Advertisement

SN Applied Sciences

, 1:645 | Cite as

Directional pixel value ordering based secret sharing using sub-sampled image exploiting Lagrange polynomial

  • Sudipta Meikap
  • Biswapati JanaEmail author
Research Article
  • 280 Downloads
Part of the following topical collections:
  1. Engineering: Digital Image Processing

Abstract

In this paper, we proposed secret sharing within the sub-sampled image using Directional Pixel Value Ordering (DPVO) for Reversible Data Hiding scheme. The original cover is partitioned into sub-sampled images which extend its size through image interpolation technique. The secret information has been converted through Lagrange interpolation polynomial techniques. These new secret data is embedded within pixels of each interpolated sub-sampled images using DPVO. In the decoder side, the secret data is taken out from the pixels of every stego image using reverse DPVO. After that, Lagrange’s interpolation is applied to generate the original secret message. The proposed scheme enhanced the security for the sharable characteristics of secrets amongst several images. It raises the data embedding capacity and improves the visual quality that is determined by peak signal to noise ratio (PSNR). The average PSNR value is over 60 dB. It shows the superiority of the new algorithm over the other existing data hiding methods regarding payload of data, quality, and security of the image. Also, the stego images have evaluated through standard deviation, regular singular analysis, correlation coefficient, normalized cross-correlation and structural similarity index in between original cover and marked (stego) image to display the robustness of our result among the several steganographic strikes.

Keywords

Directional pixel-value-ordering Image interpolation Reversible data hiding Sub-sampled image Embedding capacity RS analysis Structural similarity index Normalized cross-correlation Steganography 

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 (xf(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(ab), 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

Block diagram of sub-sampled image block

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

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

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

Block diagram of Directional PVO

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

Block diagram of Lemma 1

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

Block diagram of data embedding

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

Numerical example of data embedding

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

Block diagram of Lemma 2

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 (pqk) 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 (rsk) 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

Block diagram of data extraction

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

Numerical example of data extraction

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

Recover of cover 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 23 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

The standard cover images are used in this scheme

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)

Cover image (CI)

Data (bits)

PSNR \((ESI_{1}\) vs \(ESI_{1}^{\prime})\)

PSNR \((ESI_{2}\) vs \(ESI_{2}^{\prime})\)

PSNR \((ESI_{3}\) vs \(ESI_{3}^{\prime})\)

PSNR \((ESI_{4}\) vs \(ESI_{4}^{\prime})\)

