A Novel High-Fidelity Reversible Data Hiding Method Based on Adaptive Multi-pass Embedding

Abstract

In reversible data hiding, prediction error generation plays a crucial role, with pixel value ordering (PVO) standing out as a prediction method that achieves high fidelity. However, conventional PVO approaches select predicted pixels and their predictions independently, failing to fully exploit the inherent redundancy in ordered pixel sequences. This paper proposes a novel PVO-based prediction method that leverages the continuity and spatial correlation of ordering pixels. We first introduce a new prediction technique that exploits the redundancy of consecutive pixels. Our approach selects the most appropriate prediction method from preset prediction errors, considering both pixel position and value characteristics. Furthermore, we implement an adaptive strategy that dynamically selects multiple iteration parameters based on pixel content to obtain more expandable prediction errors and adjusts the modification of prediction errors accordingly. Unlike traditional fixed-parameter methods, our approach better utilizes the inherent structure and redundancy of image pixels, thereby improving data embedding efficiency while minimizing image distortion. We enhance performance by combining pairwise prediction-error expansion with content-based prediction error analysis. Experimental results demonstrate that the proposed scheme outperforms state-of-the-art solutions in terms of image fidelity while maintaining competitive embedding capacity, confirming the effectiveness of our method for efficient data embedding and image recovery.

1. Introduction

Reversible data hiding (RDH) is a technique that facilitates the embedding of secret information into digital media, such as images and videos, with the capability to perfectly restore the original media after data extraction. This field has garnered significant attention due to its unique advantages over traditional data hiding methods, which often result in irreversible alterations to the carrier medium [1,2,3,4,5]. The industrial relevance of RDH is vast, spanning across diverse sectors including medical imaging, forensics, intellectual property protection, and military communications. In the medical field, RDH is instrumental in embedding patient data into diagnostic images without compromising the integrity of the original images, ensuring accuracy in clinical interpretations. Similarly, in forensics and law enforcement, it plays a crucial role in embedding metadata into digital evidence, preserving the admissibility of the evidence in legal proceedings. The method also aids in safeguarding intellectual property by embedding ownership information into digital products, facilitating tracking and verification while maintaining the original content intact. Furthermore, in military and defense applications, RDH enables the secure transmission of confidential information through media files, with the assurance that the original media can be restored without any loss, which is essential for strategic operations. The importance of RDH lies in its ability to ensure data integrity, enhance security, offer versatility in application, and support compliance with data protection regulations across various industries. As the demand for secure and reliable data embedding solutions grows, the development and application of RDH techniques are becoming increasingly pertinent.

RDH methods are diverse, with some approaches fundamentally rooted in lossless compression techniques. These methods, as referenced in the seminal works [6,7,8,9,10], aim to economize storage space by employing lossless compression algorithms on the feature sets extracted from the cover image. This strategy not only preserves the original image’s quality but also creates a compact space for secret data embedding. Parallel to lossless compression-based methods, histogram shifting (HS) techniques have garnered attention in the RDH domain [11,12,13,14,15]. HS methods ingeniously exploit the statistical properties of images to embed secret data, capitalizing on the distribution of pixel values to create a covert channel for information exchange.

Furthermore, difference expansion (DE) [16,17,18] and derivation prediction error expansion (PEE) [19,20,21,22,23] stand out as innovative approaches within the RDH paradigm. DE operates on the principle of embedding one-bit data into the minute differences between adjacent pixels, a technique that is both simple and effective for certain applications. In PEE-based methods, prediction error which is the difference between the predicted pixel and the one serving as its prediction is expanded to embed data. Because the local correlations are considered in the prediction, PEE can produce prediction error with smaller magnitude, thereby significantly improving the performance. The pixel value ordering (PVO) embedding strategy [24,25,26,27,28] enables the marked image better visual quality with combining the idea of HS and PEE. These methods divide cover image into non-overlapped pixel blocks, pixels are ordered in ascending order to make prediction errors, then data are embedded through prediction error modification.

The principle of the PVO-based data hiding method is to explore the efficient utilization of predicted pixels and their predictions to produce more accurate prediction errors. To gain this advantage, content-based adaptive embedding achieves good performance. Considering the surrounding texture area of the target pixel block, content-based adaptive embedding methods are further studied [27,29,30,31,32,33]. Previous studies only focused on using the smoothness of pixel block selecting embedding method; it is more meaningful to consider the combination of pixel smoothness and prediction error generation to design adaptive prediction errors for the same embedding method. However, existing methods do not fully exploit the redundancy between consecutive pixels or equal pixels.

This paper introduces an innovative data hiding method that leverages the redundancy between consecutive pixels to enhance the efficiency of prediction error generation. Unlike methods that focus solely on pixel smoothness, our work delves into the synergistic effect of pixel continuity and prediction error accuracy. By designing an adaptive strategy that considers both the dynamic prediction error and the noise level of pixel blocks, we aim to minimize image distortion while enlarging the embedding capacity. The main contributions of this paper can be summarized as follows: This paper proposes a novel adaptive reversible data hiding method that can adaptively select multi-pass parameters based on content pixels to obtain more expandable prediction errors, and then adaptively adjust the modification of prediction errors. The ambition of this method is to formulate an optimized embedding strategy for a given cover image and a given secret data, providing new solutions for reversible data hiding. The method introduces a novel approach to adaptively select multi-pass parameters based on the content of pixels to enhance prediction error expansion. It allows for the generation of more expandable prediction errors, which is a departure from traditional fixed-parameter approaches. By dynamically adjusting these parameters, the method can better exploit the inherent structure and redundancy of the image pixel, leading to improved data embedding efficiency and reduced image distortion. A multi-pass PVO-based prediction strategy is proposed to design prediction error modification with less image distortion. It enables a more nuanced and precise control over the prediction error classification and modification, ensuring that the modifications to the prediction errors result in minimal additional distortion to the image. Experimental results show that the proposed scheme outperforms a series of state-of-the-art schemes with moderate capacity in terms of image fidelity.

The rest of this paper is organized as follows. In Section 2, the related RDH methods are firstly reviewed. In Section 3, the proposed method is introduced in three steps. Section 4 discuss the performance of the proposed method with experimental results. Finally, we summarize our work in the last section.

2. Related Work

In this section, single-pass and multi-pass PVO-based methods are briefly reviewed. Subsequently, we delve into the nuances of pairwise PVO-based prediction error modification. Finally, the motivation of this article is presented.

