Use of firefly optimization algorithm for fractional fourier transform based reversible data hiding

Abstract

This paper presents a Fractional Fourier transform based reversible data hiding technique to secure the data transmitted over communication network. The proposed algorithm modifies the cover image to improve the robustness of data hiding technique. The cover image is transformed using Fractional Fourier Transform (FrFT) into a time-frequency domain and the optimal pixel locations for hiding the secret data are found using firefly algorithm. Firefly algorithm uses multi-objective function, which is a combination of Structural Similarity Index Measure (SSIM) and Bit Error rate (BER). The histogram shifting technique embeds secret data in the optimal pixel locations. The quality of test images is analyzed under varying payload as well as under varying fractional order. Experimental results conclude that this scheme produces good quality stego image. It has also been found from the simulation results that the proposed algorithm is more robust and reversible against various attacks as it provides lower bit error rate and higher normalization coefficient.

Keywords

Reversible data hiding histogram shifting fractional fourier transform firefly algorithm imperceptibility robustness reversibility

1 Introduction

Due to the development in technologies, internet is preferred by most of the people to exchange information from one place to another across the world. However, one of the key issues while sending data over the internet is the security risk that it poses, i.e., the personal or private information may be compromised. Therefore, it is necessary to devise strategies to secure information during the process of information exchange. Data hiding involves protecting confidential data within another seemingly innocuous host media or cover media such as image, video, audio and text [1]. While secret information is embedded in a cover image, the focus is on two key issues. The first one is to create an embedded image with a tolerable level of quality in order to produce imperceptible image. The second is to create the embedded image that is tolerant to distortion, i.e., hidden data can be recovered even if the embedded image is attacked during communication. The cover image properties could be appropriately used in order to improve the robustness of data hiding techniques. By considering these aspects, embedding data in frequency domain becomes more popular than that of spatial domain techniques [2]. In frequency domain techniques, the frequency coefficients of the cover image are modified by applying a specific transformation function on the cover image. They are designed to be reversible and robust against various geometrical transformations and external attacks.

Frequency domain techniques employ transformation methods, such as integer cosine transform [3] or integer wavelet transform [4] to find frequency coefficients of the cover image. In Direct Cosine Transform (DCT) based reversible data hiding approach, one secret bit was embedded into two neighbouring DCT coefficients in an image block [5]. This approach offers a high degree of robustness towards a wide range of image processing operations. The secret payload is hidden in the high frequency coefficients of discrete wavelet transform (DWT) by using the statistical properties of the cover image [6]. The DWT works well against different image processing attacks. As it produces floating-point coefficients, embedded data is potentially lost during the reconstruction of cover image from inverse wavelet transform. This drawback is overcome by integer wavelet transform (IWT). The secret data is embedded into the middle frequency of the integer lifting wavelet domain by adjusting the histogram of the cover image [7].

Many research works are reported in the literature to find the best algorithm that can produce embedded images with minimum distortion and maximum embedding capacity and robustness. Optimization algorithms such as Genetic Algorithm (GA) [8], Particle Swarm Optimization (PSO) [9], and Bacterial foraging [10] have been widely used to identify the coefficients of the image in transform domain to embed data. In PSO based data hiding scheme in spatial domain, PSO finds the optimal pixel locations to embed secret bits in the cover image [11]. Even though the method results a good quality stego image, it is more vulnerable to various attacks during transmission and becomes less robust. In order to conceal the existence of the message, it is necessary to identify the optimal locations to hide the secret data.

As the firefly algorithm (FA) is very precise, robust and can be easily implemented, it is used for identifying optimal locations [12]. FA performs better even in noisy environments. The main advantage of firefly algorithm over other optimization algorithms is that it takes less computation time to produce optimal results. Firefly algorithm outperforms GA and PSO in terms of efficiency and success rate [13]. Fractional Fourier transform transforms the cover image from spatial to time-frequency domain. The histogram shifting technique embeds the secret bits in the optimal locations. Histogram shifting technique is preferred for data embedding due to its less computational complexity, high embedding capacity and high visual quality. It uses maximum and zero points (or minimum if no zero points are available) of histogram of image and shifts the values between these points [14]. The proposed method improves the quality of embedded image as well as the robustness of embedded payload against various attacks and the retrieves the cover image after extracting secret payload.

This paper can be organized as follows. Section 2 describes the Fractional Fourier Transform. Section 3 describes firefly algorithm. Section 4 explains the proposed algorithm. The experimental results are reported in Section 5. Finally, Section 6 concludes the paper.

2 Fractional fourier transform

The Fractional Fourier transform can be defined for all types of signals such as one dimensional and multidimensional, continuous and discrete, periodic and non-periodic signals [15]. FrFT defines the spectral content of the signal as well as the time location of the spectral components.

The Fourier transform can be elucidated as a rotation of time domain signal by an angle π/2 in the time - frequency plane, whereas FrFT of a signal can be described as rotation of the signal in the time - frequency plane by an arbitrary angle α= a π/2, where a is a fractional order and it is a real number. By making a certain angle with time axis, signal representation along this intermediate axis retains both time and frequency information. As the angle of rotation is a non-integer multiple of π/2, the transform is called Fractional fourier transform [16]. The representation of various domains corresponding to fractional order ‘a’ and rotation angle ‘α’ is shown in Fig. 1.

