SABMIS: sparse approximation based blind multi-image steganography scheme

Comparison with past work

Here, we predominately compare our SABMIS scheme with the existing steganography schemes for the embedding capacity, the quality of stego-images, and resistance to steganographic attacks. For the stego-image quality comparison, since most works have computed PSNR values only, we use only this metric for our analysis. Although we check the quality of the extracted secret images by comparing them with the corresponding original secret images (as earlier), this check is not common in the existing works. Hence, we do not perform this comparison.

In the literature, there exist multiple transform-based steganography schemes that hide one or two secret images. Hence, in Table 4 we compare our SABMIS scheme using the above mentioned metrics with such competing schemes. Recall, that like our SABMIS scheme these schemes are inherently resistant to steganographic attacks as well.

As evident from Table 4, for the case of hiding one secret image, we compare with the best work of this category (Arunkumar et al., 2019b). Here, as for us, by using a transform based approach, a grayscale secret image is hidden into a grayscale cover image. Arunkumar et al. (2019b) and our scheme both achieve an embedding capacity of 2 bpp. When comparing the stego-image and the corresponding cover image, Arunkumar et al. (2019b) achieve a PSNR value of 49.69 dB (when experimenting with eight cover images) while we achieve a lower PSNR value of 41.64 dB (when experimenting with a higher number of cover images, i.e., ten). PSNR values over 30 dB are considered good (Gutub & Shaarani, 2020; Zhang et al., 2013; Liu & Liao, 2008). Although the scheme by Arunkumar et al. (2019b) is superior than ours for hiding one secret image, it does not scale for the case of hiding multiple secret images, which we do (please see below).

For the case of hiding two secret images, we again compare with the best work of this category (Hemalatha et al., 2013). Here, using the transform based approach, two grayscale secret images are hidden into a color cover image. This setup is easier than our case where using a transform based approach, we embed two grayscale secret images into a grayscale cover image (see Table 3). Hemalatha et al. (2013) achieve an embedding capacity of 1.33 bpp while we achieve a higher embedding capacity of 4 bpp. When comparing the stego-image and the corresponding cover image, Hemalatha et al. (2013) achieve a PSNR value of 44.75 dB (when experimenting with only two cover images) while we achieve a lower PSNR value of 38.74 dB (when experimenting with a higher number of cover images, i.e., ten). To sum-up, our scheme is better than the one by Hemalatha et al. (2013) because of the below reasons.

In terms of the quality of the scheme,

1. we target a harder problem than Hemalatha et al. (2013), and

2. we achieve a higher embedding capacity than Hemalatha et al. (2013).

In terms of the validation of the scheme,

1. we experiment with a large number of cover images (ten as compared to two in Hemalatha et al. (2013)),

2. as discussed earlier, we obtain PSNR values over 30 dB of stego-images, which are considered acceptable, and

3. we check the quality of stego-image on a greater number of numerical measures (five as compared to one in Hemalatha et al. (2013)).

When using the transform-based approach, no one has hidden three or four secret images in a cover image. To demonstrate the broad applicability of our scheme, in Table 5, we compare our SABMIS scheme using the above discussed metrics with the best spatial domain-based scheme that hide three and four secret images. Recall that, unlike our SABMIS scheme, these schemes are not intrinsically resistant to steganographic attacks. Please note that in the current scenario of transmitting stego-data over the internet, security is of paramount importance.

As evident from Table 5, for the case of hiding three secret images, we compare with the best work of this category (Guttikonda, Cherukuri & Mundukur, 2018). Here, three binary secret images are hidden into a grayscale cover image. As for the above case, this setup is easier than our case of hiding three grayscale secret images into a grayscale cover image (again see Table 3). Guttikonda, Cherukuri & Mundukur (2018) achieve an embedding capacity of 1 bpp while we achieve a higher embedding capacity of 6 bpp. When comparing the stego-image and the corresponding cover image, Guttikonda, Cherukuri & Mundukur (2018) achieve a PSNR value of 46.36 dB (when experimenting with only two cover images) while we achieve a lower PSNR value of 37.17 dB (when experimenting with a higher number of cover images, i.e., ten). To sum-up, our scheme is superior than the one by Guttikonda, Cherukuri & Mundukur (2018) because of the below reasons.

In terms of the quality of the scheme,

1. we target a harder problem than Guttikonda, Cherukuri & Mundukur (2018),

2. we achieve a higher embedding capacity than Guttikonda, Cherukuri & Mundukur (2018), and

3. we further improve the security of the inherently steganographic attack resistant transform based schemes.