Lena

30,000

60.0308

60.0224

60.9691

60.0095

100,000

54.2329

54.2033

54.1779

54.1641

300,000

48.9261

48.9001

48.9143

48.9160

6,42,168

45.1081

45.1013

45.0997

45.0943

Airplane F16

30,000

59.4728

59.5083

59.4351

59.5142

100,000

54.5837

54.6013

54.5579

54.6090

300,000

50.0480

50.0639

50.0458

50.0666

7,65,010

45.4585

45.4651

45.4617

45.4513

Baboon

30,000

55.8165

55.7900

55.7392

55.6767

100,000

50.7001

50.6867

50.6597

50.7102

300,000

47.1044

47.1118

47.0013

47.0986

5,40,245

44.8035

44.8013

44.8035

44.8020

Elaine

30,000

59.1069

59.1972

58.9189

58.9341

100,000

53.8720

53.8683

53.8965

53.8905

300,000

48.1806

48.7869

48.8204

47.9237

6,31,689

45.0599

45.0510

45.0603

45.0534

Fishing boat

30,000

59.9766

59.0464

60.0011

60.0102

100,000

54.6054

54.1181

54.0575

53.9906

300,000

49.5202

49.0073

48.9970

49.1986

6,96,635

45.2234

45.2214

45.2329

45.2209

House

30,000

60.6249

60.2861

60.4119

61.0963

100,000

55.2581

55.6210

55.6402

55.4141

300,000

49.7354

49.9788

49.9166

49.9583

7,35,202

45.3607

45.3639

45.3634

45.3604

Peppers

30,000

58.4467

58.9680

58.0028

58.1158

100,000

53.4042

53.5771

53.6030

53.5919

300,000

48.5372

48.0889

48.1553

48.1973

6,40,845

45.1068

45.1091

45.1083

45.1197

Sailboat on lake

30,000

57.5059

57.7179

57.7680

57.7687

100,000

52.7810

52.2517

52.3077

53.0320

300,000

47.8125

47.7759

47.9727

47.9741

5,68,674

44.8651

44.8599

449638

44.8621

Splash

30,000

57.7015

57.8507

58.0165

57.8975

100,000

52.7727

52.3167

51.9972

52.3128

300,000

48.3853

48.4383

48.2812

48.3530

6,74,973

45.2939

45.3124

45.3927

45.3963

Tiffany

30,000

60.2009

60.1691

60.1694

60.1843

100,000

54.8347

54.8379

54.8300

54.8419

300,000

50.0615

50.0754

50.0712

50.0884

8,39,463

45.6250

45.6300

45.6355

45.6292

Table 2

Compute the PSNR (dB) for the Kodak images with the payload 30,000 bits

Cover image (CI)

PSNR \((ESI_{1}\) vs \(ESI_{1}^{\prime})\)

PSNR \((ESI_{2}\) vs \(ESI_{2}^{\prime})\)

PSNR \((ESI_{3}\) vs \(ESI_{3}^{\prime})\)

PSNR \((ESI_{4}\) vs \(ESI_{4}^{\prime})\)

kodim01

59.4209

59.4403

59.0846

59.0544

kodim02

58.5933

58.7028

58.1770

58.1733

kodim03

62.2934

62.4061

61.9761

61.8781

kodim04

61.4075

61.3699

61.1351

61.1903

kodim05

59.7293

59.6866

58.8724

58.7862

kodim06

61.9675

61.9869

61.1067

61.1065

kodim07

61.2331

61.2372

61.0314

61.1236

kodim08

60.0424

60.0359

59.9328

59.9557

kodim09

61.5466

61.3777

61.0202

61.1571

kodim10

61.1422

61.1179

60.8224

60.8478

kodim11

62.0161

62.0698

62.3510

62.3442

kodim12

61.2300

61.1838

61.2361

61.1935

kodim13

60.2932

60.2849

60.2638

60.2576

kodim14

58.0893

58.0547

57.7902

57.8439

kodim15

60.2173

60.1860

60.1543

60.1519

kodim16

63.1452

63.1632

62.2844

62.1909

kodim17

61.8148

61.7890

62.0275

62.0789

kodim18

57.1546

57.2518

57.1733

57.2758

kodim19

60.7253

60.6613

60.6020

60.6528

kodim20

65.1076

65.1481

65.5565

65.5602

kodim21

61.3161

61.4953

60.6027

60.5791

kodim22

61.6053

61.7036

61.6522

61.6385

kodim23

62.4947

62.6207

62.5790

62.5175

kodim24

62.8106

62.8101

62.7832

62.6986

Table 3

Compute the PSNR (dB) for the images of Berkeley Segmentation Dataset with the payload 30,000 bits

Cover image (CI)

PSNR \((ESI_{1}\) vs \(ESI_{1}^{\prime})\)

PSNR \((ESI_{2}\) vs \(ESI_{2}^{\prime})\)

PSNR \((ESI_{3}\) vs \(ESI_{3}^{\prime})\)

PSNR \((ESI_{4}\) vs \(ESI_{4}^{\prime})\)

