Adaptive 3D Reversible Data Hiding Technique Based on the Cumulative Peak Bins in the Histogram of Directional Prediction Error

Abstract

Reversible data hiding (RDH) is crucial in modern data security, ensuring confidentiality and tamper-proofness in various industries like copyright protection, medical imaging, and digital forensics. As technology advances, RDH techniques become essential, but the trade-off between embedding capacity and visual quality must be heeded. In this paper, the relative correlation between the pixel’s local complexity and its directional prediction error is employed to enhance an efficient RDH without using a location map. An embedding process based on multiple cumulative peak region localization (MCPRL) is proposed to hide information in the 3D-directional prediction error histogram with a lower local complexity value and avoid the underflow/overflow problems. The carrier image is divided into three color channels, and then each channel is split into two non-overlapping sets: blank and shadow. Two half-directional prediction errors (the blank set and the shadow set) are constructed to generate a full-directional prediction error for each color channel belonging to the host image. The local complexity value and directional prediction error are critical metrics in the proposed embedding process to improve security and robustness. By utilizing these metrics to construct a 3D stego-Blank Set, the 3D stego-shadow Set will be subsequently constructed using the 3D blank set. The proposed technique outperforms other state-of-the-art techniques in terms of embedding capacity, image quality, and robustness against attacks without an extra location map. The experimental results illustrate the effectiveness of the proposed method for various 3D RDH techniques.

1. Introduction

The Internet’s openness necessitates strong security measures to prevent unauthorized access to and destruction of sensitive digital information. Although secure digital information transmission is sometimes desirable, the content should be protected from unauthorized access and copying. The guarantee that the secret digital information reaches the intended destination and is received correctly is the most important issue that both the sender and the recipient are concerned about [1,2]. Secure digital information transmission is sometimes of great importance as the content can be secret or confidential and, therefore, should be protected from intruders and hackers. Information hiding is a scientific process that involves secretly broadcasting digital information by embedding it into a carrier medium. Modern digital communication technologies have become an integral part of the infrastructure of today’s world to communicate and exchange digital media such as text, audio, images, and video. One of the characteristics of digital media is that it is easy to create, store, duplicate, transmit, and modify [1,2,3,4,5]. Therefore, individuals prefer using digital information in their work and lives; enormous amounts of digital information are created and transmitted through public networks every moment. The beneficiaries and diverse industries face several obstacles to protecting digital information and maximizing its size. Secure digital information transmission is sometimes desirable; the content should be protected from unauthorized access, modification, copying, or any type of attack [2,3].

Reversible data hiding (RDH) based on color image histograms has been demonstrated in theory to be effective in disguising secret messages, as it is able to uniformly spread the data in an inconspicuous manner without any visible traces that would lead an adversary to suspect the existence of a secret message, making it more difficult to detect, achieving robust embedding locations, scattered secret data in the entire image, and high resistance against signal processing such as rotation, scattered tiles, and warping [6,7]. Existing RDH methods can accurately extract the embedded information without losing its content, making them suitable to secure several important applications such as multimedia communication, network security, medical applications, and the secure transmission of sensitive data such as financial information and medical records [6,7,8]. RDH techniques must meet certain requirements, such as undetectability, imperceptibility, carrier image visual quality, embedding distortion, accurate information extraction, security, and attack resistance. The size of the hidden message must be balanced to ensure the quality of the technique. To meet these requirements, researchers have proposed various RDH techniques, including cryptographic-based, perception-based, pixel clustering, LSB-based, histogram-based, and feature extraction-based [6,9,10,11,12]. RDH based on the histogram has limitations, including susceptibility to statistical attacks such as histogram analysis and a relatively small embedding capacity compared to other methods; thus, increasing embedding capacity reduces image quality and requires a location map that increases the amount of information to be embedded. Although most existing RDH methods can accurately extract the embedded information, the embedding process causes noticeable distortion to the carrier image. An attacker can identify any irregularities or inconsistencies in the image histogram and quality distortions, which would give away the presence of hidden information [12,13,14,15,16,17]. Furthermore, traditional histogram shifting RDH techniques [11,12,18,19,20,21] shift the peak points to the zero point of the image histogram while ignoring the amount of invalid shifting pixels (ISPs), resulting in redundant locations that cause significant image distortion; the pixels are shifted but not used in the embedding process, making it more vulnerable to attacks. Usually, the prediction error histogram is used in conjunction with histogram modification to create prediction-error histograms (PEHs), from which the appropriate expansion bins for histogram shifting and histogram expansion are chosen. To address these flaws, researchers proposed various techniques and methods [20,21,22,23,24,25,26,27,28,29,30,31,32,33,34] capable of reducing the risks associated with RDH based on a histogram in which the number of valid shifting pixels is determined for embedding based on embedding capacity and increasing embedding capacity with low distortion by enhancing the use of color image features such as color, contrast, edges, and textures, but they solved each problem separately, which leads to exacerbation of another one.

In this paper, we propose an adaptive reversible data hiding technique based on reducing the invalid shifting of the directional prediction error histogram using the histogram cumulative peak region without the need for a location map and overcoming the challenges and limitations of existing RDH methods based on histograms. The proposed RDH technique can efficiently hide a large amount of data within a 3D color image while improving the visual quality with low distortion and ensuring reversibility. The proposed technique is designed to enable multiple security requirements to be efficiently and effectively met, such as the secrecy of the transmitted data by hiding its existence in a spoof carrier image histogram based on the histogram’s cumulative peak regions, robustness to signal processing attacks by reducing the modification of the carrier image histogram so the histogram of the stego image is relatively close to the original image histogram, a large payload size because the proposed technique employs cumulative peak regions of the carrier image histogram instead of a single peak point, and good quality of the resulting stego images. The main contributions of this paper are as follows:

The embedding process utilizes multiple cumulative peak regions (CPR). The number of the multiple prediction error cumulative peak regions (MPE-CPR) is adaptively calculated based on payload size and the number of bits per pixel (NBPP) to calculate vacant bins for embedding a variable length payload, which reduces invalid pixel shifting, redundant locations, and significant image distortion using the proposed Adaptive MPE-CPR Selection technique.