In terms of the validation of the scheme,

1. we experiment with a large number of cover images (ten as compared to two in Guttikonda, Cherukuri & Mundukur (2018)),

2. as discussed earlier, we obtain PSNR values over 30 dB of stego-images, which are considered acceptable,

3. we check the quality of stego-image on a greater number of numerical measures (five as compared to one in Guttikonda, Cherukuri & Mundukur (2018)),

4. and we demonstrate the good quality of extracted secret images, which Guttikonda, Cherukuri & Mundukur (2018) do not.

Next, we compare with the best scheme that hides four secret images in a cover image, i.e., Hu (2006). As for our case, all images (secret and cover) are grayscale. Hu (2006) achieve an embedding capacity of 12 bpp while we achieve a lower embedding capacity of 8 bpp. When comparing the stego-image and the corresponding cover image, Hu (2006) achieve a PSNR value of 34.80 dB (when experimenting with five cover images) while we achieve a higher PSNR value of 35.66 dB (when experimenting with a higher number of cover images, i.e., ten). To sum-up, our scheme is better than the one by Hu (2006) because of the below reasons.

In terms of the quality of the scheme,

1. our embedding capacity, although lower than Hu (2006), is on the higher side,

2. we obtain higher PSNR values of stego-images as compared to those in Hu (2006),

3. and we further improve the security of the inherently steganographic attack resistant transform based schemes.

In terms of the validation of the scheme,

1. we experiment with a large number of cover images (ten as compared to five in Hu, 2006),

2. we check the quality of stego-image on a greater number of numerical measures (five as compared to one in Hu, 2006),

3. and we demonstrate the good quality of extracted secret images, which (Hu, 2006) do not.

Hiding secret images

First, we perform sub-sampling of the cover image to obtain four sub-images. This type of sampling is done because we are hiding up to four secret images. Let CI be the cover image of size r × r. Then, the four sub-images each of size $\frac{r}{2}\times \frac{r}{2}$ are obtained as follows (Pan et al., 2015): (1a)${\displaystyle {CI}^1\left(n_1,n_2\right)=CI\left(2n_1-1,2n_2-1\right),}$ (1b)${\displaystyle {CI}^2\left(n_1,n_2\right)=CI\left(2n_1,2n_2-1\right),}$ (1c)${\displaystyle {CI}^3\left(n_1,n_2\right)=CI\left(2n_1-1,2n_2\right),}$ (1d)${\displaystyle {CI}^4\left(n_1,n_2\right)=CI\left(2n_1,2n_2\right),}$

where CI^k, for k = {1, 2, 3, 4}, are the four sub-images; $n_1,n_2=1,2,\dots ,\frac{r}{2}$ (in our case, r is divisible by 2); and CI(⋅, ⋅) is the pixel value at (⋅, ⋅). A cover image and the corresponding four sub-sampled images are shown in Fig. 1.

Originally, these sub-images are not sparse; hence, next, we perform block-wise sparsification of each of these images. For this, we divide each sub-image into blocks of size b × b and obtain $\frac{r^2}{4\times b^2}$ blocks for each sub-image (in our case, b divides r). Now, we apply discrete cosine transformation to each block. That is, (2)${\displaystyle s_i=DCT\left(x_i\right),}$

where $i=1,2,\dots ,\frac{r^2}{4\times b^2}$, x_i and s_i are the ith original and sparse blocks of the same size, i.e, b × b, respectively, and DCT is the Discrete Cosine Transform. Further, we pick the final sparse blocks using a zig-zag scanning order as used in our earlier work (Pal, Naik & Agrawal, 2019), and obtain corresponding sparse vectors each of size b² × 1. The zig-zag scanning order for a block of size 8 × 8 is shown in Fig. 2. This order helps us to arrange the DCT coefficients with the set of large coefficients first, followed by the set of small coefficients, which assists in the preservation of a good quality stego-image.