3096

58.6916

58.6497

58.5925

58.4657

8023

56.2949

56.3324

56.3137

56.2708

21077

61.0109

60.8583

60.8933

61.0014

35058

56.7286

56.7037

56.7252

56.6267

35091

60.0031

59.2180

59.2037

59.1979

41069

59.8362

59.8686

59.8845

59.8292

69020

57.3592

57.3699

57.3876

57.3295

69040

56.0710

56.0636

56.0857

56.0452

85048

58.9009

59.0262

58.9468

58.9885

87046

59.0203

59.7342

58.9968

58.8593

105025

57.7594

57.8242

57.8054

57.8496

109034

55.3072

56.3124

55.1947

55.3374

109053

58.5136

59.2002

59.0011

58.9970

113044

57.8196

57.8137

57.9879

58.0170

157055

59.4276

59.2559

59.4356

59.3173

160068

57.9579

58.0623

58.0478

57.6848

163085

59.3265

58.9918

59.2908

59.2343

236037

59.2985

59.2908

59.2590

59.3120

253055

58.3930

57.9051

58.1994

58.4380

299086

58.8721

59.0365

58.8628

58.1069

Table 4

Compute the PSNR (dB) for the image of National Library of Medicine Dataset with the payload 30,000 bits

Cover image (CI)

PSNR \((ESI_{1}\) vs \(ESI_{1}^{\prime})\)

PSNR \((ESI_{2}\) vs \(ESI_{2}^{\prime})\)

PSNR \((ESI_{3}\) vs \(ESI_{3}^{\prime})\)

PSNR \((ESI_{4}\) vs \(ESI_{4}^{\prime})\)

CXR1000_IM-0003-1001

61.5486

61.5690

61.5570

61.5570

CXR1000_IM-0003-3001

60.5041

60.4758

61.5467

60.5469

CXR1025_IM-0020-1001

59.4676

58.9183

58.5996

59.0134

CXR1037_IM-0029-1001

60.0501

60.9341

61.2166

60.1045

CXR1074_IM-0054-1001

59.2476

59.5074

60.0428

59.3276

CXR1109_IM-0076-1001

61.3409

60.9678

61.1367

60.9134

CXR1114_IM-0079-1001

59.0756

59.0322

58.9349

58.9609

CXR1119_IM-0080-1001

58.3439

58.3369

58.1208

58.2302

CXR1120_IM-0080-1001

56.7626

57.6120

56.7328

57.6761

CXR1151_IM-0102-1001

59.1244

59.1164

58.7986

59.0914

CXR1188_IM-0127-1001

61.9989

62.2274

62.2311

62.1107

MPX1355_synpic55464

60.6871

60.7803

60.2471

60.7141

MPX1430_synpic60423

59.3604

59.3780

60.0071

60.0142

MPX1459_synpic43454

60.1727

59.8446

60.2539

59.9883

MPX1521_synpic52763

57.6684

57.9687

58.2519

58.3073

MPX1726_synpic16586

56.6520

56.7505

56.7132

56.6637

MPX1726_synpic16587

58.3329

58.4372

58.4823

58.4913

MPX1855_synpic23323

61.1139

61.1915

61.1285

61.2235

MPX2720_synpic3995

62.8506

62.6011

62.7091

62.6497

MPX2750_synpic3136

57.2539

57.2317

57.3276

57.3324

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

Cover image (CI)

Li et al. [13]

Peng et al. [24]

Ou et al. [22]

Qu et al. [25]

Meikap et al. [19]

Proposed

Airplane F 16

61.6

62.9

63.3

63.7

63.1

64.2

Baboon

53.5

53.5

54.5

54.2

55.0

60.6

Barbara

59.9

60.5

60.6

59.8

60.4

61.6

Elaine

56.8

57.3

57.4

58.7

58.1

64.1

Fishing boat

57.8

58.2

58.1

58.4

59.2

64.9

House

61.8

64.4

63.7

64.6

66.4

65.0

Lena

59.8

60.4

60.6

60.3

59.8

65.6

Peppers

58.5

58.9

59.2

58.8

57.8

62.7

Sailboat on lake

58.2

58.8

59.4

59.8

59.1

61.9

Tiffany

60.1

60.7

60.3

60.6

59.7

64.9

Average

58.8

59.5

59.7

59.8

59.9

63.6

Fig. 12

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

Fig. 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

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

Cover image (CI)