The 3D-PEH localization technique based on MPE-CPR shifts the bins of the prediction error histogram to the first bin within their cumulative peak region. For instance, if the cumulative peak region consists of four bins starting from pixel values of 50 to 53, then the frequencies of the bins 51, 52, and 53 are shifted to the first bin, 50, such that the frequency of bin 50 equals the cumulative sum of the other bins within the same region. MPE-CPR of 3D-PEH is utilized to avoid the underflow/overflow problem during the embedding process and provide increased robustness against attacks as the embedded information is distributed across multiple peak regions. The localized PEH is modified in the proposed embedding process to allocate a variable length of payload bits within each color channel, with half of the bits embedded in the shadow set and the other half in the blank set, using pixels with a lower local complexity value. The stego image is created by fusing the two generated sets (stego-blank and stego-shadow).

The rest of this paper is structured as follows: Basic background information and preliminary information are provided in Section 2. The related works and comparisons between them are presented in Section 3. In Section 4, the proposed techniques are presented. Evaluations and experimental results are presented in Section 5 and Section 6, along with conclusions and future work.

2. Preliminaries

In this section, we briefly introduce the main concepts behind the histogram cumulative-peak regions of Al-Omari et al. [35], pixel local complexity, and directional enclosed prediction and expansion. DEPE, Chen et al. [36].

2.1. Histograms’ Cumulative Peak Regions

The histogram cumulative peak regions are utilized to hide multiple bits of the secret message by performing histogram bin localization Al-Omari et al. [35] Based on the size of the secret message, the algorithm determines the ideal window size and the ideal number of cumulative peak regions. The window size designates how many consecutive histogram bins hold the most pixels from the host image. The window size (w) is chosen to be a power of 2, for which log2 w secret message bits are embedded each time. The minimum number of embedding regions is determined by the cumulative peak regions. In other words, histogram bin substitution, which modifies the least rightmost bit of the image’s histogram peak point regions, is used to hide bits of the secret message based on window sizes of 2, 4, 8, and 16. The ability to conceal information and the hiding capacity of this algorithm can be calculated using the following formula:

$$Hiding capacity=\left(log2 w\right) * {{\displaystyle \sum }}_{p=1}^{P}h\left(p\right) bits$$

(1)

where, w represents the window size; P represents the maximum number of peak regions; p represents the current peak region; and h(p) represents the frequency of the p-value in the image histogram.

The concepts of the histogram cumulative-peak regions algorithm are described through a numerical example supported by pixel values. Consider an 88-tile image with 8 bits per pixel in the red color channel. Each pixel value is represented by one of the 64 integer numbers that make up the tile. Figure 1 shows the pixel values for the tile and the corresponding histogram with two embedding regions and four bins per region.

Figure 1 shows two peaks (17 and 13) and two cumulative peak regions with four bins per region. The first region for the peak value 17 consists of four consecutive histogram pixels (16, 17, 18, and 19), which achieve a maximum cumulative bin value equal to the sum of 9, 3, 3, and 1. The second region for the peak value 13 consists of pixels 11, 12, 13, and 14, which achieve the second maximum cumulative bin value equal to the sum of 1, 2, 8, and 2. Figure 2 shows the process of histogram bin localization and the updated tile after histogram localization.

The process of histogram bin localization shifts the last three pixels of the current region to the first pixel. Finally, the tile pixels are modified based on the histogram bin localization by sequentially accessing each pixel, and if the pixel value belongs to one of the regions, then its value is updated to the first pixel in the current region.

The data embedding process sequentially scans each row of the tile from top to bottom. Two bits from the secret message are taken, and the pixel value is increased by the bit’s binary value if the current pixel value is equal to the start pixel of any embedding region. After embedding a 56-bit secret message, the stego tile and histogram are displayed in Figure 3; 00 11 01 00 11 10 10 01 00 00 11 01 10 is the secret message bits stream. The hiding capacity of the stego tile is 56 bits, and its PSNR value is 48.48 bB.

2.2. Local Complexity

The local complexity Ω P(C,v,h) can be used to identify the best prediction error, allowing for more accurate predictions by properly adjusting the prediction process. Pixels with smaller local complexity values have a high level of smoothness, which implies fewer details in the image. In contrast, a high level of smoothness suggests more intricate visual elements. Smooth pixels are ideal for images requiring a continuous gradient or subtle tone shifts. A message embedded in a pixel with high local complexity values could lead to distortion and loss of the stego image detail. The context of pixel local complexity and calculation pattern is shown in Figure 4.

Local complexity Ω P_(C,v,h) is calculated as follows:

$$\begin{gathered} \mathsf{\Omega } {\mathit{P}}_{\left(\mathit{C},\mathit{v},\mathit{h}\right)} =\left|(P_{\left(C,v-1,h\right) }-P_{\left(C,v,h-1\right) })\right|+\left|(P_{\left(C,v,h-1\right) }-P_{\left(C,v+1,h\right) })\right|+ \\ \left|(P_{\left(C,v+1,h\right) }-P_{\left(C,v,h+1\right) })\right|+\left|(P_{\left(C,v,h+1\right) }-P_{\left(C,v-1,h\right) })\right| \end{gathered}$$

(2)

where P denotes the pixel value, C ε (R, G, B) color channel, the horizontal and vertical directional coordinates of pixel P are v and h, respectively, such that v = 0, 1, 2... V − 1 and h = 0, 1, 2, … H − 1.

The pixels on the borders have two or three adjacent pixels that will be used to determine their local complexity. For example, P_{(C, 0, 0)}, P_{(C, 0, h − 1)}, P_{(C, v − 1, 0)}, and P_{(C, v − 1, h − 1)}, which are located in the color channel four corners, are calculated using two adjacent pixels: P_{(C, 1, 0)} and P_{(C, 0, 1)}, P_{(C, 0, h − 2)} and P_{(C, 1, h − 1)}, P_{(C, v − 1, 1)} and P_{(C, v − 2, 0)}, and P_{(C, v − 2, h − 1)} and P_{(C, v − 1, h − 2)} respectively. while the remaining pixels on the border of the image are calculated using the local complexity of three adjacent pixels.