Fig. 1

Representation of Various domains corresponding to the Fractional order ‘a’ and rotation angle ‘α’.

One of the special cases in FrFT with α=π/2 corresponds to the FT, and an FrFT with α= 0 corresponds to the identity operator. When α is not equal to a multiple of π/2, the FrFT is equivalent to perform α/(rmπ/2) times of FT [17, 18].

The a^th fractional order Fractional Fourier transform of a signal x (t), with a rotation angle α= aπ/2, is defined by, $X_{a} (u) = \int_{- \infty}^{\infty} x (t) K_{α} (t, u) dt$ (1) where K_α (t, u) is the transformation kernel of FrFT and is given by, $K α (t, u) = {\begin{matrix} \sqrt{\frac{(1 - j cot α)}{2 π}} e^{j (t^{2} + u^{2} / 2) cot (α) - jut csc (α),} if α \neq n π \\ δ (t - u), if α is a multiple of 2 π \\ δ (t + u), if α + π is a multiple of 2 π \end{matrix}$ (2)

The inverse FrFT is defined as, $x (t) = \int_{- \infty}^{\infty} X_{α} (u) K_{α} (t, u) du$ (3)

For analysis of two-dimensional (2D) signals such as images, a 2D version of the FrFT is used [19]. For an M×N matrix, the 2D FrFT is obtained by applying 1D FrFT to each row of the matrix and then to each column of the result. Thus, the 2D FrFT is given by, $F_{α, β} (u, v) = \int_{- \infty}^{\infty} f (x, y) K_{α, β} (u, v; x, y) dxdy$ (4) where K_α,β (u, v ; x, y) is the transformation kernel of FrFT and $K_{α, β} (u, v; x, y) = K_{α} (u, x) K_{β} (v, y)$ (5) $\begin{matrix} K α, β (u, v; x, y) = \frac{1}{2 π} \sqrt{(1 - j cot α) (1 - j cot β)} \times \\ e^{j (x^{2} + u^{2} / 2) cot α - jux csc (α)} \times \\ e^{j (y^{2} + v^{2} 2) cot β - jvy csc (β)} \end{matrix}$ (6)

In 2D FrFT, there are two angles of rotation α = aπ/2 and β = bπ/2 considered.

3 Firefly algorithm

Firefly Algorithm is a population based nature-inspired metaheuristic optimization technique. The algorithm exploits the flashing behavior of fireflies. The bioluminescence process is responsible for flashing light of fireflies. The pattern of these rhythmic flashes is unique based upon the flashing rate and the amount of time for which flashes are perceived. The intensity (I) of flashing light decreases as the distance (r) increases and thus most fireflies can communicate over several hundred meters. While implementing the algorithm, the objective function associated with the flashing light has to be optimized [20].

3.1 Light intensity and attractiveness

The brightness is proportional to the objective function for a maximization problem. Firefly algorithm is based on two important things, first is the variation in light intensity and second is the attractiveness. The attractiveness of firefly is determined by its brightness which is connected with the objective function.

At particular location x, the brightness I of a firefly can be chosen as $I (x) \propto f (x)$ (7) where f (x) is the objective function.

The light intensity I (r) varies according to the inverse square law as given below: $I (r) = \frac{I_{s}}{r^{2}}$ (8)

Where I_s is the intensity of the source.

For the fixed light absorption coefficient γ, the relative fluorescence intensity of firefly is given as, $I = I_{0} exp (- γ r_{ij}^{2})$ (9) where I₀ is the maximum fluorescence intensity of a firefly, r_ij is the spatial distance between fireflies i and j.

The attractiveness η of a firefly is proportional to the light intensity seen by adjacent fireflies, so it will differ with distance r_ij between firefly i and firefly j. $η = η_{0} exp (- γ r_{ij}^{2})$ (10) where η₀ is the maximum degree of attraction at r = 0

The light intensity decreases with distance from its source and light is also absorbed by air, so attractiveness should be allowed to vary with varying degree of absorption.

3.2 Distance and movement

The spatial distance between any two fireflies i and j at position x_i and x_j is $r_{i} j = ∥ x_{i} - x_{j} ∥ = \sqrt{\sum_{k = 1}^{d} {(x_{ik} - x_{ik})}^{2}}$ (11) where x_ik is the i^th firefly’s k^th component of the spatial coordinate x_i

The dimmer firefly i’s movement towards another brighter firefly j is calculated by $x_{i} = x_{i} + η (x_{j} - x_{i}) + δ (rand - \frac{1}{2})$ (12) where x_i and x_j are the spatial locations of fireflies i and j, δ is the step size factor, δ € [0, 1], and rand is the random number that obey uniform distribution [0, 1], η is degree of attraction.

4 Proposed data hiding scheme

The proposed data hiding scheme attempts to find optimal locations adaptively in the cover image in time-frequency domain to hide a secret data so that the resultant stego image is of good quality. The Fractional Fourier Transform transforms the cover image from the spatial domain to time-frequency domain.

