A Novel Reversible Data Hiding Scheme Based on Asymmetric Numeral Systems

Throughout this paper, the host sequence $X=\left(x_{1},x_{2},\cdots,x_{N}\right)$ is composed of i.i.d. samples drawn from the probability mass function (pmf) $P_{X}=\left\{P_{X}(s)|s\in\Sigma_{B}\right\}$, where the set $\Sigma_{B}=\left\{0,1,\cdots,B-1\right\}$ is an alphabet of size $B$. Let $P_{CX}=\left\{P_{CX}(s)=\sum_{i=0}^{s}P_{X}(i)|s\in\Sigma_{B}\right\}$ denote the cumulative pmf of $X$. It is noted that we have $P_{CX}(-1)=0$ and $P_{CX}(B-1)=1$. Let $f_{X}(s)$ denote the number of occurrences of signal $s$ in the sequence $X$, and $c_{X}(s)\triangleq\sum_{i=-1}^{s-1}f_{X}(i)$ is the cumulative frequency of signal $s$, where $f_{X}(-1)=0$ by default. The entropy function $H(X)$ of a discrete random variable $X$ is defined as $H(X)=-\sum_{i=0}^{B-1}P_{X}(i)\log_{2}{P_{X}(i)}$. The random variable $W$ represents a message uniformly distributed in $\Sigma_{M}=\left\{0,1,\cdots,M-1\right\}$. This paper considers embedding binary secret data into gray-scale signals, so we have $B=256$ and $M=2$.

Given a binary stream $S$, we construct a LIFO (Last In First Out) stack accordingly. The operation

pops $i$ bits $\left\{S_{i}\right\}_{i=0}^{i-1}$ from the bottommost position in the stack that contains $S$, and these $i$ bits form an integer $Q=S_{0}\times 2^{i-1}+\cdots+S_{i-2}\times 2+S_{i-1}$. The operation $Write_{S}(Q,i)$ pushes the least significant $i$ bits of an integer $Q$ to the stack.

In RDH, $L$ bits of message $W=\left(w_{1},w_{2},\cdots,w_{L}\right)$ are embedded by the encoder into the host sequence $X=\left(x_{1},x_{2},\cdots,x_{N}\right)$, by slightly modifying its elements to produce a stego sequence $Y=\left(y_{1},y_{2},\cdots,y_{N}\right)$, where $w_{i}\in\Sigma_{M}$ and $x_{j}\in\Sigma_{B}$, $y_{j}\in\Sigma_{Z}$ for $1\leq i\leq L$, $1\leq j\leq N$. This paper considers the convention that host signals and stego signals draw from the same set, i.e., $\Sigma_{Z}=\Sigma_{B}$. We denote the embedding capacity, also called embedding rate, by $R=L/N$. Schemes are usually constructed to minimize some distortion measure $D(s,y)$ between the host sequence $X$ and the stego sequence $Y$ for a given embedding rate $R$. The distortion metric $D(s,y)$ in this paper is defined as the square error distortion, i.e., $D(s,y)=(s-y)^{2}$. The decoder can losslessly reconstruct the message $W$ and the host sequence $X$ from the stego sequence $Y$. Figure 1 shows the general framework of RDH on a gray-scale image $X$, where $Y$ is the stego image with some distortion after embedding $W$.