2.3. Directionally-Enclosed Prediction and Expansion

Directionally-Enclosed Prediction and Expansion DEPE, Chen et al. [36], predict pixel values by expanding upon the information of four adjacent pixels and enclosing them in a certain direction. The pixel prediction value can be computed as:

$${\widehat{P}}_{\left(C,v,h\right)}=\lfloor \left( {\widehat{P}}_{\left(C,v-1,h\right)}+{\widehat{P}}_{\left(C,v+1, h\right) }+{\widehat{P}}_{\left(C,v, h-1\right) }+{\widehat{P}}_{\left(C,v, h+1\right) }\right)/4\rfloor$$

(3)

$${\widehat{P}}_{\left(C,V\right)}=\lfloor \left( {\widehat{P}}_{\left(C,v-1,h\right)}+{\widehat{P}}_{\left(C,v+1, h\right) }\right)/2\rfloor ,$$

(4)

$${\widehat{P}}_{\left(C,H\right)}=\lfloor \left( {\widehat{P}}_{\left(C,v, h-1\right) }+{\widehat{P}}_{\left(C,v, h+1\right) }\right)/2\rfloor$$

(5)

where ⌊·⌋ denotes the floor function and C Є (R, G, B) color channel; the horizontal and vertical directional coordinates of pixel P are v and h, respectively, such that v = 0, 1, 2, … V − 1 and h = 0, 1, 2, … H − 1. The vertical prediction is denoted as ${\widehat{P}}_{\left(C,V\right)}$ and the horizontal as ${\widehat{P}}_{\left(C,H\right)}$. The Directional Prediction Error DPE denoted by ${е}_{\left(C,v,h\right)}$ is calculated using:

$${е}_{\left(C,v,h\right)}=P_{\left(C,v,h\right) }-{\widehat{P}}_{\left(C,d\right)}$$

(6)

3. Related Work

Several histogram-based RDH techniques have been proposed to hide secret information based on image histograms or prediction error histograms [13,14,15,17,18,19,24,25,26,27,35,36,37]. The reversible histogram shifting method embeds secret digital information into the least rightmost bit of the cover image histogram peak point. Because the image histogram’s pairs of peaks and zero-point differences determine the embedding payload capacity, a larger difference results in a higher capacity, as determined by Zhicheng et al. [21]. To improve the Ni algorithm and increase payload, Fallahpour et al. [37] propose a histogram block-based embedding method in which the cover image is divided into blocks and the Ni algorithm is applied to each block histogram. A multilevel histogram-modification hiding technique was proposed by C. Lin et al. [14] to increase embedding capacity with high image quality. Differences in the image histogram are used to determine the peak point and associated pixels to embed the secret message. A cumulative-peak histogram region was used to embed the secret message bits based on window sizes of 2, 4, 8, and 16 by performing histogram bin substitution, which changed the least rightmost bit of the image’s histogram peak point region, as determined by Mowafi et al. [38].

To distribute pixel differencing in the local area of the natural digital image, a reversible data hiding scheme based on the two hybrid methodologies of “ripple strategy” and “histogram shifting” was proposed by Chou et al. [39] The main goal of the ripple strategy was to subtract the pixels from the outermost ripple and the pixels from the nearby inner ripple, where the biggest pixel differences occurred in three different locations (−1, 0, and 1). To determine the nearly ideal peak and zero bins to achieve a high embedding rate, a rate-distortion model was proposed by Wang et al. [26] for a multiple-layered HS-PEE embedding approach. To lessen embedding distortion, HS-PEE makes use of numerous pairs of different peaks and zero bins. The proposed model and an evolutionary optimization algorithm are used to derive a nearly optimal embedding instead of the global optimal solution due to the large size of the solution space.

To effectively reduce embedding distortion for low capacity, pairwise PEE was proposed by Ou et al. [40] to use every two pixels jointly for data embedding. Using this method, every pair of adjacent pixels in a diagonal or anti-diagonal direction is combined. To determine how similar two pixels are, the spatial separation between them can be used, which assumes that adjacent pixels typically have comparable intensities. By utilizing the similarities between the nearby pixels, a more efficient modification method can be developed for the two-dimensional prediction-error histogram (2D-PEH), which has a lower entropy than the one-dimensional (1D) PEH. In the work by Chang et al. [22], a high computational complexity RDH technique was proposed based on adaptive three-dimensional PEE and a double-deep Q-network (DQN). In which DDQN was adopted to direct the modification directions, identify the best PEE mapping paths, and give a scheme for action selection to guide modification directions so that it could quickly discover the reversible mapping paths.

The proposed RDH scheme by Jia et al. [41] minimizes invalid pixel shifting during the histogram shifting process using image texture properties. The suggested checkerboard pattern divides the image into two sections, which has the advantage of making the auxiliary information less dense. To determine the degree of smoothness, the fluctuation value of each pixel in each section is calculated and sorted in ascending order. Additionally, when combined with the prediction error, the additional data is preferentially embedded in the position of the smooth pixel, effectively reducing the invalid shifting of pixels. Skewed histogram shifting (SHS), which makes use of two extreme predictions (PVO and rhombus prediction) that take advantage of the redundancy of an embedding primitive, was proposed by Kim et al. [42] as a method to lessen distortion. Each pixel is predicted twice using a pair of extreme prediction values, creating two skewed histograms whose asymmetry is used to lessen distortion. It initially assumes a “rhombus” context. As the difference between two extreme predictions widens, the prediction accuracy then increases at the expense of fewer and fewer predicted pixels. To identify regions where local complexity is not proportional to the size of prediction error and minimize embedding distortion, the DEPE RDH schema was proposed by Chen et al. [36] First, directional DPEs are derived using vertical and horizontal predictors. Then, a single PEH is produced using these DPEs. To make data embedding reversible, it is only permitted in regions of the pixels where LC shows a proportional relationship to the size of PE. To estimate the predicted value and complexity of pixels, an RDH schema based on directional prediction was proposed by Song et al. [43]. By calculating various levels of prediction error and embedding data in accordance with the algorithm of the prediction error histogram, multiple histogram modifications are produced.

To compare related works in terms of approach, Merits, Demerits, and Performance criteria in a clear and concise manner and assess their effectiveness, a summary of the related work is provided and is listed in

Table 1. Comparison between related work in terms of Approach, Merits, Demerits, and Performance criteria.

