1 Introduction

Reversible data hiding [2, 8, 15, 29, 31] is a technique that not only recovers the original cover image, but also provides the exact secret data. During the extraction process, the pixel of the original cover image is exactly recovered from the stego image without any distortion. So reversible data hiding [7, 9, 14, 16, 28] has very important application in protecting the images that contain highly confidential information such as medical, and military images etc. where there is a need of recovering the extract original cover image and secret data. Data hiding that provides a single stego image can hold fewer amount of secret data, but the data hiding in dual stego image can hold a huge amount of secret data. Moreover the dual stego image approach is more secure than the single stego image approach since the data extraction has been possible only if, both the stego images are to be known (data extraction is not possible from one stego image). Since there are two stego images, it requires more space to store the dual stego images. However, dual stego images have a very positive advantage of having high embedding rate when compared to single image based data hiding.

Different reversible data hiding techniques have been proposed which include histogram shifting, difference expansion and prediction error expansion. Initially, the histogram shifting technique was introduced by Ni et al. [19], where the bins less than a particular intensity are shifted towards the left by one position to create a carrier bin for embedding the message bits. Tian [22] proposed the difference expansion (DE) method to provide high embedding capacity. This DE method estimates the difference between the two adjacent pixels and ‘1’ bit data is embedded in the difference value. Thodi and Rodriguez [21] first proposed the prediction error expansion (PEE) method that combines the advantage of difference expansion and histogram shifting techniques. The PEE method embeds the data on the prediction error estimated between the pixel and the neighborhood pixel.

The DE technique was improved by Weng et al. [26] by using pair-wise difference adjustment (PDA) scheme and the invariability of the sum of pixel pairs. Dragoi and Coltuc [5] proposed local prediction based difference expansion technique where the image has divided into sub-blocks and a predictor of least square was estimated on each sub-block that centered on the pixel. The prediction error was expanded to embed the data. Histogram shifting method was improved by Luo et al. [32] by estimating the difference between the corresponding pixel value and the interpolation value.

Inorder to reduce the distortion between the cover image and stego image, Li et al. [13] proposed multiple histograms based prediction error expansion where the local complexity measurement is used to estimate the multiple histograms. Pixel value ordering (PVO) based prediction error expansion was introduced by Li et al. [12] where the image was first divided into sub-blocks. The pixels belong to a particular sub-block was sorted for embedding the data. This PVO method was further improved by Wang et al. [27] by using local complexity analysis.

Inorder to preserve the privacy of cover image in cloud computing, several RDH techniques in the encrypted domain has been proposed. Since the distortion is not a concern in embedding data on an encrypted image, a high embedding rate is achieved in encrypted images. Li et al. [11] proposed a histogram shifting based scheme by using homomorphic multiplication with a public key histogram of the image. Dual image RDH was introduced to increase the embedding capacity that produces two similar stego images instead of generating a single stego image. Dual stego image based data hiding is highly secure than the single stego image based data hiding, since the unauthorized user cannot extract the data from a single stego image. Exploiting modification direction (EMD) in dual stego image was introduced by Wang et al. [30] which calculates a modulus function for the intensities from 0 to 255. The data hiding was done based on the value of the modulus function.

The remainder of the paper is arranged as follows. Few of the related works are shown in section 2. Section 3 shows the proposed two-level data encoding approach. Section 4 discusses the experimental results of the paper and finally, the conclusion is presented in section 5.

2 Related works

This section shows the few related works of dual image RDH such as exploiting modification direction, Authenticable RDH, LSB Matching and our previous embedding method based on Data Encoding.

2.1 EMD scheme

Exploiting modification direction (EMD) [30] for hiding data was initially proposed for single stego image by Zang and Wang (2006). It estimates a modulus function M using the relation in eqn (1),

$$ M\left({x}_1,{x}_2,\dots \dots {x}_l\right)=\left[\sum \limits_{i=1}^l{x}_i\times i\right]\mathit{\operatorname{mod}}\ \left(2\times l+1\right) $$
(1)

Where x1, x2, ……xlis the group of l number of pixels. Let d be the data in decimal form. If d and M differs then the group of pixels are modified as eqn (2).

$$ r=\left(d-M\right)\mathit{\operatorname{mod}}\ \left(2\times l+1\right) $$
(2)

Where r resembles the position of the pixels to be incremented or decremented.

