determine the bit error probability for this ML decoder, a statistical characterization of r is required. Here r issum of |ζ| i.i.d. random variables. By applying the central limit theorem (CLT), r can be approximated by a Gaussian random

4

variable whose mean and variance are computed as follows:

E{r} =∑

i∈ζ

βiE (|xi + 2αiwi| − |xi|) (13)

Var{r} =∑

i∈ζ

βiVar (|xi + 2αiwi| − |xi|) (14)

E{r} =∑

i∈ζ

(e−2

√2/λi +

2√

2λi

− 1

)(15)

Var{r} =∑

i∈ζ

(3− e−4

√2/λi − e−2

√2/λi

(1 +

4√

2λi

))(16)

The decoding bit error probability is given as:

Pe = Q

(|E{r}|√Var{r}

)(17)

where Q (x) = 1√2π

∫∞x

e−t2/2 dtEq. (17) shows that the decoding error probability Pe is non-zero even in the absence of attack-channel distortion, and Pe

is a function of λi.In other words, this analysis shows that the detection/decoding performance of a detector for SS-based watermarking

schemes is inherently bounded below by the host-signal interference at the detector. The main objective of this paper is todesign a watermark detector for SS-based watermarking schemes with improved watermark detection, decoding, and maximumwatermarking-rate performances by rejecting the host-signal interference at the detector. Towards this end, we use the theory ofindependent component analysis (ICA) by posing watermark estimation as a blind source separation problem. The fundamentalsof the ICA theory are briefly outlined in the following section followed by the details of the proposed ICA-based detector.

III. INDEPENDENT COMPONENT ANALYSIS

Independent Component Analysis (ICA) is a statistical framework for estimating underlying hidden factors or componentsof multivariate statistical data. In the ICA model, the data variables are assumed to be linear or nonlinear mixtures of someunknown latent variables, and the mixing system is also unknown [19]–[22]. Moreover, these hidden variables are assumedto be non-Gaussian and mutually independent. The ICA model can be considered as an extension of the principal componentanalysis (PCA) and factor analysis [20]–[22]. In fact, ICA can be treated as non-Gaussian factor analysis, since data is modeledas a linear mixture of underlying non-Gaussian factors. The ICA framework has been used in diverse application scenariosincluding blind source separation (BSS), feature extraction, telecommunication, and economics [19]–[22]. In the following wewill review only the linear ICA framework since that is applicable to the SS-based watermarking model.

In general, the linear ICA model can be defined for noise-free as well as noisy scenarios as follows.Noise-free ICA model: ICA of a random vector x consists of estimating the following generative model of the data:

x = As (18)

where x represents the observed m-dimensional random vector, s is a vector of hidden variables and the matrix A is a m×n1mixing matrix. The hidden variables (components) si in the vector s = [s1, · · · , sn1]T are assumed statistically independent.

Noisy ICA model: ICA of a random vector x consists of estimating the following generative model of the data:

x = As + n (19)

where n is m-dimensional random noise vector , while x, s, and A are the same as in the noise-free model.In this paper, we use the noisy ICA generative model to design an ICA-based watermark detector for SS-based watermarking

schemes. The proposed ICA-based watermark detector attempts to estimate the embedded watermark from the watermarkedsignal while reducing the host-signal interference at the watermark detector.

Before estimating the underlying independent components from observed data using ICA framework, the generative modelshould meet certain conditions to ensure the identifiability of the ICA model. The identifiability constraints defined in [20],[22], [24], [26] for a noise-free ICA model are outlined below:

1) Statistical independence: The hidden (latent) variables/ sources are statistically independent.2) Non-Gaussianity: At most one of the independent components si : i = 1, 2 · · ·n1, is normally distributed.3) More sensors than sources: The number of observed linear mixtures (sensors) m must be greater or equal to the number

of independent components (sources) n1 i.e. m ≥ n1 .

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4) Full rank mixing matrix: The rank of the mixing matrix, A, must be full column rank.5) Whitening: The vectors x and s are assumed to be normalized and centered.Here, the third condition (i.e. more sensors than sources) is a relatively loose restriction, as ICA model is identifiable even

when latent components (sources) are more than observations (sensors), i.e. m ≤ n1 [31], [34].The identifiability of the noisy ICA model (Eq. (19)) requires the same set of restrictions except the third condition i.e.,

m ≥ n1, assuming that noise is independent from the underlying independent components [29], [33]. In other words, a noisyICA model can be treated as a special case of noise-free ICA model where noise is treated as one of the independent sourcesand the number of sources is more than the number of sensors (degenerate case) i.e. m ≤ n1. Further details on identifiabilityof noisy ICA model can be found in [29], [33] and citation therein.