Approach	Merits	Demerits	Performance
Adaptive 3D prediction error expansion method (Chang et al. [22]). The creation of 3D-PEH is performed using directional enclosed prediction.	Enhance the performance of embedding. efficiently find the reversible mapping paths. An approach called double-deep Q-network selection is used to find the best reversible mapping path for 3D-PEH.	Extra places to share the embedded position map and auxiliary information. Lack of modern steganalysis evaluation. Invalid shifting of pixels. high computational complexity	EC = 3.75 KB PSNR = 59.50
HS-based multiple embedding rate and distortion model using the genetic algorithm (J. Wang et al. [26]).	The ideal peak and zero bins that reduce distortion can be determined effectively.	Lack of modern steganalysis evaluation. Invalid shifting of pixels.	EC = 3.125 KB PSNR = 53.188
A block histogram-based multiple cumulative bin region technique (Al-Omari et al. [35]).	Higher embedding capacity, high imperceptibility, and lower distortion. Reduce the invalid shifting of pixels.	Need to test over modern steganalysis attack.	PSNR > 42.96 EC 66.67 MB RAS: Histogram
High-Fidelity RDH Using Directionally-Enclosed Prediction (DEPE) schema (Chen et al. [36]).	Enhancing image quality and fidelity with a significant payload	Data embedding is limited to pixels, where local complexity exhibits a linear relationship with PE. Invalid shifting of pixels. lack of modern steganalysis evaluation.	EC = 1.25 KB PSNR = 60.45
RDH technique based on ripple strategy and shifting the pixel difference histogram (Chou et al. [39]).	Reduce the pixel difference over a narrow range.	Lack of modern steganalysis evaluation. Limited payload size. Invalid shifting of pixels	EC = 1.25 KB PSNR = 44.51
Novel RDH framework based on Pairwise prediction-error expansion (Ou et al. [40]).	Enhance performance by better utilizing image redundancy. Compared to the one-dimensional 1D-PEH, the 2D-PEH has a lower entropy.	Lack of modern steganalysis evaluation. Invalid shifting of pixels.	EC = 2.5 KB PSNR = 55.00
RDH scheme using the texture of the images (Jia et al. [41]).	Minimize invalid pixel shifting in the histogram. Achieve higher embedding capacity with good VQ.	Lack of modern steganalysis evaluation. Restricted to texture images with smooth regions	EC = 3.77 KB PSNR = 53.16
Skewed histogram shifting (SHS) (Kim et al. [42]).	Reduce distortion.	Lack of modern steganalysis evaluation. Invalid shifting of pixels.	EC = 3.75 KB PSNR = 54.154
RDH schema based on directional prediction was proposed by Song et al. [43].	Reduce image distortion.	Lack of modern steganalysis evaluation. Parameter selection based on directional prediction traits is not adaptive. Invalid shifting of pixels.	EC = 2.5 KB PSNR = 55.40

. To provide a comparison of related works in terms of approach, Merits, Demerits, and Performance criteria, Table 1 presents them in a clear and concise manner to evaluate their effectiveness. In addition, the analysis reveals that each technique has advantages and disadvantages of its own that can be benefited from in this study to achieve its objectives. Therefore, it is important to carefully consider the specific requirements and objectives of the presented techniques to choose the most appropriate approach for this study. Additionally, it is crucial to acknowledge that a combination of different techniques may be necessary to fully address the research objectives and ensure comprehensive results. Overall, the analysis in Table 1 emphasizes the need for a thoughtful and tailored selection of techniques based on the specific needs and goals of this study.

Most of the reversible data hiding techniques in Table 1, which are based on prediction error histograms, achieve good image quality with high embedding capacity. However, some of these techniques require additional information hiding related to embedding locations to achieve reversibility and extract the original hidden information without loss. This additional information is necessary because prediction error histograms alone may not provide sufficient details for a complete reversal of the data-hiding process. By incorporating embedding locations, these techniques ensure that the hidden information can be accurately extracted, preserving both image quality and reversibility, but they also increase the payload size.

Shifting the histogram’s peak regions to hide data can result in redundant displacement, affecting the carrier image’s visual quality. This technique requires careful consideration of data hiding capacity and visual distortion and relies on the viewer’s perception. The process can lead to a loss of image details and decreased quality. Shifting peak regions requires complex algorithms and is time-consuming, making it less practical for real-time applications or large-scale data embedding. Balancing data hiding capacity and carrier image integrity is crucial for effective and inconspicuous data embedding.

Some techniques in the literature employ adaptive algorithms that dynamically adjust the embedding process based on the characteristics of the carrier image, minimizing any potential degradation in visual quality caused by redundant displacement. Moreover, they also utilize error correction codes to enhance the reliability of data extraction, ensuring accurate retrieval even in the presence of noise or other distortions. Solving each issue independently leads to the emergence of a new issue. In addition to the lack of contemporary steganalysis evaluation in most of these technologies, there is a concern about the potential degradation in visual quality caused by redundant displacement. This means that while these technologies aim to hide information effectively and ensure accurate retrieval, there is a possibility that the visual quality of the carrier image may suffer. Furthermore, the lack of contemporary steganalysis evaluation in most of these technologies raises concerns about their overall effectiveness and ability to remain undetected. Overall, these issues highlight the need for further research and development to address the trade-off between hiding information securely and maintaining its visual quality.

4. The Proposed Technique

In this paper, an adaptive reversible data hiding technique based on optimizing the valid localization of cumulative peak bins in the prediction error histogram for color images is proposed. The key strategy of our technique focuses on using cumulative peak bin localization to reduce invalid shifting and increase valid shifting of the prediction error histogram, increasing the security of the embedding process, and embedding a high payload with low distortion. The proposed technique selects the highest cumulative peak regions from the prediction error histogram’s smooth pixels based on the local complexity of the prediction error histogram. Each color channel is broken down into two separate, non-overlapping groups called the shadow set and the blank set. The proposed technique starts with embedding data in the blank set for three color channels to generate the 3D stego-Blank Set that will be used to construct the 3D stego-Shadow Set. Both generated 3D stego sets, stego blank, and stego shadow, are then combined to generate the 3D stego-image, which is an image with secret information embedded in its color channels. The proposed Reversible Data Hiding Technique is divided into three sub-techniques on the sender side:

1:

The Image Preparation and Segmentation technique is an image preparation and preprocessing step that involves segmenting the carrier image into three color channels (red (R), green (G), and blue (B)), and then each color channel is broken down into two separate, non-overlapping sets: blank and shadow, something like a checkerboard pattern.

2:

Constructing a 3D-PEH Technique Based on 3D Local Complexity (3D-LC) and 3D Directionally Enclosed Prediction and Expansion (3D-DEPE).

3:

The proposed Embedding Technique uses a 3D-PEH with a lower local complexity value. To extract the embedded payload from the recipient side without losing its content, the reverse process is applied using the proposed extraction technique.

The main stages of the proposed technique on the sender side are illustrated in Figure 5.