Figure 1 considers the original pixels {48,86,72,94} where l = 4 and message d = 59. Using eqn (1) M(48,86,72,94) = (1 × 48 + 2 × 86 + 3 × 72 + 4 × 94) mod (2 × 4 + 1) = 2. Here the value of d and M differes. Therefore r = (5 − 2) mod (2 × 4 + 1) = 3. Ifr is less than l, increment xr by 1 otherwise decrement x2l, − r + 1 by 1. Therefore decrement x3 by 1, which gives the stego pixels {48,86,73,94}. The EMD scheme was extended on dual stego images [3] by Chang et al. 2007. The modulus function was estimated for all intensities from 0 to 255. A block of 5 × 5 pixel was selected to embed the data. The modulus function was estimated by eqn (3).

$$ M\left({x}_i,{x}_{i+1}\right)=\left({x}_i+2\times {x}_{i+1}\right)\mathit{\operatorname{mod}}\ 5 $$
(3)
Fig. 1
figure 1

Data embedding using EMD in the single stego image

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Fig. 2 assumes that an original pixel pair {26, 87} and data das {1, 3}. The stego pixel pair corresponding to the first data ‘1’ is {11, 40} and the second data ‘3’ is {9, 42}. The EMD scheme was extended on dual stego images reversibility [4] by Chang et al. 2013. The block size is increased to 9 × 9. Here the modulus function is calculated using eqn (4).

$$ M\left({x}_i,{x}_i\right)=\left({x}_i+3\times {x}_i\right)\mathit{\operatorname{mod}}\ 9 $$
(4)
Fig. 2
figure 2

Data Embedding using EMD in the dual stego image

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Fig. 3 assumes that the original image pixel is 23 and the data is 49. The modulus function for the pixel 23 is calculated as M(23, 23) = (23 + 3 × 23) mod 9 = 2. The stego pixel for the original pixel 23 is calculated as 24 and 22. Qin et al. (2015) [17] converted the binary data to base 5 values before embedding. The stego pixel pair \( {x}_i^1 \) and \( {x}_{i+1}^1 \) for the first stego image is obtained by embedding two symbols d1 and d2 on the original image pixel pairs xi and xi + 1 by using EMD embedding scheme.

Fig. 3
figure 3

Data embedding using EMD with 9 × 9 modulus function

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Fig. 4
figure 4

Data embedding using Authenticable RDH

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The procedure for estimating the second stego image pixel pair \( {x}_i^2 \) and \( {x}_{i+1}^2 \) is as follows.

Step (i): The second stego image pixel pairs \( {x}_i^2 \) and \( {x}_{i+1}^2 \) was estimated by EMD embedding process if \( {x}_i^1={x}_i \) and \( {x}_{i+1}^1={x}_{i+1.} \)

Step (ii): Find the value of W1 and W2 as eqns (4) and (5)

$$ {W}_1=\left\{\begin{array}{c}5, if\ {d}_2=M\left[{x}_i-5\times \mathit{\operatorname{sign}}\left({x}_i^1-{x}_i\right),{x}_{i+1}\right]\\ {}4, if\ {d}_2=M\left[{x}_i-4\times \mathit{\operatorname{sign}}\left({x}_i^1-{x}_i\right),{x}_{i+1}\right]\\ {}3, if\ {d}_2=M\left[{x}_i-3\times \mathit{\operatorname{sign}}\left({x}_i^1-{x}_i\right),{x}_{i+1}\right]\\ {}2, if\ {d}_2=M\left[{x}_i-2\times \mathit{\operatorname{sign}}\left({x}_i^1-{x}_i\right),{x}_{i+1}\right]\\ {}1, if\ {d}_2=M\left[{x}_i-1\times \mathit{\operatorname{sign}}\left({x}_i^1-{x}_i\right),{x}_{i+1}\right]\end{array}\right. $$
(4)
$$ {W}_2=\left\{\begin{array}{c}5, if\ {d}_2=M\left[{x}_i,{x}_{\mathrm{i}+1}-5\times \mathit{\operatorname{sign}}\left({x}_{i+1}^1-{x}_{i+1}\right)\right]\\ {}4, if\ {d}_2=M\left[{x}_i,{x}_{i+1}-4\times \mathit{\operatorname{sign}}\left({x}_{i+1}^1-{x}_{i+1}\right)\right]\\ {}3, if\ {d}_2=M\left[{x}_i,{x}_{i+1}-3\times \mathit{\operatorname{sign}}\left({x}_{i+1}^1-{x}_{i+1}\right)\right]\\ {}2, if\ {d}_2=M\left[{x}_i,{x}_{i+1}-2\times \mathit{\operatorname{sign}}\left({x}_{i+1}^1-{x}_{i+1}\right)\right]\\ {}1, if\ {d}_2=M\left[{x}_i,{x}_{i+1}-1\times \mathit{\operatorname{sign}}\left({x}_{i+1}^1-{x}_{i+1}\right)\right]\end{array}\right. $$
(5)