2.1. Single-Pass PVO-Based Reversible Data Hiding

PVO [24] involves arranging the pixel values within a non-overlapping block in ascending order. Consequently, the largest and smallest pixel are predicted by the second largest and second smallest valued pixels, respectively, with the introduction of a single prediction error. Later, considering the spatial location between the predicted pixel and its prediction, the improved pixel value ordering (IPVO) method proposed in [25] constructed a steeper prediction error histogram (PEH). A steeper PEH facilitates more expandable prediction errors, thereby improving both embedding capacity and accuracy. However, the IPVO method still faces limitations, as each pixel block can carry a maximum of two bits of data. The pixel-based PVO method (PPVO) [34] gains the most significant capacity improvement, where block-by-block prediction is abandoned, and each pixel is predicted and modified according to its sorting context. However, the prediction mechanism limits the performance of embedding, and the embedding capacity still needs to be improved. In [26], PVO is extended to PVO-k by taking the k largest (or smallest) pixels as a prediction unit. However, the fact that embedding one bit of data requires modifying k pixels causes unnecessary pixel distortion. Later, many works focused on embedding k bits data into the largest (or smallest) k pixels, called multi-pass embedding. It is beneficial to raise the utilization rate of pixels after segmenting the image block-by-block and enlarge the embedding capacity.

2.2. Multi-Pass PVO-Based Reversible Data Hiding

Compared to the utilization of a maximum of two pixels in a block, the use of multiple pixels effectively increases the pixel utilization rate. In MIPVO [35], the largest pixel is used as prediction for the remaining $n-1$ pixels, where n is the pixel number of a pixel block. Then, the smallest pixel is used as prediction for the remaining $n-1$ pixels. In addition, [35] also adopts a flexible spatial location method to optimize the location of the pixels participating in the prediction. Thus, MIPVO enlarges the embedding capacity and the fidelity of marked image. However, MIPVO takes the same treatment for all prediction errors and does not deeply discuss the correlation among different prediction errors. The GMP-PVO integrates the PVO framework with a multi-predictor mechanism [32], dynamically adjusting the prediction strategy based on the content of the image blocks. RSP-PVO [36] uses the relationship of the largest three pixels to perform multi-pass embedding, and re-collects the pixels to obtain pixel positions that are more conducive to embedding. However, the predicted pixels and their predictions are fixed, which limits the variety of predictions for the reversibility. Although existing methods have achieved certain effects in utilizing pixel positions, they often neglect the improvement of adaptive prediction mechanisms. We propose a new method that focuses on improving accuracy through adaptive prediction, thereby increasing the utilization rate of pixels while reducing the loss of information. This approach is expected to achieve more efficient data embedding and image recovery in the field of reversible data hiding.

2.3. Pairwise PVO-Based Reversible Data Hiding

Some prominent works have applied the pairwise PEE strategy to the PVO scheme, thereby integrating the advantages of two-dimensional prediction error modification within the PVO framework, which is referred to as pairwise PVO. Some fascinating methods include KPPE [37] which combined K-pass PVO with improved PEE, pairwise MIPVO [35] which combined multi-pass PVO with hybrid PEE, and LPVO [38], which is a self-learning mechanism to select the 2D mapping adaptively according to the image content. Despite these innovations, current methods often generate prediction errors based on static observational laws, which may not fully adapt to the nuances of different images. The EP-PEE [39] proposes an enhanced pairwise PEE, which designs a more efficient 2D mapping by predicting the largest/smallest two pixels depending on the content-selected expansion prediction error. Nevertheless, the discussion of the pixel relationship stops at the complex utilization of the pixel position. However, existing discussions on pixel relationships in these methods tend to focus on the complex utilization of pixel positions, with less emphasis on developing adaptive prediction mechanisms. Future research could benefit from exploring more flexible and adaptive prediction strategies that can better accommodate the diverse characteristics of various images, potentially leading to further advancements in RDH techniques.

3. Proposed Method

In this section, adaptive PVO-based pixel prediction methods along with the corresponding prediction error modification are presented. Next, the pairwise modification and adaptive framework of the proposed method is determined. For the sake of clarity, the detailed embedding and extraction procedure of reversible data hiding is presented. To facilitate a more comprehensive understanding of our approach,

Table 1. Notations and descriptions of key parameters and variables in the proposed embedding method.

Symbol	Description	Symbol	Description
$p_{\sigma \left(i\right)}$	Pixel value at position $\sigma \left(i\right)$ in the sorted sequence	$\sigma \left(i\right)$	Mapping function that maps sorted index i to original pixel position
$n_1\times n_2$	Size of non-overlapped blocks	n	Total number of pixels in a block $(n=n_1\times n_2)$
$e_n$	Prediction error for the largest pixel	$e_1$	Prediction error for the smallest pixel
$e_L$	Prediction error for pixel at position $\sigma \left(L\right)$	$e_S$	Prediction error for pixel at position $\sigma \left(S\right)$
$e_n^{\prime }$, $e_1^{\prime },e_L^{\prime }$, $e_S^{\prime }$	Modified prediction errors after data embedding	$p_{\sigma \left(i\right)}^{\prime }$	Modified pixel value after data embedding
$p_i$	Pixel value at position i in the unsorted sequence	$w_1$, $w_2$	Binary secret data bits (0 or 1) to be embedded
L	Index in range $\{n-k+1,\dots ,n-1\}$	S	Index in range $\{2,\dots ,k\}$
k	Parameter controlling number of pixels processed in each pass	$k_m$	The number of classes for pixel blocks based on noise level NL
$\nu$	Ideal k in a pixel block	$h\left(\nu \right)$	Embedding parameter for noise level $\nu$
$Tk$	The collection of noise level thresholds for pixel blocks	$T{k}^{*}$	Optimal thresholds collection for noise level classification

of important symbols and their definitions is provided below.

