Directional pixel value ordering based secret sharing using subsampled image exploiting Lagrange polynomial
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Abstract
In this paper, we proposed secret sharing within the subsampled image using Directional Pixel Value Ordering (DPVO) for Reversible Data Hiding scheme. The original cover is partitioned into subsampled 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 subsampled 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 crosscorrelation 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 pixelvalueordering Image interpolation Reversible data hiding Subsampled image Embedding capacity RS analysis Structural similarity index Normalized crosscorrelation Steganography1 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 subsampled 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 subimage 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 pixelbased 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 subsampled images has been developed by Jana [6] in 2017. According to him, this hiding process is done through Lagrange’s interpolation polynomial on subsampled 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 subsampled image using DPVO. Here, any user can conceal hidden data bits in between subsampled 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, subsampled 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 \((h1)\) can do so. Secret data communication using DPVO on several subsampled 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 subsampled 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:
 (i)
Secret data concealing through Lagrange interpolation polynomial was irreversible. Using Directional PVO (DPVO) method within interpolated subsampled images, it is possible to insert and take out secret message to cover image and from marked (stego) image respectively.
 (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 subsampled images which are served as numerous images for improve visual quality, data security. This scheme also accomplish reversibleness.
 (iii)
Use of the Lagrange interpolation function, it produces new secret data bit from the original secret message which is distributed among the interpolated subsampled 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.
 (iv)
So far, the stego image quality is calculated in PSNR dB through RDH schemes within subsampled 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.
 (v)
If any subsampled 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.
 (vi)
So far, data embedding capacity (EC) within subsampled 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 subsampled 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
2.2 Directional pixel value ordering (DPVO)
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 subsampled 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 \((w2)\) respectively.
Lemma 1
If the subsampled image size is \(((2w1)\times (2w1))\) and the ranked pixel is \((p_{1},p_{2},\ldots ,p_{2w2},\, p_{2w1})\) , then, the adjusted value of pixel will be \((p_{1}\alpha _{(w2)},p_{2} \alpha _{((w2)1)},\ldots ,p_{w1}\alpha _{((w2)(w2))}, p_{w},p_{w+1}+\alpha _{((w2)(w2))},\ldots ,p_{2w2} +\alpha _{((w2)1)},p_{2w1}+\alpha _{(w2)})\) , where, \(\alpha _{((w2)(w2))}=((w2)(w2))\) . The minimal and maximal value of \(\alpha\) will be 0 and \((w2)\) respectively.
Example 1
Assume that, the original block is \((w\times w)=(2\times 2)\), then, interpolated subsampled image block will be \((3\times 3)\). The \(\alpha\)’s maximum value \((w2)=0\), 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. \(\square\)
After that, the secrete message will placed according to the following procedure.
2.3.1 Minimummodificationbased data embedding

\(p=\sigma ((1)+k)\) 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 \(x_{\sigma ((1)+k)}<x_{\sigma ((2)+k)}\).
2.3.2 Maximummodificationbased data embedding

\(r=\sigma ((n1)k)\) and \(s=\sigma ((n)k)\) when \(\sigma ((n)k)>\sigma ((n1)k)\). Here, \(d_{\text{ max }_{k}}\le 0\).

\(r=\sigma ((n)k)\) and \(s=\sigma ((n1)k)\) when \(\sigma ((n)k)<\sigma ((n1)k)\). Here, \(d_{\text{ max }_{k}}>0\) and \(x_{\sigma ((n)k)}>x_{\sigma ((n1)k)}\).
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 subsampled 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 subsampled 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 \(\text{ fix }((100+102+107+104)/4)=103\).
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 subsampled 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 subsampled stego image block is \(((2w1)\times (2w1))\) and the ranked pixel is \((p_{1}^{'},p_{2}^{'},\ldots ,p_{2w2}^{'},p_{2w1}^{'})\), then the adjusted pixel will be \((p_{1}^{'}+\alpha _{(w2)},p_{2}^{'} +\alpha _{((w2)1)},\ldots ,p_{w1}^{'}+ \alpha _{((w2)(w2))}, p_{w}^{'}, p_{w+1}^{'}\alpha _{((w2)(w2))}, \ldots ,p_{2w2}^{'} \alpha _{((w2)1)},p_{2w1}^{'}\alpha _{(w2)})\), where the minimum value of \(\alpha =0\), the maximum value of \(\alpha =(w2)\) and the \(\alpha _{((w2)(w2))}=((w2)(w2))\).
Example 2
Assume that, the interpolated subsampled stego image block size is \((3\times 3)\), then the subsampled 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. \(\square\)
2.5.1 Minimummodificationbased data extraction
The data extraction process is occurs from the interpolated subsampled 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).

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 \(x_{\sigma ((1)+k)}=(y_{p}+\alpha )+1\).

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 Maximummodificationbased data extraction
The image restoration procedure is conducted from the interpolated subsampled 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).

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 \(x_{\sigma ((n)k)}=(y_{s}\alpha )1\).

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 \(x_{\sigma ((n)k)}=(y_{r}\alpha )1\).
2.6 Numerical illustration of extraction process
3 Experimental results and comparisons
The data bit embedding capacity (EC) with different pay load (bits) of USCSIPI 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 
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 
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 
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_IM00031001  61.5486  61.5690  61.5570  61.5570 
CXR1000_IM00033001  60.5041  60.4758  61.5467  60.5469 
CXR1025_IM00201001  59.4676  58.9183  58.5996  59.0134 
CXR1037_IM00291001  60.0501  60.9341  61.2166  60.1045 
CXR1074_IM00541001  59.2476  59.5074  60.0428  59.3276 
CXR1109_IM00761001  61.3409  60.9678  61.1367  60.9134 
CXR1114_IM00791001  59.0756  59.0322  58.9349  58.9609 
CXR1119_IM00801001  58.3439  58.3369  58.1208  58.2302 
CXR1120_IM00801001  56.7626  57.6120  56.7328  57.6761 
CXR1151_IM01021001  59.1244  59.1164  58.7986  59.0914 
CXR1188_IM01271001  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 
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 
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 
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
RS analysis for stego image
Image Database  Cover Image (CI)  Stego (Image)  \(R_{M}\)  \(R_{M}\)  \(S_{M}\)  \(S_{M}\)  RS value 

USCSIPI \((512\times 512)\)  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_IM00 761001  \(ESI_{1}^{\prime}\)  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_IM00 791001  \(ESI_{1}^{\prime}\)  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
The correlation coefficient(CC)between the original and marked image
Image database  Cover image (CI)  Data (bits)  CC  

\(ESI_{1} \& \ ESI_{1}^{\prime}\)  \(ESI_{2} \& \ ESI_{2}^{\prime}\)  \(ESI_{3} \& \ ESI_{3}^{\prime}\)  \(ESI_{4} \& \ ESI_{4}^{\prime}\)  
USCSIPI \((512\times 512)\)  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_IM00 761001  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_IM00 791001  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 
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  
USCSIPI \((512\times 512)\)  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 crosscorrelation
The data embedding capacity (EC) with different pay load (data bits) of four different image databases with structural similarity(SSIM) index and normalized crosscorrelation (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  
USCSIPI \((512\times 512)\)  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 subsampled image using Directional Pixel Value Ordering (DPVO) is proposed. The subsampled 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 subsampled image, not the entire subsampled 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)
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