Where M[a, b] is the modulus function which is calculated using the expression in eqn (6)

$$ M\left[a,b\right]=\left(a+2b\right)\mathit{\operatorname{mod}}\ 5 $$
(6)

Step (iii): If \( {x}_i^1\ne {x}_i \) and \( {x}_{i+1}^1={x}_{i+1} \) then \( {x}_i^2={x}_i-{W}_1\times \mathit{\operatorname{sign}}\left({x}_i^1-{x}_i\right) \) and \( {x}_{i+1}^2={x}_{i+1} \)

Step (iv): If \( {x}_i^1={x}_i \) and \( {x}_{i+1}^1\ne {x}_{i+1} \) then \( {x}_i^2={x}_i \) and \( {x}_{i+1}^2={x}_{i+1}-{W}_2\times \mathit{\operatorname{sign}}\left({x}_{i+1}^1-{x}_{i+1}\right) \)

2.2 Authenticable reversible data hiding

An Authenticable Reversible data hiding was proposed by K.H Jung (2017) [10] using the same modulus function is shown in eqn (3). Instead of estimating the stego pixels in a diagonal direction, this method estimates the stego pixels in the horizontal and vertical direction. Figure 4 considers an example, where the message 7 s {1, 3} and the original pixel pair be {10,41}. The vertical and horizontal pairs are estimated and one pair is chosen from it, which has less distortion using a gap function.

Fig. 5
figure 5

Block diagram of data embedding in the proposed algorithm

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2.3 LSB matching

LSB matching based data hiding was proposed by Y.L.Wang et al. (2018) [23]. In LSB matching scheme, the cover pixel is converted to 8-bit binary and it is replicated to obtain two sets of eight-bit numbers. Let the eight bit binary numbers be B7B6B5B4B3B2B1B0. Let the 3-bit message be m2m1m0. The message m1m0 is embedded in such a way that B1B0 are replaced by m1m0 to obtain the binary sequence B7B6B5B4B3B2m1m0. The message m2is embedded in the 3rd position of the 8-bit binary B7B6B5B4B3m2B1B0. The two sets of eight bit binary numbers are converted to decimal to obtain the two stego pixels.

2.4 Data encoding approach

In our previous paper [18], a new data encoding method for reversible data hiding in dual stego images based on maximum to the minimum histogram is introduced. This method works on the principle that the conversion of higher histogram data from the secret image to lower intensity and conversion of lower histogram data from the secret image to higher histogram provides quality stego images. While embedding, the data from the secret image are grouped in K bits and converted to decimal. This decimal value forms the decimal intensities. The message intensities are converted to encoded intensities using a data encoding process which works on maximum to minimum histogram values. In the data encoding process, the histograms of message intensities are estimated and the histogram is sorted from maximum to minimum. By using the histogram values, assigned intensities are estimated. Based on the initial intensity, assigned intensity and sorted histogram index, encoding is performed by traversing the message intensity sequence in the form of an adjacent pair. The encoded intensity thus obtained after data encoding is embedded in the cover pixel to obtain the stego pixel pair.

In the extraction process, the encoded intensities are obtained from the two stego images using the difference between pixels from stego image1 and stego image 2. The data decoding process converts the encoded intensities, which is extracted from the stego images are converted back to message intensities. The encoded intensities are traversed from the starting value in pair to decode the message intensity. The message intensity thus obtained are converted to K bit binary numbers to obtain the secret data which forms the secret image.

In the proposed work, we aim to improve the PSNR and embedding capacity further by including one more level encoding process to our previous work. For the first level, we uses the algorithm of our previous work, but for the second level the encoded intensities of first level are folded by identifying the negative values to obtain the folded intensities. The folded intensities are embedded in the cover image to obtain the dual stego-images.

3 Proposed method

This section shows the working of proposed two-level data encoding for reversible data hiding in dual stego image. The proposed method has two phases ie., data embedding and data extraction. The data embedding phase aims to produce the two stego images using the cover image and secret data, while the data extraction phase aims to recover the cover image and secret data using the two stego images.

3.1 Data embedding

The data embedding is performed in two levels of encoding such as Level 1 encoding and Level 2 encoding. The block diagram of the proposed data embedding process is shown in Fig. 5. Let S(x, y) be the cover image and ‘b’ be the binary data. Let S1(x, y) and S2(x, y) are the two stego images.

Fig. 6
figure 6

Histogram for message intensities mi

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Initially, the binary data b is grouped into L bits and are converted to decimal to obtain the message intensities mi, where mi lies in the interval [0, 2L − 1]. The Fig. 6 message intensities mi are encoded using Level 1 encoding to obtain the encoded intensities ei. The encoded intensities are further encoded using Level 2 encoding to obtain the folded intensities fi.