Independence and maximum non-Gaussianity are two fundamental ingredients of the ICA theory. Independence of theunderlying components is one of the assumptions that is made to estimate components from the linear mixture. Note thatindependence of the underlying components is a much stronger condition than uncorrelatedness, e.g., for the BSS problem,there might be many dependent but uncorrelated representations of the observed signals and these uncorrelated but dependentrepresentations of the observed signals cannot separate the mixed sources [20]. Therefore, uncorrelatedness itself is insufficientto solve the BSS problem. In fact, independence implies nonlinear uncorrelatedness [20], that is, if s1 and s2 are two independentcomponents then any nonlinear transformations of these components, say, φ1(s1) and φ2(s2) , are uncorrelated as well (i.e.their covariance is zero). On the other hand, if s1 and s2 are assumed to be just uncorrelated then in general, the correspondingnonlinear transformations do not necessarily have zero covariance. Thus to perform ICA, a stronger form of decorrelation ofthe underlying components is required, that is, nonlinear decorrelation. A suitable selection of nonlinearities, i.e. φ1(·) andφ2(·), can be achieved by using tools like maximum likelihood and mutual information from estimation theory and informationtheory [20], [22].

Maximum non-Gaussianity is another important requirement of ICA-based hidden components estimation [19]–[22], [27],[28], [30]. A quantity kurtosis defined in terms of the fourth-order central moment κ is generally used as a non-Gaussianitymeasure of a random variable. Kurtosis of a real r.v. s can be defined as,

κ =(E{(s− E(s))4}/E2{(s− E(s))2}

)− 3 (20)

where E{·} denotes expected value of a random variable.A normal random variable has zero kurtosis; therefore, kurtosis is a measure of the ”distance” of a random variable from a

Gaussian distribution. Distributions that are peakier (flatter) about the mean than a Gaussian distribution generally have positive(negative) kurtosis. Random variables with positive kurtosis, i.e. κ > 0, are generally called super-Gaussian. The Laplaciandistribution is a typical example of this case. Random variables with negative kurtosis value, i.e., κ < 0 are generally calledsub-Gaussian, e.g., the uniform distribution.

A. Blind Source Separation (BSS) using ICABlind source separation is one of the most widely explored applications of the ICA model [19]–[21]. The BSS problem in

ICA framework is generally modelled as,The observation x corresponds to K realizations of an m-dimensional random vector xj(t) : j = 1, 2, · · · ,m, t = 1, 2, · · · ,K

which follow the linear statistical model

x(t) = As(t) + n (21)

where A ∈mathcalRm×n1 is unknown mixing matrix, s(t) are realizations of n1-dimensional underlying independent components orsources, and n(t) is a noise which is assumed to be independent of x(t). The goal is to recover the underlying independentsources s, from the observations x. The cocktail party problem is a classical example of BSS, where several people aresimultaneously speaking in the same room and the problem is to separate the voices of different speakers (sources), usingrecordings of several microphones (observations/sensors) in the room.

In the BSS problem, the underlying variables are independent (e.g. speaker voices in the case of cocktail party problem).Therefore, the ICA model is a suitable framework to find solutions for this problem. In case of SS-based watermarking, thewatermarked signal x at the detector can be treated as an observation which is a linear mixture of the underlying hiddenindependent components, i.e., the host signal s and the watermark αwb.

In the ICA framework, the problem is to find a linear representation in which the underlying components are statisticallyindependent. In other words we intend to estimate the demixing matrix B from the observed data x. The estimated demixingmatrix B is the inverse (or generalized inverse) of mixing matrix A, i.e., B = A† = (AHA)−1AH . Most of the existingBSS schemes using ICA model are based on the information-theoretic framework. For example, Bell et al’s [23] ICA schemeis based on the idea of information maximization, or ”infomax” among the estimated independent components, Comon in[21] has used higher-order cumulants, and Gaeta et al [32] used ML estimation. Other methods are variant and extensions ofinfomax, higher-order cumulants, and ML method [19]–[22].

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IV. PROPOSED ICA BASED WATERMARK DETECTOR

The proposed ICA-based watermark detector consists of two stages: 1) watermark estimation stage, and, 2) watermarkdecoding and/or detection stage. The watermark estimation stage estimates watermark w from the received watermarkedaudio x using ICA framework, whereas, the watermark decoding and/or detection stage decodes and/or detects the embeddedwatermark using the ML approach. The block diagram of the proposed watermark detector is given in the Fig. 2.

W ATERMARK

E STIMATION

USING ICA W ATERMARK

D ECODER

S ECRET K EY

R ECEIVED

W ATERMARKED

S IGNAL

ˆ w

ˆ b

W ATERMARK

D ETECTOR

1 H

K

x

0 H

Fig. 2. Block Diagram of the Proposed ICA-based Watermark Detector

In general the ICA model for BSS estimates demixing matrix B from the observed data x, hence the underlying independentcomponents si. This model is extendable to the watermark estimation problem from the watermarked signal, assumingidentifiability conditions of the ICA model are satisfied (as discussed in section 3). In order to examine whether the SS-based watermarking satisfies the identifibility constraints of an ICA model, we consider an additive SS-based watermarkembedding scenario. Let us assume that the watermark is embedded into the host signal by a linear combination of the hostsignal and a mutually independent watermark. Recalling Eq. (1) with b = 1, we have,

x = s + αwb

where s is the host signal, x is watermarked signal, w is the watermark, and α is the watermark shaping factor to ensureinaudibility of the embedded watermark.

