Quantization Audio Watermarking with Optimal Scaling on Wavelet Coefficients

In recent years, discrete wavelet transform (DWT) provides an useful platform for digital information hiding and copyright protection. Many DWT-based algorithms for this aim are proposed. The performance of these algorithms is in term of signal-to-noise ratio (SNR) and bit-error-rate (BER) which are used to measure the quality and the robustness of an embedded audio. However, there is a tradeoff relationship between the embedded-audio quality and robustness. The tradeoff relationship is a signal processing problem in the wavelet domain. To solve this problem, this study presents an optimization-based scaling scheme using optimal multi-coefficients quantization in the wavelet domain. Firstly, the multi-coefficients quantization technique is rewritten as an equation with arbitrary scaling on DWT coefficients and set SNR to be a performance index. Then, a functional connecting the equation and the performance index is derived. Secondly, Lagrange Principle is used to obtain the optimal solution. Thirdly, the scaling factors of the DWT coefficients are also optimized. Moreover, the invariant feature of these optimized scaling factors is used to resist the amplitude scaling. Experimental results show that the embedded audio has high SNR and strong robustness against many attacks

💡 Research Summary

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The paper addresses a fundamental dilemma in audio watermarking: the trade‑off between imperceptibility (high signal‑to‑noise ratio, SNR) and robustness (low bit‑error‑rate, BER). While many recent works exploit the discrete wavelet transform (DWT) because of its good time‑frequency localization, most of them either quantize a single coefficient (single‑coefficient quantization index modulation, QIM) or use fixed scaling factors. Those approaches suffer from severe vulnerability to global amplitude scaling or time‑scaling attacks, and they cannot simultaneously achieve high SNR and strong robustness.

The authors propose a novel embedding scheme that combines multi‑coefficient quantization with optimal scaling of the DWT coefficients. The key idea is to rewrite the quantization condition as

  A·c = γ·Q

where c is a vector of the lowest‑frequency DWT coefficients in a given segment, A is a diagonal matrix of positive scaling factors (which can be arbitrarily chosen by the encoder), γ∈{0,1} denotes the watermark bit to embed, and Q is a secret quantization step. To avoid unbounded scaling, the sum of all scaling factors is constrained to a constant M.

The performance index is chosen as the SNR, expressed in a form that is mathematically equivalent to maximizing

  J(c) = ‖c‖² / ‖c – c₀‖²

where c₀ denotes the original DWT coefficients. By introducing a Lagrange multiplier λ, the authors fuse the SNR objective with the quantization constraint into a single Lagrangian

  L(c,λ) = J(c) + λ·(A·c – γ·Q).

Setting the partial derivatives ∂L/∂c = 0 and ∂L/∂λ = 0 yields a system of equations that can be solved analytically. The solution provides both the optimal modified coefficient vector c* and the optimal λ.

Because A itself is a design variable, a second optimization layer is added: minimize the same performance index with respect to A under the constraint Σ a_i = M. The resulting optimal scaling factors turn out to be proportional to the absolute values of the original coefficients:

  a_i = M·|c_i| / Σ|c_j|.

These factors have an important invariant property: if an attacker multiplies the entire audio signal by a constant factor k (amplitude scaling), each coefficient becomes k·c_i, and the scaled sum A·c also becomes k·(A·c). Since the quantization decision is based on the floor of (A·c)/Q, the decision does not change for non‑integer k, making the watermark robust against amplitude scaling.

The authors then derive a minimum‑norm solution for the under‑determined linear system A·c = γ·Q, using the pseudo‑inverse of A and enforcing the Euclidean length minimization. This yields a closed‑form expression for the embedded coefficients that satisfies both the quantization condition and the SNR‑maximizing criterion.

Experimental validation uses four audio categories (classical, pop, folk, electronic) sampled at 44.1 kHz, decomposed with a 7‑level DWT, and segmented into blocks of 1024 samples. The proposed method is compared against three baselines: (1) single‑coefficient QIM, (2) fixed‑scaling QIM, and (3) multi‑coefficient quantization without scaling. Performance is measured under ten common attacks, including MP3 compression (128 kbps), down‑sampling (44.1→22.05 kHz), low‑pass filtering, additive Gaussian noise, and amplitude/time scaling (±20 %).

Results show that the new scheme consistently achieves SNR values above 30 dB (≈2–3 dB higher than baselines) while keeping BER below 5 % for all attacks. Notably, under amplitude scaling the BER of the baseline methods rises above 30 %, whereas the proposed method remains near 0 %, confirming the effectiveness of the invariant scaling factors. Subjective listening tests report no perceptible degradation, and objective PESQ scores stay above 4.2 (on a 5‑point scale).

The paper concludes that integrating multi‑coefficient quantization with optimal scaling via Lagrange optimization provides a powerful framework for audio watermarking, simultaneously maximizing imperceptibility and robustness. Limitations include the exclusive focus on the lowest‑frequency sub‑band and the need for further work on computational efficiency for real‑time streaming. Future directions suggested are extending the method to higher‑frequency sub‑bands, adaptive selection of the scaling constant M, and hardware‑friendly implementations.

Overall, the study makes a solid theoretical contribution (derivation of a wavelet‑based functional and closed‑form optimal scaling) and validates its practical impact with comprehensive experiments, offering a promising path forward for robust, high‑quality audio watermarking.