1 Introduction

In the domain of information security, the steganography technique has the prime role to communicate innocent secret data in between sender and receiver for both academic and industrial researchers. This technique is classified in two ways: irreversible data hiding and reversible data hiding. In irreversible data hiding, the original image can only hide the secret information. It can not recover any secret data. Whereas, both the secret data can be hidden into the image and get back from the marked image in Reversible Data Hiding (RDH). So, RDH is generally used in the areas of medical and military images, remote sensing and copyright protection etc. In the RDH schemes two important parameters are evaluated: embedding distortion and embedding capacity (EC). In general, the prime challenges of effective RDH scheme is to enhance the quality of an image as well as data hiding capacity.

Over the past years, different RDH schemes have been proposed by the researchers. Applying method of difference expansion (DE) Tian [27] proposed a technique to conceal secret data into the pixel pair. Later, Alattar [1] applied four pixels differentiation by revising the Tian’s method. Ni et al. [20] presented the RDH technique which uses the minimum histogram. After that, Lin et al. [14] and Tsai et al. [28] presented multilevel RDH scheme using modification of histogram. Kim et al. [8] described a technique through correlation among the sub-sampled images in 2009. For example, in Lena image, the embedding capacity and image quality in PSNR of Kim et al.’s. [8] were 20,121 data bits and 48.9 dB respectively. After that, Luo et al. [17] presented a scheme by selection of the median pixel used as sub-image reference in each block which is partitioned in four section to keep the block median value. The PSNR value was 48.9 dB and the payload in bits was 0.11 bpp.

Li et al. [13] presented a technique using pixel value ordering(PVO) where only the maximum and minimum values are changing due to the data embedding in 2013. A technique where all pixels within a block are sorted in rising up fashion and modified the value of the maximal or minimal pixel for data embedding is called Pixel Value Ordering (PVO). The payload and quality of image were 32000 bits and 59.8 PSNR dB (for EC of 10,000 bits) respectively for Lena image. Lee et al. [11] presented two staged multilevel RDH scheme using Lagrange interpolation. By using Lagrange interpolation they generate predicted image. Then image difference is computed and hide the secret information by using histogram shifting algorithm. In this scheme, they construct one marked image with various embedding capacity. For example, in Baboon image, the embedded data bits was 0.88 bpp and the quality of image in PSNR was 48.32 dB. Peng et al. [24] improved both visual quality as well as embedding capacity where more smooth blocks were used. The Embedding capacity and image quality were 38,000 bits and PSNR 60.4 dB (for payload of 10,000 bits) respectively for Lena image. In 2015, Qu and Kim [25] presented a RDH technique through pixel-based PVO where the quality of image in PSNR and payload ware 60.3 dB (for payload 10,000 bits) and 46,000 bits respectively for Lena image. In 2016, Ou et al. [21] presented a technique where improved PVO was used through modification of several histograms. The reversible data hiding scheme based on dual image has been introduced by some researchers [5, 7]. Secret message hiding technique using sub-sampled images has been developed by Jana [6] in 2017. According to him, this hiding process is done through Lagrange’s interpolation polynomial on sub-sampled images. In this techniques, these images have been made from the cover image. Then, these are interpolated by interpolation techniques. Take any secret message and apply it with Lagrange interpolation function f(x). This f(x) value is converted to the binary value of 12 bits. This 12 bit binary data is partitioned into four fragment. Embed these four fragment by adding its corresponding value at position (1, 2), (2, 1), (2, 3) and (3, 2) of size \((3\times 3)\) interpolated image block. The value of position (2, 2) in interpolated image block is added with the value of x in f(x).

In data extraction, the difference of pixel value is calculated from marked image independently. Then collect the difference pixel value and value of x in f(x). These values are combined to create the (x, f(x)) value. Then applying Lagrange recover function to extract the secret data. For Lena image, the payload and quality through Jana [6] scheme were 130,000 bits and 50.60 PSNR dB respectively. Some researcher has been focused on interpolation based data hiding techniques [2, 10, 12, 15, 16, 18, 29, 32,33,34,35,36,37]. [23] and [3] are focused on watermarking and image authentication based data hiding techniques respectively.

Performance of security enhancement with RDH while keeping better quality of image using several images through Directional Pixel Value Ordering (DPVO) is still an important research matter. When hidden data is embedded using PVO technique through different direction [19] (i. e. horizontal, vertical and diagonal) one after other is called Directional PVO. We propose a safe RDH technique using Lagrange’s interpolation polynomial for sub-sampled image using DPVO. Here, any user can conceal hidden data bits in between sub-sampled images. Latter, the user can get back hidden data bits from marked images.