Li et al. [13]

Peng et al. [24]

Ou et al. [22]

Qu et al. [25]

Jana [6]

Meikap et al. [19]

Proposed

Airplane F 16

38,000

52,000

47,000

69,000

-

1,70,334

7,65,010

Baboon

13,000

13,000

13,000

15,000

-

1,44,767

5,40,245

Barbara

27,000

29,000

31,000

33,000

1,30,000

1,30,056

6,17,418

Elaine

21,000

24,000

23,000

29,000

-

81,302

6,31,689

Fishing boat

24,000

26,000

26,000

30,000

-

80,504

6,96,635

House

30,000

46,000

37,000

64,000

-

1,55,622

7,35,202

Lena

32,000

38,000

37,000

46,000

1,30,000

1,18,986

6,42,168

Peppers

28,000

30,000

31,000

33,000

1,30,000

95,312

6,40,845

Sailboat on lake

23,000

26,000

26,000

29,000

-

80,974

5,68,674

Tiffany

33,000

40,000

40,000

52,000

1,30,000

1,39,618

8,39,463

Average

26,900

32,400

31,100

40,000

1,30,000

1,19,747

6,67,734

Fig. 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

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

Image Database

Cover Image (CI)

Stego (Image)

\(R_{M}\)

\(R_{-M}\)

\(S_{M}\)

\(S_{-M}\)

RS value

USC-SIPI \((512\times 512)\)

Lena

\(ESI_{1}^{\prime}\)

34402

32843

30495

32054

0.0480

\(ESI_{2}^{\prime}\)

34357

32887

30540

32010

0.0453

\(ESI_{3}^{\prime}\)

34528

32763

30369

32134

0.0544

\(ESI_{4}^{\prime}\)

34461

32814

30436

32083

0.0508

Airpla ne F16

\(ESI_{1}^{\prime}\)

34124

31455

30773

33442

0.0823

\(ESI_{2}^{\prime}\)

34228

31353

30669

33544

0.0886

\(ESI_{3}^{\prime}\)

34178

31442

30719

33455

0.0843

\(ESI_{4}^{\prime}\)

34219

31360

30678

33537

0.0881

BSDS \((481\times 321)\)

69040

\(ESI_{1}^{\prime}\)

19724

19952

18237

18009

0.0120

\(ESI_{2}^{\prime}\)

19673

19960

18288

18001

0.0151

\(ESI_{3}^{\prime}\)

19765

19913

18196

18048

0.0078

\(ESI_{4}^{\prime}\)

19723

19875

18238

18086

0.0080

87046

\(ESI_{1}^{\prime}\)

19976

19389

17985

18572

0.0309

\(ESI_{2}^{\prime}\)

20261

19189

17700

18772

0.0565

\(ESI_{3}^{\prime}\)

20070

19367

17891

18594

0.0370

\(ESI_{4}^{\prime}\)

20055

19331

17906

18630

0.0381

Kodak \((512\times 768)\)

kodim18

\(ESI_{1}^{\prime}\)

52616

52593

44793

44816

0.0005

\(ESI_{2}^{\prime}\)

52576

52825

44833

44584

0.0051

\(ESI_{3}^{\prime}\)

52329

52628

45080

44781

0.0061

\(ESI_{4}^{\prime}\)

52720

52427

44689

44982

0.0060

kodim19

\(ESI_{1}^{\prime}\)

52274

50246

45135

47163

0.0416

\(ESI_{2}^{\prime}\)

51801

50798

45608

46611

0.0206

\(ESI_{3}^{\prime}\)

51950

50610

45459

46799

0.0275

\(ESI_{4}^{\prime}\)

52019

50571

45390

46838

0.0297

National Library of Medicine \((512\times 420)\)

CXR110 9_IM-00 76-1001

\(ESI_{1}^{\prime}\)

29122

27226

24091

25987

0.0713

\(ESI_{2}^{\prime}\)

29006

27287

24207

25926

0.0646

\(ESI_{3}^{\prime}\)