4.1 Embedding algorithm

The embedding technique finds optimal pixel locations in time-frequency domain using firefly Algorithm. The firefly algorithm uses multi objective function which is defined in such a way that both imperceptibility and robustness of the stego image are within an acceptable level. The imperceptibility of the stego image is obtained using Structural Similarity Index Measure (SSIM) [11] which is given as follows: $SSIM (x, y) = \frac{(2 μ_{x} μ_{y} + C_{1}) (2 σ_{xy} + C_{2})}{(μ_{x}^{2} + μ_{y}^{2} + C_{1}) (σ_{x}^{2} + σ_{y}^{2} + C_{2})}$ (13) where x and y represent cover and stego images and μ_x and μ_y are the means of x and y respectively. $σ_{x}^{2}$ and $σ_{y}^{2}$ are the variances of x and y and σ_xy is the covariance of x and y. C₁ = (k₁ L) ² and C₂ = (k₂ L) ² L = 2^{no.ofbits/pixel} - 1. [k₁, k₂] = [0.01, 0.03]

The robustness of the stego image is measured in terms of Bit Error Rate (BER) [11] which is defined as: $BER = \frac{\sum_{k = 0}^{n} (b_{k} \oplus b_{k}^{'})}{N}$ (14) where b_k and $b_{k}^{'}$ are the embedded and extracted bits respectively, N is the total number of secret bits embedded and ⊕ represents the XOR operation. The value of BER lies between 0 and 1. If BER is nearer to 1, then the error value of extracted data will be higher. BER is calculated after retrieving the secret data from the stego block. The proposed technique aims is to minimize the BER measured between the embedded and the extracted data and to maximize the SSIM index between the cover and the stego images. The objective function F to be minimized by firefly algorithm is given as: $F = λ (1 - SSIM (B, B^{'})) + (1 - λ) BER$ (15) where λ is the weighting constant and B and B’ are the cover image and stego image blocks respectively. If the weighting constant λ is equal to 0.5, then both SSIM and BER will contribute equally in the objective function.

The optimal locations obtained in the FA module are then used to embed the secret data. The proposed framework for embedding a secret data in the cover image is shown in Fig. 2.

Fig. 2

Proposed framework for embedding secret data in the cover image.

The steps to embed the secret data into cover image are presented below,

Cover image of size M×N, Secret payload bits.

Stego Image

Read the cover image and the secret payload to be embedded in the cover image.

Convert the secret data into binary form and then concatenate the binary bits to form secret word W. Let the length of W be L_W.

Divide the cover image into a number of blocks of size 8×8 as B_i, i = 1,2, ... ., t. Thus, the number of secret bits to be hidden in each block is found as $r = \frac{L_{w}}{t}$ , where t is the number of blocks.

Each spatial block is converted into time-frequency domain using Fractional Fourier Transform with the angle of rotation α ranges from 0 to 2π radians.

Firefly Algorithm finds the best pixel locations to embed data in each block. The steps are as follows,

Set the initial parameters:

n (Number of fireflies or Population size),

iter (Maximum number of iterations)

γ (weighting constant)

δ (step size factor, between 0 & 1)

C₁ andC₂ (constants)

rand (random number)

The dimension of each firefly is equal to the number of pixels that must be hidden in each block. If r is 5 then each firefly represents the group of 5 pixels which is chosen from the 64 pixel block.

In each iteration, for each firefly, the r secret bits are embedded in the firefly locations and the stego image block is obtained. Once the stego block is obtained, the SSIM of the original and stego block are calculated. During the extraction phase, the secret bits are extracted and BER is measured. Using SSIM and BER, the fitness value of the firefly with respect to the objective function as in Equation (15) is calculated. The minimum value of the objective function is stored as the best value. Then, the fireflies will choose random location and this process is repeated until one of the following conditions occur:

•Maximum number of iterations is reached

•No improvement in the successive iterations.

•An acceptable result is found.

Once the terminating condition occurs, then the corresponding location of the fireflies’ is the final location. The resulting location is the optimal location which, when used for data hiding, will provide high quality and robust image.

The r bits from secret word W is embedded in the final location of each block. Histogram shifting method is used for embedding process. First, the entire pixel value of final location is decreased by 1. Then, the maximum value is found among the final locations. Now, the secret bits are embedded by modifying the maximum value. If the secret bit is 1, then the maximum value is increased by 1. If the secret bit is 0, then the maximum value is left unchanged. In this manner, entire data is embedded in the corresponding block.

To ensure the reversibility of embedding, location map is used. The index of the final location and its maximum value has been stored in the location map. Location map will be transmitted as side information along with the stego image.

Once the embedding process is over, all sub blocks are subjected to inverse Fractional Fourier transform to get the stego image.

4.2 Extraction algorithm

Embedded Image of size M×N.

Secret payload bits, Original cover image

Divide the embedded image into a number of blocks of size 8×8.

Apply Fractional Fourier transform to each stego block to convert into time-frequency domain.

The maximum value of each stego image block is compared with the maximum value of the corresponding final location in the location map. If the maximum value of the stego image block is greater than the maximum value of the final location, then bit 1 is extracted, and if the maximum value of the stego image block is equal to the maximum value of final location, then bit 0 is extracted.