Also recall that watermark sequence, w, and the host signal, s, are assumed to be mutually independent for all SS-basedwatermarking schemes. We only need to show that the underlying components are non-Gaussian and the observation satisfiesthe full-rank condition. Real-world audio coefficients in DWT domain can be approximated by the Laplacian distribution (seeappendix) thereby meeting the requirement of non-Gaussianity. If a watermark, w, is generated based on some non-Gaussiandistribution then non-Gaussianity of an ICA model is satisfied. The full-rank condition can be relaxed if ICA framework formore sources than sensors [29], [33] is used for watermark estimation. However, in our watermark estimation stage we usea standard ICA model for BSS which requires full-rank constraint along with three other constraints for its identifiability. Inorder to meet the full-rank condition of the observation, at least one observation independent of the received watermarked audiois required. In case of SS-based watermarking, the received watermarked signal is a mixture of two independent components,i.e., host signal and watermark (under zero attack-channel distortion). Therefore, in order to meet the full-rank requirement,an observation independent of the received watermarked audio is generated locally, i.e. at the detector, a linear combinationof the watermarked signal and an independent pseudo-random noise is generated.

Once identifiability conditions of the standard ICA model are satisfied at the watermark detector, both noise free ICAmodel as well as noisy ICA model can extended to solve the watermark detection problem for SS-based watermarking. Beforeapplying an ICA model for watermark estimation, Let us first show the equivalence between SS-based watermark detectionmodel and BSS using ICA.

Lemma:

SS-based watermark estimation model and ICA model are equivalent

Proof: Let s1 be the host signal to be watermarked based on additive SS-based watermarking, then the watermarkedsignal can be expressed as (recalling Eq. 1 with b = 1)

x1 = s1 + α1 ·w (22)

where α1 is the estimated perceptual mask of the host signal, s1, based on the HAS. The observed watermarked signal,x1,at the detector in the presence of additive attack-channel distortion nattack1 can be expressed as

x1 = x1 + nattack1

= s1 + α1 ·w + nattack1 (23)

The locally-generated observation xg , to meet the full-rank requirement for the identifiability of ICA model is given as follow

xg = x1 + ν

= x1 + nattack1 + ν (24)

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where ν is the pseudo-random noise, independent of the received watermarked signal, x1. Here ν can be expressed as,

ν = ε ·w + ν (25)

where 0 ≤ ε ≤ 1.Therefore xg can be expressed as,

xg = s1 + α1 ·w + nattack1 + ν

= s1 + (α1 + ε) ·w + nattack1 + ν

= s1 + α ·w + n (26)

where α = α1 + ε and n = nattack1 + νThe received watermarked signal and locally-generated observation in matrix form,

[x1

xg

]=

[1 α1

1 α

]·[

s1

w

]+

[nattack1

nq

](27)

Let,

x =[

x1

xg

],

A =[

1 α1

1 α

],

s =[

s1

w

], and

n =[

nattack1

ng

]

rewriting Eq. ( 27) we have,x = A · s + n (28)

therefore, Eq. (28) is equivalent to Eq. (19), hence, SS-based watermark estimation model and the noisy ICA model areequivalent. Therefore, an ICA model can be used to solve SS-based watermark estimation problem

In the above proof, we assumed that watermark is embedded into the entire host audio clip (non-segmentation scenario).However, the equivalence between ICA model and SS-based watermarking estimation model will also hold under segmentationbased watermarking scenario. For example, consider the case where same watermark is embedded into q independent non-overlapping segments of the host data, where q ≥ 1. The observation matrix,x, in Eq. (28) for q-times repeated watermarkembedding case, can be expressed as,

x1

...xq

xg

=

α1

Iq

...αq

1 · · · 1 α

·

s1

...sq

w

+

nattack1

...nattackq

ng

(29)

where, Iq is q × q identity matrix, α =∑q

i=1 αi + ε, and ng =∑q

i=1 αattacki + νHere Eq. (29) is still equivalent to Eq. (19), hence ICA models are still extendable for watermark estimation for segmentation-

based watermarking. However, this should be noted that as the value of q (number of repeated watermark embedding factor)increases the watermark estimation performance improves accordingly but at the cost of lower embedding capacity, i.e.,1q × log2(M) bits per frame along with more demanding computation.

A. Watermark Estimation

For watermark estimation, the proposed watermark detector first estimates the watermark-mixing matrix A which is used toestimate the underlying independent components (i.e., the host signal s and the watermark s). An estimate of the watermark-mixing matrix, A, is obtained by optimizing some highly nonlinear function also known as contrast function. The pseudo-inverseof the estimated watermark-mixing matrix A† is applied to the observed mixture to estimate the host signal s and the watermarkw. However, as noted earlier, in the case of blind detectors for SS-based watermarking schemes, watermark estimation usingICA framework is a degenerate case, i.e., m < n1. Therefore, just the estimation of watermark-mixing matrix is insufficientto separate the underlying independent components perfectly. In the case of SS-based watermarking, the equation x = Ashas an affine set of solutions [36]. A preferred solution in this affine set is generally selected using probabilistic prior model