Our recent ANS-Variant coding (see Section II-B2 for details) can provide the compression ratio close to that of arithmetic coding. More importantly, it does not suffer from the computing precision problem that may occur in arithmetic coding. As a result, we adopt ANS-Variant as the coding framework to embed a message $W$ into the host sequence $X=\left(x_{1},x_{2},\cdots,x_{N}\right)$. The framework consists of a combination of an encoder and a decoder of ANS-Variant coding, where the encoder is used to embed the data, and the decoder returns the corresponding stego signals. From the description of ANS-Variant coding, one can see that in order to encode a host signal $x_{i}$ into the current state $x$, the ANS-Variant encoder outputs an $n$-bit value $c_{x_{i}}+(x\bmod{f_{x_{i}}})$, and if the state $x$ is less than the lower bound of the selected fixed interval, the encoder reads bits to increase it. This observation can be exploited to realize data embedding, i.e., one can choose to read bits from the message $W$ to increase the value of the state if necessary. In addition, the BFI algorithm (described in Section II-B1) can estimate the optimal pmf of the stego sequence for a given host sequence. Therefore, we can apply the ANS-Variant decoder on the statistics of the stego sequence according to the $n$-bit value $c_{x_{i}}+(x\bmod{f_{x_{i}}})$ to decode the corresponding stego signal $y_{i}$. That is, the $n$-bit value $c_{x_{i}}+(x\bmod{f_{x_{i}}})$ produced by the ANS-Variant encoder serves as the temporary information to decode the corresponding stego signal $y_{i}$.

The details of the embedding procedure are as follows. First, we require knowledge of the host statistics $F_{X}=\left\{f_{X}(s)|s\in\Sigma_{B}\right\}$ and $C_{X}=\left\{c_{X}(s)|s\in\Sigma_{B}\right\}$, and get the cumulative pmf $P_{CX}=\left\{P_{CX}(x)|x\in\Sigma_{B}\right\}$, where $P_{CX}(x)=\sum_{i=0}^{x}P_{X}(i)$ and $P_{X}(i)=\frac{f_{X}(i)}{N}$. Then, we apply the BFI algorithm on the $P_{CX}$ to estimate the optimal cumulative pmf $P_{CY}$ of the stego sequence $Y$. Once the distribution $P_{CY}=\left\{P_{CY}(y)|y\in\Sigma_{B}\right\}$ is known, we can further get the stego statistics $F_{Y}=\left\{f_{Y}(y)|y\in\Sigma_{B}\right\}$ and $C_{Y}=\left\{c_{Y}(y)|y\in\Sigma_{B}\right\}$, where $c_{Y}(y)=P_{CY}(y)\times N$ and $f_{Y}(y)=c_{Y}(y+1)-c_{Y}(y)$. During the embedding, we process the host signals forward, i.e., from the first to the last signal. Given a start state $x$, for each host signal $x_{i}$, we first use an ANS-Variant encoder to encode it. Let $I:=\left[2^{T-v\times n},2^{T}\right)$ and $I_{x_{i}}:=\left[f_{X}(x_{i})\times 2^{T-v\times n},2^{T}\right)$. During the encoding, we have the following two cases. In the first, if the current state $x<f_{X}(x_{i})\times 2^{T-v\times n}$, we read $v\times n$ bits from the message $W$ to increase $x$. That is, if the decimal representation of the $v\times n$ bits read is $D$, we have $x\leftarrow(x\ll v\times n)+D$. After that, we maintain an $n$-bit temporal information $\eta_{X}=c_{X}(x_{i})+(x\bmod{f_{X}(x_{i})})$ and update the state $x$ by $x\leftarrow\mathcal{C}\left(x,x_{i}\right)=\lfloor x/f_{X}(x_{i})\rfloor$. It is noted that the $n$-bit temporary information produced by the host signal $x_{i}$ will be used in the following steps to decode the stego signal $y_{i}$ at the corresponding position. In the second, if the current state $x$ is within $I_{x_{i}}$, we maintain an $n$-bit temporal information $\eta_{X}=c_{X}(x_{i})+(x\bmod{f_{X}(x_{i})})$ and produce a new state $x$ by $x\leftarrow\mathcal{C}\left(x,x_{i}\right)$. Next, we utilize the temporary information $\eta_{X}$ maintained in the previous step and the stego statistics $F_{Y}=\left\{f_{Y}(y)|y\in\Sigma_{B}\right\}$ and $C_{Y}=\left\{c_{Y}(y)|y\in\Sigma_{B}\right\}$ to decode the corresponding stego signal $y_{i}$. Precisely, we transmit the temporary information $\eta_{X}$ to an ANS-Variant decoder to decode the stego signal $y_{i}$, i.e., $y_{i}\leftarrow\tilde{c}_{Y}(\eta_{X})$. Then, we update the state $x$ by $x\leftarrow f_{Y}(y_{i})\times x+r_{y_{i}}$, where $r_{y_{i}}=\eta_{X}-c_{Y}(y_{i})$. From the description of ANS-Variant decoding in Section II-B2, one can see that the updated state $x$ has the following two cases. In the first, If the value of state $x$ exceeds the upper bound $2^{T}$ chosen during the encoding, we write the least significant $v\times n$ bits of $x$ to the message $W$ to make $x$ smaller, i.e., $x\leftarrow x\gg v\times n$. In the second, if the updated state is exactly within the interval $I$, no further changes to the state are needed.