3.1. Adaptive k-Pass PVO-Based Prediction

The cover image is first divided into non-overlapped blocks with the size of $n_1\times n_2$, where $n=n_1\times n_2$. For each block, pixels are sorted in ascending order as $\left\{p_{\sigma \left(1\right)},p_{\sigma \left(2\right)},....,p_{\sigma \left(n\right)}\right\}$, where $\sigma :\left\{1,2...,n\right\}\to \left\{1,2...,n\right\}$ denotes the unique one-to-one mapping such that: $p_{\sigma \left(1\right)}\le p_{\sigma \left(2\right)}\le \cdots \le p_{\sigma \left(n\right)}$, $\sigma \left(i\right)<\sigma \left(j\right)$ if $p_{\sigma \left(i\right)}=p_{\sigma \left(j\right)}$ and $i<j$. As for pixel prediction, $p_{\sigma \left(n\right)}$ is predicted first by $p_{\sigma (n-k)}$ to obtain the preset prediction error $e_n$.

$$e_n=p_u-p_v$$

(1)

where $u=min\left(\sigma \right(n),\sigma (n-k\left)\right),v=max\left(\sigma \right(n),\sigma (n-k\left)\right)$. Next, $e_L$ ($L\in \{n-k+1,..,n-1\}$) is determined by $e_n$ to predict the $p_{\sigma \left(L\right)}$. The prediction error $e_L$ is similarly calculated as

$$e_L=p_a-p_d$$

(2)

$$a=\left\{\begin{array}{cc}min\left(\sigma \right(L),\sigma (n-k\left)\right),\hfill & \mathit{if} {\mathit{e}}_{\mathit{n}}=\mathit{0} \mathit{or} {\mathit{e}}_{\mathit{n}}=\mathit{1}\hfill \\ min\left(\sigma \right(n),\sigma (L\left)\right),\hfill & \mathit{if} {\mathit{e}}_{\mathit{n}}<\mathit{0} \mathit{or} {\mathit{e}}_{\mathit{n}}>\mathit{1}\hfill \end{array}\right.$$

(3)

$$d=\left\{\begin{array}{cc}max\left(\sigma \right(L),\sigma (n-k\left)\right),\hfill & \mathit{if} {\mathit{e}}_{\mathit{n}}=\mathit{0} \mathit{or} {\mathit{e}}_{\mathit{n}}=\mathit{1}\hfill \\ max\left(\sigma \right(n),\sigma (L\left)\right),\hfill & \mathit{if} {\mathit{e}}_{\mathit{n}}<\mathit{0} \mathit{or} {\mathit{e}}_{\mathit{n}}>\mathit{1}.\hfill \end{array}\right.$$

(4)

In case of $e_n=0$ or $e_n=1$, $p_{\sigma \left(L\right)}$ is predicted by $p_{\sigma (n-k)}$ as Equation (2). If $e_n>1$ or $e_n<0$, $p_{\sigma \left(L\right)}$ is predicted by $p_{\sigma \left(n\right)}$. Then the prediction error is modified as follows.

$$e_n^{\prime }=\left\{\begin{array}{cc}{e}_n+w_1,\hfill & \mathit{if} {\mathit{e}}_{\mathit{n}}=\mathit{1}\hfill \\ e_n-w_1,\hfill & \mathit{if} {\mathit{e}}_{\mathit{n}}=\mathit{0}\hfill \\ e_n+1,\hfill & \mathit{if} {\mathit{e}}_{\mathit{n}}>\mathit{1}\hfill \\ e_n-1,\hfill & \mathit{if} {\mathit{e}}_{\mathit{n}}<\mathit{0}\hfill \end{array}\right.$$

(5)

$$e_L^{\prime }=\left\{\begin{array}{cc}{e}_L+w_2,\hfill & \mathit{if} {\mathit{e}}_{\mathit{L}}=\mathit{1}\hfill \\ e_L-w_2,\hfill & \mathit{if} {\mathit{e}}_{\mathit{L}}=\mathit{0}\hfill \\ e_L,\hfill & \mathit{if} {\mathit{e}}_{\mathit{L}}<\mathit{0} \mathit{or} {\mathit{e}}_{\mathit{L}}>\mathit{1}\hfill \end{array}\right.$$

(6)

$w_1$ and $w_2$ are binary secret data. From Equation (7), $p_{\sigma \left(n\right)}$ has been increased by 1 when $e_n>1$ or $e_n<0$. Otherwise, one-bit data are embedded into $p_{\sigma \left(n\right)}$. Following, $p_{\sigma \left(L\right)}$ is modified according to Equation (8). One bright spot is that distortion occurs in the largest $n-L$ pixel only when secret data are embedded.

$$p_{\sigma \left(n\right)}^{\prime }=\left\{\begin{array}{cc}{p}_{\sigma \left(n\right)}+w_1,\hfill & \mathit{if} {\mathit{e}}_{\mathit{n}}=\mathit{0} \mathit{or} {\mathit{e}}_{\mathit{n}}=\mathit{1}\hfill \\ p_{\sigma \left(n\right)}+1,\hfill & \mathit{if} {\mathit{e}}_{\mathit{n}}<\mathit{0} \mathit{or} {\mathit{e}}_{\mathit{n}}>\mathit{1}\hfill \end{array}\right.$$

(7)

$$p_{\sigma \left(L\right)}^{\prime }=\left\{\begin{array}{cc}{p}_{\sigma \left(L\right)}+w_2,\hfill & \mathit{if} {\mathit{e}}_{\mathit{L}}=\mathit{0} \mathit{or} {\mathit{e}}_{\mathit{L}}=\mathit{1}\hfill \\ p_{\sigma \left(L\right)},\hfill & \mathit{if} {\mathit{e}}_{\mathit{L}}<\mathit{0} \mathit{or} {\mathit{e}}_{\mathit{L}}>\mathit{1}\hfill \end{array}\right.$$

(8)

The modification on large pixels can be directly applied to small pixels. For the smallest k pixels, adaptive modification is implemented as follows. First, $p_{\sigma \left(1\right)}$ is predicted by $p_{\sigma (1+k)}$ to obtain preset prediction error $e_1$.

$$e_1=p_s-p_t$$

(9)

where $s=min\left(\sigma \right(1), \sigma (1+k\left)\right), t=max\left(\sigma \right(1), \sigma (1+k\left)\right)$. Based on $e_1$, the rest prediction error $e_S$ is calculated in two ways as Equation (10).

$$e_S=p_b-p_c$$

(10)

where $b=min\left(\sigma \right(S), \sigma (1+k\left)\right), c=max\left(\sigma \right(S), \sigma (1+k\left)\right)$ and $S\in \{2,..,k\}$.

$$b=\left\{\begin{array}{cc}min\left(\sigma \right(S),\sigma (1+k\left)\right),\hfill & \mathit{if} {\mathit{e}}_{\mathit{1}}=\mathit{0} \mathit{or} {\mathit{e}}_{\mathit{1}}=\mathit{1}\hfill \\ min\left(\sigma \right(S),\sigma (1\left)\right),\hfill & \mathit{if} {\mathit{e}}_{\mathit{1}}<\mathit{0} \mathit{or} {\mathit{e}}_{\mathit{1}}>\mathit{1}.\hfill \end{array}\right.$$