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of the independent components. The performance of the proposed ICA-based watermark estimator depends on the separationquality of the separated (estimated) watermark . The separation quality of the separated source is generally measured in termsof source-to-interference ratio (watermark-to-interference ratio (WIR), in case of watermark estimation), source-to-noise ratioand source-to-artifact ratio (for further details on these separation quality measures please see [37] and references therein). Forperformance analysis of the proposed ICA detector, only WIR distortion measure is considered here; therefore, the estimatedwatermark can be expressed as

wi = η1iαiwib + sinterfi (30)

where η1i ∈ R : 0 ≤ η1i ≤ 1 and sinterfi is interference due to the host signal.Let sinterfi = η2isi : η2i ∈ R : 0 ≤ η2i ≤ 1 then Eq. (30) can be rewritten as,

wi = η1iαiwib + η2isi (31)

The relative distortion due to interference in the estimated watermark is defined as,

Dinterf = (η1/η2)2 (32)

where WIR = 10 log10 (Dinterf)In general, Dinterf for most of the existing BSS schemes that use ICA framework. Several researchers have proposed elegant

BSS algorithms based on ICA model for noisy data [?], [28], [29], [33], [35] and these algorithms can be used for watermarkestimation. However, we have selected the FastICA for noisy data [28] in our implementations due to its better computationalperformance and separation quality over the existing algorithms [19]–[22], [36], [37].

In the following subsections we analyze the performance of the proposed ICA-based detector in terms of three parameters: (a)detection rate in terms of false negatives and true positives, (b), decoding error probability and (c) maximum watermarking rate.These parameters define salient characteristics of a detector: low false negatives and high true positives provide the performanceassessment of a detector in the presence of noise and other digital attacks; low decoding error probability characterizes the abilityof a detector to decode the detected information in the presence of modifications; maximum watermarking rate characterizesthe ability of a detector to extract watermark in the presence of minimum host media information thus allowing to embed largeamount of the data.

B. Watermark Detection: Performance Analysis

A watermark detector is generally characterized by two performance measures: the probability of false alarm PF and theprobability of detection PD. The probability of detection indicates the probability of detecting a watermark when the receivedaudio indeed contains a watermark. The probability of false alarm represents chances of detecting a watermark when in factthe received audio does not contain a watermark. The watermark detection process can be treated as a binary decision problemin the presence or absence of attack-channel distortions.

We first consider the case where the received watermarked audio has not suffered attack-channel distortion. The estimatedwatermark is given by

wi = η1iαiwib + η2isi : i ∈ ζ (33)

In this scenario, the watermark detection can be formulated as hypotheses test:

H1 : wi = η1iαiwib + η2isi

H0 : wi = η2isi : i ∈ ζ (34)

If the probability distribution function, fs(s) , of si is known, then the optimum decision tests are given by the Neyman-Pearson rule [5]. In the watermark detection process, an optimum detector that maximizes the PD for any value of PF isdetermined by

L(w) = lnfw(w|H1)fw(w|H0)

= ln∑

l

(p(bl)fs(w − η1αwbl)

fs(w)

)H1

≷H0

ξ (35)

where, l ∈ {±1}, ξ is a threshold and defined as si = η2isi. Assuming equally probable messages and Laplacian distributionfor random variable si, the log-likelihood function can be written as:

9

L(w) =∑

i∈ζ

βi (|wi| − |wi − η1iαiwi|) (36)

where βi =√

2/η2iσsiIn order to determine the statistical characterization of L(w) under hypothesis H0 and zero attack-

channel distortion, we have:

L(w) =∑

i∈ζ

βi (|wi| − |wi − η1iαiwi|) (37)

L(w) def= βi (qi) (38)

qidef= βi (|wi| − |wi − η1iαiwi|) (39)

Here L(w) is a sum of |ζ| statistically independent random variables that can be approximated by Gaussian random variablebased on the CLT with mean and variance of L(w) is calculated as follows, assuming wi ∈ {±1} with probability 1

2 :

E{L(w|H0)} =∑

i∈ζ

βiE{qi} (40)

m0def= E{L(w|H0)} (41)

Var{L(w|H0)} =∑

i∈ζ

β2i Var{qi} (42)

σ20

def= Var{L(w|H0)} (43)

E{qi|si,H0} = |si| − 12

(|si|+ η1iαi + ||si| − η1iαi|) (44)

Var{qi|si,H0 =14

(|si|+ η1iαi + ||si| − η1iαi|)2(45)

these equation can be rewritten as

E{qi|si,H0} =

{|si| − η1iαi |si| ≤ η1iαi

0 |si| > η1iαi

(46)

Var{qi|si,H0} =

{|s2

i | |si| ≤ η1iαi

η21iα

2i |si| > η1iαi

(47)

Averaging over si, we have

E{qi|H0} = Esi (E{qi|si,H0})=

1

βi

(1− e−βiη1iαi − βiη1iαi

)(48)

Var{qi|H0} = Esi (Var{qi|si,H0})

=1

β2i

(3− e−2βiη1iαi − 2e−βiη1iαi

(1 + 2βiηiαi

))(49)

Substituting E{qi|H0}, and Var{L(qi|H0)} in Eq. (41) and Eq. (43) we have

m0 =∑

i∈ζ

(1− e

− η1i√

2η2iλi − η1i

√2

η2iλi

)(50)