We repeat the above steps for the next host signal $x_{i+1}$ until there is no next host signal. As can be seen, an embedding cycle consists of a combination of an encoder and a decoder of ANS-Variant coding. Among them, the encoder aims to embed the data, and the decoder is designed to obtain the corresponding stego signal. Algorithm 2 presents the details. In Algorithm 2, Line $1$ aims to get the host statistics. Lines $2$–$3$ first apply the BFI algorithm to estimate the optimal cumulative pmf $P_{CY}$ and then get the stego statistics from the $P_{CY}$. In Line $4$, the start state $x$ is chosen to be small enough, e.g., set to $1$, in order to make the encoder read the bits from the message $W$ as early as possible, thus increasing the amount of embedded data. Lines $6$–$10$ perform the encoding of ANS-Variant coding. In particular, Lines $7$–$8$ read bits from the message $W$ to enlarge the state if the current state is too small. Lines $11$–$15$ perform the decoding of ANS-Variant coding to generate the corresponding stego signal $y_{i}$. Finally, we get a stego sequence $Y=\left(y_{1},y_{2},\cdots,y_{N}\right)$ and a final state $x$. The final state $x$ needs to be transmitted along with the stego sequence $Y$, so that the extractor knows what state to start with. In addition, the shortened length (in bits) of the message $W$ is the amount of data embedded.

In RDH techniques, data extraction includes extracting the host sequence and the embedded message from the received stego sequence. In the proposed static data embedding (see Section III-A for details), we first perform ANS-Variant encoding based on the known host statistics, and then perform ANS-Variant decoding under the premise that the optimal cumulative pmf of the stego sequence has been estimated. The data extraction is just the inverse of data embedding. Therefore, in order to achieve lossless data extraction, we first perform the ANS-Variant encoding under the premise that the optimal cumulative pmf of the stego sequence is known. Then, we perform the ANS-Variant decoding based on the host statistics to decode the host signal. It is worth noting that the data extraction requires transmitting the cumulative pmfs $P_{CX}$ and $P_{CY}$ as parameters. Another strategy is to only transmit $P_{CX}$ and the extractor generates the $P_{CY}$ with the BFI algorithm. The latter saves space but requires a computational cost to run the BFI algorithm. This section takes the latter as an example to introduce the process of static data extraction.