28953

27382

24260

25831

0.0590

\(ESI_{4}^{\prime}\)

29010

27264

24203

25949

0.0656

CXR111 4_IM-00 79-1001

\(ESI_{1}^{\prime}\)

29564

29031

23649

24182

0.0200

\(ESI_{2}^{\prime}\)

29470

28964

23743

24249

0.0190

\(ESI_{3}^{\prime}\)

29710

28965

23503

24248

0.0280

\(ESI_{4}^{\prime}\)

29562

28951

23651

24262

0.0230

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

Image database

Cover image (CI)

Data (bits)

CC

\(ESI_{1} \& \ ESI_{1}^{\prime}\)

\(ESI_{2} \& \ ESI_{2}^{\prime}\)

\(ESI_{3} \& \ ESI_{3}^{\prime}\)

\(ESI_{4} \& \ ESI_{4}^{\prime}\)

USC-SIPI \((512\times 512)\)

Lena

30,000

1.0000

1.0000

1.0000

1.0000

100,000

0.9999

0.9999

0.9999

0.9999

300,000

0.9998

0.9998

0.9998

0.9998

6,42,168

0.9995

0.9995

0.9995

0.9995

Airpla ne F16

30,000

1.0000

1.0000

1.0000

1.0000

100,000

1.0000

1.0000

1.0000

1.0000

300,000

0.9999

0.9999

0.9999

0.9999

7,65,010

0.9998

0.9998

0.9998

0.9998

BSDS \((481\times 321)\)

69040

30,000

0.9998

0.9998

0.9998

0.9998

100,00

0.9993

0.9993

0.9993

0.9993

199,617

0.9987

0.9987

0.9987

0.9987

87046

30,000

0.9999

0.9999

0.9999

0.9999

100,000

0.9998

0.9998

0.9998

0.9998

300,000

0.9996

0.9996

0.9996

0.9996

3,59,889

0.9996

0.9996

0.9996

0.9996

Kodak \((512\times 768)\)

kodim18

30,000

0.9999

0.9999

0.9999

0.9999

100,000

0.9998

0.9998

0.9998

0.9998

300,000

0.9994

0.9994

0.9994

0.9994

6,70,304

0.9989

0.9989

0.9989

0.9989

kodim19

30,000

1.0000

1.0000

1.0000

1.0000

100,000

0.9999

0.9999

0.9999

0.9999

300,000

0.9998

0.9998

0.9998

0.9998

9,01,696

0.9994

0.9994

0.9994

0.9994

National Library of Medicine \((512\times 420)\)

CXR110 9_IM-00 76-1001

30,000

1.0000

1.0000

1.0000

1.0000

100,000

0.9999

0.9999

0.9999

0.9999

300,000

0.9996

0.9996

0.9996

0.9996

6,64,532

0.9994

0.9994

0.9994

0.9994

CXR111 4_IM-00 79-1001

30,000

1.0000

1.0000

1.0000

1.0000

100,000

0.9998

0.9998

0.9998

0.9998

300,000

0.9996

0.9996

0.9996

0.9996

6,50,888

0.9992

0.9992

0.9992

0.9992

Table 9

The standard deviation (SD) between the original and marked image

Image database

Cover image

Data (bits)

SD

\(ESI_{1}\)

\(ESI_{1}^{\prime}\)

Difference

\(ESI_{2}\)

\(ESI_{2}^{\prime}\)

Difference

\(ESI_{3}\)

\(ESI_{3}^{\prime}\)

Difference

\(ESI_{4}^{\prime}\)

\(ESI_{4}\)

Difference

USC-SIPI \((512\times 512)\)