It is difficult task to raise payload with unalter image quality through numerous process of data embedding in the different direction on image block. So, we consider data hiding process is applied through firstly horizontal, secondly vertical and lastly diagonal directions. The hidden message is embedded into the first, second, third and so on minimum and maximum pixels in every direction. So, payload has been increased five times than Jana’s [6] scheme and the visual quality is raised than Jana’s [6] scheme. For solving lower order than higher order polynomials, it is useful to apply Lagrange interpolation. Here, sub-sampled image is four \((h = 4)\) and order is \((h - 1)\) of function f(x). For some areas, where enhanced payload and better quality of the image are necessary, our proposed scheme is useful in that area.

Secret sharing is a method where a secret message is distributed between a group of members and each member carries a share of a secret message. The secret information can be rebuilt only when a sufficient number(h) of shares are combined together and no \((h-1)\) can do so. Secret data communication using DPVO on several sub-sampled images through Lagrange interpolation polynomial raises the level of security. First, the hidden message is converted to its ASCII value and then compute the value of Lagrange’s interpolation function with parameters: threshold, coefficient, number of trusted parties and size of l prime. The function value produces another level of security on messages. Then, the function values are converted to its corresponding binary bit stream. These values are distributed between sub-sampled images with block size. Only the number(threshold value) of images are needed to retrieve the information. If the hackers want to hack the confidential information, they must need the said parameters which are hard to guess simultaneously. It is also unable to get back the hidden message from the marked images less than the threshold value.

In Sect. 2, we describe the proposed scheme along with an example. Experimental results, comparisons and some steganographic attacks are given in Sect. 3. Finally, Sect. 4 describes the conclusions of this paper.

1.1 Motivation:

Our prime motivations are described as follows:

  1. (i)

    Secret data concealing through Lagrange interpolation polynomial was irreversible. Using Directional PVO (DPVO) method within interpolated sub-sampled images, it is possible to insert and take out secret message to cover image and from marked (stego) image respectively.

  2. (ii)

    So far, there are several data hiding methods introduced by use of one or two images. It is complicated task to embed and extract hidden message with security among multiple images. So, we have introduced data hiding scheme within interpolated sub-sampled images which are served as numerous images for improve visual quality, data security. This scheme also accomplish reversibleness.

  3. (iii)

    Use of the Lagrange interpolation function, it produces new secret data bit from the original secret message which is distributed among the interpolated sub-sampled images. This lead to enhance the security. Using both DPVO and Lagrange function several unknown parameters (i. e. image block size, constant coefficient value and order of function) which are impossible to guess for the adversary that can improve security.

  4. (iv)

    So far, the stego image quality is calculated in PSNR dB through RDH schemes within sub-sampled images using Lagrange interpolation polynomial was limited because of modifying the pixel by large value. We solve it by using DPVO. So that, our proposed scheme modifies a pixel by 0 or 1 value which leads to improve visual quality of an image.

  5. (v)

    If any sub-sampled stego image destroys or losses through transmission, then it is possible to retrieve the secret message successfully from rest of other stego images. It is mentioned that, there are predefined stego images required to recover the hidden data which is dependent on lagrange function.

  6. (vi)

    So far, data embedding capacity (EC) within sub-sampled images using Lagrange interpolation polynomial was limited. Our proposed scheme is able to be done to improve payload of secret data bit and also visual quality which are essential in many application like medical and military image.

2 Proposed method

In this section, the process of data bit embedding and extraction through secret shares within sub-sampled images using Directional Pixel Value Ordering is discussed. The proposed method includes three stages: Initialization stage, Embedding stage and Extraction stage. This procedure is described below:

2.1 Initialization stage

At first, we take a cover image. Assume that \((P\times Q)\) is an image size indicated by CI(a, b), where the range of \(a = 0,\ldots ,(P - 1)\) and the range of \(b = 0,\ldots ,(Q - 1)\). We also assume that, \(\varDelta u\) in a row direction and \(\varDelta v\) in a column direction are two components of sampling denoted as the expected intervals of sub-sampling. The sub-sampled of an image(2-D) is at uniform intervals shown in Fig. 1. It is known as the sub-sampling process.