The maximum value of the stego image block is decreased by 1 whenever secret bit 1 is extracted.

Similarly, all secret bits are extracted from the corresponding block.

To reproduce the cover image, inverse Fractional Fourier transform is applied to all the blocks.

5 Experimental results and discussions

An optimal reversible data hiding system must satisfy the properties such as imperceptibility, robustness and reversibility. In order to investigate the performance of the proposed data hiding algorithm, several experiments are conducted in a computer system equipped with Intel core i5 processor with 8 GB RAM and a clock speed of 2.5 GHz. Matlab R2018a. As shown in Fig. 3, four standard grayscale images of size 512×512, (a) Baboon, (b) House, (c) Lake, and (d) Lena, obtained from USC-SIPI (Image database) [21], are used as cover images.

Fig. 3

(a) Cover Images, (a) Baboon (b) House (c) Lake (d) Lena.

5.1 Effect of population size on fitness values

The important issue while using the firefly algorithm for optimization is to select the appropriate population size (Number of fireflies). The firefly algorithm finds a more optimal solution as the population size grows, but it converges slowly due to its computational complexity (O (n²)), where n is the population size. If the population size is very small, then there is a chance of premature convergence. Hence, an optimal population size should be identified. In this experiment, the population size is varied as 5, 10, 15, 20 and 25. The number of iterations is fixed at 25. It has been observed from Table 1 that for Baboon and House images, lower BER and higher SSIM obtained at the population size of 15 and for Lake and Lena images, at the population size of 20. Further analysis has been carried out using this population size and it is discussed in the section below.

Table 1
Effect of population size on fitness values

Population Baboon House Lake Lena

Size (PS) BER SSIM BER SSIM BER SSIM BER SSIM

5 0.0012 0.9954 0.0014 0.9906 0.0008 0.9965 0.0011 0.9966

10 0.0010 0.9956 0.0011 0.9915 0.0007 0.9961 0.0009 0.9960

15 0.0009 0.9961 0.0010 0.9936 0.0007 0.9962 0.0009 0.9968

20 0.0009 0.9956 0.0010 0.9924 0.0007 0.9965 0.0009 0.9969

25 0.0009 0.9955 0.0010 0.9925 0.0007 0.9962 0.0009 0.9967

Population	Baboon	House	Lake	Lena
5	0.0012	0.9954	0.0014	0.9906	0.0008	0.9965	0.0011	0.9966
10	0.0010	0.9956	0.0011	0.9915	0.0007	0.9961	0.0009	0.9960
15	0.0009	0.9961	0.0010	0.9936	0.0007	0.9962	0.0009	0.9968
20	0.0009	0.9956	0.0010	0.9924	0.0007	0.9965	0.0009	0.9969
25	0.0009	0.9955	0.0010	0.9925	0.0007	0.9962	0.0009	0.9967

Effect of number of iterations on fitness values

With the population size of 15 for Baboon and House images, 20 for Lake and Lena images, the number of iterations has been varied between 5 and 30 with the interval of 5 to optimize pixel locations. From Table 2, it is inferred that for baboon and house images, lower BER and higher SSIM are obtained at 25th iterations, for Lake image at 20th iterations and for Lena image at 30th iterations. These results are carried over to next set of experiments.

Table 2

Effect of number of iterations on fitness values

No.of iterations (#iter)	Baboon (PS = 15)		House (PS = 15)		Lake (PS = 20)		Lena (PS = 20)
	BER	SSIM	BER	SSIM	BER	SSIM	BER	SSIM
5	0.0016	0.9956	0.0015	0.9915	0.0013	0.9963	0.0011	0.9966
10	0.0014	0.9962	0.0014	0.9927	0.0011	0.9961	0.0010	0.9960
15	0.0013	0.9959	0.0013	0.9934	0.0010	0.9962	0.0010	0.9951
20	0.0011	0.9953	0.0011	0.9931	0.0009	0.9965	0.0009	0.9956
25	0.0009	0.9961	0.0010	0.9936	0.0009	0.9965	0.0009	0.9969
30	0.0009	0.9961	0.0010	0.9896	0.0009	0.9965	0.0008	0.9972

5.3 Imperceptibility

Imperceptibility means that the visual quality of the original cover image should not be degraded by the embedding process. It measures the amount of distortion produced by embedding data in the original cover. In order to measure the imperceptibility, Mean Squared Error (MSE), Peak Signal to Noise Ratio (PSNR) and Structural Similarity Index Measure (SSIM) are used.

Once the suitable values for population size and number of iterations are obtained using FA, the performance of the algorithm is investigated in terms of imperceptibility. Table 3 and Table 4 depict the PSNR (dB) and SSIM values respectively, of cover images after embedding the secret payload at various embedding rates. The perceptual quality of embedded image drops down when the number of bits embedded per block in the cover image increases. The Lena image offers a maximum PSNR of 53.40 dB and SSIM of 0.9988 at the embedding rate of 0.125 bpp. The PSNR obtained for all stego images is greater than 40 dB and SSIM is about 0.98. Hence, it is clear that the embedded payload is imperceptible to the human eye.