3.1.1 Level 1 encoding

Let mi = {m1, m2, ……mn} be the ‘n’ number of data in the decimal form obtained by grouping L bits from the binary data b. The histogram of the message intensity mi is estimated and it is sorted in descending order to obtain the sorted index Ci which is represented as eqn (7),

$$ {C}_i=\left\{{C}_0,{C}_1,{C}_2,\dots \dots {C}_{N-1}\right\}\kern2.25em 0\le {C}_i\le N-1 $$
(7)

The sorted index Ci contains Nnumber of samples,whereN = 2L. The sorted indices C0 to CN − 1 are the message intensities that has maximum to minimum histogram values. Consider a histogram plot for the message intensities mi is shown in Fig. 6.

In the example, shown in Fig. 7 the sorted index is represented as Ci = {6, 5, 7, 3, 1, 4, 2, 0}.After estimating the sorted index, Form the assigned intensity table Bi as eqn (8),

$$ {B}_i=\left\{0,1,-1,2,-2,3,-3,\dots \dots .\left(\frac{N}{2}-2\right),-\left(\frac{N}{2}-2\right),\left(\frac{N}{2}-1\right),-\left(\frac{N}{2}-1\right)\right\} $$
(8)
Fig. 7
figure 7

An example of Histogram sorting

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Where, Bi contains alternate positive and negative numbers starting from zero. From, initial intensity table Ai as eqn (9),

$$ {A}_i=\left\{\frac{N}{2},-\frac{N}{2}\left(\frac{N}{2}+1\right),-\left(\frac{N}{2}+1\right)\dots \dots \left(N-2\right),-\left(N-2\right),\left(N-1\right),-\left(N-1\right)\right\} $$
(9)

The assigned intensity table Bi and initial intensity table Ai have N number of elements. The procedure for Level 1encoding is shown below.

  • Step 1: Traverse any message intensity mi from starting to end of the sequence.

  • Step 2: While traversing m0 through mn replace lonely message intensity mi by its corresponding initial intensity Ai using Table 1 till pair of mi is found.

  • Step 3: While traversing m0 through mn if there is a pair of mi ie. (mi, mi) is found, replace the first element of the pairmi by its corresponding initial. Intensity Ai also replace the second element of the pair mi by the assigned intensity Bi corresponding to the sorted index of di. ie., the pair (mi, mi) will be replaced as (Ai, Bi) if the sorted index of mi is other than n − 1. The pair (mi, mi) will be replaced by (Ai, −(N − 1)) if the sorted index of mi is N − 1.

  • Step 4: While traversing m0 through mn, the message intensity mi found after the first pair must be replaced by Bi, if the sorted index mi is other than N − 1 and replaced by −(N − 1) if the sorted index of mi is N − 1 .

  • Step 5: Finally, encoded intensityei is obtained by repeating step 1 to 4 for all message intensity.

Table 1 Level 1 Encoding table
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3.1.2 Level 2 encoding

The level 2 encoding converts the encoded intensities ei to folded intensities fi. Let the encoded intensity be ei = {e0, e1, ……eN}. Note the position of negative numbers. Let the position of negative numbers be represented as li = {l0, l1, ……lu − 1}. Therefore the number of negative numbers in eiis u. Estimate the magnitude intensity (Mai) of encoded intensities (ei) as an eqn (10)

$$ {Ma}_i=\left|{e}_i\right| $$
(10)

Estimate the new sorted index of Mai as shown in Fig. 7. Let the new sorted index be gi.

Based on the New sorted index (gi) of the magnitude intensity (Mai) replace the magnitude intensity (Mai) by the New Assigned intensity (Vi) to obtain the transformed intensity (Ti) as shown in Table 2. After estimating the transformed values (Ti) insert ‘1’ just after the elements present in the location li to obtain the folded intensity fi which are embedded in the cover image S(x, y) as eqns (11) and (12),

$$ {S}_1\left(x,y\right)=S\left(x,y\right)+\left\lfloor \frac{f_i}{2}\right\rfloor $$
(11)
$$ {S}_2\left(x,y\right)=S\left(x,y\right)-\left\lceil \frac{f_i}{2}\right\rceil $$
(12)
Table 2 Level 2 Encoding table
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The accessorial information is AI ∈ {0, 1, −1, 2, −2, 3, −3……….}which corresponds to the new sorted index {0,1,2,3,4,5,6……..}. For example if the new sorted index- isgi = {7, 6, 3, 2, 5, 1, 0, 4} then the accessorial information is AI = {4, −3, 2, −1, 3, 1, 0, −2} as eqn (13).

$$ AI=\left\{\begin{array}{c}\frac{-{g}_i}{2}\kern0.5em if\ {g}_i\ is\ even\ \\ {}\frac{g_i+1}{2}\ if\ {g}_i\ is\ odd\end{array}\right. $$
(13)

The procedure for embedding the data mi using the proposed two-level data encoding algorithm is summarized as follows.

