# Compressive Sampling with Multiple Bit Spread Spectrum-Based Data Hiding

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## Abstract

**:**

## 1. Introduction

## 2. Sparse Singular Value and CS Technique

## 3. Data Hiding Model

#### 3.1. An Overcomplete Dictionary with SS-Based Content

#### 3.2. Data and Dictionary Detection

#### 3.3. Security Model

#### 3.4. Signal Reconstruction

#### 3.5. Noisy Environment

#### 3.6. Feasible Parameters

## 4. Experimental Result

#### 4.1. Audio Quality Performance in Relation to r, M, Payload, and Compression Ratio

#### 4.2. Complexity and Computational Time

#### 4.3. Security Analysis

#### 4.4. Noisy Environment

#### 4.5. Method Comparison to References

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CS | Compressed Sensing/Compressive Sampling |

WHT | Walsh Hadamard Transform |

DCT | Discrete Cosine Transform |

LSB | Least Significant Bit |

TLC | Karhunen-Loeve Transform |

MP3 | Motion Picture Experts Group Audio Layer 3 |

SVD | Singular Value Decomposition |

CR | Compression Ratio |

IDCT | Inverse DCT |

SS | Spread Spectrum |

BER | Bit Error Rate |

OMP | Orthogonal Matching Pursuit |

SNR | Signal-to-Noise power Ratio |

ODG | Objective Difference Grade |

PEAQ | Perceptual Evaluation of Audio Quality |

AMD | Advanced Micro Devices |

RAM | Random Access Memory |

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**Figure 6.**BER in relation to ${N}_{l}$ and r using a different Hadamard matrix between encoding and decoding.

Step 1: | Read a host signal $\mathit{x}\left(\mathit{n}\right)$, and transform it into the frequency domain by DCT L-point obtaining $\mathit{X}\left(\mathit{k}\right)$ |

Step 2: | Reshape $X\left(k\right)$ in L and sample it to a 2D square matrix producing $\mathbf{X}$ with size $M\times M$ |

Step 3: | Decompose $\mathbf{X}$ to $\mathbf{U}$, $\mathbf{S}$, and $\mathbf{V}$ using SVD |

Step 4: | Reduce the matrix size of $\mathbf{U}$, $\mathbf{S}$, and $\mathbf{V}$ with rank r to ${\mathbf{U}}_{\mathbf{r}}$, ${\mathbf{S}}_{\mathbf{r}}$, and ${\mathbf{V}}_{\mathbf{r}}$ |

Step 5: | Generate the $\mathbf{A}$ matrix containing p Hadamard sequences by mapping each multi-watermark bit to an associated random Hadamard sequence using (13) |

Step 6: | Apply CS acquisition to $\mathbf{A}$ and ${\mathbf{S}}_{\mathbf{r}}$ by (4), producing $\mathbf{Y}$ |

Step 7: | Transmit the compressed signal with hidden data represented using ${\mathbf{U}}_{\mathbf{r}}$, $\mathbf{Y}$, and ${\mathbf{V}}_{\mathbf{r}}$ |

Index (${\mathit{t}}_{\mathit{i}}$) | Watermark Bits (${\mathbf{w}}_{\mathit{i}}$) | Hadamard Sequence (${\mathbf{p}}_{{\mathit{t}}_{\mathit{i}}}$) |
---|---|---|

1 | {−1,−1,−1} | {+1,+1,+1,+1,+1,+1,+1,+1} |

2 | {−1,−1,+1} | {+1,−1,+1,−1,+1,−1,+1,−1} |

3 | {−1,+1,−1} | {+1,+1,−1,−1,+1,+1,−1,−1} |

4 | {−1,+1,+1} | {+1,−1,−1,+1,+1,−1,−1,+1} |

5 | {+1,−1,−1} | {+1,+1,+1,+1,−1,−1,−1,−1} |

6 | {+1,−1,+1} | {+1,−1,+1,−1,−1,+1,−1,+1} |

7 | {+1,+1,−1} | {+1,+1,−1,−1,−1,−1,+1,+1} |

8 | {+1,+1,+1} | {+1,−1,−1,+1,−1,+1,+1,−1} |

Step 1: | Detect ${\mathit{t}}_{\mathit{i}}$ from ${\mathit{Y}}^{\prime}$ using (22) for extracting the hidden data |

Step 2: | Associate detected ${t}_{i}$ with ${\mathbf{p}}_{ti}$, and form $\widehat{\mathbf{A}}$ using (13) |

Step 3: | Reconstruct ${\mathbf{Y}}^{\prime}$ using $\widehat{\mathbf{A}}$ by (37), (39), (40), and (41) to obtain ${\mathbf{S}}_{\mathbf{r}}^{\prime}$ |