Table 3
PSNR of stego images under varying payload at the fractional order a = 0.5

Secret payload (bits) Embedding Rate (bpp) PSNR (dB)

Baboon House Lake Lena

(PS = 15, #iter = 25) (PS = 15, #iter = 25) (PS = 20, #iter = 20) (PS = 20, #iter = 30)

32,768 (8 bits/ block) 0.125 52.63 51.24 52.68 53.40

65,536 (16 bits/block) 0.250 50.27 50.42 50.34 51.73

98,304 (24 bits/block) 0.375 48.92 48.46 48.86 49.29

1,31,072 (32 bits/block) 0.500 46.77 46.02 46.64 48.40

1,63,840 (40 bits/block) 0.625 45.27 44.98 45.14 47.02

1,96,608 (48 bits/block) 0.750 43.78 43.46 44.18 45.86

2,29,376 (56 bits/block) 0.875 41.14 41.72 42.78 43.56

2,62,144 (64 bits/block) 1.000 40.12 40.24 41.25 41.62

Secret payload (bits)	Embedding Rate (bpp)	PSNR (dB)
32,768 (8 bits/ block)	0.125	52.63	51.24	52.68	53.40
65,536 (16 bits/block)	0.250	50.27	50.42	50.34	51.73
98,304 (24 bits/block)	0.375	48.92	48.46	48.86	49.29
1,31,072 (32 bits/block)	0.500	46.77	46.02	46.64	48.40
1,63,840 (40 bits/block)	0.625	45.27	44.98	45.14	47.02
1,96,608 (48 bits/block)	0.750	43.78	43.46	44.18	45.86
2,29,376 (56 bits/block)	0.875	41.14	41.72	42.78	43.56
2,62,144 (64 bits/block)	1.000	40.12	40.24	41.25	41.62

Table 4

SSIM of stego images under varying payload at the fractional order a = 0.5

Secret payload (bits)	Embedding Rate (bpp)	SSIM
		Baboon	House	Lake	Lena
		(PS = 15, #iter = 25)	(PS = 15, #iter = 25)	(PS = 20, #iter = 20)	(PS = 20, #iter = 30)
32,768 (8 bits/ block)	0.125	0.9984	0.9978	0.9985	0.9988
65,536 (16 bits/block)	0.250	0.9980	0.9968	0.9974	0.9986
98,304(24 bits/block)	0.375	0.9961	0.9936	0.9965	0.9972
1,31,072 (32 bits/block)	0.500	0.9924	0.9920	0.9940	0.9932
1,63,840 (40 bits/block)	0.625	0.9924	0.9920	0.9940	0.9932
1,96,608 (48 bits/block)	0.750	0.9864	0.9854	0.9872	0.9888
2,29,376 (56 bits/block)	0.875	0.9824	0.9834	0.9852	0.9865
2,62,144 (64 bits/block)	1.000	0.9788	0.9802	0.9821	0.9852

The same is illustrated in Fig. 4. As the number of bits embedded in cover image increases, the PSNR and SSIM values drop down and hence the quality of embedded image gets reduced.

Fig. 4

Performance comparison of proposed method in terms of PSNR between original images and stego images (Baboon, House, Lake, Lena) under given embedding rate. a) PSNR b) SSIM.

Table 5 and Table 6 list the PSNR and SSIM values, respectively, of cover images after embedding a fixed payload of 12 KB at various fractional orders. The FrFT transform provides frequency coefficients with variations of its fractional order ‘a’. It is found that by using FrFT, high visual quality embedded image can be obtained. By adjusting ‘a’ to different values, FrFT achieves higher PSNR and SSIM, while preserving fidelity of the embedded images. The quality of embedded image gets changed as the fractional order changes. The Lena image offers a maximum PSNR of 49.29 dB and SSIM of 0.9972 for the fractional order of 0.5. From Table 5, it is understood that the algorithm provides better PSNR and SSIM when a = 0.5 (α = 45°).

Table 5

PSNR after embedding a fixed payload of 12 KB in various cover images at different fractional order

Fractional order (a)	Angle of Rotation (α deg)	PSNR (dB)
		Baboon	House	Lake	Lena
0.2	18°	46.30	46.34	46.76	47.05
0.4	36	47.34	47.36	47.10	48.78
0.5	45°	48.92	48.46	48.86	49.29
0.6	54°	45.64	46.35	46.16	46.14
0.8	72°	45.68	46.38	46.25	46.48
1.0	90°	45.72	46.41	46.28	46.88

Table 6

SSIM after embedding a fixed payload of 12 KB in various cover images at different fractional order

Fractional order (a)	Angle of Rotation (α deg)	SSIM
		Baboon	House	Lake	Lena
0.2	18°	0.9959	0.9928	0.9924	0.9952
0.4	36	0.9960	0.9929	0.9945	0.9968
0.5	45°	0.9961	0.9936	0.9965	0.9972
0.6	54°	0.9940	0.9929	0.9948	0.9944
0.8	72°	0.9942	0.9932	0.9964	0.9952
1.0	90°	0.9942	0.9934	0.9964	0.9969

Table 7 and Table 8 list the PSNR and SSIM values, respectively, of the Lena image for fractional order varying between 0.2 and 1.0 under varying payload. The quality of the stego image gets degraded as the embedding rate increases. The fractional order a = 0.5 gives better PSNR and SSIM compared to the other fractional orders. The embedded payload is highly imperceptible to the human eye as the PSNR is greater than 40 dB. The maximum amount of embedded payload without affecting the perceptual quality is about 32 KB.