Lena

30,000

39.1769

39.1985

0.0216

39.1842

39.2059

0.0217

39.1847

39.2063

0.0217

39.1872

39.2087

0.0215

100,000

39.2545

0.0776

39.2626

0.0784

39.2630

0.0783

39.2656

0.0784

300,000

39.4549

0.2780

39.4628

0.2786

39.4622

0.2775

39.4646

0.2774

6,42,168

39.7881

0.6112

39.7966

0.6124

39.7965

0.6118

39.7994

0.6122

Airpla ne F16

30,000

48.4544

48.4851

0.0307

48.4892

48.5194

0.0302

48.4345

48.4649

0.0304

48.4694

48.4993

0.0300

100,000

48.5589

0.1045

48.5931

0.1038

48.5389

0.1044

48.5732

0.1038

300,000

48.7699

0.3155

48.8027

0.3135

48.7497

0.3152

48.7833

0.3139

7,65,010

49.2497

0.7953

49.2832

0.7940

49.2287

0.7942

49.2631

0.7937

BSDS \((481\times 321)\)

69040

30,000

25.7045

25.7514

0.0469

25.7308

25.7773

0.0465

25.7074

25.7539

0.0465

25.7153

25.7610

0.0457

100,000

25.9376

0.2331

25.9583

0.2274

25.9394

0.2319

25.9463

0.2309

199,617

26.4275

0.7230

26.4563

0.7255

26.4319

0.7245

26.4398

0.7244

87046

30,000

41.2540

41.2884

0.0344

41.2508

41.2855

0.0347

41.3117

41.3460

0.0343

41.3051

41.3399

0.0348

100,000

41.3888

0.1348

41.3856

0.1348

41.4464

0.1347

41.4410

0.1359

300,000

41.8094

0.5554

41.8068

0.5560

41.8688

0.5572

41.8625

0.5574

3,59,889

41.9537

0.6997

41.9514

0.7006

42.0129

0.7012

42.0072

0.7021

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)

Image database

Cover image (CI)

Data (bits)

\(ESI_{1}\) vs \(ESI_{1}^{\prime}\)

\(ESI_{2}\) vs \(ESI_{2}^{\prime}\)

\(ESI_{3}\) vs \(ESI_{3}^{\prime}\)

\(ESI_{4}\) vs \(ESI_{4}^{\prime}\)

SSIM

NCC

SSIM

NCC

SSIM

NCC

SSIM

NCC

USC-SIPI \((512\times 512)\)

Lena

30,000

1.0000

1.0000

1.0000

1.0000

0.9999

1.0000

0.9999

1.0000

100,000

0.9997

0.9999

0.9997

0.9999

0.9997

0.9999

0.9997

0.9999

300,000

0.9987

0.9998

0.9987

0.9998

0.9987

0.9998

0.9986

0.9998

6,42,168

0.9961

0.9995

0.9961

0.9995

0.9961

0.9995

0.9961

0.9995

Airpla ne F16

30,000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

100,000

0.9999

1.0000

0.9999

1.0000

0.9999

1.0000

0.9999

1.0000

300,000

0.9997

0.9999

0.9997

0.9999

0.9997

0.9999

0.9997

0.9999

7,65,010

0.9990

0.9998

0.9990

0.9998

0.9990

0.9998

0.9990

0.9998

BSDS \((481\times 321)\)

69040

30,000

0.9989

0.9998

0.9990

0.9998

0.9989

0.9998

0.9990

0.9998

100,000

0.9968

0.9993

0.9969

0.9993

0.9969

0.9993

0.9969

0.9993

199,617

0.9944

0.9987

0.9944

0.9987

0.9944

0.9987

0.9944

0.9987

87046

30,000

0.9995

0.9999

0.9995

0.9999

0.9996

0.9999

0.9995

0.9999

100,000

0.9991

0.9998

0.9991

0.9998

0.9991

0.9998

0.9991

0.9998

300,000

0.9978

0.9996

0.9978

0.9996

0.9978

0.9996

0.9978

0.9996

3,59,889

0.9976

0.9996

0.9976

0.9996

0.9976

0.9996

0.9976

0.9996

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.

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no competing interests.