σ20 =

∑

i∈ζ

(3− e

− η1i2√

2η2iλi − 2e

− η1i√

2η2iλi

(1 +

2η1i

√2

η2iλi

))(51)

When hypothesis H1 is true, i.e.,wi = η1iαiwib + si, the log-likelihood function can be written as:

L(wi) = βi (|wi + η1iαiwi| − |wi|) (52)

10

Here L(wi) can be approximated by a Gaussian random variable with the same set of assumptions as under hypothesis H0.In addition, the distribution of L(wi) under hypothesis H1 is symmetrical to the distribution of under H0 with respect to theorigin. Therefore,

m1def= E{L(w)|H1}= −E{L(w)|H0} (53)

σ21

def= Var{L(w)|H1}= Var{L(w)|H0} (54)

The probabilities of false alarm PF and detection PD are given as:

PF = Q

(ξ + m1

σ1

)(55)

PD = Q

(ξ −m1

σ1

)(56)

If we define the watermark-to-noise ratio (WNR1)

WNR1 def=m2

1

σ21

(57)

If we denote Q−1(PF ) the value x ∈ R such that Q(x) = PF then receiver operating characteristics (ROC) of the proposeddetector is given as:

PD = Q(Q−1(PF )− 2

√WNR1

)(58)

Eq. (58) shows that the detection performance of the proposed detector exclusively depends on the WNR1. As the proposedICA-based detector cancels the host-signal interference before detection, the ICA-based detector is expected to perform betterthan the traditional blind detectors [36], [37] operating without reducing the host-signal interference for watermark detectionprocess. Fig. 3 gives the theoretical ROC performance of the proposed ICA-based detector for different values of WIR andalong with ROC performance of an optimal detector proposed in [17], [18]. We can see from Fig. 3 that the ROC performanceof the proposed ICA-based detector is better than the ROC performance of the optimal detector proposed in [17], [18], andthis improvement is a function of WIR in the estimated watermark. This should be noted that the ROC performance of theproposed ICA-based detector approaches to the ROC the optimal detector proposed in [17], [18] as host-signal interference inthe estimated watermark increases and at 0 dB WIR the proposed detector performs as the optimal detector proposed in [17],[18].

10−15

10−10

10−5

100

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PF

PD

P. Gonzalex et al’s Detector [17]WIR = 6.0 dBWIR = 14.0 dBWIR = 20.0 dB

Fig. 3. ROC Performance of the Proposed Detector and the Detector proposed in [17], [18], for different values of WIR, WSR = 13 dB, and Number ofAudio Samples per Bit, i.e. |ζ| = 5 (theoretical values)

C. Watermark Decoding: Performance AnalysisIn order to evaluate the performance of the proposed detector in terms of decoding error probability, let us consider SS-based

watermark embedding and decoding framework discussed in Section 2. Consider the case when watermarked sample xi isobtained by adding watermark αiwib. In this case, the decoding error probability for the estimated watermark wi (given byEq. ( 31)) can be expressed as:

Pe ICA =12e−{ η1i

η2i}(√

2/λi) (59)

11

Eq. (59) shows that ICA-based detector improves decoding error performance compared to the optimal decoder that doesnot reduce the host-signal interference at the detector (discussed in Section 2). The decoding error performance gain for theproposed ICA-based watermark detector over the optimal decoder proposed in [17] is given as:

G =Pe

Pe ICA= e

−√

2λi{1− η1i

η2i} (60)

Note that, in general, BSS schemes using ICA have relatively small interference distortion or large WIR value, i.e., η1i/η2i

[36], therefore, G ≥ 0 for WIR > 0dB. The improvement in the decoding error probability due to ML decoder used in theproposed ICA-based detector over that of the optimal decoder given by Eq. (7) is plotted in Fig. 4, for different values ofWIR and SWR. It shows that for a fixed value of SWR, decoding error probability of the proposed detector improves with theincrease in the separation quality of the ICA scheme used for watermark estimation.

0 2 4 6 8 10 12 14 16 18 20

100

101

102

103

104

105

106

107

108

WIR (dB)

Gai

n: G

SWR = 0.0 dBSWR = 20.0 dBSWR = 13.0 dBSWR = 6.0 dB

Fig. 4. Decoding Bit Error Probability Performance Improvement due to Host-Signal-Interference Cancelation at the Detector (theoretical values)

Let us now consider the case when an antipodal binary information bit b is embedded into the selected indices ζ of the hostaudio s, i.e., watermarked audio signal is given as:

xi = si + αiwib : i ∈ ζ (61)

In this case, the estimated watermark w using proposed ICA-based watermark detector, under zero attack-channel distortion,can be expressed as,

wi = η2isi + η1iαiwib : i ∈ ζ (62)

If we assume that all hidden messages have same priori probability, then an ML decoder will minimize the decoding errorprobability satisfying the following condition:

lnf(w|b+)fw(w|b−)

= lnfs(w − η1αwb+)fs(w − η1αwb−)

> 0 (63)

The ML sufficient statistics assuming s obeys Laplacian distribution and can be written as:

r =∑

i∈ζ

βi (|wi + η1iαiwi| − |wi − η1iαiwi|) (64)

if b = 1 was embedded, then r can be expressed as

r =∑

i∈ζ

βi (|si + 2η1iαiwi| − |si) (65)

r def=∑

i∈ζ

βizi (66)

zi = (|si + 2η1iαiwi| − |si) (67)