(11)

$$c=\left\{\begin{array}{cc}max\left(\sigma \right(S),\sigma (1+k\left)\right),\hfill & \mathit{if} {\mathit{e}}_{\mathit{1}}=\mathit{0} \mathit{or} {\mathit{e}}_{\mathit{1}}=\mathit{1}\hfill \\ max\left(\sigma \right(S),\sigma (1\left)\right),\hfill & \mathit{if} {\mathit{e}}_{\mathit{1}}<\mathit{0} \mathit{or} {\mathit{e}}_{\mathit{1}}>\mathit{1}.\hfill \end{array}\right.$$

(12)

In case of $e_1=0$ or $e_1=1$, one-bit data are embedded into $p_{\sigma \left(1\right)}$. Otherwise, the smallest pixel is decreased by 1 when $e_1<0$ or $e_1>1$.

$$p_{\sigma \left(1\right)}^{\prime }=\left\{\begin{array}{cc}{p}_{\sigma \left(1\right)}-w_1,\hfill & \mathit{if} {\mathit{e}}_{\mathit{1}}=\mathit{0} \mathit{or} {\mathit{e}}_{\mathit{1}}=\mathit{1}\hfill \\ p_{\sigma \left(1\right)}-1,\hfill & \mathit{if} {\mathit{e}}_{\mathit{1}}<\mathit{0} \mathit{or} {\mathit{e}}_{\mathit{1}}>\mathit{1}\hfill \end{array}\right.$$

(13)

By modifying only $p_{\sigma \left(S\right)}$ according to the Equation (14), one-bit data can be embedded by referring to $e_S$.

$$p_{\sigma \left(S\right)}^{\prime }=\left\{\begin{array}{cc}{p}_{\sigma \left(S\right)}-w_2,\hfill & \mathit{if} {\mathit{e}}_{\mathit{S}}=\mathit{0} \mathit{or} {\mathit{e}}_{\mathit{S}}=\mathit{1}\hfill \\ p_{\sigma \left(S\right)},\hfill & \mathit{if} {\mathit{e}}_{\mathit{S}}<\mathit{0} \mathit{or} {\mathit{e}}_{\mathit{S}}>\mathit{1}\hfill \end{array}\right.$$

(14)

In the context of extracting embedded data from marked images, it is imperative that the data can be accurately retrieved and that the cover image can be fully restored following the extraction process. Taking the smallest set of k pixels as an example, the prediction errors $e_1^{\prime }$ and $e_S^{\prime }$ for the marked image are computed according to Equations (9) and (10). Conversely, the conduction of prediction errors and the recovery of image pixels diverge from the process of data embedding. Subsequently, the procedures for data extraction and image pixel recovery are delineated as follows.

• If $e_1^{\prime }\in \{-1,0,1,2\}$, the embedded data are extracted as $w_1=e_1^{\prime }-1$ as $e_1^{\prime }\in \{1,2\}$ and $w_1=-e_1^{\prime }$ as $e_1^{\prime }\in \{0,-1\}$ and the pixel value is restored as $p_{\sigma \left(1\right)}=p_{\sigma \left(1\right)}^{\prime }+w_1$.

  (a)

  If $e_S^{\prime }>2$ or $e_S^{\prime }<-1$, there is neither data extraction nor pixel restoration.

  (b)

  If $e_S^{\prime }\in \{1,2\}$, the embedded data are extracted as $w_2=e_S^{\prime }-1$ and the pixel value is restored as $p_{\sigma \left(S\right)}=p_{\sigma \left(S\right)}^{\prime }+w_2$.

  (c)

  If $e_S^{\prime }\in \{0,-1\}$, the embedded bit is extracted as $w_2=-e_S^{\prime }$ and the pixel value is restored as $p_{\sigma \left(S\right)}=p_{\sigma \left(S\right)}^{\prime }+w_2$.

• If $e_1^{\prime }<-2$, there is no data extraction and the pixel value is restored as $p_{\sigma \left(1\right)}=p_{\sigma \left(1\right)}^{\prime }+1$.

  (a)

  If $e_S^{\prime }\in \{-1,0\}$, the embedded bit is extracted as $w_1=e_S^{\prime }+1$ and the pixel value is restored as $p_{\sigma \left(S\right)}=p_{\sigma \left(S\right)}^{\prime }+w_1$.

  (b)

  If $e_S^{\prime }\in \{1,2\}$, the embedded bit is extracted as $w_1=2-e_S^{\prime }$ and the pixel value is restored as $p_{\sigma \left(S\right)}=p_{\sigma \left(S\right)}^{\prime }+w_1$.

The restoration process for large pixels is analogous to that of small pixels, and thus, it will not be further detailed in this section. Typically, during the data embedding phase, larger pixels are processed prior to smaller ones, whereas the sequence is inverted during the data extraction phase.

Specially, Figure 1 gives an example on the largest three pixels to illustrate the proposed method clearly. For the given pixel block, six pixels $\{130,118,130,119,120,$$129\}$ are sorted in ascending order to obtain sequences $\{118,119,120,129,130,130\}$ and $\left\{\sigma \right(1),\sigma (2),\cdots ,\sigma (6\left)\right\}=\{2,4,5,6,1,3\}$. Then, we have $e_6=p_3-p_6=1$ and $e_5=p_1-p_6=1$. As a result, the one-bit data point $w_1=1$ is embedded into $p_1$ while another bit data point $w_2=1$ is embedded into $p_3$. For the modification on small pixels, the prediction errors are $e_1=p_2-p_5=-2$ and $e_2=p_2-p_4=-1$. Next, the smallest pixel is decreased by 1. The marked pixel sequence is $\{131,117,131,119,120,$$129\}$. During extraction, small pixels are restored first and then large pixels. The two prediction errors $e_1^{\prime }=p_2^{\prime }-p_5^{\prime }=-3$ and $e_2^{\prime }=p_1^{\prime }-p_6^{\prime }=-2$ are calculated. Following, the smallest pixel is restored as $p_2=p_2^{\prime }+1=118$. As for large pixels, $e_6^{\prime }=p_3^{\prime }-p_6^{\prime }=2$ and $e_5^{\prime }=p_1^{\prime }-p_6^{\prime }=2$ are calculated. Then, $w_1=e_5^{\prime }-1=1$ is extracted and $p_1$ is restored as $p_1=p_1^{\prime }-w_1=130$, $w_2=e_6^{\prime }-1=1$ is extracted and $p_3$ is restored as $p_3=p_3^{\prime }-w_2=130$. In general, data embedding follows a position-based strategy that processes pixels sequentially from smaller to larger positions, guided by prediction errors. Data extraction leverages these same positional relationships. This systematic approach mathematically guarantees complete reversibility.