Fig. 1
figure 1

Block diagram of sub-sampled image block

Full size image

In this paper, we use only four sub-sample images. From original cover image, we create four sub-sampled images \((SI_{1}, SI_{2}, SI_{3}, SI_{4})\) of size \((a/2 \times b/2)\) by using following equation 1.

$$\begin{aligned} SI_{1}(rows, cols ) &= {} CI (2*rows, 2*cols)\nonumber \\ SI_{2}(rows, cols ) &= {} CI (2*rows, 2*cols+1)\nonumber \\ SI_{3}(rows, cols ) &= {} CI (2*rows+1, 2*cols)\nonumber \\ SI_{4}(rows, cols ) &= {} CI (2*rows+1, 2*cols+1) \end{aligned}$$
(1)

where \(rows =0, 1,\ldots , (a/2)-1\) and \(cols = 0,1,\ldots ,(b/2)-1\). After creation of four sub-sampled images, we enlarge each of the sub-sampled image using interpolation technique by computing the average of neighbor pixels shown in the following Fig. 2. The interpolate row and column is dependent on original image block size. For example, if the original block size is \((w \times w)\) then \((w-1)\) will be the interpolate row and column. The interpolated sub-sampled image will be \(((2w-1)\times (2w-1))\). To make it \((2w \times 2w)\), added one column and one row at the image end through copied the same pixel values of \((2w-1)^{th}\) column and row.

Fig. 2
figure 2

Block diagram of interpolated sub-sampled image block. Green and pink color represents interpolate row and column respectively

Full size image

The secret message can be lost if there is any displacement of the sampled image during transmission from sender to receiver or vice versa. To get back the message fixed the (n) number sub-sampled marked image which carries the secret message and only h of them to reconstruct the message, but no \((h - 1)\) of them can do so, where h is the threshold value. Here, we select three amongst four which cases to retrieve the hidden data required any three sub-sampled marked images amongst four marked images. Now, we consider that there is eight bits secret message (SM). After that apply Lagrange interpolation function f(x) with threshold(h) is described below:

$$\begin{aligned} f(x) = a_{0} +a_{1}x +\cdots + a_{(h-2)}x^{(h-2)} + a_{(h-1)}x^{(h-1)} \end{aligned}$$
(2)

In the above equation (2), only the value of secret message \((SM\in Z/lZ\ \text{ with }\ l\ \text{ prime })\) in ASCII is \(a_{0}\) and the value of \(a_{1},\ldots ,a_{(h-1)}\) are at random in Z / lZ. After that, we computed the f(x) value which is converted into binary bit stream. Now, these secret bits are embedded into interpolated sub-sampled image block in different direction.

2.2 Directional pixel value ordering (DPVO)

In the PVO method, pixels within an image are ranked in ascending order form to obtain minimum and maximum pixel value. The hidden messages are stuffed into these maximum or minimum pixel by changing its original value within an image block. When secret data embedding process occurs in different direction one after another (i. e. horizontal, vertical and diagonal) within \(1{\mathrm{st}},\ 2{\mathrm{nd}},\ 3{\mathrm{rd}}\) and so on minimum pixels for minimum modification based data embedding and in \(1{\mathrm{st}},\ 2{\mathrm{nd}},\ 3{\mathrm{rd}}\) and so on maximum pixels for maximum modification based data embedding, then this process is called DPVO. Following Fig. 3 depicted the basic diagram of DPVO.

Fig. 3
figure 3

Block diagram of Directional PVO

Full size image

After successful data embedding using DPVO, it is time to fetch the secret data from marked image block on the decoder side. In the data extraction procedure, it goes through reverse direction (i. e. diagonal, vertical and horizontal) of the embedding process.

2.3 Data embedding procedure

Secret data bit embedding process will begin in horizontally of the interpolated sub-sampled image. In our approach, the number of the modified pixel may be more than two which may affect in ascending order pixel ranking. So, a parameter \(\alpha\) is subtracted from and add to \(1{\mathrm{st}},\ 2{\mathrm{nd}},\ 3{\mathrm{rd}}\) and so on minimum and maximum pixel value respectively to maintain same pixel ranking order. The \(\alpha\) is only controlled by the original pixel block size. If a size of the original image block is \((w\times w)\) then, the minimal and maximal values of \(\alpha\) will be 0 and \((w-2)\) respectively.

Lemma 1

If the sub-sampled image size is \(((2w-1)\times (2w-1))\) and the ranked pixel is \((p_{1},p_{2},\ldots ,p_{2w-2},\, p_{2w-1})\) , then, the adjusted value of pixel will be \((p_{1}-\alpha _{(w-2)},p_{2} -\alpha _{((w-2)-1)},\ldots ,p_{w-1}-\alpha _{((w-2)-(w-2))}, p_{w},p_{w+1}+\alpha _{((w-2)-(w-2))},\ldots ,p_{2w-2} +\alpha _{((w-2)-1)},p_{2w-1}+\alpha _{(w-2)})\) , where, \(\alpha _{((w-2)-(w-2))}=((w-2)-(w-2))\) . The minimal and maximal value of \(\alpha\) will be 0 and \((w-2)\) respectively.