Table 7

PSNR values of Lena image after embedding data at different rates and at different fractional order

Fractional order (a)	Angle of Rotation (deg)	PSNR (dB)
		0.25 bpp	0.5 bpp	0.75 bpp	1.0 bpp
0.2	18°	48.14	45.58	40.74	40.24
0.4	36	48.36	46.12	41.64	40.94
0.5	45°	51.73	48.40	45.86	41.62
0.6	54°	48.28	46.04	41.02	41.02
0.8	72°	48.34	46.28	41.28	37.46
1.0	90°	48.68	46.30	41.30	37.78

Table 8

SSIM values of Lena image after embedding payload at different rates and at different fractional orders

Fractional order (a)	Angle of Rotation (deg)	SSIM
		0.25 bpp	0.5 bpp	0.75 bpp	1.0 bpp
0.2	18°	0.9962	0.9920	0.9814	0.9714
0.4	36	0.9978	0.9926	0.9828	0.9728
0.5	45°	0.9986	0.9932	0.9838	0.9756
0.6	54°	0.9972	0.9924	0.9824	0.9724
0.8	72°	0.9974	0.9926	0.9826	0.9726
1.0	90°	0.9974	0.9927	0.9828	0.9728

Table 9 compares the PSNR and SSIM values of the proposed method with the existing methods in the literature. It is understood that the proposed method is better than the other approaches as it results in reduced distortion in the cover image by embedding the secret data in locations generated FA and thus produces a good quality image. The techniques proposed by Khodaei & Faez [22] and Bedi et al. [11] that are based on particle swarm optimization algorithm (PSO) and Abbasi et al. [23] which is based on Multilayered n-Bit Localization have good data embedding capacity, but they are less resistant to various attacks as LSB substitution is used to embed data.

Table 9

Comparison of PSNR & SSIM of the proposed method and the existing methods in the literature

Host image	Khodaei &Faez [22] (based on PSO)		Bedi et al. [11] (based on PSO)		Abbasi et al. [23] (based on n-bit localization)		Proposed method (based on FA)
	PSNR (dB)	SSIM	PSNR (dB)	SSIM	PSNR (dB)	SSIM	PSNR (dB)	SSIM
Baboon	42.71	0.9792	44.31	0.9926	45.93	0.9940	46.92	0.9961
House	43.46	0.9782	46.28	0.9905	47.06	0.9925	47.46	0.9936
Lake	42.76	0.9566	45.67	0.9882	46.84	0.9962	47.86	0.9965
Lena	42.87	0.9554	45.19	0.9892	47.25	0.9960	48.29	0.9972

The proposed method is also compared with data hiding scheme, presented by Amsaveni & Arunkumar [12], which is based on firefly algorithm in spatial domain. Even though this work yields a PSNR of 52.86 dB (for Lena image) which is greater than proposed work, the resultant stego image is more vulnerable to intentional and unintentional attacks and also the cover image is irreversible. However, the proposed scheme is more robust and reversible as the secret payload is embedded into FrFT (reversible transform) frequency co-efficients.

5.4 Robustness

Robustness is defined as the ability of the secret payload to withstand against various intentional attacks such as image processing operations and unintentional attacks such as addition of noise. Higher the degree of robustness, better the efficiency of the embedding algorithm. The metric used to test the robustness of a data hiding technique against various attacks is Bit error rate of the extracted payload against the original payload.

Generally, the stego image gets degraded and distorted due to addition of noise. The unintentional attacks such as gaussian noise, poisson noise, impulse noise, speckle noise and intentional attacks such as rotation, scaling, blurring, and cropping are applied to embedded images and robustness of the algorithm is tested. The Gaussian noise with a variance of 0.2 and 0.4, Speckle Noise with a variance of 0.05 and 0.10 and impulse noise with a variance of 0.05 and 0.1 are added to the embedded images. The image processing attacks such as rotation (5° and 10°), scaling (200% and 400%), blurring and cropping (10% and 25%) are also applied. Table 10 summarizes the experimental results against various attacks. As the BER is about 0.15 to 0.25 % of embedded payload, the proposed algorithm is more robust.