3.2. Content-Based IPVO-Based Pairwise PEE

In this section, the pairwise PEE is adopted to optimize the use of expandable prediction error. The superiority of pairwise PEE over conventional PVO is attributed to the improved modification of prediction error pairs, each comprising two prediction errors. The PVO-based modification embeds two bits of information into two expandable prediction errors, resulting in a distortion of 2. This approach, while effective, leaves room for improvement in terms of embedding efficiency. To address this, PEE-based modification ingeniously embeds a ternary number ‘0’, ‘1’, or ‘2’ into a single paired prediction error $(e_h,e_v)$. It reduces the distortion to a minimal value of 1 but also embeds ${log}_{2}3$-bits data, thereby enhancing the embedding capacity within the prediction errors.

In the proposed method, the pairwise PEE [22] is adopted to optimize the embedding when $e_n\in \{0,1\}$ and $e_1\in \{0,1\}$. In this case, the prediction errors are suitable to be combined into prediction error pairs (0, 0), (0, 1), (1, 0) and (1, 1) for 2D prediction mapping. For a pixel block, when $e_n=1$ or $e_n=0$, k prediction errors can be obtained. When k is an even number, $k/2$ prediction error pairs are formed. When k is odd, $(k-1)/2$ prediction error pairs are formed, and the remaining one prediction error is processed in a 1D prediction error modification. For example, when the collected prediction errors are 0, 1, 1. The first two prediction errors form a prediction error pairs (0, 1). The prediction error pairs follow the modification in Figure 2. The final prediction error ’1’ is modified according to the modification of the 1D histogram. In addition, it is not suitable to adapt pairwise PEE embedding as $e_n\notin \{0,1\}$. Pairwise PEE introduces additional computational overhead due to the pairing and 2D mapping of prediction errors, resulting in higher constant factors in both time and space complexity compared to conventional PEE. However, this added complexity significantly improves image quality metrics. Specifically, pairwise PEE achieves higher PSNR values, and larger embedding capacity, due to its ability to reduce overall distortion while embedding more data. This makes pairwise PEE a more effective approach for reversible data hiding, despite its increased computational demands.

Our method balances the embedding quantity for data hiding and the quality of the marked image by adaptively selecting the parameter k, achieving the desired embedding effect. As shown in Figure 3, with an increase of k, there is a corresponding increase in both the total number of prediction errors and the subset of expandable prediction errors. As $k=1$ and $k=2$, bin 0 and bin 1 represent the primary and secondary peak points, respectively. When $k=3$, bin 2 emerges as the peak point, yet the counts for bins 0 and 1 continue to rise. However, bin 0 and bin 1 are no longer peak points when $k>3$. The number of bin 0 and bin 1 is smaller than that under the case $k=3$. In essence, the challenge is not about changing k, but a way to choose an appropriate k. This selection is contingent upon the characteristics of the given embedding payload and the cover image, which dictate the classification of pixel blocks. The challenge lies not in changing k, but in selecting an appropriate value based on the given embedding payload and cover image characteristics. For a pixel block, the optimal k is set to $\nu$ according to:

$$h\left(\nu \right):\nu =k, \forall p_n-p_{n-k}<=1$$

(15)

In other words, $\nu$ enables the method to embed up to $(\nu -1)$ bits of data with larger pixel embedding in a pixel block.

To achieve adaptive embedding, we use the complexity of pixel blocks to determine the optimized k. For a block of size $n_1\times n_2$, its closed circle pixel blocks form a boundary region. We use the texture of this boundary region to estimate the smoothness of the target pixel block. We define the noise level ($NL$) as the sum of absolute differences between consecutive pixels in the boundary region, following the approach in [27]. For pixel blocks with the same $NL$, we use the most common $\nu$ as parameter k, expressed as $h\left(\nu \right)$.

3.3. Content-Based Adaptive k

For a given cover image and a specified amount of secret data, the parameters k and $h\left(\nu \right)$ are adaptively assigned based on the content complexity to achieve adaptive embedding. Using $NL$, a threshold T is used to fulfill block classification and embedding. Secret data are embedded into the pixel blocks with complexity less than or equal to the threshold T, where the optimal thresholds $T^{*}$ is determined by satisfying the embedding capacity and causing minimum image distortion. Thus, embedding in the selected pixel blocks with noise level less than or equal to $T^{*}$ ensures that the image distortion produced by the given embedding payload is minimized. For pixel blocks with noise levels $NL\le T^{*}$, they are categorized into $k_m$ distinct classes. Each class of blocks is characterized by its own $h\left(\nu \right)$, which is tailored to the specific blocks. To achieve these settings, the noise level thresholds for pixel blocks are represented as $Tk=\{T{k}_1,...,T{k}_{k_m-1},T^{*}\}$.

$$\left\{\begin{array}{c}T{k}^{*}={arg}_{Tk}minD\left(Tk\right)\hfill \\ subject to E\left(Tk\right)\ge EC\hfill \end{array}\right.$$

(16)

where $D\left(Tk\right)$ quantifies the distortion introduced by partitioning pixel blocks into $(k_m+1)$ distinct sets based on $k_m$ different $h\left(\nu \right)$ values. Furthermore, $E\left(Tk\right)$ represents the actual embedding payload, while $EC$ denotes the required embedding capacity. To ensure the image distortion remains below a predetermined threshold while satisfying the required embedding capacity, we precisely calculate the threshold $Tk$ according to Equation (16). This optimization framework effectively balances the trade-off between data hiding security and visual quality preservation, allowing for adaptive parameter selection that maintains the imperceptibility of the marked image while maximizing embedding efficiency.

3.4. Implementation of the Proposed Method

To implement data embedding, auxiliary information are set to embed pure payload. The framework of the proposed embedding method is shown in Figure 4. At first, the cover image is divided into non-overlapped blocks with a size of $n_1\times n_2$, where $n_1,n_2\in \{2,3,4,5\}$. Subsequently, these pixel blocks are categorized into two distinct sets, referred to as the X-set and the O-set. The secret data are embedded in two stages: first, half of the data are concealed within the X-set, followed by the embedding of the remaining half into the O-set. During the decoding phase, the pixels in the O-set are restored prior to those in the X-set. Given that the embedding technique is uniform across both sets, the X-set is employed as a representative example for illustration purposes. When processing the pixel blocks within the X-set, the adjacent pixel blocks from the O-set serve as the content blocks to calculate noise level.

For eight-bit gray-scale pixels, pixels valued 255 or 0 are susceptible to overflow/underflow when the pixel modification is over 1. To address this, pixels with values of 255 or 0 are adjusted to 254 or 1, respectively, and a location map ($LM$) is built to record these modifications. A bit 1 in $LM$ indicates a pixel modification while a bit 0 indicates no pixel modification. By using arithmetic coding, $LM$ is losslessly compressed into a shorter bit stream. Then, our re-collecting and sorting pixel model [36] is adopted to re-collect pixels in a block. Next, the least significant bits (LSBs) of the first two lines of pixels, the compressed location map and the pure payload (denoted as $S_{LSB}$) are embedded as a whole. Upon completion of the data embedding process, the auxiliary information is also ascertained. This information comprises six distinct data segments as follows.