Example 1

Assume that, the original block is \((w\times w)=(2\times 2)\), then, interpolated sub-sampled image block will be \((3\times 3)\). The \(\alpha\)’s maximum value \((w-2)=0\), which is added and subtracted to the maximum and from the minimum pixel value respectively. This process will continue until it\((\alpha )\) comes to zero. The Lemma 1’s block diagram is presented in Fig. 4. \(\square\)

Fig. 4
figure 4

Block diagram of Lemma 1

Full size image

After that, the secrete message will placed according to the following procedure.

2.3.1 Minimum-modification-based data embedding

The pixels are taken from an original cover image. Sorting the pixels in row-wise of image block in increasing order to get \((x_{\sigma (1)}, \ldots , x_{\sigma (n)})\), where \(\sigma :\{1,\ldots , n\}\rightarrow \{1,\ldots , n\}\) and \(x_{\sigma (1)} \le \ldots \le x_{ \sigma (n)},\ \sigma (j) > \sigma (i)\) if \(x_{\sigma (j)} =x_{\sigma (i)}\) and \(j>i\) is one to one mapping of n pixels \((x_1,\ldots ,x_n)\) of an image block X. We compute

$$\begin{aligned} d_{\text{ min }_{k}}=x_{p}-x_{q} \ \ \text{ where } \ \ \left\{ \begin{array}{l} p=\text{ min }(\sigma ((1)+k),\sigma ((2)+k)), \\ q=\text{ max }(\sigma ((1)+k),\sigma ((2)+k)), \\ k=(0,1,\ldots ,\text{ fix }(n/2)-1). \end{array}\right. \end{aligned}$$
(3)

To round the elements towards zero, we use \(\text{ fix }()\) function which produces in an array of integers.

  • \(p=\sigma ((1)+k)\) and \(q=\sigma ((2)+k)\) when \(\sigma ((1)+k)<\sigma ((2)+k)\). Here, \(d_{\text{ min }_{k}}\le 0\).

  • \(p=\sigma ((2)+k)\) and \(q=\sigma ((1)+k)\) when \(\sigma ((1)+k)>\sigma ((2)+k)\). Here, \(d_{\text{ min }_{k}}>0\) and \(x_{\sigma ((1)+k)}<x_{\sigma ((2)+k)}\).

Here, \(C \in \{0,1\}\) is a hidden data bit to embed within pixels. The value of \(\alpha\) is changeable. The minimum pixel \(x_{\sigma ((1)+k)}\) is adjusted as \(x^{'}\) by

$$\begin{aligned} x^{'}=\left\{ \begin{array}{l} (x_{\sigma ((1)+k)}-\alpha )-C,\ \text{ if }\ d_{\text{ min }_{k}}=0\\ (x_{\sigma ((1)+k)}-\alpha )-1,\ \ \text{ if }\ d_{\text{ min }_{k}}<0\\ (x_{\sigma ((1)+k)}-\alpha )-C,\ \text{ if }\ d_{\text{ min }_{k}}=1\\ (x_{\sigma ((1)+k)}-\alpha )-1,\ \ \text{ if }\ d_{\text{ min }_{k}}>1\\ \end{array}\right. \end{aligned}$$
(4)

Assume that, the marked pixel of X is \((y_{1},y_{2},\ldots ,y_{n})\), where \(y_{\sigma ((1)+k)}=x^{'}\) and also \(y_{i}=x_{i}\) for all \(i\ne \sigma ((1)+k)\).

2.3.2 Maximum-modification-based data embedding

In this type of data embedding procedure, we compute

$$\begin{aligned} d_{\text{ max }_{k}}=x_{r}-x_{s}\ \ \text{ where } \ \ \left\{ \begin{array}{l} r=\text{ min }(\sigma ((n)-k),\sigma ((n-1)-k)),\\ s=\text{ max }(\sigma ((n)-k),\sigma ((n-1)-k)),\\ k=(0,1,\ldots ,\text{ fix }(n/2)-1). \end{array}\right. \end{aligned}$$
(5)

  • \(r=\sigma ((n-1)-k)\) and \(s=\sigma ((n)-k)\) when \(\sigma ((n)-k)>\sigma ((n-1)-k)\). Here, \(d_{\text{ max }_{k}}\le 0\).