The ML decoder is a bit-by-bit hard decoderb = sgn(r) (68)

In order to determine the bit error probability for this ML decoder, a statistical characterization of r is required. Here r issum of |ζ| i.i.d. random variables. Therefore, by applying CLT, r can be approximated by a Gaussian random variable, with

12

mean and variance computed as:

E{r} def=∑

i∈ζ

βiE{zi} (69)

Var{r} def=∑

i∈ζ

β2i Var{zi} (70)

In zi, variables wi and si are the only random variables and if wi ∈ {±1} with probability 12 , then by integrating zi over

zi condition to the selected host indices si : i ∈ ζ we have:

E{qi|si} =12

(|si|+ 2η1iαi + ||si| − 2η1iαi|)− |si| (71)

Var{L(qi|si,H0)} =14

(|si|+ 2η1iαi + ||si| − 2η1iαi|)2 (72)

these equation can be rewritten as

E{zi|si} =

{−|si|+ 2η1iαi |si| ≤ 2η1iαi

0 |si| > 2η1iαi

(73)

Var{zi|si} =

{|s2

i | |si| ≤ 2η1iαi

4η21iα

2i |si| > 2η1iαi

(74)

Averaging over si, we have

E{zi} = Esi (E{zi|si})=

1

βi

(e−2βiη1iαi + 2βiη1iαi − 1

)(75)

Var{zi} = Esi (Var{zi|si})

=1

β2i

(3− e−4βiη1iαi − 2e−2βiη1iαi

(1 + 4βiηiαi

))(76)

Substituting E{zi}, and Var{zi} in Eq. (70) and Eq. (70) we have

E{r} =∑

i∈ζ

(e− η1i2

√2

η2iλi +2η1i

√2

η2iλi− 1

)(77)

var{r} =∑

i∈ζ

(3− e

− η1i4√

2η2iλi − 2e

− η1i2√

2η2iλi

(1 +

4η1i

√2

η2iλi

))(78)

Thus decoding bit error probability for an ICA-based detector is given as:

Pe ICA = Q

(|E{r}|√Var{r}

)(79)

Eq. (77), (78) and (79) show that the decoding error probability of the ML decoder applied to the estimated watermark isa function of WIR and SWR.

The decoding error probability performance of the ML decoder used for the proposed ICA-based detector is compared againstdecoding error probability performance of the ML decoder operating without reducing the host-signal interference given byEq. (17). The performance of the proposed ICA-based detector given by Eq. (79) and the optimal decoder given by Eq. (17)for different values of WIR and SWR is plotted in Fig. ?? and 5.

The analytical results shown in Fig. 5 depict that the proposed detector exhibits superior decoding error probability comparedto the existing optimal detectors [7]–[14]. This superior performance can be attributed to the host-signal interference cancelationcapability of the proposed ICA-based detector. Note that the decoding performance of the proposed ICA-based detector is afunction of the WIR performance of the BSS scheme used for watermark estimation. When WIR = 0 dB in the estimatedwatermark, the proposed detector is equivalent to the optimal detector proposed in [17], [18]. The proposed ICA-based detectoris expected to show similar improvements in terms of decoding error probability when compared with traditional watermarkdetectors that operate without reducing the host-signal interference.

13

0 2 4 6 8 10 1210

−300

10−250

10−200

10−150

10−100

10−50

100

WSR (dB)

Pe

P. Gonzalez et al’s Detector [17]WIR = 6.0 dBWIR = 14.0 dBWIR = 20.0 dB

0 2 4 6 8 10 1210

−300

10−250

10−200

10−150

10−100

10−50

100

WSR (dB)

Pe

P. Gonzalez et al’s Detector [17]WIR = 6.0 dBWIR = 14.0 dBWIR = 20.0 dB

Fig. 5. Decoding Bit Error Probability Performance of the Proposed ICA-based Detector for different values of WIR and Number of Audio Samples per Bit,i.e. |ζ| = 5 (left) , 10 (right) (theoretical values)

D. Maximum Watermarking-Rate: Performance Analysis

The maximum watermarking-rate is another watermarking performance measure which indirectly depends on the detectorstructure. In order to evaluate the maximum watermarking-rate performance of the proposed ICA-based watermark detectoragainst informed and blind watermark detectors, we assume a watermarked sample is obtained by adding αiib to the host-audiocoefficient si. Let the received watermarked sample is corrupted by independent additive white Gaussian noise, with zero meanand σ2

nivariance. In this case, the received watermarked sample xi can be approximated by a Gaussian random variable based

on the CLT. Therefore, based on the independence of s,w, andn, variance of the received watermarked signal xi (given byEq. (4)) is given as:

σ2xi

= σ2wi

+ σ2si

+ σ2ni

(80)

In this case, the estimated watermark sample at the detector is given as:

wi = η1iαiwi + η2isi + η3ini (81)

where η3ini is noise contribution (interference) in the estimated watermark, and η3i ∈ R : 0 ≤ η3i ≤ 1.By applying CLT the watermark estimate wi can be approximated by zero-mean Gaussian random variable with variance

given as:

σ2wi

= η21iσ

2wi

+ η22iσ

2si

+ η23iσ

2ni

(82)

The maximum watermarking-rate of SS-based watermarking using blind watermark detector, operating without reducing thehost-signal interference, can be calculated by the capacity of an additive white Gaussian noise channel [3], i.e.