• The length of binary auxiliary information (10 bits).
• Block size $n_1,n_2\in \{2,3,4,5\}$ (4 bits).
• Thresholds $T{k}^{*}$ ($\left(k_m\right)\times 10$ bits).
• Parameter k ($k_m\times 4$ bits).
• Embedding capacity (16 bits).
• Length of the compressed location map $l_{LM}$ (16 bits).

The length of binary auxiliary information and auxiliary information are recorded as binary data. Finally, the above seven parts of data are embedded into the first two lines of pixels by LSB replacement for blind extraction.

The process of data extraction is briefly described as three steps:

• Auxiliary information extraction The LSBs of the first two lines pixels of the marked image are read to retrieve auxiliary information. First, the length of the binary auxiliary information is obtained according to the first 10 bits of the extracted data. Following, the auxiliary information is extracted.
• Data extraction and image restoration The marked image is divided into non-overlapped blocks with the extracted block size. The noise level of pixel block is calculated. If $NL\le T_1^{*}$, extraction is performed with the help of the extracted parameter k and $Tk$. Otherwise, the block is skipped.
• Restoration of the rest pixels The first two lines of bits data of $S_{LSB}$ are used to recover the first two lines of pixels. Then, the location map $LM$ is restored by decompressing the compressed location map. Finally, pixels valued 0 or 255 are restored according to the location map.

4. Experimental Results

In this section, the performances of the three proposed methods are first compared, then some important parameters are given to verify the embedding. Next, the proposed method is compared with state-of-the-art methods on general images. Finally, the discrimination of the proposed method is proved by t-test.

The indicators for evaluating the performance of similar methods mainly include embedding capacity and visual quality (peak signal-to-noise ratio, also denoted PSNR) of the marked image. Specifically, the embedding payloads were handled from 5000 bits to the EC. General standard images (Airplane, Baboon, Barbara, Boat, Elaine, Lake, Lena, and Peppers in Figure 5) are from the USC-SIPI database http://sipi.usc.edu/database/database.php?volume=misc (accessed on 30 November 2020). Additionally, we also utilized the Kodak dataset https://www.kaggle.com/datasets/sherylmehta/kodak-dataset (accessed on 19 May 2025) to validate the performance.

4.1. Analysis of Three Proposed Methods

When the parameter k is fixed at a stable value of 2, the method described in Section 3.1 is referred to as TP-IPVO. In contrast, when k is dynamically selected, the method is identified as KP-IPVO. Furthermore, the KP-IPVO method, which incorporates pairwise prediction error modifications, is termed PairwiseKP-IPVO. In data embedding, the TP-IPVO (fixed-parameter prediction) and KP-IPVO (dynamic parameter prediction) methods have been proposed to enhance embedding efficiency and image quality. The former achieves rapid embedding through fixed predictive parameters, while the latter optimizes predictive outcomes by dynamically selecting parameters to adapt to different image content. The comparative performance of TP-IPVO, KP-IPVO, and PairwiseKP-IPVO is shown in Figure 6. At low embedding capacities, KP-IPVO exhibits a more pronounced PSNR improvement over TP-IPVO. For example, in reversible data hiding on the Lena image, the PSNR gain is 0.51 dB at an embedding capacity of 10,000, and 0.19 dB at EC of 30,000. This observation indicates that as the embedding capacity increases and smooth pixel blocks are fully exploited, the subsequent utilization of coarser pixel blocks inevitably results in more significant image distortion.

Additionally, the proposed method generates more efficient predictions due to the presence of more consecutive or identical pixels in smooth images. In smooth images, it is easier for adjacent pixel values to be similar. This similarity allows the predictor to produce smaller prediction errors during processing, which is beneficial for obtaining expandable prediction errors rather than shifting errors, thereby increasing the embedding capacity and image quality. This effect is particularly notable for the Airplane image, where a greater number of prediction errors contribute to 2D PEH mapping. This observation demonstrates that the proposed content-based selection can optimize predictions. Since 2D PEH mapping is only applied to part-prediction errors, its main advantage is that it can enhance PSNR under the same embedding capacity. PairwiseKP-IPVO, due to its requirement for multiple pixel block threshold corrections, has a much higher computational load than TP-IPVO, thus taking the longest time under the same conditions. Although this feature reduces processing speed, it effectively improves the PSNR performance of the final image. The results indicate that adaptive prediction not only increases embedding capacity but also enhances PSNR, as content-based optimization prioritizes more accurate predictions.

4.2. Analysis of Parameters

Table 2. Embedding parameters for specific capacities on the image Lena, the unit of PSNR is dB.

	EC = 10,000				EC = 20,000				EC = 30,000
	${\mathit{t}}_{\mathbf{1}}\times {\mathit{t}}_{\mathbf{2}}$	${\mathit{Tk}}^{*}$	$\mathit{k}$	PSNR	${\mathit{t}}_{\mathbf{1}}\times {\mathit{t}}_{\mathbf{2}}$	${\mathit{Tk}}^{*}$	$\mathit{k}$	PSNR	${\mathit{t}}_{\mathbf{1}}\times {\mathit{t}}_{\mathbf{2}}$	${\mathit{Tk}}^{*}$	$\mathit{k}$	PSNR
KP-IPVO	$4\times 2$	(20,26,57)	$\{3,2,1\}$	61.47	$2\times 4$	(25,31,43,82)	$\{4,3,2,1\}$	57.66	$4\times 2$	(35,51,119)	$\{3,2,1\}$	54.90
PairwiseKP-IPVO	$4\times 4$	(4,6,44,44,49)	$\{5,4,3,1\}$	61.49	$4\times 2$	(3,46,51)	$\{3,2,1\}$	57.86	$4\times 2$	(4,4,82)	$\{3,1\}$	55.32

delineates the embedding parameters for KP-IPVO and PairwiseKP-IPVO in varying EC in the standard test image, Lena. The embedding parameters for KP-IPVO and PairwiseKP-IPVO are presented in Table 2. It is noteworthy that the parameters vary for pixel blocks of different sizes, reflecting the inherent complexities associated with each block size. In both methods, the complexity threshold tends to increase with the payload. However, the use of pixel blocks with higher complexity introduces additional image distortion, which in turn leads to a rapid decline in the PSNR. Due to the reduced distortion in PairwiseKP-IPVO with 2D prediction error pairs mapping, the advantage in PSNR is more pronounced under larger embedding payloads. Furthermore, the table reveals that PairwiseKP-IPVO benefits from processing more pixels within a block, which enhances the pairing of prediction errors. This is particularly advantageous for larger payloads, where the method’s performance surpasses that of KP-IPVO. Additionally, the relative best PSNR is observed to decrease in smaller pixel blocks under large payloads exceeding 10,000 bits. This decline is attributed to two factors: on one hand, an increase in prediction errors within smaller pixel blocks, and on the other hand, a reduction in pixel continuity that results in lower prediction accuracy. In summary, the $Tk$ directly changes the key indicators. Their appropriate values are obtained after embedding the data.

