Progressive Hartley image secret sharing for high-quality image recovery

Abstract This paper introduces a highly efficient approach for progressive image secret sharing through the Discrete Hartley Transform (DHT), offering distinct advantages over existing methods. Leveraging DHT in the color domain, our approach achieves progressive sharing of color images while maintaining robustness and minimal visual impact. By optimizing quantization table selection, our method focuses on maximizing image recovery fidelity. Our extensive experiments showcase the efficacy of our proposed method, highlighting its potential for secure image sharing and superior image quality enabled by the real-number DHT transformation. Notably, our approach demonstrates strong security robustness and high-quality image reconstruction through histogram analysis. Objective assessment values of $88.106$88.106 (PSNR), $0.884$0.884 (NCC), $0.095$0.095 (NAE), and $0.895$0.895 (SSIM) underscore the superiority of our method. These results, coupled with its unique contributions and key factors, position our work as a promising solution in the field of image security and distribution, motivating further exploration and inspiring related research endeavors.


Introduction
In the context of the rapid growth of the Internet of Things (IoT) and the demand for quick software development cycles, security requirements often take a back seat.However, ensuring the safety and trustworthiness of information is crucial, especially in an era where the value of information lies in its security.As social media platforms gain prominence, colored images have become a significant focal point.Human brains tend to respond more quickly and attentively to visual information, including images and colors, compared to other types of data.This growing importance of visual information has given rise to the emerging field of Image Secret Sharing (ISS), ABOUT THE AUTHOR Raviraja M Holla is an Assistant Professor at Manipal Institute of Technology, part of Manipal Academy of Higher Education.He holds a Ph.D. from the National Institute of Technology Karnataka, with expertise in Information Security, High-Performance Computing, and Semantic Web.Suma D, also an Assistant Professor at Manipal Institute of Technology, specializes in Object-Oriented Programming, High-Performance Computing, and Data Mining.Both contribute to advancing technology and education at the institute.
where techniques are developed to securely share and protect colored images.In the realm of secure information sharing and image protection, ISS schemes have gained considerable attention.These schemes allow for the distribution of a secret image among multiple shares, ensuring that the original secret can only be revealed when the shares are combined.
The primary objective of this research is to propose an efficient approach for progressive colour secret sharing using the DHT, with a particular emphasis on achieving high-quality image recovery.By harnessing the computational capabilities of DHT, we aim to significantly enhance the performance of the secret sharing algorithm.The DHT serves as a powerful tool in this context, providing advantages in terms of computational efficiency and the ability to manipulate image data in the frequency domain.Leveraging the DHT in the colour domain enables progressive sharing of colour images while ensuring robustness and minimal visual impact.By carefully selecting the quantization table, we aim to optimize the image recovery process, resulting in the retrieval of images with high fidelity and minimal loss of visual information.
In the evolving landscape of digital security, this paper introduces an innovative approach to progressive image secret sharing, utilizing the Discrete Hartley Transform (DHT) to strike a harmonious balance between security and risk factors.This strategy harnesses the DHT in the color domain, ensuring secure sharing of color images with minimal visual disruption.Moreover, its versatility extends its applicability to diverse domains including sensitive financial accounts and voting system trust (Al-Shaarani & Gutub, 2022).It also finds utility in medical consensus, wills and inheritance, medical imaging, and confidential data distribution (Gutub et al., 2019).By illuminating the nuanced relationship between security and risk in image sharing, this work not only advances image security but also offers insights into broader information security decisions, promising a holistic perspective on managing the delicate interplay between security measures and associated risks.
The structure of this study is as follows: Section 2 offers an overview of the relevant literature that influenced the development of our novel ISS scheme, elaborated in Section 3. The experimental findings and results are discussed in Section 4, and lastly, Section 5 presents the concluding remarks.