As shown in Figure 5, the proposed technique divides the carrier image into three color channels: B-channel, G-channel, and B-channel, and each color channel is divided into two non-overlapping sets, “blank and shadow”, in the first stage. The result of this stage is a 3D Blank Set and a 3D Shadow set, which are the main inputs for the second stage. To achieve reversibility, the second stage starts with a 3D Blank Set until generating a 3D stego-Blank Set, which will be used later to construct a 3D stego-Shadow Set. To achieve reversibility, the second stage starts with a 3D Blank Set until generating a 3D stego-Blank Set, which will be used later to construct a 3D stego-Shadow Set. In the second stage, 3D local complexity and 3D Directionally Enclosed Prediction and Expansion are calculated using Equation (2) in Section 2.2 and Equations (3)–(5) in Section 2.3 for the 3D Blank set and then the 3D Shadow Set.

The LC and their associated directional prediction error are sorted in ascending order before constructing the PEH. A 3D-PEH consists of each set histogram within a three-color channel. PHE is a graphical representation of each set’s DEPE distribution within the color channel that is used to represent the frequency of each prediction error value within a set, which is between −255 (dark color “black”) and 255 (light color “white”). PEH allows us to understand and compare the distribution of prediction errors between colors. In the final stage, the 3D Embedding Process utilizes multiple cumulative peak regions (CPR). The number of the multiple prediction error cumulative peak regions (MPE-CPR) is adaptively calculated based on payload size using the proposed Adaptive MPE-CPR Selection. Then the 3D-PEH localization is applied to generate free bins to hide the message using the proposed Embedding Procedure into the localized PEH for each color channel.

The rest of this section is divided into the following categories: The suggested image preparation and segmentation techniques are presented in Section 4.1. The proposed 3D-PEH construction technique is covered in Section 4.2, and Section 4.3 introduces the proposed embedding technique. The suggested method for 3D extraction is presented in Section 4.4.

4.1. Image Preparation and Segmentation

The carrier image is a 24-bit color image of size V × H that is made up of three color channels: red (R), green (G), and blue (B), which represents the main input for this technique as shown in Figure 6.

In which A pixel P _{(C, v, h)} has a C Є (R, G, B) color channel as well as the vertical and horizontal directional coordinates v and h, such that v = 0, 1, 2, … V − 1 and h = 0, 1, 2, … H − 1. It appears each color channel is split into two sets, blank and shadow, in a checkerboard-like arrangement. Thus, the pixel P _{(C, v, h)} placed in the blank set when the following conditions are met v mod 2 equals zero, and h mod 2 equals zero. If both v mod 2 and h mod 2 are not equal to zero, the pixel is consequently added to the shadow set.

4.2. Constructing the 3D-PEH Technique Based on 3D Local Complexity (3D-LC) and 3D Directionally Enclosed Prediction and Expansion (3D-DEPE)

The term “3D-PEH” refers to the prediction error histogram for blank or shadow sets of size $\frac{V}{2}$ × $\frac{H }{2}$within the red, green, and blue channels. Notation in 3D-PEH: A pixel P _{(C, v, h)} has a C Є (R, G, B) color channel and the vertical and horizontal directional coordinates v and h. Each color channel is divided into two non-overlapping sets: the blank and shadow sets, such that v = 0, 2, … V − 2 and h = 0, 2, … H − 2, belong to the blank set, while the shadow set consists of pixels in v = 1, 3, … V − 1 and h = 1, 3, … H − 1. The directionally enclosed prediction and expansion (DEPE) and measurement of the pixel local complexity of all the pixels belonging to each set within the C color channel are used to create the three-dimensional prediction error histogram (3D-PEH). The 3D-PEH is a valuable tool for characterizing the smoothness level of the color channel’s histogram, in which the range of the PEH for each color channel is extended from [0, 255] to [−255, 255]. For instance, the red color channel histogram range [0, 255] using the prediction error histogram is extended to the range [−255, 255]. This technique consists of three main stages:

Stage 1.

Computing a 3D-LC, in which the local complexity value LC for each set within the C color channel is calculated to accurately measure the smoothness of each set within color channel C. For example, the local complexity value LC for the shadow set belonging to the red channel is calculated to accurately measure its smoothness.

Stage 2.

Computing a 3D-DEPE, in which the directional prediction error for each set within the C color channel is calculated based on directionally enclosed prediction and expansion (DEPE). For example, the directional prediction error for the shadow set belonging to the red channel is calculated based on directionally enclosed prediction and expansion (DEPE).

Stage 3.

Constructing a 3D prediction error histogram (3D-PEH) for each set based on 3D-LC and 3D-PE. For example, a 3D-PEH for the shadow set is constructed by combining the PEH of the shadow set belonging to the red channel with the PEH of the shadow set belonging to the green channel and the PEH of the shadow set belonging to the blue channel.

4.2.1. Computing 3D Local Complexity (3D-LC)

The calculation of the local complexity value LC of the current pixel P _{(C, v, h)} Є {blank set} is obtained according to four adjacent pixels Є {shadow set} using Formula (2). On the other hand, the current P _{(C, v, h)} Є {shadow set}is obtained according to four adjacent pixels Є {Stego blank set} using Formula (2). The LC P _{(C, v, h)} provides an accurate measure of pixel smoothness, as this approach takes into consideration not only the complexity of the adjacency pixels but also how much the current pixel deviates from that complexity.

4.2.2. Computing 3D Directionally Enclosed Prediction and Expansion (3D-DEPE)

Utilizing directionally enclosed prediction and expansion (DEPE) to predict the pixel value of P _{(C, v, h)} be ${\widehat{P}}_{\left(C,v,h\right)}$. Two half-enclosed predictions (the blank set and the shadow set) are combined to generate a full-enclosed prediction error for each color channel belonging to the host image P _{(C, v, h)} Є {blank set} is predicted as ${\widehat{P}}_{\left(C,v,h\right)}$according to four adjacent pixels Є {shadow set} using Formulas (3)–(5). On the other hand P _{(C, v, h)} Є {shadow set} is predicted as ${\widehat{P}}_{\left(C,v,h\right)}$according to four adjacent pixels Є {Stego blank set} using Formulas (3)–(5). ${\widehat{P}}_{\left(C,d\right)}$ in Equation (6) denotes the smallest magnitude prediction error for pixel P _{(C, v, h)} and d Є {V, H} such that:

$${\widehat{P}}_{\left(C,d\right)}=\mathrm{If}(\left|(P_{\left(C,v,h\right) }-{\widehat{P}}_{\left(C,V\right)})\right|<\left|(P_{\left(C,v,h\right) }-{\widehat{P}}_{\left(C,H\right)})\right|)? {\widehat{P}}_{\left(C,V\right)}:{\widehat{P}}_{\left(C,H\right)}$$

(7)

4.2.3. Constructing a 3D Prediction Error Histogram (3D-PEH)

After calculating the 3D directionally enclosed prediction and expansion (3D-DEPE) and 3D local complexity (3D-LC) for each set within the C color channel, the set pixels are associated with three parameters: the color channel, its calculated 3D-DEPE value, and the 3D-LC value. Smaller LC values correlate with higher smoothness, so the LC and their associated directional prediction error are sorted in ascending order before constructing PEH. A 3D-PEH consists of each set histogram within a three-color channel. PHE is a graphical representation of each set’s DEPE distribution within the color channel that is used to represent the frequency of each prediction error value within a set, which is between −255 (dark color “black”) and 255 (light color “white”). PEH allows us to understand and compare the distribution of prediction errors between color channels and provides the first indicator of the information-hiding process.

