Progressive Data Hiding in Integer Wavelet Transform of Electrocardiogram by Using Simple Decision Rule and Coefficient Calibration

To obtain a high hiding capacity with low computation time, and favorable perceived quality with robust performance, we embedded data bits in the IWT domain [17]. First, an input ECG host was decomposed into low subband coefficients (I_L) and high subband coefficients (I_H) by using level 1 (L1) one dimensional (1D) IWT. Then, predetermined criteria for bit embedding and bit extraction were employed to conceal secret information in the I_L and I_H subbands, respectively. However, to obtain high perceived quality, data bits can be embedded only in the I_H subband of IWT coefficients with host bundle sizes of <4. Although a multilevel IWT decomposition can be employed in our method to further promote robustness with approximate payload capability and similar perceptual quality, the cost of computation time might be an issue for real-time applications. As stated earlier, we focus on hiding secret data in L1 IWT domain. Without loss of generality, let $H_{j}=\left\{s_{j i}\right\}_{i=0}^{\mathrm{n}-1}$ be the jth host bundle of size n, as illustrated in Figure 1. The decision criterion Θ is defined as follows:

$\Theta=\left|\frac{\sum_{i=0}^{n-2} s_{j i}}{n-1}-s_{j n-1}\right| \leq \tau$     (1)

where, t is the control integer. For example, a data bit 1 or 0 can be virtually hidden in a host bundle of size 2 if $-\tau \leq\left|s_{j 0}-s_{j 1}\right|<0$ or $0 \leq\left|s_{j 0}-s_{j 1}\right| \leq \tau$ is satisfied. Then, the subsequent input data bit 1 or 0 can be embedded in the (enlarged) bundle of size 3 along with the preceding sub bundle, if either $-\tau \leq\left|\frac{s_{j 0}+s_{j 1}}{2}-s_{j 2}\right|<0 \quad$ or $\quad 0 \leq\left|\frac{s_{j 0}+s_{j 1}}{2}-s_{j 2}\right| \leq \tau$ is satisfied, and so on. Theoretically, if no violation occurs, then data bits (of length n−1) can be completely concealed in a host bundle of size n. Notably, a large value of n, large payload size, and low SNR were obtained using the proposed method.

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Figure 1. jth host bundle of size n

In our method, it is crucial to determine the values of n and $\tau$. Coefficient adjustment must be conducted if violation occurs during bit embedding. Furthermore, if the bundle (or subbundle) carries no data bit, it is considered a skipped bundle. Because the skipped bundles (or subbundles) can be easily detected at the receiver site according to the decision criterion $\Theta$ extra information is not required to recode the position of skipped bundles. The major steps of bit embedding and bit extraction for our proposed method are summarized in the following sections.

2.1 Bit embedding

The main procedure of bit embedding is described in the following algorithm.

Algorithm 1. Hiding secret bit in the ECG host.

Input: Host ECG data $\Psi=\left\{s_{k}\right\}_{k=0}^{|\Psi-1|},$ the desired size of a host bundle $n,$ a control integer $\tau,$ and a secret message $W$.

Output: Marked ECG data $\widehat{\Psi}$.

Method:

Step 0. Perform 1 D forward IWT from $\Psi$ to obtain two sets of host bundles $I_{L}=\left\{H_{j}|j=1,2, \ldots,| I_{L} |\right\}$ and $I_{H}=\left\{H_{j} | j=1,2, \ldots,\left|I_{H}\right|\right\}$.

Step 1. Input a bundle, for example $H_{j}=\left\{s_{j i}\right\}_{i=0}^{n-1}$ from $I_{L}$ (and $I_{H}$). If the end of input is encountered, then go to Step 12.

Step 2. Compute the offset $\emptyset=s_{j 0}-s_{j 1},$ if the condition $|\emptyset|>\tau$ is satisfied, then go to Step $5,$ otherwise, input a data bit $b_{p}$ from $W$.

Step 3. If both conditions of $b_{p}=1$ and $-\tau \leq \emptyset<0$ are satisfied, the sub-bundle carries data bit "1," then go to Step 5. Otherwise, if the condition $b_{p}=1$ is satisfied, then repeatedly adjust the value of $\phi$ by increasing $s_{j 1}$ by 1 and decreasing $s_{j 0}$ from 1 simultaneously until $-\tau \leq \emptyset<0$ is achieved, go to Step 5。

Step 4. If both conditions of $b_{p}=0$ and $0 \leq \varnothing \leq \tau$ are satisfied, the sub-bundle carries data bit "0," then go to Step 5. If the condition $b_{p}=0$ is satisfied, then repeatedly adjust the value of $\phi$ by increasing $s_{j 0}$ by 1 and decreasing $s_{j 1}$ from 1 simultaneously until the condition $0 \leq \varnothing \leq \tau$ is satisfied.