Related works
Based on the recovery of the secret image, ISS schemes can be categorized into three types: Random-grid, Traditional, and Progressive ISS schemes.
Traditional schemes aim to recover the secret image by stacking a minimal number of shares.Naor and Shamir (Naor & Shamir, 1994) introduced the concept of Visual Cryptography (VC), which uses basis matrices to conceal original pixels of the secret image.Hou et al (Hou & Quan, 2011) addressed the pixel expansion issue by generating uniform shares for each participant.Ateniese et al (Ateniese et al., 1996) proposed a General Access Structure (GAS) scheme where users are divided into qualified and forbidden sets, allowing only eligible users to recover the secret image.Extended Visual Cryptography Scheme (EVCS), introduced by Ateniese et al (Ateniese et al., 2001), generates meaningful shares that contain the cover image alongside the secrecy.Various VSS methods applicable to grayscale and color images have been proposed in literature.Recent advancements include Quantum Secret Sharing (QSS) for GASs in Bassirian et al (Bassirian et al., 2019) and many-core rotational VSS in Suma et al (Suma et al., 2019).
Seminal work on image encryption proposed in Kafri et al (Kafri & Keren, 1987) has evolved over the last decade.This approach uses two random-grids, where an encoded random-grid is obtained based on a master random-grid and an input image.The secret image is revealed when these grids overlap.The technique has been extended to encrypt color images in Shyu et al (Shyu, 2009).Essential features of ISS schemes, such as non-pixel expansion, color applicability, and threshold generality, are incorporated into the scheme proposed by Yan et al (Yan et al., 2018).It enhances image quality while minimizing computation.However, with advancements in mobile devices and their computing capabilities, the assumption of a computation-restricted environment becomes less significant.Huang et al (Huang & Juan, 2020) made progress in reducing communication costs by providing security to multiple images with fewer shares.
Hou et al (Hou et al., 2013) invented block-based progressive ISS (BPISS), which works like a jigsaw puzzle and supports color images.However, it faces challenges with the recovery of multitone secret images and produces images with maximum 50% contrast.To address these issues, Mhala et al (Mhala et al., 2017) proposed a Randomized scheme, combining progressive scheme with reversible Discrete Cosine Transformation (DCT) data embedding.
There are steganographic techniques that utilize the Discrete Wavelet Transformation (DWT) (Al-Shaarani & Gutub, 2022) and Discrete Hartley Transform (DHT) to enable secrecy and deceit by covertly embedding information within images (Mandal & Ghosal, 2013;Patwari et al., 2023).However, it is important to note that, to the best of our knowledge, there are currently no image secret sharing schemes that overtly ensure the privacy of the secret information, which is the primary objective of cryptography (Wang et al., 2023).Cryptography focuses on the secure communication and preservation of confidential data, while steganography aims at concealing the existence of hidden information.Therefore, while steganography techniques utilizing DHT offer covert methods of information concealment, the explicit privacy preservation objective of cryptography is not directly addressed in these schemes.
Generally, DHT is known to have lower computational complexity than DCT and DWT (Wang et al., 2023).In terms of arithmetic operations, DHT requires fewer multiplications and additions compared to DCT and DWT.DHT involves only real-valued arithmetic operations, whereas DCT and DWT involve complex-valued operations, which are computationally more intensive (Kumar et al., 2023).Moreover, DHT can be efficiently computed using data-parallel processing techniques, such as those available on GPUs, which can further enhance its performance and reduce the overall computational time (Tao & Kwan, 2012;Torres et al., 2023).In terms of arithmetic operations, DHT requires fewer multiplications and additions compared to DCT and DWT (Mir et al., 2022).DHT involves only real-valued arithmetic operations, whereas DCT and DWT involve complex-valued operations, which are computationally more intensive (Wang et al., 2022).
In conclusion, utilizing Discrete Hartley Transform (DHT) of shares in visual secret sharing schemes provides a wide array of advantages, including enhanced robustness, improved visual quality, reversibility, efficient embedding, strong security, and the potential for parallel processing.As a result, DHT emerges as a favorable choice over Discrete Cosine Transform (DCT) and Discrete Wavelet Transform (DWT) for specific visual secret sharing applications.
Based on these advantages, we propose a secure image sharing technique that leverages DHT along with an optimized quantization table.Additionally, we incorporate a reversible data concealment approach to enhance the reconstructed image quality, all while ensuring a manageable computational cost.By combining these features, our proposed technique offers an efficient and secure solution for visual secret sharing, with the potential to cater to a wide range of applications with high fidelity and confidentiality.

Proposed method
To enhance convenience and understanding, we have organized the proposed method into two distinct phases: the secret share generating phase and the secret reconstruction phase.In the secret share generating phase, we first perform block-wise share generation, followed by applying the Discrete Hartley Transform (DHT) and embedding data into the shares.On the other hand, during the secret reconstruction phase, we extract the embedded data, perform the inverse DHT, stack the shares, and finally integrate the extracted data to recreate the original image.Detailed explanations of these phases are provided below Figure 1.