Given a stego sequence $Y=\left(y_{1},y_{2},\cdots,y_{N}\right)$, a start state $x$, and the cumulative pmf $P_{CX}=\left\{P_{CX}(x)|x\in\Sigma_{B}\right\}$, the details of data extraction are as follows. First, we apply the BFI algorithm on the host $P_{CX}$ to estimate the optimal cumulative pmf $P_{CY}=\left\{P_{CY}(y)|y\in\Sigma_{B}\right\}$. According to the $P_{CX}$ and $P_{CY}$, we calculate the stego statistics $F_{Y}=\left\{f_{Y}(y)|y\in\Sigma_{B}\right\}$ and $C_{Y}=\left\{c_{Y}(y)|y\in\Sigma_{B}\right\}$, and the host statistics $F_{X}=\left\{f_{X}(s)|s\in\Sigma_{B}\right\}$ and $C_{X}=\left\{c_{X}(s)|s\in\Sigma_{B}\right\}$, where $c_{Y}(y)=P_{CY}(y)\times N$, $f_{Y}(y)=c_{Y}(y+1)-c_{Y}(y)$, and $c_{X}(s)=P_{CX}(s)\times N$, $f_{X}(s)=c_{X}(s+1)-c_{X}(s)$. These statistics are used in the subsequent encoding and decoding of ANS-Variant coding. As the encoder and decoder of ANS-Variant coding run in reverse order, we process the stego signals backwards, i.e., from the last to the first signal. For each stego signal $y_{i}$, we first perform the ANS-Variant encoding based on the obtained stego statistics $F_{Y}$ and $C_{Y}$. Specifically, let $I:=\left[2^{T-v\times n},2^{T}\right)$ and $I_{y_{i}}:=\left[f_{Y}(y_{i})\times 2^{T-v\times n},2^{T}\right)$. In the encoding, we have the following two cases. In the first, if the current state $x<f_{Y}(y_{i})\times 2^{T-v\times n}$, we read $v\times n$ bits from the message $W$ to increase $x$. Then, we hold an $n$-bit temporal information $\eta_{Y}=c_{Y}(y_{i})+(x\bmod{f_{Y}(y_{i})})$ and update the state $x$ by $x\leftarrow\mathcal{C}\left(x,y_{i}\right)=\lfloor x/f_{Y}(y_{i})\rfloor$. In the second, if the current state $x$ is within $I_{y_{i}}$, we directly maintain an $n$-bit temporal information $\eta_{Y}=c_{Y}(y_{i})+(x\bmod{f_{Y}(y_{i})})$ and produce a new state $x$ by $x\leftarrow\mathcal{C}\left(x,y_{i}\right)$. The above $n$-bit temporary information $\eta_{Y}$ produced by encoding the stego signal $y_{i}$ is used in the following steps to decode the corresponding host signal $x_{i}$.

Next, we utilize the temporary information $\eta_{Y}$ maintained in the previous step and the host statistics $F_{X}=\left\{f_{X}(s)|s\in\Sigma_{B}\right\}$ and $C_{X}=\left\{c_{X}(s)|s\in\Sigma_{B}\right\}$ to decode the host signal $x_{i}$. Precisely, we transmit the temporary information $\eta_{Y}$ to the ANS-Variant decoder to decode the host signal $x_{i}$, i.e., $x_{i}\leftarrow\tilde{c}_{X}(\eta_{Y})$. Then, we update the state $x$ by $x\leftarrow f_{X}(x_{i})\times x+r_{x_{i}}$, where $r_{x_{i}}=\eta_{Y}-c_{X}(x_{i})$. From the description of ANS-Variant decoding in Section II-B2, one can see that the updated state $x$ has the following two cases. In the first, if the value of state $x$ exceeds the upper bound $2^{T}$ chosen during the data embedding, we write the least significant $v\times n$ bits of $x$ to the message $W$ to make $x$ smaller, i.e., $x\leftarrow x\gg v\times n$. In the second, if the updated state is exactly within the interval $I$, no further changes to the state are needed. We repeat the above steps for the next stego signal $y_{i-1}$ until there is no next stego signal. Algorithm 3 gives the details. In Algorithm 3, Line $1$ uses the BFI algorithm to estimate the optimal cumulative pmf $P_{CY}$. Lines $2$–$3$ derive the host and stego statistics from the known $P_{CX}$ and $P_{CY}$, respectively. Lines $5$–$9$ perform the encoding of ANS-Variant coding on the stego signal and maintain a temporary information $\eta_{Y}$. Lines $10$–$14$ utilize the temporary $\eta_{Y}$ produced in the previous step to decode the host signal $x_{i}$. In particular, Lines $12$–$14$ write $v\times n$ bits to the message $W$ when the state exceeds a given upper bound $2^{T}$. Finally, we can reconstruct the host sequence $X=\left(x_{1},x_{2},\cdots,x_{N}\right)$ and the embedded message $W$ without loss.