4.3. Comparison with Advanced Methods

The proposed PairwiseKP-IPVO method is compared with the classic PVO-based methods [24,25,26], and competing methods [32,35,38,39] shown in Figure 7. Two comparisons of fidelity are enumerated in

Table 3. PSNR (dB) in the representative methods [25,26], the competing methods [32,35,36,39] and the proposed PairwiseKP-IPVO method under the embedding payload of 10,000 bits.

	[25]	[26]	[35]	[32]	[36]	[39]	Proposed
Lena	58.99	59.16	61.08	60.81	61.02	61.01	61.56
Airplane	62.89	63.05	63.86	63.50	63.70	63.70	64.24
Baboon	53.58	54.15	54.63	54.92	55.13	55.13	55.18
Barbara	60.47	60.45	61.08	61.11	61.08	60.96	61.22
Boat	58.88	59.23	60.51	58.73	58.83	58.78	59.23
Elaine	57.35	57.36	59.14	57.81	58.56	58.75	59.28
Lake	58.26	58.06	58.85	59.70	60.03	60.43	60.99
Peppers	60.46	60.57	60.77	59.54	59.24	59.61	59.86
Average	58.86	59.00	59.99	59.51	59.70	59.80	60.19

and

Table 4. PSNR (dB) in the representative methods [25,26], the competing methods [32,35,36,39] and the proposed PairwiseKP-IPVO method under the embedding payload of 20,000 bits.

	[25]	[26]	[35]	[32]	[36]	[39]	Proposed
Lena	56.53	56.60	57.30	57.14	57.18	57.34	57.87
Airplane	59.06	59.26	60.34	59.72	59.97	59.97	60.94
Baboon	-	-	-	-	-	-	-
Barbara	56.18	56.50	56.92	57.04	57.10	57.09	57.51
Boat	53.85	53.72	54.40	54.53	54.63	54.51	54.88
Elaine	52.60	52.71	53.87	53.62	53.85	53.87	54.19
Lake	53.63	54.28	55.34	54.92	54.96	55.41	55.73
Peppers	54.78	54.93	55.37	55.58	55.24	55.67	55.76
Average	55.23	55.43	56.22	56.08	56.13	56.27	56.70

. In general, the proposed PairwiseKP-IPVO outperforms other methods in most cases. Unlike PairwiseKP-IPVO, there are at most two prediction errors in a pixel block in PVO [24], IPVO [25] and PVO-K [26]. It limits their embedding capacity. On Lena, 15,000 bits and 10,000 bits improvements of embedding capacity are exited as compared with IPVO and PVO-K, respectively. Different from PVO, IPVO adds the pixel position to optimizes the prediction and make the prediction more accurate. IPVO outperforms PVO under all embedding payloads. The position of the pixels and the self-learning design are combined in the proposed generation of prediction error, greatly improving performance. On Lena, as the payload reaches 10,000 bits, the PSNR advantages of PairwiseKP-IPVO over IPVO and PVO-K are 1.33 dB and 1.19 dB, respectively. Obviously, our method achieves the better performance for most applications.

MIPVO [35] also generates multiple prediction errors at a time. However, each prediction of MIPVO is independent, while prediction of PairwiseKP-IPVO comes from consecutive or equal pixel values. In theory, the prediction errors produced by consecutive or equal pixels are more correlated, thereby building steeper PEH. The specific data confirm that the proposed scheme achieves 0.20 dB higher PSNR on average in Table 3. The proposed method has image distortion reduced on eight test images and the average PSNR gain of the proposed method is 0.48 dB at 20,000 bits embedding payload. It shows that the advantages of the proposed method are stable for most kinds of applications.

RSP-PVO [36] is a recently proposed method to construct an RSP model to map multiple prediction errors. Since the proposed method improves the prediction of continuous pixels, the PSNR advantage is more significant on smooth images. Likewise, the proposed method performs better on smooth images under all embedding payloads. In addition, embedding capacity is also extended due to the use of error pairs for 2D joint mapping on eight images. As for application, the proposed method has greater advantages in projects with more secret data.

In the GMP-PVO method [32], the generic multiple predictor framework based on content allows predicted pixels to be used as an aid to the PVO-based prediction. In terms of embedding capacity, the proposed method outperforms GMP-PVO on eight test images. Compared to GMP-PVO, the average PSNR values of the proposed method are 0.93 dB and 0.52 dB higher under 10,000 bits and 20,000 bits embedding payloads on eight test images. In Elaine, the PSNR of PairwiseKP-IPVO drops a lot when the actual embedding payload reaches 22,000 bits. At this time, its performance is similar to that of GMP-PVO, because as many pixels as possible are used without prioritizing smooth pixels.

In the EP-PEE method [39], the prediction errors are constructed by the redundancy among the largest three or the smallest three pixels. In order to make better use of the prediction errors, a content selection of expandable bin is proposed to improve the 2D PEH mapping. An adaptive pairwise PEE strategy is adopted in the PairwiseKP-IPVO, combining prediction errors [0,1] into prediction error pairs. On average, the PSNR of PairwiseKP-IPVO is 0.39 dB higher than that of EP-PEE in Table 3 and 0.33 dB in Table 4. In addition, the embedding capacity of PairwiseKP-IPVO is higher than that of [39] on smooth images of Airplane, Lena, Lake, and Barbara. It means that the fidelity advantage is over a wider range of payloads. On the image Lena and Peppers, PairwiseKP-IPVO obtains better PSNR than [39] when the payloads do not exceed 39,000 and 35,000 bits. In the proposed scheme, the prediction errors with low distortion are exhausted at high embedding payloads. Prediction errors in rough pixel blocks introduce more distortion. In this case, the 1D-based modification also introduces more distortion.