Next, we represent each vector in two groups based upon large (say #p₁) and small (say #p₂) coefficients, i.e., s_i,u ∈ ℝ^p₁ and s_i,v ∈ ℝ^p₂, where p₁ ≤ p₂. Each of these vectors is sparse and p₁ + p₂ = b². Further, we oversample each sparse vector using linear measurements as below. (3)${\displaystyle y_i=\left[\begin{array}{c}\hfill y_{i,u}\hfill \\ \hfill y_{i,v}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill s_{i,u}\hfill \\ \hfill \Phi s_{i,v}\hfill \end{array}\right],}$

where y_i ∈ ℝ^{(p₁+p₃)×1} is the set of linear measurements, and Φ ∈ ℝ^p₃×p₂ is the column normalised measurement matrix consisting of normally distributed random numbers with p₃ > p₂ and p₃ ∈ ℕ (i.e., the sparse coefficients are oversampled).² This oversampling helps us to perform sparse approximation. By employing this approximation (along with our novel embedding rule discussed towards the end of this subsection), we achieve a higher embedding capacity. Moreover, our approach gains an extra layer of security because the linear measurements include measurement-matrix encoded small coefficients of the sparse vectors obtained after DCT. Since the distribution of coefficients of the generated sparse vectors is almost the same for all the blocks of an image, we use the same measurement matrix for all the blocks.

Next, we perform processing of the secret images for hiding them into the cover image. Let the size of each secret image be m × m. Initially, we perform block-wise DCT of each of these images and obtain their corresponding DCT coefficients. Here, the size of each block taken is l × l, and hence, we have $\frac{m^2}{l^2}$ blocks for each secret image. In our case, l divides m, and we ensure that $\frac{m^2}{l^2}$ will be less than or equal to $\frac{r^2}{4\times b^2}$ so that the number of blocks of the secret image is less than or equal to the number of blocks of a cover sub-image. Thereafter, we arrange these DCT coefficients as a vector in the earlier discussed zig-zag scanning order. Let $t_{\hat{i}}\in R^{l^2\times 1}$, for $\hat{i}=1,2,\dots ,\frac{m^2}{l^2}$, be the vector representation of the DCT coefficients of one secret image. We pick the initial p₄ DCT coefficients with relatively larger values (out of the available l² coefficients) for hiding,³ where p₄ ∈ ℕ. Omitting the remaining coefficients (l² − p₄) does not significantly deteriorate the quality of the extracted secret image.

Here, we show the hiding of only one secret image into one sub-image of the cover image. However, in our steganography scheme, we can hide a maximum of four secret images, one in each of the four sub-images of the cover image, which is demonstrated in the experimental results section. If we want to hide less than four secret images, we can randomly select the corresponding sub-images from the available four.

Next, using our novel embedding rule (discussed below), we hide the chosen p₄ DCT coefficients of the secret image into a selected set of p₁ + p₃ linear measurements obtained from the sub-image of the cover image, leading to the generation of the stego-data (we ensure that p₄ is less than p₁ + p₃).

We hide secret image data into the cover image by taking linear combinations of each secret image coefficient with a companion linear measurement coefficient of the cover image. These linear combinations replace certain other linear coefficients of the cover image to obtain the so called stego-data (subsequently, stego-image). The three groups of index coefficients are listed in Table 6.

The data in Table 6 is based upon three design choices as below.

1. As can be seen from Table 6, we divide each group of coefficients into three ranges in a staggered manner to achieve a higher level of security.

2. The specific choice of indices in the second and fourth rows of Table 6 is made so as to hide secret image coefficients in relatively small valued cover image coefficients (companion linear measurement coefficients). This results in a relatively improved quality stego-image.

3. In Table 6, the replaced linear measurement coefficient indices differ just slightly from the chosen companion coefficient indices (fourth and sixth rows respectively). The reason for this is that we want our extraction rule (discussed in section ‘Extraction of the secret images’) to be as less lossy as possible, resulting in less deteriorated extracted secret images.

The whole process is given in Algorithm 1. Specifically, the indices discussed in Table 6 are given on line 3, lines 4–6, and lines 7–9 of this algorithm, respectively. The block diagram for this complete data embedding process is given in Fig. 3. A small numerical example, which further explains this hiding process is given in Appendix, ‘A small numerical example of our embedding process’.

Construction of the Stego-Image

As mentioned earlier, the next step in our scheme is the construction of the stego-image. Since we can hide a maximum of four secret images into four sub-images of a single cover image, we first construct four sub-stego-images and then perform inverse sampling to obtain a single stego-image. Let s_i′ be the sparse vector of the ith block of a sub-stego-image. The sparse vector s_i′ is the concatenation of $s_{i,u}^{\prime }$ and $s_{i,v}^{\prime }$. Here, the size of $s_{i,u}^{\prime }$, $s_{i,v}^{\prime }$, and s′ is the same as that of s_i,u, s_i,v, and s, respectively. Then, we have (4a)${\displaystyle s_{i,u}^{\prime }=y_{i,u}^{\prime },}$ (4b)${\displaystyle s_{i,v}^{\prime }=\underset{s_{i,v}^{\prime }\in {\mathbb{R}}^{p_2}}{argmin}\parallel s_{i,v}^{\prime }{\parallel }_1\text{subject to}\Phi s_{i,v}^{\prime }=y_{i,v}^{\prime },}$