Supplementary material

Supplementary material 1 (mp4 9628 KB)

References

  1. 1.
    Alattar AM (2004) Reversible watermark using the difference expansion of a generalized integer transform. IEEE Trans Image Process 13(8):1147–1156MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chang YT, Huang CT, Lee CF, Wang SJ (2013) Image interpolating based data hiding in conjunction with pixel-shifting of histogram. J Supercomput 66(2):1093–1110CrossRefGoogle Scholar
  3. 3.
    Chowdhuri P, Pal P, Jana B (2018) A new dual image-based steganographic scheme for authentication and tampered detection. Inf Technol Appl Math ICITAM 2017 699:163zbMATHGoogle Scholar
  4. 4.
    Fridrich J, Goljan M, Du R (2001) Reliable detection of LSB steganography in color and grayscale images. In: Proceedings of the 2001 workshop on Multimedia and security: new challenges, ACM, 27–30Google Scholar
  5. 5.
    Jana B (2016) Dual image based reversible data hiding scheme using weighted matrix. Int J Electron Inf Eng 5(1):6–19Google Scholar
  6. 6.
    Jana B (2017) Reversible data hiding scheme using sub-sampled image exploiting Lagrange’s interpolating polynomial. Multimed Tools Appl 77:1–17Google Scholar
  7. 7.
    Jana B, Giri D, Mondal SK (2018) Dual image based reversible data hiding scheme using (7, 4) hamming code. Multimed Tools Appl 77(1):763–785CrossRefGoogle Scholar
  8. 8.
    Kim KS, Lee MJ, Lee HY, Lee HK (2009) Reversible data hiding exploiting spatial correlation between sub-sampled images. Pattern Recognit 42(11):3083–3096CrossRefGoogle Scholar
  9. 9.
    Kodak lossless true color image suite, http://r0k.us/graphics/kodak/
  10. 10.
    Lee CF, Huang YL (2012) An efficient image interpolation increasing payload in reversible data hiding. Expert Syst Appl 39(8):6712–6719CrossRefGoogle Scholar
  11. 11.
    Lee CF, Chang CC, Gao CY (2013) A two-staged multi-level reversible data hiding exploiting lagrange interpolation. In: 2013 Ninth international conference on intelligent information hiding and multimedia signal processing, IEEE, (pp. 485-488)Google Scholar
  12. 12.
    Lee CF, Weng CY, Chen KC (2017) An efficient reversible data hiding with reduplicated exploiting modification direction using image interpolation and edge detection. Multimed Tools Appl 76(7):9993–10016CrossRefGoogle Scholar
  13. 13.
    Li X, Li J, Li B, Yang B (2013) High-fidelity reversible data hiding scheme based on pixel-value-ordering and prediction-error expansion. Signal Process 93(1):198–205CrossRefGoogle Scholar
  14. 14.
    Lin CC, Tai WL, Chang CC (2008) Multilevel reversible data hiding based on histogram modification of difference images. Pattern Recognit 41(12):3582–3591CrossRefGoogle Scholar
  15. 15.
    Lu TC, Chang CC, Huang YH (2014) High capacity reversible hiding scheme based on interpolation, difference expansion, and histogram shifting. Multimed Tools Appl 72(1):417–435CrossRefGoogle Scholar
  16. 16.
    Luo L, Chen Z, Chen M, Zeng X, Xiong Z (2010) Reversible image watermarking using interpolation technique. IEEE Trans Inf Forensics Secur 5(1):187–193CrossRefGoogle Scholar
  17. 17.
    Luo H, Yu FX, Chen H, Huang ZL, Li H, Wang PH (2011) Reversible data hiding based on block median preservation. Inf Sci 181(2):308–328CrossRefGoogle Scholar
  18. 18.
    Malik A, Sikka G, Verma HK (2017) An image interpolation based reversible data hiding scheme using pixel value adjusting feature. Multimed Tools Appl 76(11):13025–13046CrossRefGoogle Scholar
  19. 19.
