# Blind Audio Watermarking Based on Parametric Slant-Hadamard Transform and Hessenberg Decomposition

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## Abstract

**:**

## 1. Introduction

## 2. Related Research

## 3. Background Information

#### 3.1. Parametric Slant-Hadamard Transform (PSHT)

#### 3.2. Hesssenberg Decomposition (HD)

## 4. Proposed Watermarking Algorithm

#### 4.1. Watermark Preprocessing

Algorithm 1: Watermark Preprocessing |

Variable Declaration: |

W (i = 1, 2, …., M; j = 1, 2, …., M): the watermark image |

$y\left(i+1\right)\left(i=\text{}1,\text{}2,\text{}\dots ,\text{}M\times M\right)$: logistic mapping parameter |

a, b: real parameters |

$z\left(i\right)\left(i=\text{}1,\text{}2,\text{}\dots ,\text{}M\times M\right):$ binary sequence |

$T:$ predefined threshold value |

$r\left(i\right)\left(i=\text{}1,\text{}2,\text{}\dots ,\text{}M\times M\right):$ new one dimensional sequence from Wi |

$u\left(i\right):$ encrypted watermark sequence |

Watermark Preprocessing Procedure: |

Let $y$(1)∈ (0,1) |

for i = 1: M do |

calculate $y\left(i+1\right)$ using Equation (7) |

calculate $z\left(i\right)$ using Equation (8) |

calculate $u\left(i\right)$ using Equation (9) |

end for |

return encrypted watermark sequence |

#### 4.2. Watermark Embedding Process

_{i}is converted into two-dimensional matrix ${C}_{i}$ of size m×m, where i represents the frame number.

_{j}and ${Z}_{j}$ denotes the absolute mean of the ${j}^{th}$ block.

_{max}= max{Z

_{1}, Z

_{2}, Z

_{3}, …,E

_{N}} of the blocks {B

_{1}, B

_{2}, B

_{3}, …, B

_{N}}, where max operation returns the largest value in {Z

_{1}, Z

_{2}, Z

_{3}, …, Z

_{N}}.

_{i}denotes the orthogonal matrix and H

_{i}denotes the Hessenberg matrix.

_{i}is calculated using the following equation:

_{i}. Watermark is embedded using the following rule:

_{i}, 2) = 0, the following equation is used:

_{i}, 2) = 1, the following equation is used:

Algorithm 2: Watermark Embedding |

Variable Declaration: |

Y: host audio signal |

F: segmented non-overlapping frame |

${C}_{i}\left(i=\text{}1,\text{}2,\text{}\dots ,\text{}M\times M\right):$ frame represented in dimensional matrix with size m×m |

${T}_{i}\left(i=\text{}1,\text{}2,\text{}\dots ,\text{}M\times M\right):$ transformed matrix |

${B}_{j}\left(i=\text{}1,\text{}2,\text{}\dots ,\text{}N\right):$ non-overlapping bloc |

${Z}_{j}\left(i=\text{}1,\text{}2,\text{}\dots ,\text{}N\right):$ sum of absolute mean of the ${j}^{th}$ block |

${R}_{i}\left(i=\text{}1,\text{}2,\text{}\dots ,\text{}M\times M\right)$: block with maximum sum of absolute mean |

${H}_{i}\left(i=\text{}1,\text{}2,\text{}\dots ,\text{}M\times M\right)$: Hessenberg matrix |

${n}_{i}\left(i=\text{}1,\text{}2,\text{}\dots ,\text{}M\times M\right)$: the 2nd order Euclidean normalization |

${x}_{i}\left(i=\text{}1,\text{}2,\text{}\dots ,\text{}M\times M\right)$: quantization coefficient for embedding |