Step 4: | Reconstruct ${\mathbf{U}}_{\mathbf{r}}$, ${\mathbf{S}}_{\mathbf{r}}^{\prime}$, and ${\mathbf{V}}_{\mathbf{r}}$ by SVD reconstruction to obtain the decompressed signal in 2D matrix ${\mathbf{X}}_{\mathbf{r}}^{\prime}$ by (7) |

Step 5: | Reshape 2D matrix ${\mathbf{X}}_{\mathbf{r}}^{\prime}$ to a 1D matrix, obtaining ${X}^{\prime}\left(k\right)$ |

Step 6: | Transform ${X}^{\prime}\left(k\right)$ to the time domain by the IDCT L-point, obtaining the reconstructed signal ${x}^{\prime}\left(n\right)$ |

r | M | ODG | C | CR |
---|---|---|---|---|

2 | 8 | −0.02 | 2756.25 | 1.60 |

2 | 7 | −0.02 | 2756.25 | 1.60 |

2 | 5 | −0.03 | 4833.33 | 1.11 |

2 | 10 | −0.03 | 1764 | 2.08 |

2 | 9 | −0.03 | 1764 | 2.08 |

2 | 6 | −0.03 | 4900 | 1.12 |

3 | 12 | −0.04 | 5512.50 | 1.45 |

3 | 11 | −0.04 | 5512.50 | 1.45 |

3 | 10 | −0.04 | 5512.50 | 1.45 |

2 | 12 | −0.05 | 1225 | 2.57 |

r | M | ODG | C | CR |
---|---|---|---|---|

5 | 20 | −0.29 | 8268.75 | 1.23 |

5 | 19 | −0.29 | 8268.75 | 1.23 |

5 | 18 | −0.29 | 8268.75 | 1.23 |

5 | 17 | −0.29 | 8268.75 | 1.23 |

5 | 16 | −0.29 | 8268.75 | 1.23 |

6 | 24 | −0.32 | 8268.75 | 1.14 |

6 | 23 | −0.32 | 8268.75 | 1.14 |

6 | 22 | −0.32 | 8268.75 | 1.14 |

6 | 21 | −0.32 | 8268.75 | 1.14 |

6 | 20 | −0.32 | 8268.75 | 1.14 |

r | M | ODG | C | CR |
---|---|---|---|---|

2 | 30 | −0.99 | 196 | 7.03 |

2 | 29 | −0.99 | 196 | 7.03 |

2 | 27 | −0.99 | 225 | 6.53 |

2 | 28 | −0.99 | 225 | 6.53 |

2 | 25 | −0.94 | 260.95 | 6.04 |

2 | 26 | −0.94 | 260.95 | 6.04 |

2 | 23 | −0.82 | 306.25 | 5.54 |

2 | 24 | −0.82 | 306.25 | 5.54 |

2 | 22 | −0.68 | 364.47 | 5.04 |

2 | 21 | −0.68 | 364.46 | 5.04 |

SNR | BER (%) | Detected Watermark |
---|---|---|

0 | 24.1 | |

5 | 18.7 | |

10 | 17.4 | |

15 | 14.3 | |

20 | 11.7 | |

25 | 8.8 | |

30 | 6.1 | |

35 | 5.7 | |

40 | 5.6 | |

45 | 3.6 | |

50 | 0.4 | |

55 | 0 |

Ref. | Hiding Method | Audio Reconstruction | Audio Quality | Robustness | Payload | Compression Ratio |
---|---|---|---|---|---|---|

[1] | Watermark Projection | × | × | √ | × | × |

[2] | Semi-Fragile Zero Watermarking | × | × | √ | × | × |

[3] | - | √ | √ | × | × | √ |

[4] | Basis Pursuit Denoising | √ | √ | √ | √ | × |

Proposed | Multi-Bit Spread Spectrum | √ | √ | √ | √ | √ |

Ref. | Clips | Audio Quality | BER | C (bps) | CR |
---|---|---|---|---|---|

[4] | 1 | SNR = 28 dB | 0–3% | 11–344 | not reported |

Proposed | 50 | ODG = [−0.94 −0.74] | 0–13% | 729–5292 | 1.47–4.84 |

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## Share and Cite

**MDPI and ACS Style**

Budiman, G.; Suksmono, A.B.; Danudirdjo, D. Compressive Sampling with Multiple Bit Spread Spectrum-Based Data Hiding. *Appl. Sci.* **2020**, *10*, 4338.
https://doi.org/10.3390/app10124338

**AMA Style**

Budiman G, Suksmono AB, Danudirdjo D. Compressive Sampling with Multiple Bit Spread Spectrum-Based Data Hiding. *Applied Sciences*. 2020; 10(12):4338.
https://doi.org/10.3390/app10124338

**Chicago/Turabian Style**

Budiman, Gelar, Andriyan Bayu Suksmono, and Donny Danudirdjo. 2020. "Compressive Sampling with Multiple Bit Spread Spectrum-Based Data Hiding" *Applied Sciences* 10, no. 12: 4338.
https://doi.org/10.3390/app10124338