Table 10
Effect of various attacks on BER when a secret payload of 12 KB is embedded in the cover images

Attacks Bit Error Rate (BER)

Baboon House Lake Lena

Gaussian Noise (σ² = 0.2) 0.0015 0.0012 0.0016 0.0013

Gaussian Noise (σ² = 0.4) 0.0018 0.0014 0.0010 0.0014

Poisson Noise 0.0011 0.0014 0.0012 0.0010

Impulse Noise (σ² = 0.05) 0.0016 0.0017 0.0014 0.0014

Impulse Noise (σ² = 0.10) 0.0018 0.0020 0.0016 0.0016

Speckle Noise (σ² = 0.05) 0.0014 0.0011 0.0014 0.0012

Speckle Noise (σ² = 0.1) 0.0018 0.0015 0.0019 0.0018

Rotation (5°) 0.0015 0.0012 0.0015 0.0016

Rotation (10°) 0.0018 0.0016 0.0019 0.0019

Scaling (200%) 0.0009 0.0010 0.0011 0.0009

Scaling (400%) 0.0011 0.0011 0.0012 0.0011

Blurring (5) 0.0014 0.0011 0.0012 0.0011

Blurring (10) 0.0016 0.0015 0.0014 0.0016

Cropping (10%) 0.0018 0.0015 0.0015 0.0014

Cropping (25%) 0.0020 0.0021 0.0024 0.0021

Attacks	Bit Error Rate (BER)
Gaussian Noise (σ² = 0.2)	0.0015	0.0012	0.0016	0.0013
Gaussian Noise (σ² = 0.4)	0.0018	0.0014	0.0010	0.0014
Poisson Noise	0.0011	0.0014	0.0012	0.0010
Impulse Noise (σ² = 0.05)	0.0016	0.0017	0.0014	0.0014
Impulse Noise (σ² = 0.10)	0.0018	0.0020	0.0016	0.0016
Speckle Noise (σ² = 0.05)	0.0014	0.0011	0.0014	0.0012
Speckle Noise (σ² = 0.1)	0.0018	0.0015	0.0019	0.0018
Rotation (5°)	0.0015	0.0012	0.0015	0.0016
Rotation (10°)	0.0018	0.0016	0.0019	0.0019
Scaling (200%)	0.0009	0.0010	0.0011	0.0009
Scaling (400%)	0.0011	0.0011	0.0012	0.0011
Blurring (5)	0.0014	0.0011	0.0012	0.0011
Blurring (10)	0.0016	0.0015	0.0014	0.0016
Cropping (10%)	0.0018	0.0015	0.0015	0.0014
Cropping (25%)	0.0020	0.0021	0.0024	0.0021

5.5 Reversibility

Reversibility is the ability to restore the original cover image from the embedded image. The algorithm is said to be reversible if the extracted cover image is similar to the original cover image. The metric used to measure the similarity between the two images is Normalized Cross-correlation Coefficient (NCC). NCC will approach to one if the extracted cover image resembles the original cover image. NCC [24] is defined as, $NCC = \frac{\sum_{x, y} (f (x, y) - f_{mean}) (g (x, y) - g_{mean})}{\sqrt{\sum_{x, y} (f (x, y) - f_{mean})^{2} \sum_{x, y} (g (x, y) - g_{mean})^{2}}}$ (16) where f (x, y) is the original cover image, g (x, y) is the extracted cover image.

Various attacks like Gaussian noise, Poisson Noise, Impulse Noise, Speckle Noise, Rotation, Scaling, Blurring and Cropping are applied on embedded image. The original cover image is extracted and NCC of extracted cover image against original cover image is computed. Table 11 summarizes the experimental results for the proposed data hiding scheme against various attacks. As the NCC values are greater than 0.98, it is concluded that the algorithm exactly restores the original cover image and hence it is said to be reversible.

Table 11

Effect of various attacks on NCC when secret payload of 12 KB is embedded in the cover images

Attacks	Normalized Cross Correlation Co-efficient (NCC)
	Baboon	House	Lake	Lena
Gaussian Noise (σ² = 0.2)	0.9867	0.9853	0.9845	0.9839
Gaussian Noise (σ² = 0.4)	0.9850	0.9842	0.9862	0.9805
Poisson Noise	0.9849	0.9864	0.9842	0.9872
Impulse Noise (σ² = 0.05)	0.9879	0.9883	0.9866	0.9881
Impulse Noise (σ² = 0.10)	0.9820	0.9845	0.9846	0.9844
Speckle Noise (σ² = 0.05)	0.9853	0.9822	0.9871	0.9874
Speckle Noise (σ² = 0.1)	0.9841	0.9805	0.9824	0.9826
Rotation (5°)	0.9830	0.9828	0.9828	0.9822
Rotation (10°)	0.9820	0.9812	0.9814	0.9820
Scaling (200%)	0.9886	0.9892	0.9889	0.9898
Scaling (400%)	0.9910	0.9916	0.9921	0.9928
Blurring (5)	0.9864	0.9850	0.9869	0.9866
Blurring (10)	0.9855	0.9846	0.9848	0.9842
Cropping (10%)	0.9832	0.9834	0.9829	0.9828
Cropping (25%)	0.9810	0.9804	0.9808	0.9818

6 Conclusion

In this paper, a reversible data hiding technique using firefly algorithm for embedding secret data in cover image in time-frequency domain is proposed. In this algorithm, the cover image has been transformed into time-frequency domain using FrFT and the optimal pixel locations for embedding the secret data are found by using firefly algorithm. The objective function for firefly algorithm is selected in such a manner that both visual quality and robustness of the embedded image are lying within an acceptable value. The histogram shifting technique embeds secret bits in the optimal pixel locations. The resultant embedded image is better in quality as the PSNR is greater than 40 dB. As BER and NCC are 0.25% of embedded payload and greater than 0.98 respectively, the stego image can able to withstand certain noise and image processing attacks during transmission. Due to the advancement in integrated circuit technology, reversible data hiding techniques may be implemented on field programmable gate arrays (FPGA) technology. This results in reduction in space, cost and embedding and extraction time in real time applications.