Watermark Embedding Procedure: |

for i = 1: $M\times M$ do |

convert the ${i}^{th}$ frame coefficients into two dimensional matrix ${C}_{i}$ |

apply PSHT on ${C}_{i}$ to obtain ${T}_{i}$ |

for j = 1: N do |

subdividing into non-overlapping block ${B}_{j}$ |

calculate the sum of absolute mean ${Z}_{j}$ of each block ${B}_{j}$ using Equation (10) |

end for |

select block ${R}_{i}$ with maximum sum of absolute mean ${Z}_{max}$ |

apply HD on matrix ${R}_{i}$ using Equation (11) |

calculate ${n}_{i}$ using Equation (12) |

calculate ${d}_{i}$ and ${x}_{i}$ |

update ${n}_{i}$ into ${n}_{i}\prime $ using Equations (13) and (14) |

modify the largest Hessenberg coefficient ${{{H}_{i}}_{\left(k,1\right)}}_{largest}$ using Equation (15) |

apply inverse HD on matrix ${R}_{i}^{\ast}$ using Equation (16) |

apply inverse PSHT on ${T}_{i}^{\ast}$ |

reshape ${C}_{i}$ properly |

reshape ${F}_{i}^{\ast}$ properly. |

end for |

return watermarked audio ${Y}^{\ast}$ |

#### 4.3. Watermark Extraction Process

Algorithm 3: Watermark Extraction |

Variable Declaration: |

${Y}^{\ast}$: attacked watermarked audio signal |

F: attacked watermarked frame |

${{C}_{i}}^{\ast}(\left(i=\text{}1,\text{}2,\text{}\dots ,M\times M\right):$ watermarkedframe represented in two dimensional matrix with size m×m |

${{T}_{i}}^{\ast}\left(i=\text{}1,\text{}2,\text{}\dots ,M\times M\right):$ modified transformed matrix |

${{B}_{j}}^{\ast}\left(i=\text{}1,\text{}2,\text{}\dots ,N\right):$ modified non-overlapping block |

${{Z}_{j}}^{\ast}\left(i=\text{}1,\text{}2,\text{}\dots ,N\right):$ sum of absolute mean of modified the ${j}^{th}$ block |

${{R}_{i}}^{\ast}\left(i=\text{}1,\text{}2,\text{}\dots ,M\times M\right)$: modified block with maximum sum of absolute mean |

${{H}_{i}}^{\ast}\left(i=\text{}1,\text{}2,\text{}\dots ,M\times M\right)$: modified Hessenberg matrix ${{n}_{i}}^{\ast}\left(i=\text{}1,\text{}2,\text{}\dots ,M\right)$: modified the 2ndorder |

Euclidean normalization |

${{x}_{i}}^{\ast}\left(i=\text{}1,\text{}2,\text{}\dots ,M\right)$: quantization coefficientfor extraction |

Watermark Extraction Procedure: |

for i = 1: $M\times M$ do |

convert the coefficients of the ${i}^{th}$ frame into two dimensional matrix ${{C}_{i}}^{\ast}$ |

apply PSHT on ${{C}_{i}}^{\ast}$ to obtain |

for j = 1: N do |

subdividing into non-overlapping block ${B}_{j}$ |

calculate the sum of absolute mean ${Z}_{j}$ of each block ${B}_{j}$ |

end for |

select block ${R}_{i}^{\ast}$ with maximum sum of absolute mean ${{Z}_{max}}^{\ast}$ |

apply HD on matrix ${R}_{i}^{\ast}$ |

calculate ${{n}^{\ast}}_{i}$ |

calculate ${{d}^{\ast}}_{i}$ and ${{x}^{\ast}}_{i}$ |

calculate ${u}^{\ast}\left(i\right)$ using the Equation (17) |

calculate ${r}^{\ast}\left(i\right)$ using the Equation (18) |

reshape ${r}^{\ast}\left(i\right)$ |

end for |

return watermark ${W}^{\ast}$ |

## 5. Experimental Results and Discussion

_{i}with size 8×8 for better computation cost of space and time.