4.3. Embedding Technique

The proposed embedding process is divided into three major stages that are applied to each set separately, beginning with the blank set until the 3D stego-Blank Set is constructed and progressing to the shadow set until the 3D stego-Shadow Set is constructed, then combining them to generate a stego image. The main stages are:

Stage 1.

Adaptive MPE-CPR Selection to calculate the MPE-CPR required parameters based on the payload size distribution in each set within each color channel and the number of bits per pixel (NBPP).

Stage 2.

A 3D-PEH localization is based on the multiple prediction error cumulative peak region (MPE-CPR) technique.

Stage 3.

The proposed Embedding Procedure into the localized PEH for each color channel.

4.3.1. Adaptive MPE-CPR Selection

Let 3D-PEH denote the probability density distribution for pixels’ prediction errors belonging to the host color image with three color channels. The “number of bits per pixel” (NBPP) denotes the number of bits to be embedded per pixel. From a histogram perspective, the NBPP is used to determine the number of cumulative peak bins (CRB), which represent nonoverlapping and consecutive bins in the prediction error histogram that contain the maximum cumulative prediction error. CRB is assigned a power of 2 NBPP. For example, to embed 2 bits per pixel, we need 22 cumulative peak bins. The prediction error cumulative peak region (PE-CPR) denotes the region in the PEH where NBPP bins achieve the maximum cumulative PE. The multiple prediction error cumulative peak region (MPE-CPR) is selected to satisfy the minimum number of cumulative peak regions in PEH to embed the payload size based on NBPP values, such that the first cumulative peak region within any color channel must achieve the following requirements:

$${{\displaystyle \sum }}_{i=FCPB}^{FCPB+CRB-1}PEH\left(i\right) >{{\displaystyle \sum }}_{i=SB}^{SB+CRB-1}PEH\left(i\right)$$

(8)

where FCPB determines the first bin of the current cumulative peak region not equal to SB; PEH(i) determines the value of bin i in the prediction error histogram PEH; and CRB determines the number of bins within the current cumulative peak region. The same requirements apply for the second cumulative peak region after excluding the first region…and so on until all cumulative peak regions (MPE-CPR) are covered. The concept of multiple prediction error cumulative peak region (MPE-CPR) with four regions and four bins per region is illustrated in Figure 7.

Figure 7 shows four multiple cumulative peak bin regions with four bins per region. The first region consists of four consecutive histogram pixels (153, 154, 155, and 156), which achieve a maximum cumulative bin value equal to the sum of 200, 210, 250, and 190. The second region consists of pixels 149, 150, 151, and 152, which achieve the second maximum cumulative bin value equal to the sum of 160, 200, 230, and 165. The third region consists of pixels 159, 160, 161, and 162, which achieve the third maximum cumulative bin value equal to the sum of 120, 200, 180, and 190. The fourth region consists of pixels 145, 146, 147, and 148, which achieve the fourth maximum cumulative bin value equal to the sum of 140, 180, 240, and 120.

4.3.2. 3D-PEH Localization

A 3D-PEH localization based on multiple prediction error cumulative peak region (MPE-CPR) technique is proposed to offer additional advantages over the conventional single peak shifting technique since it utilizes the cumulative peak regions of 3D-PEH to generate vacant bins for embedding variable-length payloads, reduces invalid pixel shifting, avoids the underflow/overflow problem during the embedding process, and provides increased robustness against attacks as the embedded information is distributed across multiple peaks based on lower local complexity values. Let CPR denote the number of cumulative peak regions in MPE-CPR per color channel, CRB denote the number of bins within each cumulative peak region that will be used for the embedding process, and FCPB denote the first bin in the current cumulative peak region. The color channel PEH localization technique moves PEH bins within the current cumulative peak regions from FCPB + 1 to FCPB + CRB to the first cumulative peak bin FCPB, such that the bins in PEH from FCPB + 1 to FCPB + CRB are assigned to value zero, and the bin FCPB is assigned to the sum of the cumulative bin values within the current region, that is:

$$PEH\left(FCPB\right)={\displaystyle \sum }_{i=FCPB}^{FCPB+CRB-1}PEH\left(i\right)$$

Using the previous Figure 7, four multiple cumulative peak regions in the PEH localization process are illustrated in Figure 8, and the prediction error histogram after applying bin localization is illustrated in Figure 8 and Figure 9.

4.3.3. Embedding Procedure

The proposed embedding process allocates a variable length of payload bits within each color channel by modifying the localized PEH based on the calculated cumulative peak bins (CPB), such that half of the bits are embedded in the blank set and the other half in the shadow set. The stego image is constructed by combining the two generated marked sets {Stego-blank set, Stego-shadow set} for each color channel.

Let ${é}_{\left(C,v,h\right)}$ denote the modified PEH within the color channel; BPP denotes the number of bits to be embedded per pixel x to be a sequence of the embedding payload bits of size equal to BPP; and CPn denotes the cumulative peak n in PEH, where n Є {1, … CPR }. The embedding process is carried out by modifying e to é with the knowledge that the MPE-CPR sequence per color channel is prioritized for embedding, as follows:

$${é}_{\left(C,v,h\right)} = \left\{{е}_{\left(C,v,h\right)}+ConvertToBinarry\left(x\right)\right\}$$

(9)

Thus, the corresponding original $P_{\left(C,v,h\right) }$value is modified by:

$$P‴{ }_{\left(C,v,h\right)}={\widehat{P}}_{\left(C,v,h\right)}+{é}_{\left(C,v,h\right)}$$

(10)

where ${\widehat{P}}_{\left(C,v,h\right)}$ denote the predicter value of $P_{\left(C,v,h\right) }$, $P{‴ }_{\left(C,v,h\right)}$ denote the marked pixel value, and ${é}_{\left(C,v,h\right)}$ the corresponding modified prediction error ${е}_{\left(C,v,h\right)}$.