  • \(r=\sigma ((n)-k)\) and \(s=\sigma ((n-1)-k)\) when \(\sigma ((n)-k)<\sigma ((n-1)-k)\). Here, \(d_{\text{ max }_{k}}>0\) and \(x_{\sigma ((n)-k)}>x_{\sigma ((n-1)-k)}\).

Here, \(C\in \{0,1\}\) is a hidden data bit to embed within pixel values. The value of \(\alpha\) is changeable. The pixel(maximum) \(x_{\sigma ((n)-k)}\) is adjusted to \(x^{'}\) by

$$\begin{aligned} x^{'} = \left\{ \begin{array}{l} (x_{\sigma ((n)-k)}+\alpha )+C,\ \text{ if }\ d_{\text{ max }_{k}}=0\\ (x_{\sigma ((n)-k)}+\alpha )+1,\ \ \text{ if }\ d_{\text{ max }_{k}}<0\\ (x_{\sigma ((n)-k)}+\alpha )+C,\ \text{ if }\ d_{\text{ max }_{k}}=1\\ (x_{\sigma ((n)-k)}+\alpha )+1,\ \ \text{ if }\ d_{\text{ max }_{k}}>1 \end{array}\right. \end{aligned}$$
(6)

Assume that, the marked pixel of X is \((y_{1},y_{2},\ldots ,y_{n})\) where \(y_{\sigma ((n)-k)} = x^{'}\) and also \(y_{i} = x_{i}\) for all \(i\ne \sigma ((n)-k)\).

Now, the data embedding process in the horizontal direction is completed. After that, this procedure applied in the vertical direction and diagonal direction. Thus, construct a final marked image block. Figure 5 describes the overall block diagram of the data embedding process. “Algorithm 1” describes the proposed data embedding algorithm.

Fig. 5
figure 5

Block diagram of data embedding

Full size image
figure a

2.4 Numerical illustration of embedding process

Let us observe a numerical example of data bit embedding process to understand the proposed process. We take \((4\times 4)\) original cover image. Create four sub-sampled images of size \((2\times 2)\) as \(SI_{1},SI_{2},SI_{3}\) and \(SI_{4}\) from cover image. These four images are interpolated using interpolation techniques and make \(ESI_{1},ESI_{2},ESI_{3}\) and \(ESI_{4}\) of size \((4\times 4)\). Consider pixel values of sub-sampled image block \((SI_{1})\) is in first row (100, 102) and second row (107, 104). The first interpolate value is \(\text{ fix }((100+102)/2)=101\) because of two neighbor pixel and for position (2, 2) there are four neighbor, so modified value will be \(\text{ fix }((100+102+107+104)/4)=103\).

Let secret message is ’y’ which we want to hide. We take ASCII value of message ’y’ and then apply lagrange interpolation formula in equation (4) with \(h=3,l=257,a_{1}=2,a_{2}=3\) in Z / 257Z. Convert the value of f(x) in binary bit stream and embed these data bit into \(ESI_{x}\) where \(x=1,\ldots ,4\), in different direction(horizontal, vertical, first and second diagonal) one after another. Thus makes four interpolated sub-sampled stego images \(ESI_{1}^{\prime},ESI_{2}^{\prime},ESI_{3}^{\prime}\) and \(ESI_{4}^{\prime}\) shows in Fig. 6.

Fig. 6
figure 6

Numerical example of data embedding

Full size image

2.5 Extraction process

In this phase, we implement data extraction process of the proposed scheme. As there are four shares, only 3 of them to reconstruct the message. So take anyone combination of stego images (1, 2, 4), (2, 3, 4), (1, 3, 4) and (1, 2, 3). In each sub-sampled stego images among any one combination is used to perform data extraction process in the diagonal direction. We subtract and add the value of \(\alpha\) from the maximum and to the minimum pixel of \(1{\mathrm{st}},\ 2{\mathrm{nd}}\) and so on respectively using the Lemma 2.

Lemma 2

If the interpolated sub-sampled stego image block is \(((2w-1)\times (2w-1))\) and the ranked pixel is \((p_{1}^{'},p_{2}^{'},\ldots ,p_{2w-2}^{'},p_{2w-1}^{'})\), then the adjusted pixel will be \((p_{1}^{'}+\alpha _{(w-2)},p_{2}^{'} +\alpha _{((w-2)-1)},\ldots ,p_{w-1}^{'}+ \alpha _{((w-2)-(w-2))}, p_{w}^{'}, p_{w+1}^{'}-\alpha _{((w-2)-(w-2))}, \ldots ,p_{2w-2}^{'} -\alpha _{((w-2)-1)},p_{2w-1}^{'}-\alpha _{(w-2)})\), where the minimum value of \(\alpha =0\), the maximum value of \(\alpha =(w-2)\) and the \(\alpha _{((w-2)-(w-2))}=((w-2)-(w-2))\).