#### 5.1. Imperceptibility Analysis

#### 5.1.1. Subjective Analysis

#### 5.1.2. Objective Analysis

#### 5.2. Robustness Analysis

- Noise addition: Additive white Gaussian noise (AWGN) was added with a watermarked signal until the signal had an SNR of 20 dB.
- Cropping: A number of 1000 samples of the watermarked audio were removed from different positions, and then, these samples were replaced with the watermarked audio signal attacked by additive white Gaussian noise.
- Re-sampling: The watermarked signal with a sample rate of 44.1 kHz was sampled to 22.05 kHz and again resampled by a rate of 44.1 kHz.
- Re-quantization: The watermarked audio was quantized from 16 bit to 8 bit.
- Compression: The watermarked signal was compressed using MPEG-1 layer 3 compression (128 kbps).
- Noise Reduction: Noise reduction was successfully done from the watermarked audio with the help of “Hiss removal” function.
- Echo addition: Echo signal containing a delay time of 150 ms and decay rate of 35% was applied to the watermarked signal.
- Distortion: The watermarked audio signal was distorted within a range of 0 dB to −10 dB.
- Amplification: The watermarked audio was amplified (enlarged) by 1.25 times of its original amplitude.
- Delay: A delay time of 150 ms was used and the volume of the delayed signal contains 3% of the original signal.
- Invert: The watermarked audio signal was fully inverted to obtain the inverted form of the actual watermark signal.
- Low-Pass Filter: A low-pass filter with a cut-off frequency of 15,000 Hz was applied to the watermarked audio.

#### 5.3. Data Payload

#### 5.4. Security Analysis

#### 5.5. Computation Time Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**Extracted watermark against different attacks for pop audio signal: (