Step 5. Compute the offset $\gamma=\frac{s_{j 0}+s_{j 1}}{2}-s_{j 2},$ if the condition $|\gamma|>\tau$ is satisfied, then go to Step $8,$ otherwise, input next input bit $b_{q}$ from $W$。

Step 6. If both conditions of $b_{q}=1$ and $-\tau \leq \gamma<0$ are satisfied, indicating the subbundle carries data bit " 1 ," then go to Step $8 .$ Otherwise, if the condition $b_{q}=1$ is satisfied, then repeatedly adjust the value of $\gamma$ by increasing $s_{j 2}$ by 1 and decreasing both $s_{j 0}$ and $s_{j 1}$ from 1 simultaneously until the condition $-\tau \leq \gamma<0$ is satisfied, go to Step 8。

Step 7. If both conditions of $b_{q}=0$ and $0 \leq \gamma \leq \tau$ are satisfied, indicating the subbundle carries data bit "0," then go to next step. Otherwise, if the condition $b_{q}=0$ is satisfied, then repeatedly adjust the value of $\gamma$ by increasing both $s_{j 0}$ and $s_{j 1}$ by 1 and decreasing $s_{j 2}$ from 1 simultaneously, until the condition $0 \leq \gamma \leq \tau$ is met.

Step 8. If $n>3,$ then set parameter $m=4$, otherwise go to Step 1.

Step 9. Compute the offset $\beta=\frac{\sum_{i=0}^{m-2} s_{j i}}{m-1}-s_{j m-1}$ if both conditions of $|\beta|>\tau$ and $m \leq n$ are satisfied, then set $m=m+1$ and repeat this step. Otherwise, if the condition $m>n$ is satisfied, then go to Step 1.

Step 10. Input next bit $b_{s}$ from $W .$ If both conditions of $b_{s}=$ 1 and $-\tau \leq \beta<0$ are satisfied, then set $m=m+1$ and go to Step $9 .$ Otherwise, if the condition $b_{q}=1$ is satisfied, then change the value of $s_{j m-1}$ to $s_{j m-1}=s_{j m-1}+|\beta|+1,$ set $m=m+1$ and go to Step 9.

Step 11. If both conditions of $b_{s}=0$ and $0 \leq \beta \leq \tau$ are satisfied, then set $m=m+1$ and go to Step $9 .$ Otherwise, if the condition $b_{s}=0$ is satisfied, then change the value of $s_{j m-1}$ to $s_{j m-1}=s_{j m-1}-|\beta|-1,$ set $m=m+1,$ and go to Step 9.

Step 12. Perform $1 \mathrm{D}$ inverse IWT for $I_{L}\left(\text { and } I_{H}\right)$ to obtain mark ECG data $\widehat{\Psi}$.

Step 13. Stop.

2.2 Bit extraction

In the proposed method, the bit extraction is considerably simpler than bit embedding. The major steps of bit extraction are listed in the following algorithm.

Algorithm 2. Extracting hidden message from marked ECG.

Input: Marked ECG data $\widehat{\Psi}$, the desired size of a host bundle $n,$ and an integer $\tau$.

Output: Secret message W.

Method:

Step 0. Perform $1 \mathrm{D}$ forward IWT by using $\hat{\Psi}$ to obtain two sets of marked bundles $\hat{I}_{L}=\left\{\widehat{H}_{j}|j=1,2, \ldots,| \hat{I}_{L} |\right\}$ and $\hat{I}_{H}=\left\{\hat{H}_{j}|j=1,2, \ldots,| I_{H} |\right\}$, respectively.

Step 1. Input a bundle $\widehat{H}_{j}=\left\{\hat{s}_{j i}\right\}_{i=0}^{n-1}$ from $\hat{l}_{L}\left(\text { and } \hat{I}_{H}\right) .$ If the end of input is encountered, then go to Step 9.

Step 2. Compute the offset $\emptyset=\hat{s}_{j 0}-\hat{s}_{j 1},$ if the condition $|\emptyset|>\tau$ is satisfied, then go to Step 4.

Step 3. If the condition $-\tau \leq \emptyset<0$ is satisfied, then data bit " 1 " is recognized, otherwise, data bit "0" is identified.

Step 4. Compute the offset $\gamma=\frac{\hat{s}_{j 0}+\hat{s}_{j 1}}{2}-\hat{s}_{j 2},$ if the condition $|\gamma|>\tau$ is satisfied, then go to Step 6.

Step 5. If the condition $-\tau \leq \gamma<0$ is satisfied, then data bit " 1 " is recognized, otherwise, data bit "0" is identified.

Step 6  If $n>3,$ then set parameter $m=4,$ otherwise, go to Step 1.

Step 7. Compute the offset $\beta=\frac{\sum_{i=0}^{m-1} \hat{s}_{j i}}{m-1}-\hat{s}_{j m-1},$ if both conditions of $|\beta|>\tau$ and $m \leq n$ are satisfied, then set $m=m+1$ and repeat this step. Otherwise, if the condition $m>$ $n$ is satisfied, then go to Step 1.