RCor =12

log2

(1 +

σ2wi

σ2si

+ σ2ni

)(83)

Similarly, maximum watermarking-rate of SS-based watermarking using an informed detector can be expressed as:

RInformed =12

log2

(1 +

σ2wi

σ2ni

)(84)

And maximum watermarking-rate of SS-based watermarking using proposed ICA-based watermark detector can be expressedas:

RICA =12

log2

(1 +

η21iσ

2wi

η22iσ

2si

+ η23iσ

2ni

)(85)

Since the ICA scheme used for watermark estimation has reasonably good source separation performance i.e. low interferenceand noise distortions in the estimated watermark [36], [37], therefore

σ2wi

σ2ni

≤ η21iσ

2wi

η22iσ

2si

+ η23iσ

2ni

≤ σ2wi

σ2si

+ σ2ni

(86)

⇒ RCor ≤ RICA ≤ RCor (87)

14

Hence, if we assume that the estimated watermark has very low interference due to other independent sources (host signaland attack-channel noise), then theoretically speaking the proposed watermark ICA-based detector can achieve the maximumwatermarking-rate performance close to informed watermark detector as interference due to host signal in the estimatedwatermark approaches to zero.

The above results show that the proposed ICA-based watermark detector performs better than the conventional watermarkdecoders and detectors [?], [1]–[3], [7]–[14] that operate without reducing the host-signal interference.

V. CONCLUSION

An improved watermark detector for SS-based watermarking is presented in this paper. The proposed watermark detector iscapable of canceling the host-signal interference at the watermark detector using ICA framework. Bind watermark detection,lower host-signal interference at the detector, improved decoding, detections and watermarking-rate performances are thesalient features of the proposed ICA-based watermark detector. The proposed ICA-based detector can be used for SS-basedwatermarking for all types of multimedia data, e.g., audio, video, images, etc. The theoretical results show that the proposeddetector performs significantly better than the existing blind detectors used for SS-based watermarking. The simulation resultsfor the proposed detector using real-world audio clips can be accessed via [42]. Currently we are investigating the performanceof the proposed ICA-based detector for SS-based image and video watermarking. We are also looking forward to use theproposed ICA-based detector for multimedia fingerprinting for secure multimedia distribution on the Web and multimediausage tracking applications.

APPENDIX ISTATISTICAL CHARACTERIZATION OF THE WAVELET COEFFICIENTS OF AUDIO SIGNALS

To determine the statistical characterization of the subband coefficients of the real-world speech samples, the speech samples,yii = 0, 1 · · ·N − 1 are assumed to be i.i.d. Laplacian r.v. with mean zero and variance σ2

yi. The one-dimensional discrete

wavelet transform (DWT) of audio signal, y, can be calculated using Mallat’s algorithm [16]. The DWT coefficients usingMallat’s algorithm [16], e.g., approximate coefficients ak and detailed coefficients dk, at different scales can be expressed as[41],

aj−1i =

N1−1∑

k=0

hk−2iajk (88)

dj−1i =

N1−1∑

k=0

gk−2iajk (89)

where j denotes the resolution and i is the index.Eq. (88) and (89) describe linear filtering operation using filters h and g followed by down-sampling. Here h and g are

finite impulse response (FIR) quadrature-mirror filters, also known as the scaling and the wavelet filters, respectively. Thescaling filter is a lowpass filter, while the wavelet filter is a highpass filter. Moreover, the top-level coefficients aJ representthe original signal y. Eq. (88) and (89) can be expressed using a single equation,

sj−1i =

N1−1∑

k=0

δk−2isjk (90)

where δi is the weighting factor depending on the filter coefficients hi and gi, i.e. approximate coefficients or detailedcoefficients, and sj

i is the wavelet coefficient at jth-level.Here Eq. (90) states that a wavelet coefficient at an arbitrary level j− 1, is a weighted sum of N1 wavelet coefficients from

jth-level wavelet. The wavelet coefficients at J − 1 level can be expressed as

sJ−1i =

N1−1∑

k=0

δk−2isJk (91)

According to Eq. (90), each wavelet coefficient an arbitrary level j : 1 ≤ j ≤ J−1 is a weighted sum of i.i.d. r.v. (e.g. audiosamples in our case), therefore, the pdf of a wavelet coefficient sj

i at jth-level, can be determined using joint characteristicfunction Φsj

i(ω). If we assume that audio sample yi, is a Laplacian r.v., then pdf of yi can be expressed as,

fy(y) =γ

2e−γ|y| : |y| < ∞ (92)

where γ =√

2σ2

y

Here characteristic function of r.v. yi, Φyi(ω), can be expressed as [6],

15

Φyi(ω) =

γ2i

ω2(93)

Let us consider a r.v. z which is obtained by magnitude scaling of a r.v. y i.e., z = δy, the characteristic function of z,Φz(ω), in terms of Φy(ω) can be expressed as [6],