Input: Original image S(x, y), data b, Key ‘K’ and Group size L.

Output: Stego images S1(x, y) and S2(x, y)

Step 1: Select the embeddable pixels whose intensity lies between (2L − 1) and (256 − 2L − 1).

Step 2: Group ′L′ data bits from b and convert to decimal to obtain decimal intensity(mi).

Step 3: Apply level 1 encoding to convert the decimal intensity (mi) to encoded intensity (ei).

Step 4: Apply level 2 encoding to convert the encoded intensity (ei) to folded intensity (fi).

Step 5: Shuffle the folded intensity (fi) using key ‘K’.

Step 6: Embed the Accessorial information (AI) on first 2L embeddable pixels and embed the shuffled folded intensity (fi) on remaining embeddable pixels using eqn (11) and (12)

Consider an example where the message intensity mi = {7, 6, 3, 7, 1, 7, 7, 6, 6, 3, 7, 2, 6, 1}. The histogram of the message intensity mi is tabulated and it is sorted as shown in Fig. 8.

Fig. 8
figure 8

Estimation of sorted index

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Since L = 3, the assigned intensity Bi, initial intensity Aiis tabulated as Bi = {0, 1, −1, 2, −2, 3, −3} and Ai = {4, −4, 5, −5, 6, −6, 7, −7} in Table 3.

Table 3 An example of Level 1 Encoding
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Figure 9 shows consider the message intensity mi = {7, 6, 3, 7, 1, 7, 7, 6, 6, 3, 7, 2, 6, 1}. From the message mi, traverse the intensity ′7′ from left to right. There are lonely 7 ′ s in position 1 and 4. So replace the lonely 7’s by the corresponding initial intensity (−7). There is an adjacent pair of 7 in position 6 and 7. Therefore encode the pair (7, 7) as (−7, 0). i.e. replace first element of pair by initial intensity Ai and second element of pair by assigned intensity Bi. After first pair, replace all 7 ′ s by its assigned intensity ‘0’. Repeat the same process until all the message intensities are encoded.

Fig. 9
figure 9

Encoding of ‘7’ using level 1 encoding

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Figure 10 shows he encoded intensities ei isestimated as {−7,7,-5,-7,-4,-7,0,7,1,-5,0, 5,1,-4}. Estimating the location of negative elements to get li = {1, 3, 4, 5, 6,10,14}. The magnitude intensity of ei is calculated as Mai = {7, 7, 5, 7, 4, 7, 0, 7, 1, 5, 0, 5, 1, 4}. The histogram of the magnitude intensityMai is calculated as,

Fig. 10
figure 10

Histogram sorting for level 2 encoding

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Table 4 shows the transformed values (Ti) is calculated by replacing the sorted magnitude by new assigned intensity(Vi). Therefore the transformed values, Ti = {0, 0, −1, 0, 3, 0, 2, 0, −2, −1, 2, −1, −2, 3}. Inserting ′1′ just after the elements present in the location li = {1, 3, 4, 5, 6,10,14}. Figure 11 shows the estimation of the folded intensities fi = {0, 1, 0, −1, 1, 0, 1, 3, 1, 0, 1, 2, 0, −2, −1, 2, −1, −2, 3}, and the accessorial information AI is obtained from the new sorted index gi = {7, 5, 0, 1, 4, 2, 3, 6} as AI = {4, 3, 0, 1, −2, −1, 2, −3} using eqn (13) which is embedded in first L embeddable pixels. The first four folded intensities fi = {0, 1, 0, −1} is embedded in the pixels S(x, y) = {93,124,19,87} by using eqn (11) and (12) as S1(x, y) = {93,124,19,86}and S2(x, y) = {93,123,19,87}.

Table 4 Example of level 2 encoding
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Fig. 11
figure 11

An example of two-level encoding

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3.2 Data extraction

Figure 12 shows the block diagram of data extraction in the proposed algorithm. The data extraction algorithm is inverse to data embedding, where level 2 data decoding and level 1 decoding are performed to extract the hidden data.