Step 8. If the condition $-\tau \leq \beta<0$ is satisfied, then data bit "1" is recognized, otherwise, data bit "0" is identified. Set $m=m+1$ and go to Step 7.

Step 9. Assemble all extracted bits and rebuild the secret message.

Step 10. Stop.

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Figure 2. Scenarios of embedding two data bits 11 in the host bundle of size 3. (a) Initial (host) bundle, (b) the transit bundle, which carried bit 1 (without adjustment), and (c) the final form of the bundle (marked bundle), which carried the second bit 1 (after adjustment)

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Figure 3. Scenarios of embedding three data bits 111 in a host bundle of size 4

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Figure 4. Scenarios of embedding four data bits 1111 in the host bundle of size 5

Figures 2-4 present the examples of bit embedding in the host bundles with the size of 3-5, respectively, that was performed using our proposed method with t = 9. Figure 2(a) illustrates the initial (host) bundle, Figure 2(b) presents the transit bundle, which carried data bit “1” without adjustment (according to Steps 2 and 3 of Algorithm 1), and Figure 2(c) presents the final form of the bundle (marked bundle), which carried another data bit “1” after adjustment. The figure displays the aligned values as three numbers surrounded by squares. Similarly, according to Steps 8–10 of Algorithm 1, Figures 3 and 4 illustrate the scenarios of embedding data bits in a host bundle of size 4 and 5. Furthermore, the mean absolute error (MAE) computed from Figures 2(a) and 2(c), Figures 3(a) and 3(d), and Figures 4(a) and 4(e) were 1, 1, and 1.8, respectively. We assumed that no error (or data corruption) occurred during transmission at the receiver site. According to Steps 2–5 of Algorithm 2, data bits “11” could be extracted using the marked bundle illustrated in Figure 2(c). Similarly, both the hidden bits “111” and “1111” could be extracted using the marked bundles illustrated in Figures 3(d) and 4(e), respectively, which is in accordance with Steps 6–8 of Algorithm 2.

From the above two algorithms we can see that the computation complexity of the proposed decision rule and coefficient adjustment is quite simple. For example, a host bundle whose size is larger than 2, it requires the computation of (n-2) ´ (n-1) ´ addition, (n-2) ´ division, subtraction, and absolute value; if the size of a host bundle is 2, it only needs the computation of subtraction and absolute value. Furthermore, the process of coefficient calibration just uses increment and decrement to achieve the goal within a finite number of times. That is, the adjustment values would be limited between -t and t. It may take a little computation time at the Steps 3-4 and Steps 7-8 of Algorithm 1 during coefficient adjustment procedure. A possible way of reducing computation time is not use iterative adjustment, that is, we could directly force and chane the values of the target coefficients to be satisfied with decision criteria. It would result in a lower SNR value obtained by our method. That is the reason why we employ simple decision and (iterative) coefficient calibration in our proposed method. To obtain extreme hiding storage (and if one neglects perceived quality), the optimal payload of the proposed method with host bundleof size $n$ and without violation was $\frac{n-1}{n} \times|\Psi| .$ By contrast, to obtain high perceived quality, data bits can only be embedded in the $I_{H}$ sub-band of IWT coefficients. Because the distortion caused by the proposed method by embedding secret bits in $I_{H}$ was considerably less than that caused by the proposed method by embedding secret bits in $I_{L} .$ However, in this case, the payload without violation would be at most $\frac{2}{3} \times\left|I_{L}\right|$ when the size of a host bundle was $3 .$ In addition, the value of $\tau$ was not necessarily constant. When $\tau$ value was small, SNR increased and vice versa. Moreover, when the size of host bundles was large, a large payload was obtained using the proposed method. Furthermore, for a given constant value of $\tau$, the maximum payload of the proposed method with a certain size of host bundle can be observed in the ECG host. The proposed method with a large $\tau$ provides more robust performance than it provides with any smaller $\tau$. Figure 5 presents the block diagram of the proposed method.

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Figure 5. Block diagram of the proposed method

In this study, we established a cumulative data hiding approach in ECG signals based on a 1D IWT domain. According to the predetermined criterion with host bundles of various sizes, sensitive data can be successfully embedded in the IWT coefficients of ECG hosts. Experimental results confirmed that the average SNR and payload of our proposed method are superior to those of the conventional ECG steganography. Moreover, the proposed method exhibited robust performance, which has rarely been observed in existing ECG steganography techniques. The proposed method exhibited the merits of high hiding capacity and superior resultant perceived quality. Moreover, our method was resistant toward attacks such as cropping, inversion, scaling, translation, truncation, and Gaussian noise addition. Because the procedures of bit embedding and extraction are simple and the computation time is short, our method can be applied in mobile biometric devices.