    Meikap S, Jana B (2018) Directional PVO for reversible data hiding scheme with image interpolation. Multimed Tools Appl 77(23):31281–31311CrossRefGoogle Scholar
  20. 20.
    Ni Z, Shi YQ, Ansari N, Su W (2006) Reversible data hiding. IEEE Trans Circuits Syst Video Technol 16(3):354–362CrossRefGoogle Scholar
  21. 21.
    Ou B, Li X, Wang J (2016) Improved PVO-based reversible data hiding: a new implementation based on multiple histograms modification. J Vis Commun Image Represent 38:328–339CrossRefGoogle Scholar
  22. 22.
    Ou B, Li X, Zhao Y, Ni R (2014) Reversible data hiding using invariant pixel-value-ordering and prediction-error expansion. Signal Process Image Commun 29(7):760–772CrossRefGoogle Scholar
  23. 23.
    Pal P, Chowdhuri P, Jana B (2018) Weighted matrix based reversible watermarking scheme using color image. Multimed Tools Appl, 1–26Google Scholar
  24. 24.
    Peng F, Li X, Yang B (2014) Improved PVO-based reversible data hiding. Digit Signal Process 25:255–265CrossRefGoogle Scholar
  25. 25.
    Qu X, Kim HJ (2015) Pixel-based pixel value ordering predictor for high-fidelity reversible data hiding. Signal Process 111:249–260CrossRefGoogle Scholar
  26. 26.
    The National Library of Medicine presents \(\text{MedPix}^{\textregistered }\), https://openi.nlm.nih.gov/gridquery.php?q=&it=x
  27. 27.
    Tian J (2003) Reversible data embedding using a difference expansion. IEEE Trans Circuits Syst Video Technol 13(8):890–896CrossRefGoogle Scholar
  28. 28.
    Tsai P, Hu YC, Yeh HL (2009) Reversible image hiding scheme using predictive coding and histogram shifting. Signal Process 89(6):1129–1143CrossRefGoogle Scholar
  29. 29.
    Tsai Y.Y, Chen J.T, Kuo Y.C, Chan C.S (2014) A generalized image interpolation-based reversible data hiding scheme with high embedding capacity and image quality. KSII Trans Internet Inf Syst, 8(9)Google Scholar
  30. 30.
    University of Southern California, The USC-SIPI image database, http://sipi.usc.edu/database/database.php?volume=misc
  31. 31.
    University of California, Berkeley, The Berkeley segmentation dataset and benchmark, https://www2.eecs.berkeley.edu/Research/Projects/CS/vision/bsds/BSDS300-images.tgz
  32. 32.
    Wahed M.A, Nyeem H (2017, September) A simplified parabolic interpolation based reversible data hiding scheme. In: 2017 4th International conference on advances in electrical engineering (ICAEE), IEEE, (pp. 743–748)Google Scholar
  33. 33.
    Wahed M.A, Nyeem H (2017, December) Efficient Data Embedding for Interpolation based Reversible Data Hiding Scheme. In: 2017 2nd international conference on electrical and electronic engineering (ICEEE), IEEE, (pp. 1–4)Google Scholar
  34. 34.
    Wahed M.A, Nyeem H (2018) Reversible data hiding with interpolation and adaptive embedding. Multimed Tools Appl, 1–25Google Scholar
  35. 35.
    Wahed MA, Nyeem H (2019) High capacity reversible data hiding with interpolation and adaptive embedding. PloS one 14(3):e0212093CrossRefGoogle Scholar
  36. 36.
    Wang XT, Chang CC, Nguyen TS, Li MC (2013) Reversible data hiding for high quality images exploiting interpolation and direction order mechanism. Digit Signal Process 23(2):569–577MathSciNetCrossRefGoogle Scholar
  37. 37.
    Zhang X, Sun Z, Tang Z, Yu C, Wang X (2017) High capacity data hiding based on interpolated image. Multimed Tools Appl 76(7):9195–9218CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Computer ScienceHijli CollegePaschim MedinipurIndia
  2. 2.Department of Computer ScienceVidyasagar UniversityMidnaporeIndia

Personalised recommendations