An Adaptive Embedding Approach for High Imperceptible and Robust Audio Watermarking Using Framelet Transform and SVD

Abstract

This paper presents a high imperceptible and robust audio watermarking algorithm by optimizing the scaling parameter as per the imperceptible requirements with minimum bit error rate. The spread spectrum-based audio watermarking is used in this paper, where the scaling parameter is optimized to achieve trade-off between imperceptibility and robustness of watermarked audio. Primarily, the effect of scaling parameter on the perceptual quality of watermarked audio is investigated and the total error introduced in the host audio due to embedding the watermark is computed. The scaling parameter is optimized for maximizing the robustness by considering objective difference grade (ODG) score (for music signals), perceptual evaluation of speech quality (PESQ) score (for speech signals) as a constraint to meet the imperceptibility requirements. To solve this proposed optimization problem, two search algorithms are developed. The embedding is performed in low-pass framelet transform coefficients through SVD with the optimized scaling parameter. The experimental results show that the proposed algorithm achieves good imperceptibility with an average ODG score of −0.32 and PESQ score of 3.86 for music and speech signals, respectively, under various payload conditions. The proposed algorithm shows better robustness to the common signal processing attacks such as noise addition, filtering, resampling, MP3 compression, amplitude scaling, cropping, and requantization.

Appendices

Appendix A: Effect of Additive White Gaussian Noise

In the proposed algorithm, the watermark is extracted by computing the average of non-diagonal elements of the matrix $[D_w]$ and by using the decision logic in Eq. (17). It can be observed that, the extraction of watermark depends upon the orthonormal matrices $U_1$ and $V_1$. If the singular valued matrix $S_r$ at the extraction is invariant then the watermark can be extracted with minimum error.

The effect of AWGN on singular values is modeled intuitively as follows. Assume that, AWGN follows independent and identically distributed (i.i.d) random process.

$$\begin{aligned} y=x_w+n, \end{aligned}$$

(A.1)

where $x_w$ is the watermarked signal, n is the White Gaussian noise, and y is the noisy signal. As $L_2$ norm is preserved, the energy of watermarked signal $E_w$ can be written as,

$$\begin{aligned} \Vert x_w\Vert ^2=\Vert c\Vert ^2=\Vert USV^T\Vert ^2=\Vert S\Vert ^2, \end{aligned}$$

(A.2)

where c indicates the framelet transform coefficients, U and V are unitary matrices.

By using the property of i.i.d for AWGN, the energy of noisy signal can be approximated as,

$$\begin{aligned} \begin{aligned} E_y&=E_w+E_n\\ \Vert S_r\Vert ^2&=\Vert S\Vert ^2+\Vert n\Vert ^2. \end{aligned} \end{aligned}$$

(A.3)

This shows that, AWGN effects the singular values at the extraction side. If the energy of watermarked signal is high when compared to noise energy then the singular values will be invariant to the AWGN attack.

From Eq. (16), it can be observed that the energy of watermarked signal depend on the scaling parameter $\alpha$. Therefore, by maintaining good SNR at the embedding side, the effect of AWGN on watermarked signal can be minimized.

Appendix B: Effect of Amplitude Scaling

The effect of amplitude scaling on the extraction of watermark is discussed here. Consider the watermarked signal x(t) and its framelet transform coefficients are obtained by Eq. (3)

$$\begin{aligned} c_{j}(k)=\int x(t)2^{j/2}\phi (2^{j}t-k) \hbox {d}t \end{aligned}$$

(B.1)

The coefficients are arranged in a matrix [A] and SVD is performed to decompose into [U], [S], [V] matrices as below:

$$\begin{aligned} A=U\times S\times V^{T} \end{aligned}$$

(B.2)

If the amplitude of watermarked signal is scaled by a factor of $\beta$ then the corresponding framelet coefficients are also scaled by the factor of $\beta$ and is shown below:

$$\begin{aligned} \begin{aligned} c^\prime _{j}(k)&=\int \beta (x(t))2^{j/2}\phi (2^{j}t-k) \hbox {d}t\\ c^\prime _{j}(k)&=\beta \int x(t)2^{j/2}\phi (2^{j}t-k) \hbox {d}t \end{aligned} \end{aligned}$$

(B.3)

The coefficients are arranged in a matrix [B] and its SVD can be expressed as

$$\begin{aligned} B=U\times \beta (S)\times V^{T} \end{aligned}$$

(B.4)

This shows that, the decision rule in Eq. (17) doesn’t gets effected due to the amplitude scaling attack. Hence, the recovery of watermark with minimum error is possible.