 
__________________________ 
Algorithm 1 Embedding Rule_____________________________________________________________________________ 
Input: 
       ∙ yi: Sequence of linear measurements of the cover image with i = 1,2,...,  r2 __ 
4×b2 . 
         ∙ tˆi : Sequence of transform coefficients of the secret image with ˆ i = 1,2,..., m2 
 l2  . 
         ∙ The choice of our r, b, m, and l is such that m2 
 l2   is less than or equal to   r2 __ 
4×b2 . 
         ∙ p1 and p4 are lengths of certain vectors defined on pages ix and x, respectively. 
         ∙ α, β, γ, and c are algorithmic constants that are chosen based upon experience. 
            The choices of these constants are discussed in the experimental results sections. 
Output: 
       ∙ y′i: The modified version of the linear measurements with i = 1,2,...,  r2 __ 
4×b2 . 
  1:  Initialize y′i to yi, where i = 1,2,...,  r2 __ 
4×b2 . 
  2:  for ˆ i = 1 to m2 
 l2   do 
 3:       // Embedding of the first coefficient. 
          y′ˆi (p1) = yˆi (p1 − 2c) + α × tˆi (1). 
  4:       for j = p1 − c + 1 to p1 − 1 do 
 5:            // Embedding of the next c − 1 coefficients. 
              y′ˆi (j) = yˆi (j − c) + β × tˆi (j − p1 + c + 1). 
  6:       end for 
 7:       for k = p1 + p4 + 1 to p1 + 2 × p4 − c do 
 8:            // Embedding of the remaining p4 − c coefficients. 
              y′ˆi (k) = yˆi (k − p4 + c) + γ × tˆi (k − p1 − p4 + c). 
  9:       end for 
10:  end for 
11:  return y′i_______________________________________________________________________________________________________    

where $y_i^{{}^{\prime }}$ is defined in Algorithm 1, and it is equal to $\left[\begin{array}{c}\hfill y_{i,u}^{{}^{\prime }}\hfill \\ \hfill y_{i,v}^{{}^{\prime }}\hfill \end{array}\right]$ as split in Eq. (3). The second part Eq. (4b) (i.e., obtaining s_i,v′), is an ℓ₁-norm minimization problem. Here, we can observe that in the above optimization problem, the constraints are oversampled. As earlier, this oversampling helps us to do sparsification, which leads to increased embedding capacity as well as increased security because the measurement matrix is encoded. For the solution of the minimization problem Eq. (4b), we use ADMM (Boyd et al., 2010; Gabay, 1976) to solve the LASSO (Hwang, Kim & Kim, 2016; Nardone, Ciaramella & Staiano, 2019) formulation of this minimization problem.⁴ We use this method because it has a fast convergence, is easy to implement, and also is extensively used in image processing (Boyd et al., 2010; Hwang, Kim & Kim, 2016).

Next, we convert each vector s_i′ into a block of size b × b. After that, we apply inverse discrete cosine transformation (i.e., the two-dimensional Inverse DCT) to each of these blocks to generate blocks $x_i^{{}^{\prime }}$ of the image. That is, (5)${\displaystyle x_i^{{}^{\prime }}=IDCT\left(s_i^{{}^{\prime }}\right).}$

Next, we construct the sub-stego-image of size $\frac{r}{2}\times \frac{r}{2}$ by arranging all these blocks $x_i^{{}^{\prime }}$. We repeat the above steps to construct all four sub-stego-images. At last, we perform inverse sampling and obtain a single constructed stego-image from these four sub-stego-images. In the experiments section, we show that the quality of the stego-image is also very good. The block representation of these steps is given in Fig. 4. A small numerical example, which further explains this process is given in Appendix, ‘A small numerical example of our embedding process’.