Conflict of interest

The authors declare that they have no conflict of interest

References

Amsaveni

and Vanathi

P.T.

, A comprehensive study on image steganography and steganalysis techniques, Int J Information and Communication Technology 7(4/5) (2015), 406–424.

Amsaveni

and Vanathi

P.T.

, An efficient reversible data hiding approach for colour images based on Gaussian weighted prediction error expansion and genetic algorithm, Int J Advanced Intelligence Paradigms 7(2) (2015), 156–171.

Yang

, Schmucker

, Funk

, Busch

and Sun

, Integer DCT-based reversible watermarking for images using companding technique, Proceedings of Electronic Imaging, Science and Technology 5306 (2014), 405–415.

Xuan

, Zhu

, Chen

, Shi

Y.Q.

, Ni

and Su

, Distortionless data hiding based on integer Wavelet Transform, IEEE Electronics Letters 38(25) (2003), 1646–1648.

Lin

S.D.

, Shie

S.C.

and Guo

J.Y.

, Improving the robustness of DCT based image watermarking against JPEG compression, Int J Computer Standards and Interfaces 32(1) (2010), 54–60.

Xiang

, Kim

H.J.

and Huang

, Invariant Image watermarking based on statistical features in the low frequency domain, IEEE Transactions on Image Processing 14(12) (2008), 2140–2150.

Amsaveni

and Vanathi

P.T.

, Reversible Data Hiding Based on Radon and Integer Lifting Wavelet Transform, Journal of Microelectronics Electronic Components and Materials 47(2) (2017), 90–98.

Shieh

C.S.

, Huang

H.C.

, Wang

F.H.

and Pan

J.S.

, Genetic watermarking based on transform-Domain techniques, Pattern Recognition 37(3) (2003), 555–565.

and Wang

, A Steganographic method based upon JPEG and particle swarm optimization algorithm, Information Sciences 177 (2007), 3099–3109.

10.

Huang

H.C.

, Yueh

H.C.

and Ajith

, Optimized Watermarking Using Swarm-Based Bacterial Foraging, Journal of Information Hiding and Multimedia Signal Processing 1(1) (2010), 51–58.

11.

Bedi

, Bansal

and Sehgal

, Using PSO in a spatial domain based image hiding scheme with disortion tolerance, Int J Computers and Electrical Engineering 39(2) (2013), 640–654.

12.

Amsaveni

and Arunkumar

, An Efficient Data Hiding Scheme Using Firefly Algorithm In Spatial Domain, IEEE Sponsored Second International Conference on Electronics and Communication System ICECS (2015), 650–655.

13.

Yang

X.S.

, Nature-Inspired Metaheuristic Algorithms, Luniver Press (2008).

14.

, Xiong

, Zhou

and Cai

, Reversible data hiding based on multilevel histogram modification and pixel value grouping, Journal of Visual Communication and Image Representation 40 (2016), 459–469.

15.

Cariolaro

, Erseghe

, Kraniauskas

and Laurenti

, Multiplicity of fractional Fourier transforms and their relationships, IEEE Transactions on Signal Processing 48(1) (2000), 227–241.

16.

Cohen

, Daubechies

and Feauveau

, Bi-orthogonal bases of compactly supported wavelets, Journal of Communications on Pure and Applied Mathematics 45(5) (1992), 485–560.

17.

Bultheel

and Martinez

, Ashattered survey of the Fractional Fourier Transform Report TW 337 Department of Computer Science, K. U. Leuven, (2002).

18.

Candan

, Kutay

M.A.

and Ozaktas

H.M.

, Discrete fractional Fourier transform, IEEE Transactions on Signal Processing 48(5) (2000), 1329–1337.

19.

Yetik

I.S.

, Kutay

M.A.

and Ozaktas

H.M.

, Image representation and compression with the fractional Fourier transform, Journal of Optics Communications 197 (2001), 275–278.

20.

Yang

X.S.

, Firefly Algorithms for multimodal optimization, Lecture Notes in Computer Science 5792 (2009), 169–178.

21.

http://sipi.usc.edu/database/

22.

Khodaei

and Faez

, Image Hiding by using genetic algorithm and LSB substitution, Proceedings of International Conference on Signal and Image Processing (ICSIP 2010). LNCS 6134. (2010), 404–411.

23.

Abbasi

, Xu

, Amin

and Luo

, Efficient Lossless Compression Based Reversible Data Hiding Using Multilayered n-Bit Localization, Security and Communication Networks (2019), 1-13.

24.

Nemani

, Talari

and Vangala

, Estimation of performance metrics for Reversible Data Hiding before encryption, 2017 International Conference on Intelligent Computing and Control Systems (ICICCS), Madurai, (2017), 1176–1181.