Φz(ω) = Φz(δω) (94)

Therefore, the characteristic function of r.v. sJ−1i ,ΦsJ−1

i(ω), can be expressed as

ΦsJ−1i

(ω) =N1∏

k=1

Φyk(δk−2iω) (95)

=N1∏

k=1

γ2i(

γ2i + (δk−2iω)2

) (96)

=N1∏

k=1

γ2i

(γ2i + ω2)

(97)

where γi = γi/γk−2i and N1 is the length of the wavelet filter.In order to determine the pdf of wavelet coefficients sJ−1

i , fsJ−1i

(s) characteristic function ΦsJ−1i

(ω) (given by Eq. (97))is used. The pdf of a r.v. can be determined either using the uniqueness theorem or the convolution theorem [6]. The pdf ofwavelet coefficients sJ−1

i , fsJ−1i

(s) using ΦsJ−1i

(ω) based on the convolution theorem can be expressed,

fsJ−1i

(s) =γi

2e−γi|t|

(N1−1∑

k=0

cki γk

i tk

)(98)

where ck ∈ R is a real constant.For different values of N1, the polynomial coefficients, ck, are given as:N1 = 2, c0 = c1 = 1

2 , and N1 = 3, c0 = c1 = 38 , and c1 = 1

8 and so on.According to the Eq. (97) and (98), as the pdf of wavelet coefficients at jth level, fsj−1

i(s), is a obtained by convolving

the pdf of r.v. yi, therefore, based on the CLT, the pdf of the subband coefficients move to words Gaussianity as value ofN1 increases or in other words, pdf of wavelet coefficients at coarser level is closer to the Gaussianity than higher levelcoefficients. This is because at coarser level, for each wavelet coefficient more audio samples contribute in the weighted-sumequation (given by Eq. (91)) than higher level coefficients.

To verify this fact, a 4–level DWT decomposition of an arbitrary frame of the music clip ’I Want It That Way· · · ’ by”Backstreet Boys”, using ’Daubechies-8’ decomposition filter, is given in Fig. ??. The pdf (based on histogram approximation)of corresponding wavelet coefficients at different levels is plotted in Fig. ??. This is clear from Fig. ?? that the higher level,wavelet coefficients exhibit non-Gaussian distribution and distribution moves towards Gaussianity for coarser coefficients dueto longer weighted-sum effect at the coarser level.

Therefore, the pdf of each subband coefficient (at higher level) of the host signal, si, can be approximated by Laplaciandistribution, which is given as,

fsi(s) =βi

2e−βi|si| : |si| < ∞ (99)

where βi =√

2σ2

si

REFERENCES

[1] I. J. Cox, M. L. Miller, and J. A. Bloom, Digital Watermarking, Morgan Kaufmann, 2001.[2] M. Wu, and B. Liu, Multimedia Data Hiding, Springer-Verlag, 2002.[3] J. Eggers, and B. Girod, Informed Watermarking, Kluwer Academic Publisher, 2002.[4] R. E. Zwicker, and H. Fastl, Psychoacoustics: Facts and Models, Springer-Verlag, Berlin, 1999.[5] H. V. Poor, An Introduction to Signal Detection and Estimation, Springer-Verlag, 2nd–ed., 1994[6] A. Papoulis, and S. U. Pillai, Probability Random Variables and Stochastic Processes, McGraw-Hill, New York, 4thed., 2002.[7] I.J. Cox, J. Kilian, T. Leighton, and T. Shamoon, Secure Spread Spectrum Watermarking for Multimedia, IEEE Trans. Image Processing, vol. 6(12), pp.

1673–1687, 1997.[8] R. B. Wolfgang, C. I. Podilchuk, and E. J. Delp, Perceptual Watermarks for Digital Images and Video, Proc. of IEEE, vol. 87(7), pp. 1108–1126, July,

1999.[9] M. D. Swanson, B. Zhu, A. H. Tewfik, and L. Boney, Robust Audio Watermarking using Perceptual Masking, Signal Processing, Elsevier Science, vol.

66, pp. 337–355, 1998.[10] P. Bassia, and I. Pitas, Robust Audio Watermarking in the Time Domain, Proc. Int. Conf. Acoustics Speech and Signal Processing (ICASSP’99), 1999.[11] C.-P. Wu, P.-C. Su, and C.-C. J. Kuo, Robust Audio Watermarking for Copyright Protection, SPIE’s 44th AMASPAAI’99, 1999.

16

2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 1 2 0 0

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cA1

cD1

cD2

cD3

cA2

cA3

cA4

Y: Signal cD4

Level 1

Level 2

Level 3

Level 4

Detail Coefficients

App. Coefficients

cA1 cD

1

cD2

cD3

cA2

cA3

cA4

Y: SignalcD

4

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Level 2

Level 3

Level 4

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100

200

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50

100

150

200

250

300

350

400

-0.6 -0.4 -0.2 0 0.2 0.4 0.60

50

100

150

200

250

Fig. 6. Distribution of Detail and Approximate Coefficients at each level of 4-Level Wavelet Decomposition of an Audio Signal, y, based on HistogramApproximation

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