Fig. 12
figure 12

Block diagram of data extraction in the proposed algorithm

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Similar to the data embedding process, level 2 decoding converts the folded intensities fi to encoded intensities ei, while level 1 decoding converts the encoded intensities ei to message intensities ei. The folded intensities is extracted by estimating the difference between the two stego image as eqn (14)

$$ {f}_i={S}_1\left(x,y\right)-{S}_2\left(x,y\right) $$
(14)

Similarly, the Accessorial information (AI) is extracted from the first 2L embeddable pixels. From the accessorial information (AI) construct the new sorted index gi using eqn (15). For example, if the accessorial information (AI) is AI = {4, −3, 2, −1, 3, 1, 0, −2} then the new sorted index is calculated using the relation gi = {7, 6, 3, 2, 5, 1, 0, 4}

$$ {g}_i=\left\{\begin{array}{c}\kern-1.8em 2\times \mid AI\mid \kern2em if AI\le 0\ \\ {}2\times AI-1\kern1.5em if AI>0\kern1.5em \end{array}\right. $$
(15)

Note the position of 1’s in the folded intensity (fi), also replace the 1’s by N. Let the position of 1’s present in the folded intensity (fi) be represented as li = {l0, l1, ……lu − 1}. Replace the folded intensity (fi) by their corresponding new sorted index (gi) shown in Table 5. Also, put a negative sign for the intensities just before the position li i.e. in position {l0 − 1, l1 − 1, ……lu − 1 − 1}. After placing the negative sign discard the intensity N to obtain the encoded intensities (ei). Let the encoded intensities be ei = {e1, e2, ……. . en}.

Table 5 Level 2 Decoding table
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The procedure for obtaining the folded intensity (fi) from the encoded intensity (ei) is shown below.

  • Step 1: Traverse the encoded intensity (ei) from the starting value.

  • Step 2: While traversing, form a pair of intensity (ei,ei + 1), where ei + 1 is the next adjacent element of ei.

  • Step 3: If both the intensities ei andei + 1 belongs to the category initial intensity Ai, replace ei by the corresponding message intensity mi.

  • Step 4: If the first intensity ei of the pair belongs to the category initial intensity Ai and the second intensity ei + 1 of the pair belongs to the category assigned intensity Bi. Then replace ei by its corresponding message intensity mi. Also, replace all the element of the encoded intensity with the intensity value ei + 1 by the message intensity mi.

  • Step 5: Step 1 to Step 4 is repeated until all the encoded intensities are replaced, which gives the complete message intensity mi.

The original image S(x, y) is recovered from the stego images S1(x, y) and S2(x, y) using the relation as eqn (16).

$$ S\left(x,y\right)=\left\lceil \frac{S_1\left(x,y\right)+{S}_2\left(x,y\right)}{2}\right\rceil $$
(16)

The data extraction algorithm is summarized as follows.

Input: Stego images S1(x, y)and S2(x, y), Key K, Group size L

Output: Binary data, Original image S(x, y)

  • Step 1: Identify the embedded stego pixels. Embedded pixels have intensities between (2L − 1)and (256 − 2L − 1) in both the stego images S1(x, y) and S2(x, y).

  • Step 2: Find the difference between the first 2L embedded pixels of the stego images S1(x, y)and S2(x, y), using eqn (18) to extract the accessorial information.

  • Step 3: Find the difference between the remaining embedded pixels (other than first 2L embedded pixels) of the stego images S1(x, y) and S2(x, y) using eqn (14) to extract folded intensity fi. Using the key K, rearrange the folded intensity fi.

  • Step 4: Find the encoded intensity eifrom the folded intensity fi and the accessorial information (AI) using the Level 2 decoding.

  • Step 5: Apply level 1 decoding to obtain the message intensity mi from the encoded intensity ei.

  • Step 6: Find the secret data by converting the message intensity to L bit binary.

  • Step 7: Recover the original image S(x, y) from S1(x, y) and S2(x, y) using eqn (16).

Fig. 13 shows consider the two stego images, S1(x, y) = {93,124,19,86} S2(x, y) = {93,123,19,87}and L = 4. The folded intensity fi is obtained using eqn (14) as fi = {0, 1, 0, −1}. Let fi = {0, 1, 0, −1, 1, 0, 1, 3, 1, 0, 1, 2, 0, −2, −1, 1, 2, −1, −2, 3, 1} be the folded intensity obtained from the entire stego images. Let the accessorial information extracted from the first L embedded pixels

Fig. 13
figure 13

An example of the data extraction process

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are ={4, 3, 0, 1, −2, −1, 2, −3} . Using eqn (15), the new sorted index gi = {7, 5, 0, 1, 4, 2, 3, 6}. Note the position of 1’s in the folded intensity fi, Let the position of 1’s be li = {2, 5, 7, 9, 11,16,21}, Also replace the 1’s by N = 8, to obtain seq = {0, 8, 0, −1, 8, 0, 8, 3, 8, 0, 8, 2, 0, −2, −1, 8, 2, −1, −2, 3, 8}. Using the Table 6, transform the folded intensity present in the sequence seqto seq1 = {7, 8, 7, 5, 8, 7, 8, 4, 8, 7, 8, 0, 7, 1, 5, 8, 0, 5, 1, 4, 8}. Place a negative sign before the elements N = 8, To get seq2 = {−7, 8, 7, −5, 8, −7, 8, −4, 8, −7, 8, 0, 7, 1, −5, 8, 0, 5, 1, −4, 8}. Discarding N = 8, on seq2 we can obtain the encoded intensity ei = {−7, 7, −5, −7, −4, −7, 0, 7, 1, −5, 0, 5, 1, −4}.