**a**) no attack, (

**b**) noise addition, (

**c**) noise reduction, (

**d**) echo addition, (

**e**) cropping, (

**f**) re-quantization, (

**g**) compression (MP3), (

**h**) re-sampling, (

**i**) distortion, (

**j**) amplification, (

**k**) delay, (

**l**) invert, (

**m**) low-pass filter.

**Figure 5.**Extracted watermark against different attacks for classical audio signal: (

**a**) no attack, (

**b**) noise addition, (

**c**) noise reduction, (

**d**) echo addition, (

**e**) cropping, (

**f**) re-quantization, (

**g**) compression (MP3), (

**h**) re-sampling, (

**i**) distortion, (

**j**) amplification, (

**k**) delay, (

**l**) invert, (

**m**) low-pass filter.

**Figure 6.**Extracted watermark against different attacks for jazz audio signal (

**a**) no attack, (

**b**) noise addition, (

**c**) noise reduction, (

**d**) echo addition, (

**e**) cropping, (

**f**) re-quantization, (

**g**) compression (MP3), (

**h**) re-sampling, (

**i**) distortion, (

**j**) amplification, (

**k**) delay, (

**l**) invert, (

**m**) low-pass filter.

**Figure 7.**Extracted watermark against different attacks for rock audio signal (

**a**) no attack, (

**b**) noise addition, (

**c**) noise reduction, (

**d**) echo addition, (

**e**) cropping, (

**f**) re-quantization, (

**g**) compression (MP3), (

**h**) re-sampling, (

**i**) distortion, (

**j**) amplification, (

**k**) delay, (

**l**) invert, (

**m**) low-pass filter.

SDG | ODG | Description | Quality |
---|---|---|---|

5 | 0 | Imperceptible | Excellent |

4 | −1 | Perceptible, but not annoying | Good |

3 | −2 | Slightly annoying | Fair |

2 | −3 | Annoying | Poor |

1 | −4 | Very annoying | Bad |

Audio Signal | MOS | Correct Detection | SNR | ODG |
---|---|---|---|---|

Pop | 4.90 | 54% | 43.81 | −0.46 |

Classical | 5.00 | 48% | 47.75 | −0.35 |

Jazz | 5.00 | 48% | 47.08 | −0.37 |

Rock | 4.90 | 54% | 47.60 | −0.38 |

Average | 4.95 | 51% | 46.56 | −0.39 |

**Table 3.**A comparative analysis between the proposed and various methods in terms of imperceptibility.

Reference | Method | SNR | MOS |
---|---|---|---|

[4] | Energy averaging | 41.47 | - |

[5] | Localized and self-adaptive algorithm | 31.40 | 3.7 |

[6] | LRS-FFT | 44.81 | - |

[7] | DCT-SVD-ELO | 33.47 | 4.88 |

[8] | SSA-PM | 25.61 | - |

[9] | Multifunctional algorithm | 23.33 | - |

[10] | DCT-SVD-LPT | 37.20 | 4.85 |

[11] | SVD-QIM | 19.39 | - |

[12] | FS-AE | 33.6 | - |

Proposed | PSHT-HD | 46.56 | 4.95 |

Attack | Pop | Classical | Jazz | Rock |
---|---|---|---|---|

No attack | 1 | 1 | 1 | 1 |

Noise Addition | 0.9986 | 0.9995 | 0.9911 | 1 |

Noise Reduction | 1 | 1 | 1 | 1 |

Echo Addition | 1 | 1 | 1 | 1 |

Cropping | 0.9978 | 0.9977 | 0.9988 | 0.9982 |

Re-quantization | 0.9968 | 1 | 0.9992 | 1 |

Compression (MP3) | 0.9566 | 0.9459 | 0.9619 | 0.9643 |

Re-sampling | 0.9836 | 1 | 0.9943 | 0.9893 |

Distortion | 0.9766 | 1 | 0.9895 | 0.9992 |

Amplification | 0.9944 | 1 | 0.9871 | 1 |

Delay | 0.9944 | 0.9976 | 0.9895 | 1 |

Invert | 1 | 1 | 1 | 1 |

Low-Pass Filtering | 0.9649 | 0.9871 | 0.9822 | 0.9919 |

Attack | Pop | Classical | Jazz | Rock |
---|---|---|---|---|

No attack | 0 | 0 | 0 | 0 |

Noise Addition | 0.37 | 0.88 | 1.07 | 0 |

Noise Reduction | 0 | 0 | 0 | 0 |

Echo Addition | 0 | 0 | 0 | 0 |

Cropping | 0.24 | 0.026 | 0.14 | 0.20 |

Re-quantization | 0.39 | 0 | 0.09 | 0 |

Compression (MP3) | 5.18 | 6.54 | 4.59 | 4.30 |

Re-sampling | 1.67 | 0 | 0.68 | 0.88 |

Distortion | 2.83 | 0 | 1.27 | 0.09 |

Amplification | 0.68 | 0 | 1.56 | 0 |

Delay | 0.49 | 0.29 | 1.27 | 0 |

Invert | 0 | 0 | 0 | 0 |

Low-Pass Filtering | 4.54 | 1.56 | 2.15 | 0.98 |

Reference | Method | Noise Addition | Resampling | Re-Quantization | MP3 Compression |
---|---|---|---|---|---|

Proposed | PSHT-HD | 0.58(20 dB) | 0.81(22.05 kHz) | 0.12 (8 Bits/Sample) | 5.15(128 kbps) |

[4] | Energy averaging | - | 8.0(22.05 kHz) | - | 5.0(128 kbps) |

[5] | Localized and self-adaptive algorithm | 6.03(30 dB) | 0(22.05 kHz) | 0.14(8 bits/sample) | 6.20(64 kbps) |

[6] | LRS-FFT | 5.17(-) | 6.56(22.05 kHz) | 4.94(8 bits/sample) | 6.88(128 kbps) |

[7] | DCT-SVD-ELO | 0.91(-) | 0.88(22.05 kHz) | 0.23(8 bits/sample) | 6.13 (32 kbps) |

[8] | SSA-PM | 2.50(36 dB) | 6.06(22.05 kHz) | 8.83(16 bits/sample) | 9.44(128 kbps) |

[9] | Multifunctional algorithm | 4.22(-) | 0(22.05 kHz) | - | 7.48(32 kbps) |

[10] | DCT-SVD-LPT | 0.83(-) | 1.56(22.05 kHz) | 0(8 bits/sample) | 3.91(128 kbps) |

[11] | SVD-QIM | 10.25(30 dB) | 4.88(16 kHz) | - | 17.76(128 kbps) |

[12] | FS-AE | 7.23(20 dB) | - | - | 6.04(48 kbps) |

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## Share and Cite

**MDPI and ACS Style**

Dhar, P.K.; Chowdhury, A.H.; Koshiba, T.
Blind Audio Watermarking Based on Parametric Slant-Hadamard Transform and Hessenberg Decomposition. *Symmetry* **2020**, *12*, 333.
https://doi.org/10.3390/sym12030333

**AMA Style**

Dhar PK, Chowdhury AH, Koshiba T.
Blind Audio Watermarking Based on Parametric Slant-Hadamard Transform and Hessenberg Decomposition. *Symmetry*. 2020; 12(3):333.
https://doi.org/10.3390/sym12030333

**Chicago/Turabian Style**

Dhar, Pranab Kumar, Azizul Hakim Chowdhury, and Takeshi Koshiba.
2020. "Blind Audio Watermarking Based on Parametric Slant-Hadamard Transform and Hessenberg Decomposition" *Symmetry* 12, no. 3: 333.
https://doi.org/10.3390/sym12030333