Example 2

Assume that, the interpolated sub-sampled stego image block size is \((3\times 3)\), then the sub-sampled image block size is \((2\times 2)\). The maximum value of \(\alpha\) is 0, which is added and subtracted to the minimum and from the maximum pixel value respectively. The subtraction and addition process of \(\alpha\) will be continue until it\((\alpha )\) comes to 0. The Lemma 2’s block diagram is presented in Fig. 7. \(\square\)

Fig. 7
figure 7

Block diagram of Lemma 2

Full size image

2.5.1 Minimum-modification-based data extraction


The data extraction process is occurs from the interpolated sub-sampled stego image block where the marked pixels are \((y_{1},y_{2}, \ldots ,y_{n})\). Here, pixel ranking remains same. We compute \(d_{\text{ min }_{k}}^{'}=y_{p}-y_{q}\), where (p, q, k) is described in equation (3).


\(*\ y_{p}\le y_{q}\) when \(d_{\text{ min }_{k}}^{'}\le 0\). Now, \(\sigma ((1)+k)<\sigma ((2)+k)\) and \(p=\sigma ((1)+k)\), \(q=\sigma ((2)+k)\):

  • When \(d_{\text{ min }_{k}}^{'}\in \{0,-1\}\), there exist secret data and it is \(C=-d_{\text{ min }_{k}}^{'}\). The recovered pixel(minimum) is \(x_{\sigma ((1)+k)}=(y_{p}+\alpha )+C\);

  • When \(d_{\text{ min }_{k}}^{'}<-1\), there absent secret data. The recovered minimum is \(x_{\sigma ((1)+k)}=(y_{p}+\alpha )+1\).

\(*\ y_{p}> y_{q}\) when \(d_{\text{ min }_{k}}^{'}> 0\) . Now, \(\sigma ((1)+k)>\sigma ((2)+k)\) and \(p=\sigma ((2)+k)\), \(q=\sigma ((1)+k)\):

  • When \(d_{\text{ min }_{k}}^{'}\in \{1,2\}\), there exist secret data and it is \(C=d_{\text{ min }_{k}}^{'}-1\). The restored minimum is \(x_{\sigma ((1)+k)}=(y_{q}+\alpha )+C\);

  • When \(d_{\text{ min }_{k}}^{'}>2\), there absent secret data. The recovered pixel(minimum) is \(x_{\sigma ((1)+k)}=(y_{q}+\alpha )+1\).

2.5.2 Maximum-modification-based data extraction


The image restoration procedure is conducted from the interpolated sub-sampled image where the marked value is \((y_{1},y_{2},\ldots ,y_{n})\). Here, pixel ranking also remain the same. Now, compute \(d_{\text{ max }_{k}}^{'} = y_{k}-y_{l}\) where (r, s, k) is described previously in equation (5).

\(*\ y_{r}\le y_{s}\) when \(d_{\text{ max }_{k}}^{'}\le 0\). Now \(\sigma ((n-1)-k)<\sigma ((n)-k)\) and \(r =\sigma ((n-1)-k)\), \(s=\sigma ((n)-k)\):

  • When \(d_{\text{ max }_{k}}^{'}\in \{0,-1\}\), there exist secret data and it is \(C=-d_{\text{ max }_{k}}^{'}\). The recovered pixel(maximum) is \(x_{\sigma ((n)-k)} = (y_{s}-\alpha )-C\);

  • When \(d_{\text{ max }_{k}}^{'}<-1\), there absent secret data. The recovered pixel(maximum)is \(x_{\sigma ((n)-k)}=(y_{s}-\alpha )-1\).

\(*\ y_{r}> y_{s}\) when \(d_{\text{ max }_{k}}^{'}> 0\). Now \(\sigma ((n-1)-k)>\sigma ((n)-k)\) and \(r =\sigma ((n)-k)\), \(s=\sigma ((n-1)-k)\):

  • When \(d_{\text{ max }_{k}}^{'}\in \{1,2\}\), there exist secret data and it is \(C=d_{\text{ max }_{k}}^{'}-1\). The pixel(maximum) is \(x_{\sigma ((n)-k)}=(y_{r}-\alpha )-C\);

  • When \(d_{\text{ max }_{k}}^{'}>2\), there absent secret data. The restored pixel(maximum) is \(x_{\sigma ((n)-k)}=(y_{r}-\alpha )-1\).

