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Digital Watermarking is a technology, to facilitate the authentication, copyright protection and Security of digital media. The objective of developing a robust watermarking technique is to incorporate the maximum possible robustness without compromising with the transparency. Singular Value Decomposition (SVD) using Firefly Algorithm provides this objective of an optimal robust watermarking technique. Multiple scaling factors are used to embed the watermark image into the host by multiplying these scaling factors with the Singular Values (SV) of the host audio. Firefly Algorithm is used to optimise the modified host audio to achieve the highest possible robustness and transparency. This approach can significantly increase the quality of watermarked audio and provide more robustness to the embedded watermark against various attacks such as noise, resampling, filtering attacks etc.

The term digital watermarking [

According to domain, watermarking is classified as spatial domain and transform domain [

The singular value decomposition (SVD) [

A = U S V T

where S is an N × N diagonal matrix. U and V^{T} are N × N orthogonal matrices, whose column vectors are u_{i}’s and v_{i}’s, respectively. The important property of the singular values is that any modifications done on these values do not show any change in the respective matrix. Based on this property, the singular values are modified with the singular values of the watermark image. An N × N image can have N singular values that reveal various tolerances to modifications [

Singular value decomposition (SVD) comes under the category of transform domain technique of digital audio watermarking, which is akin theory of diagonalizing of symmetric matrix in linear algebra. SVD decomposes a matrix into three sub-parts: U, S and V. U and V are the orthogonal matrices while S is the diagonal matrix. These diagonal elements are called the singular values of the corresponding matrix. This decomposition can be illustrated as:

A = U S V T (1)

where A is a matrix of dimension m × n. U is made up of the eigen vectors of AA^{T} and is called left singular vector. V is formed by the orthogonal vectors of A^{T}A and is called right singular vector. S contains the square roots of either U or V in descending order in its diagonal being a diagonal matrix. Let the rank of the matrix A be r (r < n), then the diagonal elements of S will follow the following relation:

α 1 ≥ α 2 ≥ ⋯ ≥ α r ≥ α r + 1 ≥ α r + 2 ≥ ⋯ ≥ α n (2)

Now A can be derived as:

A = ∑ i = 1 r α i u i v i (3)

where α i is the diagonal element of matrix S at i^{th} position.

The singular values give the luminance of the audio at each i^{th} position, whereas singular vectors give the geometrical property. The most important property of SVD is that if any changes are applied to the singular values then will be no significant changes seen on the given matrix. Using this property the watermark image is modified by applying change in its singular values and embedded into the singular values of the host audio without getting any distortions and any perpetual change.

Properties of SVD:

1) Singular values preserve the energy as well as prevent the image from attacks.

2) The matrix in SVD can be variable. It need not be always scalar.

3) The singular values α_{i} are unique in the matrix S.

4) The rank of the matrix is given by the non-zero elements in the diagonal matrix, S.

Firefly algorithm [

In firefly algorithm, the brightest firefly is a firefly with current global best solution and it will move in random direction if no brighter firefly is found. This random movement may decrease the brightness depending on direction. As a consequence the overall performance of the algorithm is decreased in that particular iteration.

It is proved in elementary physics that intensity of light is inversely proportional to the square of the distance from the source to the object. Therefore we can formulate the light intensity, I in terms of distance, r as follows:

I ( r ) = I 0 e − λ r (4)

where λ is the light absorption coefficient and I_{0} is the light intensity at the source point.

For the sake of simplicity this can be written as:

I ( r ) = I 0 1 + λ r 2 (5)

Likewise, attractiveness can also be derived:

A ( r ) = A 0 1 + λ r 2 (6)

where A_{0} is the attractiveness at r = 0.

Steps of implementation of firefly are as follows:

1) Generate a solution set randomly.

2) Find the intensity for each of the generated firefly.

3) The movement of the firefly will be done in the direction of brighter firefly and if no such direction is found then the firefly will move in random direction.

4) Now, solution is updated.

5) End the process if termination condition holds true; else go back to step 2.

The main drawback of the FA is that if there is no such direction in which the brightness increases, it moves the firefly randomly, and this random movement may sometimes cause degradation in the performance of FA because brightness may reduce in some random direction. Now if we change this property of random movement by moving in a particular direction in which its brightness increases then it will not degrade the performance in that iteration. If such direction does not exist then the firefly will remain at its current position. This is the Modified Firefly algorithm.

The movement of the firefly will be according to the following relation:

d : = d + α μ (7)

where, d is the location of the firefly, µ is the chosen direction in which movement is to be done and α is the step length selected randomly.

Attractiveness of a firefly can be calculated as:

A 0 = I ′ 0 I 0 (8)

where A_{0} is the attractiveness of a firefly say, i at r = 0, I ′ 0 is the intensity of firefly i and I_{0} is the intensity of firefly j.

This paper proposes a SVD [

1) Steps of embedding watermark:

The steps of embedding the watermark image into the host audio are shown in

Step1: Divide the host audio (H) and watermark image (W) into n non overlapping frames of size m × m.