Secret share generating phase
This step uses the BPISS scheme proposed in (Hou et al., 2001).BPISS has been selected due to its adaptability to multitone secret images, color image compatibility, and progressive security enhancement.The number of shares equals the number of participants involved in the ISS process.
Let n be the number of participants, and I be the image having size W � H.In the proposed method, instead of using RGB (Red, Green, Blue) color space, we adopt the YUV (Y-luminance, U-chrominance blue, and V-chrominance red) color space to transform the secret image into a halftone image.The YUV Color Space's capacity to segregate luminance and chrominance data plays a pivotal role in bolstering the resilience and superior image recovery of our proposed technique.The choices of employing BPISS and the YUV color space have been directed by a systematic analytical process, underscoring a holistic comprehension of how our method adeptly tackles the existing challenges.
The Y component (luminance) in YUV contains crucial visual details (Dinh, 2023), making it an ideal choice for embedding data as it offers higher hiding capacity while preserving visual quality.Utilizing DHT on the YUV halftone image yields improved visual quality compared to the RGB halftone image.This is because DHT focuses on luminance rather than color information, resulting in reduced distortion and enhanced overall image quality.
The halftone version of the secret image, denoted as I halftone , is divided into l ¼ W�H n nonoverlapping blocks, where n represents the number of shares or participants involved in the secret sharing process.B i j represents the j th ð1 � j � lÞ block of the i th ð1 � i � nÞ share.The size of the encrypted transparencies is equal to the size of the secret image I.A total of n þ 1 basis matrices, denoted as B 0 ; B 1 ; . . .; B n , of size 2 � n, are required to generate n shares.Table 1 shows the generated basis matrices for four shares.Random row B 0 is selected for the white pixel of any block of the original image to update the corresponding pixels of the same block in all the shares.
Similarly, B 1 ; . . .; B n are used for the black pixel entries for different shares specific to the respective block.This randomness in selecting a row increases the robustness of the security.
The shares obtained are then subjected to transformation into the frequency domain using the Discrete Hartley Transformation (DHT) with 8 � 8 blocks (as per Equation 1).Drawing inspiration from (Gujjunoori & Amberker, 2013b) and (Gujjunoori & Amberker, 2013a), which proposed data embedding techniques for videos based on ceaseless zeros, our approach has been customized for both grayscale and color images.To optimize data embedding capacity in the mid frequencies, a standard quantization table (depicted in Table 2) is employed, to assign gradually increasing values to the higher frequency components while keeping lower frequency components relatively preserved.This approach aims to reduce quantization errors in the higher frequencies that are more sensitive to visual quality, resulting in better visual fidelity in the reconstructed images.
In the data embedding process within a share, we utilize the approach outlined in (Chang et al., 2007) and implement Equation 4to embed data specifically in the middle frequency region of the share.By embedding ancillary information into each share-pixel, we ensure that the original pixel value can be fully restored during the inverse phase of the process.This reversible embedding technique preserves the integrity of the original share while securely incorporating the desired data.
where 0 � u; v � 7 and casðxÞ ¼ sinðxÞ þ cosðxÞ For each block, we utilize nine distinct sets denoted as D l ð1 � l � 9Þ, as shown in Figure 2, along with their respective sizes presented in Table 3.
To embed the secret image pixels in a recoverable manner, we perform downsampling on the secret image to obtain a low-resolution image with a block size.Our target location for hiding pixel data is chosen to be the middle position within each set.This strategic choice is based on the covert idea that changes to the middle frequency elements are less discernible to the human visual system.To achieve this, we make use of ceaseless zero frequency locations for embedding data.We count the ceaseless zero frequency, denoted as z l , in the given set D l from high to low frequency.For the l th set, if the value of z l is greater than or equal to  To distinguish between the coefficients and the embedded data at a later stage, we rely on the function Amb (described in Equation 3), ensuring clear retrieval of both the coefficients and the embedded data when needed.This contributes to the robustness and reliability of our image sharing technique.After resolving the ambiguous condition, we employ the Embed function presented in Equation 4to determine the values to be embedded at the designated location.In this context, ω represents the value intended for embedding into the shares.If the position value is zero, we directly embed ω; otherwise, we incorporate a modified value Λ at that specific location.Following the completion of the embedding process, the resulting shares are assigned to separate recipients.To enhance the contrast of the recovered image, we select the pixel value ω from the ordered vector of the low-resolution image.