Extraction of the secret images

In this subsection, we discuss the process of extracting secret images from the stego-image. Initially, we perform sampling (as done in section ‘Hiding Secret Images’ using Eq. (1a)–Eq. (1d)) of the stego-image to obtain four sub-stego-images. Since the extraction of all the secret images is similar, here, we discuss the extraction of only one secret image from one sub-stego-image. First, we perform block-wise sparsification of the chosen sub-stego-image. For this, we divide the sub-stego-image into blocks of size b × b. We obtain a total of $\frac{r^2}{4\times b^2}$ blocks. Further, we sparsify each block (say $x_i^{\prime }$) by computing the corresponding sparse vector (say $s_i^{\prime }$). That is, (6)${\displaystyle s_i^{\prime }=DCT\left(x_i^{\prime }\right).}$

Next, as earlier, we arrange these sparse blocks in a zig-zag scanning order, obtain the corresponding sparse vectors each of size b² × 1, and then categorize each of them into two groups $s_{i,u}^{\prime }\in {\mathbb{R}}^{p_1}$ and $s_{i,v}^{\prime }\in {\mathbb{R}}^{p_2}$. Here, as before, p₁ and p₂ are the numbers of coefficients having large values and small values (or zero values), respectively. After that, we oversample each sparse vector using linear measurements (say $y_i^{\prime }\in {\mathbb{R}}^{\left(p_1+p_3\right)\times 1}$), (7)${\displaystyle y_i^{\prime }=\left[\begin{array}{c}\hfill y_{i,u}^{\prime }\hfill \\ \hfill y_{i,v}^{\prime }\hfill \end{array}\right]=\left[\begin{array}{c}\hfill s_{i,u}^{\prime }\hfill \\ \hfill \Phi s_{i,v}^{\prime }\hfill \end{array}\right].}$

From $y_i^{\prime }$, we extract the DCT coefficients of the embedded secret image using Algorithm 2. This extraction rule is the reverse of the embedding rule given in Algorithm 1.

 
_________________________________________________________________________________________________________________________ 
Algorithm 2 Extraction Rule______________________________________________________________________________ 
Input: 
       ∙ y′′i: Sequence of linear measurements of the stego-image with i = 1,2,...,  r2 __ 
4×b2 . 
         ∙ p1, p4, α, β, γ, and c are chosen as in Algorithm 1. 
Output: 
       ∙ t′ˆi : Sequence of transform coefficients of the secret image with ˆ i = 1,2,..., m2 
 l2  . 
  1:  Initialize t′ˆi to zeros, where ˆ i = 1,2,..., m2 
 l2  . 
  2:  for ˆ i = 1 to m2 
 l2   do 
 3:       // Extraction of the first coefficient. 
          t′ ˆ i (1) = y′′ 
ˆi (p1)−y′′ 
ˆi (p1−2c) 
           α          . 
  4:       for j = p1 − c + 1 to p1 − 1 do 
 5:            // Extraction of the next c − 1 coefficients. 
              t′ˆi (j − p1 + c + 1) = y′′ 
ˆi (j)−y′′ 
ˆi (j−c) 
         β        . 
  6:       end for 
 7:       for k = p1 + p4 + 1 to p1 + 2 × p4 − c do 
 8:            // Extraction of the remaining p4 − c coefficients. 
              t′ˆi (k − p1 − p4 + c) = y′′ 
ˆi (k)−y′′ 
ˆi (k−p4+c) 
            γ           . 
  9:       end for 
10:  end for 
11:  return t′ˆi_______________________________________________________________________________________________________    

In Algorithm 2, $t_{\hat{i}}^{{}^{\prime }}\in {\mathbb{R}}^{l^2\times 1}$, for $\hat{i}=1,2,\dots ,\frac{m^2}{l^2}$, are the vector representations of the DCT coefficients of the blocks of one extracted secret image. Next, we convert each vector $t_{\hat{i}}^{{}^{\prime }}$ into blocks of size l × l, and then perform a block-wise Inverse DCT (IDCT) (using Eq. (5)) to obtain the secret image pixels. Finally, we obtain the extracted secret image of size m × m by arranging all these blocks column wise. As mentioned earlier, this steganography scheme is a blind multi-image steganography scheme because it does not require any cover image data at the receiver side for the extraction of secret images.

Here, the process of hiding (and extracting) secret images is not fully lossless,⁵ resulting in the degradation of the quality of extracted secret images. This is because we first oversample the original image using Eq. (3), and then we construct the stego-image by solving the optimization problem (4b), which leads to a loss of information. However, our algorithm is designed in such a way that we are able to extract high-quality secret images. We support this fact with examples in the experiments section (specifically, ‘Secret Image Quality Assessment’). We term our algorithm Sparse Approximation Blind Multi-Image Steganography (SABMIS) scheme due to the involved sparse approximation and the blind multi-image steganography.