Table 6 Example of Level 2 decoding
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Table 7 shows the level 1 decoding table is formed as shown in table. From the encoded intensity ei = {−7, 7, −5, −7, −4, −7, 0, 7, 1, −5, 0, 5, 1, −4}, form the pair (−7,7). Here both the elements belongs to initial intensity category, therefore replace −7 by its message intensity 7, Again form a pair (7,-5), both 7 and − 5 belongs to initial intensity category, therefore replace 7 by its message intensity 6. Repeat the same process till the pair (−4,7) is reached. Again form the next pair (−7,0), here −7 belongs to initial intensity category, but 0 belongs to assigned intensity category, therefore replace −7 by its message intensity 7 and all 0 by the same message intensity 7. Repeat the same process to obtain the complete message intensity mi = {7, 6, 3, 7, 1, 7, 7, 6, 6, 3, 7, 2, 6, 1}. The original image S(x, y) are obtained from the stego images S1(x, y) = {93,124,19,86}, S2(x, y) = {93,123,19,87} using eqn (20) as S(x, y) = {93,124,19,87}.

Table 7 Example of Level 1 decoding
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Table 8 PSNR and Capacity comparison for different test images with a different value of L
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4 Experimental results

The data embedding and data extraction of the proposed two-level data hiding algorithm were verified using the 10 standard test images shown in Figs. 14 and 5 rarely used test images shown in Fig. 17. We have used Fig. 14(a)-(f) as cover images and Fig. 14(g)-(j) as secret images. The algorithm was implemented using MATLAB 2018a. The test ‘Original images’ are gray scale images [20, 24] each having a size of 512 × 512 and the test secret image [1, 6, 25], are gray scale images having a size of 216 × 216. The performance of the proposed algorithm was evaluated using the parameters such as embedding rate (bpp), peak signal to noise ratio (PSNR). The embedding rate (bpp) resembles the number of bits hidden in a pixel which is estimated by eqn (17),

$$ bpp=\frac{t}{2\times h\times w} $$
(17)
Fig. 14
figure 14

Test Images (Cover Images are (a) – (f) and secret images are (g) – (j))

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Here h × w is the size of the original image and t is the capacity of data (in bits) hidden in two stego images. The parameter PSNR shows the distortion level of two stego images from the original image which is calculated using the eqns (18) and (19),

$$ {PSNR}_1=10\times {\log}_{10}\left[\frac{255^2}{\frac{1}{h\times w}\sum \limits_{x=1}^h\sum \limits_{y=1}^w{\left(S\left(x,y\right)-{S}_1\left(x,y\right)\right)}^2}\right] $$
(18)
$$ {PSNR}_2=10\times {\log}_{10}\left[\frac{255^2}{\frac{1}{h\times w}\sum \limits_{x=1}^h\sum \limits_{y=1}^w{\left(S\left(x,y\right)-{S}_2\left(x,y\right)\right)}^2}\right] $$
(19)

The average PSNR (PSNRavg) and the PSNR of the two stego images PSNR1and PSNR2 are related as eqn (20),

$$ {PSNR}_{avg}=\frac{\left({PSNR}_1+{PSNR}_2\right)}{2} $$
(20)

Table 8. The performance of the proposed algorithm was evaluated for different values of group size ‘L’ (L = 3, L = 4, L = 5) for different cover images. For L = 3, the embedding capacity is around t = 54,000 bits, while the PSNR of the first stego image is around 48.7 dB, while the PSNR of the second image is around 50.6 dB. The average PSNR (PSNRavg) is around 49.6 dB. As the value of L is increased to 4 and 5, the embedding capacity is increased, but the PSNR reduces. However, there is a high increase in capacity at slight reduction in PSNR. The embedding capacity at L = 4 and L = 5 are aroundt = 80,000 bits and t = 10,00,000 bits respectively. The average PSNR for L = 4 is 47.393 dB and the average PSNR for L = 5 is 43.68105 dB. The embedding capacity is not constant for a particular value of L for different cover images. For example for L = 4, the cover image ‘Zelda’ has a embedding capacity of t = 10,21,365 bits, while the cover image ‘Man’has a embedding capacity of t = 8,40,696 bits. The reason is the number of embeddable pixels varies in different cover images. i.e. The cover image ‘Zelda’ has more embeddable pixels when compared to the cover image ‘Man’.