Step 2: Apply SVD on these blocks of host audio (H_{i}) and watermark image (W_{i}) simultaneously.

[ U i S i V i ] = SVD ( H i ) (9)

[ U w i S w i V w i ] = SVD ( W i ) (10)

Step 3: Embed the singular values of the watermark image (S_{wi}) into the singular values of host audio (S_{i}) using the following formula:

S ′ i = { S i + ρ * S w i , if S i ( x , y ) < S w i ( x , y ) S i − ρ * S w i , otherwise (11)

Step 4: Do inverse SVD on the sub-blocks to regain the H_{i}:

H i = U i S ′ i V i (12)

Step 5: Recombination of the blocks is performed to get the watermarked image with size equal to the host image.

2) Extraction process of watermark

In the extraction process watermark image is being extracted from the watermarked audio (H_{w}), which is produced as a result of the embedding process. This extraction procedure is described below in

Step 1: Divide the produced watermarked image (H_{w}) into n non overlapping blocks (H_{iw}) of equal size.

Step 2: Perform SVD on each sub-block:

[ U f i S f i V f i ] = SVD ( f i ) (13)

Step 3: Singular Values of the watermark image is extracted using:

S ′ w i = { S i − S f i ρ , if S i ( x , y ) > S f i S f i − S i ρ , otherwise (14)

Step 4: Now watermark image is recovered from the watermarked image by:

W ′ i = U w i * S w i * V w i T (15)

Step 5: To get the original size and dimension of the watermark image, the recovered blocks are recombined.

Let the host audio be H and watermark image be W of size N × N, then the following are the steps of the algorithm by which this model works:

Step 1: n no. of fireflies are generated randomly using MFA.

Where n = [ ρ 1 , ρ 2 , ρ 3 , ⋯ , ρ n ]

Step 2: for each generated firefly, ρ, perform the following operations:

1) Apply embedding process discussed in the previous section on the host audio and watermark image.

2) Induce r number of attacks on the watermarked audio (Hw); hence attacked audio (Hw') are generated.

3) Extract the watermark from the host audio and attacked images using extraction algorithm described above.

4) Compute the PSNR values of the host audio (H), watermarked audio (Hw) and attacked audio (Hw').

5) Compute the objective function (O) of the firefly (ρ) using the objective function below:

O = PSNR ( H , H w ) + PSNR ( W , H w ′ ) + ∑ i = 1 r PSNR ( W , H ′ W i ) (16)

where PSNR ( W , H w ′ ) is the peak signal to noise ratio between watermark audio and the watermark extracted from the attacked audio.

Step 3: Now take the maximum value of the objective function to choose the multiple scaling factor which in turn optimizes the trade-off between the imperceptibility and robustness of the watermarking procedure.

To verify the results of above-mentioned technique, we implemented the algorithm in the MATLAB 7.0. The audio file named “in” (

The following attacks [

Sl. No. | Name of Attack | Normalized Coefficient (NC) | Bit Error Rate (BER) | Recovered Watermark |
---|---|---|---|---|

1. | AWGN (m = 0, var = 0.000005) | 0.8934 | 0.1511 | |

2. | Resampling | 0.8526 | 0.2105 | |

3. | Low Pass Filtering | 0.8675 | 0.1532 | |

4. | Clipping | 0.8974 | 0.1611 | |

5. | MPEG-2 Compression | 0.9312 | 0.1524 |

1) Additive white Gaussian noise (AWGN):

White Gaussian noise [

2) Re-sampling:

The Watermarked signal originally sampled at 44.1 kHz is re-sampled at 22.05 kHz, and then restored by sampling again at 44.1 kHz. Correlation [

N C ( w m , w m ′ ) = ∑ i = 1 M ∑ j = 1 M W ( i , j ) w m ( i , j ) ¯ ∑ i = 1 M ∑ j = 1 M w m 2 ( i , j ) ∑ i = 1 M ∑ j = 1 M w m 2 ( i , j ) ¯

The bit error rate (BER) is used to measure the robustness of our scheme:

BER ( w m , w m ¯ ) = Errorbits TotalBits

3) Low-pass Filtering:

The low-pass filter [

Robust watermarking scheme can provide better authentication and security of the digital audio. In this rapid growing era of technology there are many tools available which can easily modify or extract the watermark from an audio, hence it is a necessary thing to have more robust watermarking scheme which can withheld these attacks and forgeries. This is a new method of robust audio watermarking based on SVD using Modified Firefly Algorithm. Modified Firefly Algorithm is used to employ optimise function that was defined by two conflicting requirements of watermarking i.e. transparency and robustness. The watermark image is embedded into the host audio by modifying the singular values of the host audio. To achieve maximum robustness without losing transparency, modifications are to be done using multiple scaling factors obtained by Modified Firefly Algorithm.

Rizvi, S.A.M. and Chauhan, S.P.S. (2018) Robust Digital Audio Watermarking Based on SVD and Modified Firefly Algorithm. Journal of Information Security, 9, 1-11. https://doi.org/10.4236/jis.2018.91001