The above extraction process is represented via a block diagram in Fig. 5. As discussed earlier, this extraction is just the reverse of the embedding process.

Embedding capacity analysis

The embedding capacity (or embedding rate) is the number (or length) of secret bits that can be hidden/ embedded in each pixel of the cover image. It is measured in bits per pixel⁷ (bpp) and is calculated as follows: (8)${\displaystyle \text{EC in bpp}=\frac{\text{Total number of secret bits embedded}}{\text{Total number of pixels in the cover image}}.}$

As motivated on the previous page, we chose the size of the length and the width of secret image to be half of the length and the width of cover image, respectively. Since our cover images are of size 1024 × 1024, our secret images are taken to be of size 512 × 512. For a grayscale image, each pixel size is 8 bits. Hence, when hiding one secret image in a cover image, we obtain the embedding capacity as below. (9)${\displaystyle \text{EC in bpp}=\frac{512\times 512\times 8}{1024\times 1024},}$

which is equal to 2 bpp. Similarly, while hiding two, three, and four secret images in a cover image, we obtain the embedding capacities of 4 bpp, 6 bpp, and 8 bpp, respectively.

Stego-image quality assessment

In general, the visual quality of the stego-image degrades as the embedding capacity increases. Hence, preserving the visual quality becomes increasingly important. There is no universal criterion to determine the quality of the constructed stego-image. However, we evaluate it by visual and numerical measures. We use Peak Signal-to-Noise Ratio (PSNR), Mean Structural Similarity (MSSIM) index, Normalized Cross-Correlation (NCC) coefficient, entropy, and Normalized Absolute Error (NAE) numerical measures.

When using the visual measures, we construct the stego-images corresponding to the different cover images used in our experiments and then check their distortion visually. We also check their corresponding edge map diagrams. Here, we present the visual comparison only for ‘Stream’ as the cover image with ‘Lake’ secret image and the corresponding stego-image. We get similar results for the other images as well. The comparison is given in Fig. 7. The cover image and its corresponding edge map are shown in parts (A) and (B) of this figure. The stego-image and its corresponding edge map are given in parts (C) and (D) of the same figure. When we compare each figure with its counterpart, we find that they are very similar.

Next, when using the numerical measures to assess the quality of the stego-image with respect to the cover image, we first evaluate the most common measure of PSNR value in section ‘Peak Signal-to-Noise Ratio (PSNR) Value’. Subsequently, we evaluate the other more rarely used numerical measures of MSSIM index, NCC coefficient, entropy, and NAE in section ‘Other Numerical Measures’.

Peak Signal-to-Noise Ratio (PSNR) value

We compute the PSNR values to evaluate the imperceptibility of stego-images (SI) with respect to the corresponding cover images (CI) as follows (Elzeki et al., 2021): (10)${\displaystyle PSNR\left(CI,SI\right)=10{log}_{10}\frac{R^2}{MSE\left(CI,SI\right)}dB,}$

where R is the maximum intensity of the pixels, which is 255 for grayscale images, dB refers to decibel, and MSE(CI, SI) represents the mean square error between the cover image CI and the stego-image SI that is calculated as (11)${\displaystyle MSE\left(CI,SI\right)=\frac{\sum _{i=1}^{r1}\sum _{j=1}^{r2}{\left(CI\left(i,j\right)-SI\left(i,j\right)\right)}^2}{r1\times r2},}$

where r1 and r2 represent the row and column numbers of the image (for us either cover or stego), respectively, and CI(i, j) and SI(i, j) represent the pixel values of the cover image and the stego-image, respectively.

A higher PSNR value indicates a higher imperceptibility of the stego-image with respect to the corresponding cover image. In general, a value higher than 30 dB is considered to be good since human eyes can hardly distinguish the distortion in the image (Gutub & Shaarani, 2020; Zhang et al., 2013; Liu & Liao, 2008).

The PSNR values of the stego-images corresponding to the ten cover images are given in Figs. 8 and 9. In Fig. 8, we show the PSNR values of all the stego-images when separately all the four secret images (mentioned above in Fig. 6) are hidden. In this figure, we obtain the highest PSNR value (46.25 dB) when the ‘Peppers’ secret image is hidden in the ‘House’ cover image, while the lowest PSNR value (37.66 dB) is obtained when the ‘Stream’ secret image is hidden in the ‘Stream’ cover image.