Figure 15 shows the PSNR1 and PSNR2 Comparison of the cover image ‘Baboon’ for different values of L. The difference between PSNR1 and PSNR2 for L = 3 is high when compared to L = 4and L = 5. For a embedding rate of bpp = 1 and L = 3, the PSNR difference between the two stego image is around2dB. For L = 4 and L = 5, the PSNR difference between the two stego images is less than 1. As the value of L increases the PSNR reduces around 3 dB, but the embedding rate increases around 0.5bpp.

Fig. 15
figure 15

PSNR1 and PSNR2 Comparison of the cover image ‘Baboon’

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The PSNR and embedding capacity of the proposed method was compared with the traditional methods such as Chang et al. (2007) [3],Chang et al. (2013) [4],Qin et al. (2015) [17], Ki-Hyun Jung (2017) [10],Wang et al. (2018) [23], Data Encoding approach (2019) [18]. The proposed method provides a low PSNR when compared to our previous Data Encoding approach (2019), however, the embedding capacity of the proposed method is around 8,00,000 bits which is very high than the traditional methods. The comparison of PSNR and Embedding capacity is tabulated in Table 9. Fig. 16 shows a graphical comparison of proposed and traditional methods for different embedding rate. While embedding the data at different embedding rate, the PSNR is found almost the same in different test images. But the maximum embedding capacity differs in different cover images because of the change in a number of embeddable pixels. The graph depicted in Fig. 16 concludes that the proposed method shows a high PSNR than the traditional methods for L = 3 and L = 4 for different test images. We have also evaluated the quality of the stego image using SSIM (structural similarity index measurement). The average SSIM (SSIMavg) can be estimated as,

$$ {SSIM}_{avg}=\left({SSIM}_1+{SSIM}_2\right)/2 $$
(21)
Table 9 PSNR and Embedding capacity comparison of proposed and traditional methods
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Fig. 16
figure 16

PSNR comparison for different test images

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Where SSIM1 and SSIM2 can be estimated between the cover image S(x, y) and stego images S1(x, y) and S2(x, y) respectively using the relation,

$$ {SSIM}_i=L\left({S}_i,S\right)\ C\ \left({S}_i,S\right)\ Sa\left({S}_i,S\right)\kern1.25em i\in \left\{1,2\right\}\kern1.75em $$
(22)

Here L(Si, S), C (Si, S) and Sa(Si, S) are the Luminance, contrast and saturation comparison function respectively. We have also used the rarely used test images shown in Fig. 17 taken from (http://decsai.ugr.es/cvg/CG/base.htm) to evaluate SSIM performance.

Fig. 17
figure 17

Rarely used test images from (http://decsai.ugr.es/cvg/CG/base.htm)

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Table 10 shows the PSNR, SSIM and capacity comparison for the test images shown in Fig. 18 with different values of L. Similar to the test images shown in Fig. 15, the average PSNR of the images shown in Fig. 18 decreases as the value of L increases. The capacity increases as L increases from 3 to 5. The SSIM for L = 4, L = 5, and L = 6 is around 0.9988, 0.9888 and 0.9788 respectively. Higher values of SSIM indicates that the proposed method produces high quality stego images.

Table 10 PSNR, SSIM and Capacity comparison for the test images shown in Fig. 18
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Fig. 18
figure 18

Dual stego images for different values of L (a) L = 3 (b) L = 4 (c) L = 5 (First row: Stego image 1, Second row: Stego image 2)

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Figure 18 shows the visual appearance of the stego images obtained for the cover image shown in Fig. 18 (a). There is only a slight change in visual quality as the value of L increases from 3 to 5. The next section shows the conclusion of the proposed work.

5 Conclusion

This paper proposed a two-level data encoding approach for reversible data hiding in dual stego images. The two-level data encoding includes level 1 encoding that converts message intensities to encoded intensities and level 2 encoding converts encoded intensities to folded intensities. This proposed method assigns lower intensities to intensities that have higher histogram and higher intensities to intensities that have a lower histogram. The aim of the proposed method is to represent the most repeated secret information by lower intensity. The experimental verification was done on standard test images for different values of group size ‘L’ and comparison were made with the traditional RDH methods. The proposed method provides a high PSNR for L = 3. As the value of L increases to 4 and 5, the PSNR reduces around 3 dB, but the embedding rate increases around 0.5 bpp. The experimental results shows that the proposed method outperforms than the traditional dual stego image based reversible data hiding schemes in terms of image quality and embedding capacity.