To verify the effectiveness of our method, extensive comparisons were conducted on Kodak test images under a 10,000-bit embedding payload, as detailed in

Table 5. Comparison of PSNR (dB) and MSE between the proposed PairwiseKP-IPVO (Proposed) method and the method of [24,25,26,39], for the embedding capacity of 10,000 bits in 24 other test images.

Image	PSNR (dB)					MSE
	Proposed	[26]	[24]	[25]	[39]	Proposed	[26]	[24]	[25]	[39]
kodim01	62.68	62.36	50.25	60.43	63.64	0.035	0.038	0.613	0.059	0.028
kodim02	62.81	62.78	52.31	61.06	64.23	0.034	0.034	0.382	0.051	0.025
kodim03	63.93	63.74	49.55	59.15	65.04	0.026	0.028	0.720	0.079	0.020
kodim04	62.76	62.75	52.74	61.59	63.79	0.034	0.034	0.346	0.045	0.027
kodim05	61.70	61.31	-	53.25	63.38	0.044	0.048	-	0.307	0.030
kodim06	64.95	64.47	-	-	66.70	0.021	0.023	-	-	0.014
kodim07	63.60	63.39	53.25	62.59	64.42	0.028	0.030	0.307	0.036	0.024
kodim08	60.03	56.79	-	-	57.04	0.065	0.136	-	-	0.128
kodim09	62.17	62.11	53.62	62.09	61.73	0.039	0.040	0.283	0.040	0.044
kodim10	62.29	61.44	48.58	57.17	60.98	0.038	0.047	0.901	0.125	0.052
kodim11	63.85	63.65	52.95	62.74	64.83	0.027	0.028	0.330	0.035	0.021
kodim12	63.36	63.43	-	54.81	64.45	0.030	0.030	-	0.215	0.023
kodim13	59.86	56.93	-	-	58.46	0.067	0.132	-	-	0.093
kodim14	61.27	61.23	48.72	57.57	62.83	0.048	0.049	0.874	0.114	0.034
kodim15	64.27	61.62	-	-	61.10	0.024	0.045	-	-	0.050
kodim16	63.68	63.56	52.39	61.67	64.81	0.028	0.029	0.375	0.044	0.021
kodim17	62.87	62.66	54.81	63.57	62.91	0.033	0.035	0.215	0.029	0.033
kodim18	60.09	59.42	46.54	54.39	59.36	0.064	0.074	1.443	0.236	0.075
kodim19	61.90	61.78	51.60	59.94	62.03	0.042	0.043	0.450	0.066	0.041
kodim20	62.31	53.76	-	-	52.26	0.038	0.274	-	-	0.387
kodim21	62.37	62.03	46.68	55.38	61.78	0.038	0.041	1.402	0.188	0.043
kodim22	61.78	61.54	-	53.55	61.43	0.043	0.046	-	0.288	0.047
kodim23	63.51	62.83	46.99	55.98	63.16	0.029	0.034	1.301	0.164	0.031
kodim24	63.10	58.67	-	-	56.93	0.032	0.088	-	-	0.132
Average	62.55	61.43	50.73	58.72	61.97	0.036	0.049	0.550	0.089	0.041

. The Mean Squared Error (MSE) was employed to quantify the average squared difference between the original and marked images, thereby providing a measure of the overall distortion introduced by the embedding process. Compared to PVO [24] and IPVO [25], our method showed significant improvements with average PSNR gains of 11.82 dB and 3.83 dB, respectively, corresponding to substantial MSE reductions from 0.550 to 0.036 and from 0.089 to 0.036. Against PVO-K [26], we achieved consistent advantages of 1.12 dB in PSNR with MSE improvement from 0.049 to 0.036. Our method also outperformed [39] with 0.58 dB higher PSNR and MSE reduction from 0.041 to 0.036. The MSE results further validate less distortion and better image quality preservation of our approach, with our PairwiseKP-IPVO method achieving the lowest average MSE of 0.036 among all compared methods. These dual-metric evaluations confirm that our proposed PairwiseKP-IPVO method delivers highly competitive performance with enhanced image quality in reversible data hiding applications.

4.4. t-Test

The paired t-test is suitable for comparing the performance differences between methods. We adopted paired t-tests to determine whether the proposed PairwiseKP-IPVO method significantly differs from the other four methods in terms of PSNR. For each test image and embedding payload, we paired the PSNR values of our method with each of the other methods to conduct the paired t-tests.

The t-statistics from our paired t-tests are shown in

Table 6. The t-statistics between the proposed PairwiseKP-IPVO method and methods PVO method [24], RSP-PVO method [36], EP-PEE method [39], and GMP-PVO method [32] in paired t-tests.

	[24]	[36]	[39]	[32]
Lena	28.72	8.01	10.84	39.37
Barbara	10.00	3.02	14.49	5.44
Boat	53.53	5.48	7.79	16.34
Elaine	15.59	3.35	5.83	6.74
Lake	46.43	7.42	10.32	18.53
Peppers	25.70	−6.25	1.32	9.52

. A positive t-value indicates that our PairwiseKP-IPVO method achieved higher PSNR values than the compared method. Based on the degrees of freedom in our experiments, t-values greater than 2.78 correspond to p < 0.01, indicating highly significant differences. The results show that our proposed method significantly outperforms the compared methods in most test cases. The only exceptions are with the RSP-PVO method on the Peppers image, where our method performed worse (negative t-value), and with the EP-PEE method on Peppers, where the difference was not statistically significant. Overall, the experimental data demonstrate that the proposed method achieves significant improvement in prediction accuracy by making full use of closely distributed pixels. The superiority is more significant on smoothed images and is prevalent for both digital images.

5. Conclusions

In this paper, we have presented a novel approach of reversible data hiding that leverages the redundancy between consecutive pixels to enhance the efficiency of prediction error generation. Our method introduces an adaptive strategy that dynamically selects multi-pass parameters based on the content of pixels, leading to more expandable prediction errors. This innovative approach departs from traditional fixed-parameter methods, allowing for better exploitation of the inherent structure and redundancy of image pixels. The result is an improved data embedding efficiency with reduced image distortion. Our experimental results demonstrate that the proposed scheme outperforms a series of state-of-the-art schemes with moderate capacity in terms of image fidelity. This confirms the efficacy of our method in achieving more efficient data embedding and image recovery. As the demand for secure and reliable data embedding solutions grows, our method could service as a promising solution for various applications where data security and image quality are paramount. The proposed method encounters limitations in computational complexity, primarily attributable to the adaptive parameter selection and multi-pass embedding strategy. Future endeavors will concentrate on developing lightweight iterations of the adaptive parameter selection mechanism. Additionally, we will persist in investigating the potential of high-dimensional histogram modification within the realm of RDH.