In Fig. 9, we show the PSNR values for the four cases of hiding one, two, three, and four secret images in the ten cover images. As we have four secret images, when hiding one secret image, we have a choice of hiding any one of them and present the resulting PSNR values. However, we separately hide all four images, obtain their PSNR values, and then present the average results. Similarly, the average PSNR values are presented for the cases when we hide two and three images. In this figure, we obtain the highest average PSNR value (45.21 dB) when one secret image is hidden in the ‘House’ cover image, while the lowest PSNR value (31.78 dB) is obtained when all four secret images are hidden in the ‘Stream’ cover image. Also, we observe that for all test cases, we obtain PSNR values higher than 30 dB which, as earlier, are considered good.

Secret image quality assessment

Since human observers are considered the final arbiter to assess the quality of the extracted secret images, we compare one such original secret image and its corresponding extracted secret image. The results of all other combinations are almost the same. In Figs. 10A and 10C, we show the original ‘Lake’ secret image and the extracted ‘Lake’ secret image (from the ‘Stream’ stego-image). From these figures, we observe that there is little distortion in the extracted image. Besides this, for these two images, we also present their corresponding edge map diagrams (in Figs. 10B and 10D, respectively). Again, we observe minimal variations between the original and the extracted secret images.

Security analysis

The SABMIS scheme is a transform domain based technique which employs an indirect embedding strategy, i.e., it does not follow the Least Significant Bits (LSB) flipping method, and hence, it is immune to statistical attacks (Westfeld & Pfitzmann, 2000; Yu et al., 2009). Moreover, in the SABMIS scheme, the measurement matrix Φ, and the embedding/ extraction algorithmic settings are considered as secret-keys, which are shared between the sender and the legitimate receiver. Even if the eavesdropper intercepting the stego-data becomes aware that the SABMIS scheme has been used to embed a secret image, he would not know these secret keys. Hence, we achieve increased security in our proposed system.

To justify this, we extract the secret image in two ways, i.e., by using correct secret-keys and by using wrong secret-keys. Here, we embed only one secret image in a cover image although these experiments can be extended to the cases of embedding two, three or four secret images. Since the measurement matrix, which we use (random matrix having numbers with mean 0 and standard deviation 1) is one of the most commonly used measurement matrices and the eavesdropper might be able to guess it, we use this same measurement matrix while building wrong secret-keys. Here, we use the same dimension of this matrix as well, i.e., p3 × p2. In reality, the guessed matrix size would be different from the original matrix size, which would make the extraction task of the eavesdropper more difficult.

The algorithmic settings that we use will be completely unknown to the eavesdropper as above. These involve using a set of cover image coefficient indices where secret image coefficients are embedded (p1 and p4) and few constants (α = 0.01, β = 0.1, γ = 1 and c = 6). While building wrong secret-keys, without changing the indices (i.e., same p1 and p4), we take the common guess of one for all constants (i.e., α = 1, β = 1, γ = 1 and c = 1). In reality, the eavesdropper would not be able to correctly guess these indices as well, resulting in further challenges during extraction.

In Figs. 11A and 11B, we compare the ‘Lake’ secret image when extracted using correct and wrong secret-keys (from the ‘Stream’ stego-image), respectively. From this figure, we see that when using correct secret-keys, the visual distortion in the extracted secret image is negligible (as evident by comparing with Fig. 6E), and when using the wrong secret-keys, the distortion in the extracted secret image is very high (it is almost black).

Further, we numerically demonstrate that the correctly and wrongly extracted secret images are very different. We compute all the earlier discussed measures, i.e., PSNR, MSSIM, NCC, Entropy, and NAE values between the correctly and wrongly extracted secret images (when all four secret images had been separately embedded in the ten cover images). The average values of all these metrics are given in Table 8. In this table, we observe that PSNR values are very low (recall over 30 dB are considered good). The MSSIM and NCC values are close to 0. The entropy values of correctly and wrongly extracted secret images are far from each other. Finally, NAE values are close to 1. Hence, two images are substantially different from each other. Therefore, in the SABMIS scheme, a change in secret-keys will lead to a shift in the accuracy between the correctly and wrongly extracted secret images, in turn, making our scheme secure.

Timing data

The time taken by our SABMIS scheme is not of great importance here because all computations are done offline, whether it is hiding of secret images, stego-image construction, or the extraction of the secret images. However, for the sake of completeness, this data, while together hiding the four secret images in the ten cover images, is given in Table 9.

It is evident that the scheme is completely executed in a few minutes. Further, hiding and the extraction steps take about the same time (which they should because of similar steps), which is 10% of the total time. The most expensive step is stego-image construction, where the optimization problem is solved, which takes 80% of the total time.