4.4. 3D Extraction Technique

At the recipient, the reverse operation is applied to calculate the local complexity value LC, and thus the current pixel’s $P_{\left(C,v,h\right)}$ Є {shadow set} is obtained according to four adjacent pixels’ Є {blank set} using Equations (1) and (2), and vice versa. Then the $P_{\left(C,v,h\right) }$ Є {shadow set} is predicted as ${\widehat{P}}_{\left(C,v,h\right)}$according to four adjacent pixels, Є {blank set}, and the blank set pixels are predicted using four adjacent pixels, Є {shadow set}, using Equations (3)–(6). The 3D-PEH was also constructed using our proposed technique. Finally, the proposed adaptive MPE-CPR selection technique is applied with the knowledge that the required MPE-CPR parameters for each color channel are passed to the recipient, such as the prediction error cumulative peak regions (PE-CPR) count and the number of bits per pixel (NBPP).

The extraction process is applied sequentially, pixel by pixel, in the stego image, and each modified pixel $P{‴ }_{\left(C,v,h\right)}$ is associated with its corresponding prediction error value ${\widehat{P}}_{\left(C,v,h\right)}$ and modified prediction error ${é}_{\left(C,v,h\right)}$. The embedded bits (EB) are extracted as follows:

$${é}_{\left(C,v,h\right)}=P‴{ }_{\left(C,v,h\right)}-{\widehat{P}}_{\left(C,v,h\right)}$$

(11)

$$x={é}_{\left(C,v,h\right)}-{е}_{\left(C,v,h\right)}$$

(12)

EB = ConvertToBit(X).

(13)

where x denotes the extracted bits in the binary value, and EB denotes extracted bits.

5. Evaluation and Experimental Results

The proposed technique is assessed empirically or visually to evaluate its performance. The RGB color model is used for visual assessment to analyze carrier image properties and characteristics from visual quality and image histograms. Stego image visual quality accomplished on a particular set of testing images is the primary variable that is typically measured in empirical investigations of RDH techniques. This section focuses on an empirical performance evaluation of the proposed RDH technique using an RGB color model to optimize the valid shifting of the prediction error histogram and overcome the difficulties and constraints of existing RDH methods based on histograms and prediction errors that accomplish the desired goals. To evaluate the performance of the proposed technique and demonstrate its practicality, qualitative and quantitative analyses were conducted on the carrier images within the RGB color model. The organization of this section begins with a detailed description of the RDH benchmarks and testing images listed in Section 5.1. Section 5.2 provides a detailed discussion and lists the RDH evaluation matrices. Section 5.3 presents the proposed technique’s experimental results and comparison with some state-of-the-art works.

5.1. Benchmark

The first version of the USC-SIPI image database was made available in 1977 to aid research in machine vision, image analysis, and image processing. The main characteristics of the test images are used to divide the USC-SIPI database into volumes. The image sizes in each volume range from 256 × 256 to 1024 × 1024 pixels. In contrast to the color image, which has 24 bits per pixel, the grayscale image has 8 bits per pixel. Lena, Baboon, Pepper, Lake, Barbara, Tiffany, Boats, and an Airplane were used as the cover images. These standardized 512 × 512 RGB color images from the USC-SIPI image database are frequently used in RDH technique literature. These testing images are shown in Figure 10.

5.2. Evaluation Matrices

RDH technique performance evaluated and analyzed depends on several factors related to embedding capacity and stego image visual quality. Human vision should not be affected by stego image distortion, and the amount of payload that can be embedded also matters a lot. Every algorithm undergoes extensive testing to determine how much information can be embedded. The common evaluation matrices for RDH techniques and the measuring tools that are widely used in image quality assessment are peak signal to noise ratio (PSNR), the bits per pixel to denote the number of embedded bits in each pixel, and MSE, which denotes the mean square errors, which represent the difference between the original image and the stego image.

The embedding capacity is measured by the pure payload, which determines the number of pixels used in the embedding process. The image’s visual quality is measured by PSNR. PSNR represents the similarity between the stego image and the original image. The pure payload denotes the maximum hiding capacity size of bitstream sequences that can be embedded into converted-color image channels’, which are three color channels of size 512 × 512 8-bit pixels per color channel. A higher PSNR value means higher visual quality, which improves performance. The PSNR value can be calculated using the following equation:

$$\mathit{PSNR} = 10 \mathit{log}10\frac{Ma{x}^2}{MSE} \mathrm{dB}$$

(14)

where the difference between the original image I and the stego image I′ is represented by the mean square error or MSE. The formula below can be used to determine MSE:

$$MSE=\left(\frac{1}{3\times M\times N}\right){\displaystyle \sum }_{x=0}^{M-1}{\displaystyle \sum }_{y=0}^{N-1}{\left(I\left(x,y\right)-I’\left(x,y\right)\right)}^2$$

where: M × N is the dimension of the cover image measured in pixels. I(x,y) is the pixel value of the cover image. I′(x,y) is the pixel value of the stego image.

The PSNR value depends on the embedding bitstream sequence; it is measured against all types of bitstream sequences using the 1’s bits PSNR, 0’s bits PSNR, and average PSNR. The 1’s bit PSNR and 0’s bit PSNR were determined after embedding a sequence of the 1’s bitstream, respectively. The average PSNR was determined by averaging the PSNR of 100 random embeddings of the sequence of the 0’s and 1’s bitstream.

5.3. Experimental Results

The experiments were conducted, implemented, and tested using C# .NET 2022 to study, analyze, and evaluate the proposed technique’s performance from the perspectives of image quality and embedding capacity. The peak-signal-to-noise ratio (PSNR) in dB serves as a gauge for the quality of the image. The embedding capacity evaluates how much pure payload, expressed in bits, can be embedded inside a cover image. The proposed method is compared with five related techniques: Li et al. [40], Kouhi et al. [44], Al-Omari et al.’s [35], Mowafi et al.’s [38], and Kim et al.’s [42] using eight typical 512 × 512 RGB color images (Lena, Baboon, Lake, Airplane, Boats, Pepper, Tiffany, and Barbara) as shown in Figure 11.