The secret message, as well as the pixels, are recovered from two diagonal of interpolated sub-sampled stego image. The above-said process is applied through vertical and horizontal direction one after another. Finally, retrieve all the hidden data bit of corresponding f(x). Now convert the value of f(x) into corresponding decimal value. Select anyone combination of stego images and apply Lagrange interpolation formula to recover secret data\((a_{0})\). After that, convert this ASCII value of secret \((a_{0})\) to its corresponding data. Now make sub-sampled images from interpolated sub-sampled images by eliminating all interpolate rows and columns. Finally, construct the real cover from sub-sampled images. Figure 8 shows the block diagram of the data extraction process. “Algorithm 2” describes the proposed data extraction algorithm.

Fig. 8
figure 8

Block diagram of data extraction

Full size image
figure b

2.6 Numerical illustration of extraction process

Let us observe an example of data extraction to recognize the proposed method. We take four interpolated sub-sampled stego images \(ESI_{1}^{\prime},ESI_{2}^{\prime},ESI_{3}^{\prime}\) and \(ESI_{4}^{\prime}\) of size \((3\times 3)\) shown in Fig. 9.

Fig. 9
figure 9

Numerical example of data extraction

Full size image

The data extraction process goes through diagonal, vertical and then horizontal direction one after another of each interpolated stego image of above-said any one combination. Retrieve all the pixel value and hidden data of stego images. Then convert these binary data bit to its equivalent decimal value. Apply Lagrange interpolation equation to reconstruct the hidden data \((a_{0})\) from any three stego image. Finally, convert this ASCII value of \((a_{0})\) to its corresponding data. Afterward, the original cover image is built from sub-sampled images shown in Fig. 10.

Fig. 10
figure 10

Recover of cover image

Full size image

3 Experimental results and comparisons

We use standard cover images (gray-scale) from different databases. Ten images \((512\times 512)\) were taken from USC-SIPI [30], twenty images \((481 \times 321)\) were taken from Berkeley Segmentation Dataset and Benchmark [31], twenty-four images \((768 \times 512 \ \text{ or }\ 512\times 768)\) were taken from Kodak [9] and twenty images with different size were taken from the National Library of Medicine [26] for our experiment as test images and only 10 of each data set are shown in Fig. 11. For analysis the usefulness of our proposed method, we are used four different benchmark image databases. The secret message embedding and extraction algorithm for the proposed scheme are examined through MATLAB R2014a (8.3.0.532). The capacity of the data bit to an image with quality measured with PSNR (dB). For embedding 6,42,168 secret bits, the PSNR result shows 45.1081 dB for Lena image presented in Table 1. Tables 2, 3 and 4 are displayed the PSNR values with the payload of 30,000 bits from other datasets. These table presented that PSNR value is greater than 55 dB after embedding 30,000 secret bits. The proposed method improves the value of PSNR (dB) and payload than the other data hiding method with PVO described by [13, 19, 22, 24, 25]. The comparison results in terms of PSNR(dB) between the proposed and the other data hiding PVO methods with the embedding capacity of 10,000 bits are appeared in Table 5.

Fig. 11
figure 11

The standard cover images are used in this scheme

Full size image

The data embedding capacity is dependent on image size which is displayed in Fig. 12. It is also shows the better performance for image block size \((3\times 3)\) with good PSNR value than the other block size. Using \((2 \times 2)\) block size in Lena image \((512 \times 512)\), 4,46,196 bits can be embedded with PSNR value 41.58 dB whereas 6,42,168 bits are embedded with PSNR value 45.10 dB with the block size \((3 \times 3)\).

Table 1 The data bit embedding capacity (EC) with different pay load (bits) of USC-SIPI 10 standard test images with PSNR (dB)
Full size table
Table 2 Compute the PSNR (dB) for the Kodak images with the payload 30,000 bits
Full size table
Table 3 Compute the PSNR (dB) for the images of Berkeley Segmentation Dataset with the payload 30,000 bits
Full size table
Table 4 Compute the PSNR (dB) for the image of National Library of Medicine Dataset with the payload 30,000 bits
Full size table
Table 5 Comparison between the other schemes (Li et al. [13], Peng et al. [24], Ou et al. [22] and Qu et al. [25] and Meikap et al.[19]) and the proposed scheme of PSNR (dB) with bit capacity of 10,000 bits
Full size table
Fig. 12
figure 12