Secret reconstruction phase
Once the data embedding process is completed, the resulting shares are transmitted to the respective participants.Each share is divided into blocks of size 8 � 8 to facilitate further processing.In the inverse process, the embedded data is extracted from these blocks using the function Extract, as described in Equation 5.The extracted secret data is then upsampled to restore its original size.
To reconstruct the original coefficient value, the function Restore, as presented in Equation 6, is applied.Each block is subsequently de-quantized by multiplying it with the standard quantization  table shown in Table 2. Finally, the inverse DHT (Equation 2) is performed to restore the original shares.
To obtain the high-quality recovered image, the upsampled data is XORed with the corresponding pixels of all the shares.This process ensures the proper reconstruction of the secret image with minimal distortion, thus preserving the image's visual quality.To reconstruct pixel values from the halftone image, we employ the inverse halftone technique.This process enables us to estimate the original pixel values, which were transformed into the halftone representation during the embedding phase.
To further enhance the quality of the recovered image and reduce noise artifacts, we employ the bilateral filter as a post-processing step.The bilateral filter is a widely used technique for noise reduction and signal enhancement in image processing applications.Utilizing the bilateral filter allows us to improve the visual quality of the recovered image while mitigating any distortions introduced during the embedding process.

Experimental findings and results
The proposed model was developed and evaluated using Python and OpenCV.The experiments were conducted on the image dataset available at (Index of/ychou/BPVSS, 2012; SIP Image Database), consisting of 44 images, including 16 color images, 28 grayscale images, and two binary images.Given the limited space available, Figures 3 and 4 showcase only four color and grayscale images, respectively.
In this study, we evaluate the performance of the proposed scheme through various methodologies.Section 4.1 presents the analysis of image transitions using histogram-based techniques.Objective parameters are utilized in Section 4.2 to quantitatively assess the scheme's performance.Additionally, Section 4.3 focuses on highlighting the advantages of the proposed scheme, emphasizing key contributing factors.
In Figure 5, the transformation process of the original secret image is depicted, following the overall proposed pipeline described in Section 3. The original image is first converted to a halftone image in the YUV color space, displayed as Figure 5(a) and (b) respectively.The downsampled version of the image, suitable for embedding in shares, is represented in Figure 5(c), which is then scaled to 32 times its original size for visualization in Figure 5(d).Subsequently, Figures 5(e-h) display the four shares generated.Extracted secret information is shown in Figure 5(i) after upsampling.XORing the images from Figures 5(e-i) yields the recovered image displayed in Figure 5(j).Finally, the high-quality reconstructed image is obtained through bilateral filtering of the recovered image.The transformation process of input images with different sizes (Figures 5(a,b)) is visually depicted in Figure 5.The figure demonstrates the steps of the forward and backward phases, with the progression from left to right.

Analysis of histogram comparisons
In the histogram comparisons, we present four distinct plots, each depicting a comparison between different image pairs.Figures 6 and 7 exhibit a noticeable misalignment between the share image and the embedded image with respect to the original image.This observation reinforces the security robustness of the proposed method, as it demonstrates the intentional distortion of the share and embedded images to ensure the protection of the secret information.Figure 8 provides evidence of the similarity between the share image and the embedded image, validating the covertness of the proposed scheme.This covertness is achieved through the embedding of secret information in the share image, necessary for the high-quality recovery of the hidden content during the secret  reconstruction stage.In Figure 9, we observe a remarkable overlap of the reconstructed image curve with the original image curve, indicating the superior quality of the recovered image.This outcome highlights the effectiveness of the reconstruction process, which successfully restores the original image's visual fidelity during the secret recovery phase.

Objective assessment of the proposed model's performance
In order to showcase the enhanced performance of the proposed model over two advanced models, RVSS (Mhala et al., 2018) and SR-PVSS (Mhala & Pais, 2019), we utilized the following objective performance analysis parameters.Let W and H represent the width and height of the image, respectively.The input (I) and reconstructed (R) images are denoted by equations 7 and 8, where iðm; nÞ and rðm; nÞ correspond to the intensities of the pixels at coordinates ðm; nÞ in images I and R, respectively.

Peak Signal to Noise Ratio (PSNR)
The Mean Square Error (MSE), as defined in Equation 9, serves as an effectiveness metric.A lower MSE indicates a higher level of similarity between the reference and target images, reflecting improved effectiveness.Additionally, the Root Mean Squared Error (RMSE) and PSNR, defined in Equations 10 and 11, respectively, contribute to the assessment of image quality.A higher PSNR value signifies superior image quality, highlighting the positive alignment between the two.The proposed scheme achieved a maximum PSNR of 88:106, as demonstrated in Table 4.  aiding in the evaluation of median differences between methods.Non-overlapping notches suggest statistically significant differences in medians.
In this context, the plot enables the comparison of PSNR distribution across the methods.The proposed method exhibits both higher median PSNR values and a broader interquartile range, depicted by its box position and size.Non-overlapping notches between the"Proposed" method's box and the other two boxes indicate a statistically significant median difference, providing evidence of the"Proposed" method's superior image quality.

Normalized Cross-Correlation (NCC)
The NCC as defined in Equation 12, offers a metric to assess the similarity between two images.A higher NCC value indicates a stronger degree of resemblance between the images, acting as an indicator of their likeness.Table 4 demonstrates a NCC value of 0:884, showcasing the performance of the proposed scheme.

Normalized Absolute Error (NAE)
The NAE, defined in Equation 13, quantifies the discrepancy between the secret and recovered images.A lower NAE value indicates a higher quality of the recovered image.Thus, minimizing the NAE serves as a measure of the effectiveness of the recovery process in achieving superior image quality.
As indicated by the data presented in Table 4, the proposed scheme demonstrates an NAE value of 0:095.

Structural Similarity Index (SSIM)
The SSIM is a spatial measure that evaluates the luminance, contrast, and structure of two images originating from the same source.It yields a numerical value between −1 and 1, where an SSIM value of 1 signifies a perfect match between the compared images.Equation ( 14) provides the calculation for SSIM, considering various components to assess the similarity between the images.As shown in Table 4, the proposed scheme achieves an SSIM value of 0:895, highlighting its efficacy in producing similar images.
In this context, σ I and σ R stand for the local standard deviations of the input image I and the recovered image R, respectively.Likewise, μ I and μ R represent the means of I and R, while σ IR indicates the cross-covariance between the two images.

Comparison of positive features in the proposed scheme with state-of-the-art approaches
This section emphasizes the distinctive positive features of the proposed scheme compared to the existing approaches outlined in Table 5.The proposed (n, t) scheme stands out due to its remarkable flexibility, where 'n' represents the number of secret images and 't' can vary, either being equal to, greater than, or less than 'n'.This inherent flexibility allows the scheme to be applicable to both grayscale and colored images.The sharing capability is determined by dividing the number of secret images by the number of shared images, represented by 'n/t'.This sharing capacity metric indicates the scheme's efficiency in handling various scenarios.Importantly, the proposed approach efficiently manages secret images with fixed dimensions, making it a robust choice for such scenarios.To recover the secret images, the scheme utilizes the computationally inexpensive XOR method, contributing to its computational efficiency.Moreover, the proposed scheme leverages DHT (Discrete Hartley Transform), a less computationally intensive approach, further enhancing its computational performance.

Conclusion
In conclusion, this research introduces an innovative approach to progressive image secret sharing, capitalizing on the Discrete Hartley Transform (DHT) for robust image recovery.The integration of DHT within the color domain ensures secure sharing of color images while minimizing visual impact.The optimized quantization table selection enhances image recovery precision.Extensive experimentation validates the scheme's efficacy, offering secure image sharing and superior quality via the real-number DHT transformation.To enhance the method, refining quantization table optimization and exploring DHT's synergy with advanced cryptography is promising.Further, delving into diverse domains like sensitive bank accounts, trust in voting systems, medical consensus, and inheritance distribution extends the method's applicability.This work bridges image security and cryptography, inviting researchers to explore intricate security risk interdependency dynamics, opening avenues for advancements in image secret sharing techniques and information security decisions understanding.
While the proposed method comes with significant benefits, it's essential to acknowledge its limitations.These encompass susceptibility to noise and the computational demands associated with processing large images in real-time scenarios.These aspects warrant careful consideration when contemplating the practical implementation of this approach in real-world applications.

Figure 1 .
Figure 1.Block diagram of the proposed ISS using DHT and YUV color Space.
middle element as the location for embedding data.Consider 'ω' as the middle value within the given set in downsampled secret image, and let μ ¼ D l ðl; KðlÞ 2 � � Þ represent the coefficient value at the middle location in the specified set.

Figure 2 .
Figure 2. The defined sets and locations for embedding the data.

Figure
Figure 5. Visual representation of image transformations in the proposed model.

Figure
Figure 6.Histogram comparison -original image vs. Share image.

Figure
Figure 8. Histogram comparison -share image vs. Embedded share.
The plot (Figure10) combines a box plot with error bars to effectively support statistical hypothesis testing and enable a direct comparison of different methods in relation to the PSNR metric.The box plot offers several crucial insights for hypothesis testing.The boxes symbolize the interquartile range of PSNR values for each method, indicating data variability.The line within each box represents the median PSNR, a measure of central tendency.Extending from the boxes, the whiskers portray the range of PSNR values, while any data points beyond the whiskers are considered outliers.The error bars incorporated visually represent the uncertainty associated with median PSNR values.Notches in the boxes serve as confidence intervals around the medians,

Figure
Figure 9. Histogram comparison -original image vs. Reconstructed image.

Figure
Figure 10.Comparison of PSNR values using box plot and error bars.

Table 1 .
Basis matrices for n ¼ 4 shares