Application of our scheme on real-life data

In the two subsections below, we experiment on hiding mammograms and brain images (in cases where some loss is acceptable) in nondescript cover images. Sending these images safely across the internet is useful in breast cancer and brain related disease diagnosis, respectively. For the first case, we do not have reference steganographic data to compare against, while for the second case, we do have such data.

Hiding mammograms

Here, we hide one through four mammograms (Heath et al., 1998; Heath et al., 2001) (see two in Figs. 12A and 12C) into all the cover images used in our experiments. These mammograms are freely available for research purposes. In Table 10, we present the embedding capacity and PSNR values from these experiments. As evident, we obtain good embedding capacity and average as well as maximum PSNR values. The other image comparison metrics turn out to be similar as well.

In Fig. 13, we present the visual comparison for ‘Stream’ as the cover image and the corresponding stego-image. We see that the cover and its corresponding stego-image are very similar. We get analogous results for the other images as well. We also check their edge maps (as discussed in section ‘Stego-Image Quality Assessment’) and obtain good results.

Next, we assess the quality of the extracted secret mammograms. In Figs. 12A and 12C, we show two original mammograms, and in Figs. 12B and 12D, we show the two respective extracted mammograms (from the ‘Stream’ stego-image). From these figures, we observe that there is very little distortion in the extracted mammograms. We get similar results for the other two mammograms as well.

Hiding brain images

Arunkumar et al. (2019b) hide a brain image into a cover image. Since the original brain image as used in Arunkumar et al. (2019b) is not publicly available, we work with an image that is quite similar to the image used in Arunkumar et al. (2019b), and is available in free public domain with no copyright (see Fig. 14A) (Rawpixel, 2022; Creative Commons, 2022). By using SABMIS, we hide one through four copies of this image into all cover images (presented earlier), and compare with the results of Arunkumar et al. (2019b).

Table 10:

Results of applicability of our scheme on real-life data (i.e., mammograms).

No. of secret images	Steganography scheme	Type of secret image	Type of cover images	EC (in bpp)	(Avg. PSNR, No. of cover images)	Max. PSNR
1	SABMIS	Grayscale	Grayscale	2	(44.30, 10)	49.41
2	SABMIS	Grayscale	Grayscale	4	(35.54, 10)	39.90
3	SABMIS	Grayscale	Grayscale	6	(34.87, 10)	39.10
4	SABMIS	Grayscale	Grayscale	8	(34.32, 10)	38.56

DOI: 10.7717/peerjcs.1080/table-10

This comparison is given in Table 11. As evident, we are not competitive with Arunkumar et al. (2019b) for the case of hiding one secret image (also discussed in ‘Comparison with Past Work’). However, Arunkumar et al. (2019b)’s scheme can hide only one secret image while our scheme can hide multiple secret images. We observe that using SABMIS to hide four secret images in a cover image, we obtain a good embedding capacity of 8 bpp and a good average PSNR value of 33.56. The other image comparison metrics turn out to be similar as well.

As mentioned above, Arunkumar et al. (2019b) do not hide more than one secret image, and hence, we have no reference data to compare against in the rest of our results (quality of stego-image, quality of secret image, and resistant to steganographic attacks). In Fig. 15, we present the visual comparison of ‘Stream’ as the cover image and the corresponding stego-image while hiding four copies of this brain image. As evident, the cover and its corresponding stego-image are very similar. We get analogous results for the other cover images as well. We also check their edge maps (as discussed in section ‘Stego-image Quality Assessment st2’) and obtain good results.

In Fig. 14, we show the original brain secret image and one of the extracted brain images (from the ‘Stream’ stego-image). From these figures, we observe that when compared with the original secret image, the quality of the extracted secret image is good. Finally, like (Arunkumar et al., 2019b), our scheme is inherently resistant to steganographic attacks. Our design makes our scheme more robust.

Table 11:

Application of our scheme on real-life data (brain image), and its comparison with one scheme.

No. of secret images	Steganography scheme	Type of secret image	Type of cover images	EC (in bpp)	(Avg. PSNR, No. of cover images)	Max. PSNR
1	Arunkumar et al. (2019b)	Grayscale	Grayscale	2	(49.69, 8)	50.15
1	SABMIS	Grayscale	Grayscale	2	(41.54, 10)	44.58
4	SABMIS	Grayscale	Grayscale	8	(33.56, 10)	37.74

DOI: 10.7717/peerjcs.1080/table-11