As shown in Figure 12, regardless of the embedding capacity or type of the cover image, the proposed technique outperforms other techniques. The proposed method provides the highest image quality under all payload conditions. The PSNR value for payload conditions is significantly improved for all images compared to other technologies, with a slight change in the PSNR value with increasing the size of the embedded payload. On the other hand, the properties of the load images and the local complexity of the pixel values play an important role in the recorded results. Embedding the data in the image with lower local complexity values recorded a significant improvement in the value of PSNR compared to other techniques. For example, compared with Kouhi et al. [44], the average PSNR gains for the proposed technique on the images Lena, Boat, and Lake are 9.82 dB, 11.91 dB, and 11.61 dB, respectively, while the average PSNR gain across all images for various capacities is 9.66 dB. This improvement can be attributed primarily to the proposed method’s use of an adaptive number of cumulative peak regions in the prediction error histogram, which decreases the number of invalid shifting pixels.

The proposed embedding process, in the prediction error histogram, outperforms Mowafi et al.’s [38] and AlOmari et al.’s [35] techniques, which are based on embedding messages in cumulative peak regions of the image histogram, with average PSNR values of 12.59 and 17.14, respectively. The cumulative peak region value calculated from the PEH is much higher than the cumulative peak region value calculated from the original image histogram.

On the other hand, the results obtained using eight images with embedding capacities ranging from 0.5 × 104 bits to 5 × 104 bits are shown in Figure 12. Here, the complexity of the embedding capacity varies from 0.5 × 104 bits to the maximum capacity of 5 × 104 bits. The stego image’s visual quality, determined by PSNR, is used to evaluate the performance. to demonstrate the effectiveness of the proposed technique and its performance based on image characteristics in terms of smoothness and local Complexity.

The proposed method produced significantly superior results using Tiffany and Airplane images, with average PSNR values of 74.08 dB and 73.42 dB, respectively, while Baboon achieved the lowest average PSNR value of 60.75 dB. This indicates that the proposed method can be more effective in enhancing image quality, especially for images with lower local complexity and higher smoothness values, which achieve our target. Tiffany and Airplane images have smooth textures, minimal local complexity, and few dominant RGB colors, making them more successful as carrier images with our proposed technique. The cumulative peak region value from the PEH is significantly higher than the image histogram, increasing embedding capacity.

6. Discussion

RDH techniques based on the prediction error histogram shift the schema shift bins from the peak point to the zero point, which significantly affects the image quality when many invalid shifts are generated and aren’t used for the embedding process. Therefore, the proposed technique uses cumulative peak region localization to localize the bins to the first bin within its region that solves overflow/underflow problems, with the maximum number of localized bins within a region equaling two to the power of the number of bits per pixel, and adaptively determines the number of cumulative peak regions based on payload size, which reduces invalid shifting pixels and improves image quality. Additionally, the cumulative peak regions are sorted based on the cumulative local complexity value for each region in ascending order before embedding, so there is no need to construct a location map for the extraction process, and thus no extra information will be embedded or exchanged with the recipient.

Tiffany and Airplane images have smooth textures, little local complexity, and only a few dominant RGB colors to produce a comparable hue value with smooth color transitions and, as a result, higher cumulative peak values for their prediction error histogram. This indicates that it may be more successful to use smooth images as carrier images with our proposed technique without significant loss of visual quality compared to images with more complex textures and color distributions, such as the Baboon image. However, it is important to note that RDH techniques should always be carefully selected based on the specific characteristics of the image being used as a carrier image. It is more effective to embed a message based on peak regions into a PEH than it is to embed a message based on peak regions into an image histogram. We observe that the cumulative peak regions value calculated from the PEH is significantly higher than the cumulative peak regions value calculated from the image histogram by comparing the stego image histogram constructed after applying the proposed embedding process to the cumulative peak regions of the directional prediction error histogram and the cumulative peak regions of the image histogram, which increases embedding capacity.

7. Future Work Directions

In future work, we seek to develop the proposed technology using image texture properties, study the performance of the proposed technique based on both properties, and develop more effective ways to minimize and calculate the total number of invalid shifting pixels. Additionally, we aim to investigate the impact of different image resolutions on the performance of the proposed technique. Furthermore, we plan to explore potential optimizations to enhance the computational efficiency of minimizing and calculating the total number of invalid shifting pixels.

In addition to studying the effects of using different color models on the proposed technique, enhance the usage of different color models based on how humans perceive colors. For instance, to achieve data embedding by taking advantage of the human eye’s ability to perceive color, it can be relied upon to embed secret information into the chrominance component of YUV, YCbCr, YIQ, and YCoCg color models while using the luminance component of HSL, HSI, and HSV color models.

8. Conclusions

In this paper, we propose an adaptive RDH technique for localizing the directional prediction error histogram for color images based on multiple cumulative peak regions. The proposed technique aims to improve the robustness and efficiency of reversible data hiding (RDH) in color images. It utilizes multiple cumulative peak regions of the directional prediction error histogram to reduce invalid shifting in the prediction error histogram and enable adaptive embedding of secret data. Additionally, the technique considers the perceptual characteristics of color images to ensure minimal distortion and a high visual quality based on the relative correlation between the pixel’s local complexity and its directional prediction error, which is employed to enhance an efficient RDH without using a location map.

The proposed technique is based on multiple cumulative peak regions for localization of the directional prediction error histogram for color images. Each carrier image color channel is divided into two nonoverlapping, separated sets, like a checkerboard pattern. The embedding process for each color channel starts with its blank set until the stego blank set construction is complete, which will be used to construct a stego shadow set. Both sets will be combined to generate a stego channel, while three stego channels are combined to construct a stego image. The local complexity value of each pixel in each set is calculated to determine the degree of smoothness to be associated with cumulative peak regions within each set, in which the message is preferentially embedded in the cumulative peak region with a lower cumulative smoothness value, effectively reducing the invalid shifting of pixels, eliminating the need for a location map, and ensuring the full distribution of the embedded message within the carrier image; therefore, it achieves embedding process security and robustness against attacks.

The experimental results have demonstrated the outstanding performance of the proposed technique over existing RDH techniques based on embedding data into the prediction error histogram in terms of visual quality. In addition, the cumulative peak region value calculated from the PEH is much higher than the cumulative peak region value calculated from the original image histogram. This suggests that the PEH captures more detailed information about the distribution of pixel intensities in the image. It is likely that the PEH considers additional factors or features, like local complexity, that contribute to a higher cumulative peak region value compared to the original histogram.