An Adaptive Audio Watermarking Scheme Method Based on Kernel Fuzzy C-means Clustering

Digital audio watermarking technique provides efficient tools for ensuring that product ownership of audio multimedia is preserved, even if multimedia data is attracted by attackers. For an audio watermarking system, imperceptibility and robustness are its two basic requirements. Imperceptibility refers to the perceptual quality of the data being protected. For audio data, digital watermark should be inaudible. The digital watermark should also be robust to audio signal processing. Ideally, the amount of signal distortion necessary to remove the watermark can degrade the desired audio quality to point of becoming commercially valueless. Typical audio processing operations include lossy compression (such as MP3), additive noise, filtering, resampling, etc. Currently, many watermarking techniques have been developed, such as quantization-based techniques, relationship-based techniques, physiological model-based techniques, adaptive techniques, etc.

Recently, some machine learning methods, such as neural networks and support vector machines, are introduced into digital audio watermarking technique, and can simultaneously improve robustness and audible quality of the watermarked audio signal. In [1], neural networks were introduced into a nonblind audio watermarking scheme, which was used to estimate the watermark scaling factor intelligently from the host audio

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signal, and make the watermark detection more robust against common attacks. Wang et al.[2] introduced SVM for audio watermark detection in DWT domain, which considered the watermark extraction as a two-class problem. In [3], Serap et al. presented an audio watermark decoding process based on SVM, which combined the watermark decoding and detection problems into a single classification problem. Xu et al. [4] proposed a DWT-based audio watermarking method using support vector regression (SVR) which was used to learning the correlation between the sub-audios obtained by sub-sampling technique. Peng et al.[5] proposed an audio watermarking method in multiwavelet domain based on SVM, where SVM was used to learn nonlinear relationship between local audio features in order to extract watermark signal.

In this paper, we propose an adaptive audio watermarking scheme. Firstly, we divide audio signal into audio frames, and these audio frames are transformed into DWT domain respectively. Secondly, we extract its local features for each audio frame, that is, its local energy and the maximal peaks of its all subbands. Next, through running kernel fuzzy c-means (KFCM) clustering algorithm on these features, we obtain the maximal fuzzy membership degree for each audio frame. According the fuzzy membership degree, we can select the audio frames to be embedded watermark and determine the embedding strength of each audio frame adaptively. The proposed method can simultaneously improve the robustness and audible quality of the watermarked audio signal.

The rest of this paper is organized as follows: Section II introduces the concept of kernel fuzzy c-means clustering algorithm. In Section III, we present the proposed audio watermarking approach. Simulation results for several watermarked audio manipulations are presented in Section IV. Finally, we draw our conclusions in Section V.

• 2.    Kernel Fuzzy C-means Algorithm

Given an unlabeled data set X ={ x 1 , x 2 ,…, x m } ⊆ R^d , and a nonlinear mapping φ : R^d → F from this input space to a high-dimensional feature space F . By applying the nonlinear mapping φ , the dot product x_i ⋅ x j in the space is mapped to φ ( x i ) ⋅φ ( x j ) in the feature space. The key notion in kernel-based learning is that the mapping φ need not be explicitly specified. The dot product φ ( x_i ) ⋅φ ( x j ) in the high-dimensional feature space can be calculated through the kernel function K ( x i , x j ) in the input space. The kernel fuzzy c-means algorithm in the feature space F by a mapping φ minimizes the function J_r [6],[7]:

cm

J r ( X ) = ∑∑ ( µ ij ) ^r || φ ( x j ) - ν i ^φ ||² i = 1   j = 1

where µij is the membership degree of data point xj to the ith fuzzy cluster, and r is a fuzziness coefficient. The ith cluster centroid is νiφ =ni-1∑mj=1(µij)rφ(xj) and ni=∑mj=1(µij)r

The key notion in the kernel fuzzy c-means algorithm lies in the calculation of the distance in the feature space. The distance between φ(xj) and νiφ in the feature space is calculated through the kernel in the input space:

II Ф(Xj) -vt II2

^{ф (} x j ) • ^{ф (} x j ) ^- 2 Ф ⁽x j ) •

Е mm=1(^ik) rф( xk) е mj m.) r

E m i⁽ M ik ) r ^{ф (} x k )   Е mi- 1⁽ Mu ) r ф ⁽ x i )

+---

е m = i ( M ik ) r        е m j M il ) r

= ^ф ( x j ) • ^ф ( x j ) -

2E m=1(Mik) r ф(xk) •ф( xj-)

е mm = i ( M ik ) r

+ Е m=1 E L(Mik) r(Mu) rф( xk) •ф( x)

= K ( x j , xj) -

( Е m = i ( M ik ) r ) 2

² E m j M ik ) r K ( X k , X j )

е m = i ( M ik ) r

+ е "=1 е mj Mik) r (Mil) rK (Xk, xi)

( Е m = = i ( M ik ) r )²

,

Where K ⁽ x_k , X i ) = ф ⁽x_k ) • ф ⁽X i ) By using n₍ = E mm _{= 1}( M ij ) r , we have

2               2m ll ^xXj)-vfii = K(Xj, Xj) - n~ EM)) k(Xk, Xj)

mm

⁺ EE ⁽ M k ) r ( Mil ) r ^{K ( x} ) , X )

n i k =1 i =1

Therefore, the objective function can be rewritten as follows:

cm

J t ( X ) = EE ( M ij ) T ( K ( X j , X j )

i =1 j =1

mm

• -    ~EE ⁽ M ik ) T ( M ii ) T ^K ( ^X k , ^X i ) n i   k =1 i =1

  
  
  
  

The kernel fuzzy c-means algorithm iteratively updates the new membership degree M ij at each iteration. The update of M ij in the feature space is defined through the kernel in the input space as follows:

M j

Г

c

E k=1 V

( II Ф (X j ) - v f II² V^( X j ) - v k II²

1/( r -1) A ¹

From (3), the kernel fuzzy c-means algorithm does not need to calculate the cluster centroids because the centroid information is considered in updating the membership degree M ij ■

The kernel fuzzy c-means algorithm can be summarized as follows:

Step 1: Fix c , t _ma x , r > 1 and e > 0 for some positive constant.

Step 2: Initialize the memberships µ _i ⁰ _j .

Step 3: For $t=1,2, , do: