可逆数据隐藏：通用扩展移位模型研究

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"这篇研究论文探讨了一种用于可逆数据隐藏的通用扩展移位模型，由北京大学计算机科学技术研究所的李小龙和郭宗明提出。该模型引入了可逆嵌入函数（Reversible Embedding Function, REF），将每个Zn域中的点映射到非空的Zn子集，并确保不同点的映射子集互不重叠，以保证数据隐藏的可逆性。通过REF，可以设计RDH方法并建立相应的率失真公式，从而有可能优化可逆嵌入性能。论文还给出了REF的一些实例，并进行了初步的理论研究。" 在可逆数据隐藏（Reversible Data Hiding, RDH）领域，技术的目标是在隐藏信息的同时，确保原始数据和嵌入的数据都能从标记后的数据中精确地提取出来。传统的数据隐藏方法可能会影响原始数据的质量，而RDH则解决了这一问题，使得隐藏过程对数据的改变是可恢复的。本文提出的通用扩展移位模型通过定义可逆嵌入函数（REF）来实现这一目标。REF是一个映射函数，它将Zn域中的每个元素映射到非空的Zn子集，且不同的输入点映射到的子集相互独立。这种特性确保了在解码时能够准确地还原原始数据和隐藏的信息，因为每个点的映射是唯一的，不会导致信息混淆或丢失。利用REF，研究者可以构建RDH算法，并建立相应的率失真理论框架。率失真理论是信息论的一个分支，关注在给定的失真水平下，如何最大化信息传输的速率。在RDH中，这意味着在保持可逆性的前提下，寻找最佳平衡点，既尽可能多地隐藏数据，又尽可能减少对原始数据的影响。论文中，作者提供了REF的若干示例，这些示例可能包括不同的映射策略和结构，以展示如何实际应用这个模型。同时，还进行了一项初步的理论分析，这可能涉及到REF的性质、效率以及可能存在的限制等方面的研究。这项工作为RDH提供了新的理论工具和方法，对于改进数据隐藏的效率和可逆性具有重要的意义，特别是在保护数字内容的版权、数据安全和隐私等领域有着广阔的应用前景。

General expansion-shifting model for reversible data

hiding

Xiaolong Li and Zongming Guo

Institute of Computer Science and Technology, Peking University, Beijing 100871, China

lixiaolong@pku.edu.cn, guozongming@pku.edu.cn

Abstract—Reversible data hiding (RDH) is a speciﬁc informa-

tion hiding technique in which both the embedded data and

the original cover medium can be exactly extracted from the

marked data. In this paper, we present a general expansion-

shifting model for RDH by introducing the so-called reversible

embedding function (REF) which maps each point of Z

to a non-

empty subset of Z

. Moreover, to guarantee the reversibility, the

mapped subsets of distinct points should be disjoint from each

other. With REF, RDH can be designed and the corresponding

rate-distortion formulations can be established, providing a

possibility to optimize the reversible embedding performance.

Some examples of REF are given, and a preliminary theoretical

investigation on REF is conducted as well.

I. INTRODUCTION

Reversible data hiding (RDH) is a special type of informa-

tion hiding. Being reversible, the original digital content can

be completely restored after data extraction [1]. Early RDH

methods are mainly based on lossless compression [2]–[4],

in which certain features of the cover image are losslessly

compressed to save space for embedding the payload. These

methods usually provide a low capacity and may lead to

severe degradation in image quality. Later on, more efﬁcient

algorithms have been devised which emphasize increasing

the embedding capacity and keeping the embedding distor-

tion low. Meanwhile, several valuable reversible embedding

techniques are proposed, e.g., histogram shifting (HS) [5],

[6], difference expansion (DE) [7]–[9], prediction-error ex-

pansion (PEE) [10]–[18], and integer transform [19]–[22],

etc. Among them, the expansion-shifting based ones such as

HS and PEE have attracted considerable attention due to the

efﬁciency in reversible data embedding. A common feature

of expansion-shifting based RDH is that, by generating ﬁrst a

histogram (e.g., the pixel-intensity histogram, the prediction-

error histogram, etc), some histogram bins are shifted to create

vacancies, while some other bins are expanded to embed data.

In this paper, we present a general expansion-shifting model

for RDH by introducing the so-called reversible embedding

function (REF) which maps each point of Z

to a non-empty

subset of Z

. In other word, REF is a function deﬁned as

f : Z

7→ P(Z

) − {∅} where P(Z

) is the power set

of Z

. Moreover, to guarantee the reversibility, the mapped

subsets of distinct points of Z

should be disjoint from each

other. With the proposed REF, RDH can be designed and the

corresponding rate-distortion formulations can be established,

This work is supported by National Natural Science Foundation of China

under contract No. 61572052.

providing a possibility to optimize the reversible embedding

performance. This paper is an extension of our previous work

[23].

In the rest of this paper, the deﬁnition of REF is ﬁrst

introduced in Section II. Some examples of REF are given in

this section as well. Then, some theoretical results of REF and

discussions are presented in Section III. Finally, we conclude

our work in the last section.

II. REF: DEFINITION AND EXAMPLES

REF is deﬁned as a function

f : Z

7→ P(Z

) − {∅} (1)

where P(Z

) is the power set of Z

, i.e., P(Z

) is the set

consisting of all subsets of Z

. By this deﬁnition, for each

x ∈ Z

, f (x) is a non-empty subset of Z

. Moreover, to

guarantee the reversibility of the corresponding RDH, the REF

f should satisfy the following property

x = y =⇒ f(x) ∩ f(y) = ∅. (2)

For x ∈ Z

satisfying |f (x)| = 1, it will be shifted in data

embedding, where |A| means the number of elements of a

set A. On the other hand, if |f(x)| > 1, it will be expanded

to embed data. More speciﬁcally, take f(x) = {y

, ..., y

}

where N = |f(x)|, in the data embedding process,

• if N = 1, x will be directly shifted as y

;

• if N > 1, x will be randomly expanded as one of y

Notice that, with the condition (2), the original cover recovery

is obvious: for a given marked data y, it can be recovered as

−1

(y). Then, we see that a RDH scheme can be derived

based on a given REF.

With REF, the rate-distortion formulations can be estab-

lished for the corresponding RDH scheme. Consider here the

n-dimensional histogram of the cover data which is a function

h : Z

7→ Z

(3)

Notice that, the histogram maybe the pixel-intensity histogram,

the prediction-error histogram, etc. Then, based on the afore-

mentioned data embedding process, we see that the embedding

capacity and the embedding distortion denoted respectively as

EC(f) and ED(f) can be formulated as

EC(f) =



x∈Z

h(x) log

|f(x)| (4)

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