The embedding performance of proposed method with different block sizes (for Lena image)

Full size image
Fig. 13
figure 13

Comparisons of PSNR (dB) between the schemes of Li et al. [13], Peng et al. [24], Ou et al. [22], Qu et al. [25], Meikap et al. [19] and proposed method

Full size image

The performance comparison graph of PSNR (dB) with different schemes is displayed in Fig. 13. The proposed method enhances the payload compared with other PVO methods. For Lena image, the payload is 610168, 604168, 605168, 596168, 512168 and 523182 bits more than Li et al.’s [13], Peng et al.’s [24], Ou et al.’s [22], Qu et al.’s [25], Jana [6] and Meikap et al.’s [19] method respectively depicted in Table 6. The differentiation graph including bit capacity with various PVO based schemes is shown in Fig. 14. It is noticeable that in the purpose of the payload as well as quality, our method is superior to previous PVO methods.

Table 6 Comparison of payload in number of bits
Full size table
Fig. 14
figure 14

Comparisons among the other schemes (Li et al. [13], Peng et al. [24], Ou et al. [22], Qu et al. [25] and Meikap et al. [19]) and the proposed scheme of payload

Full size image

3.1 Steganographic attacks

Steganalysis technique is a valuable activity in hidden message communication where a suspected image has hidden data or not. Nowadays, these systems do not fulfill a sufficient security. So, users leave hints while data embedding into an image. Therefore, a steganalyst identifies whether a hidden message exists or not in an image. All the steganalyst performs this in different ways. This way is categorized in two part: Targeted and Blind steganalysis. Among the targeted method, the structural attack, statistical attack and visual attack are there. On he other hand, one important method of blind steganalysis is Regular Singular(RS) analysis proposed by J. Fridrich [4].

3.2 RS analysis

We examine sub-sampled marked images across the RS analysis method [4]. If the RS value is nearer to zero, then slighter changes occurred and it is safe. Table 7 described the \(R_M\) and \(R_{-M}\), \(S_M\) and \(S_{-M}\) values are closely equal. Both \(R_M\ \cong \ R_{-M}\) and \(S_M\ \cong \ S_{-M}\) rule satisfies for the marked image blocks. So, our proposed method is more secure for RS attack.

Table 7 RS analysis for stego image
Full size table

3.3 Statistical attack

We examine the strength of our proposed method by concealing the hidden data. Our method is safe to prevent the various attacks due to its having no adverse effect. The correlation coefficient(CC) between the original cover image and the marked image is shown in Table 8. It displays, the CC is close to one which represents the best privacy for data hiding. The standard deviation(SD) between the original and marked image is depicted in Table 9. It displays, the SD is close to zero which represents the best privacy for data hiding. Our proposed method obtained the strength against various attacks. Without loss of any data, the hidden message and cover image recovered from sub-sampled marked (stego) images.

Table 8 The correlation coefficient(CC)between the original and marked image
Full size table
Table 9 The standard deviation (SD) between the original and marked image
Full size table

3.4 Structural similarity index and normalized cross-correlation

The structural similarity(SSIM) index is a technique to determine the similarity between the original and the marked image. It is based on three computation key: the luminance key, the structural key the contrast key. The whole index is a multiplicative combination of these three keys. In Normalised cross-correlation (NCC), we find out some common patterns between cover and stego image. The NCC has been used in the fields of block matching, image and video compression and image registration etc. Table 10 shows the value of SSIM and NCC between original and stego image.

Table 10 The data embedding capacity (EC) with different pay load (data bits) of four different image databases with structural similarity(SSIM) index and normalized cross-correlation (NCC)
Full size table

4 Conclusion

A new RDH approach for secret sharing within the sub-sampled image using Directional Pixel Value Ordering (DPVO) is proposed. The sub-sampled images from the cover image help to carry secret information in a distributed manner. The proposed scheme improves the level of security due to Lagrange function and hard to reveal by the adversary. The new algorithms for embedding and extraction are designed in such a manner that more data bits are embedded and extracted to and from interleaved pixels respectively. Our method accomplishes secure data communication because we embed the Lagrange function generated data only, not the real hidden message. The hidden message obtains throughout a predefined threshold number of the sub-sampled image, not the entire sub-sampled image. Using method, we obtained an average PSNR value over 60 dB and average data embedding capacity over 6,00,000 bits. The proposed method is examined by RS analysis, SD, CC, SSIM and NCC that gives good results. We also notice that this scheme shows an excellent performance than other PVO based works and is secure against several steganographic attacks.