# Compressive Sampling with Multiple Bit Spread Spectrum-Based Data Hiding
## 3 Data Hiding Model
### An Overcomplete Dictionary with SS-Based Content
For example, if we have three bits watermarked in a Hadamard sequence, or \(N_{s}=3\) bits, then \(N_{p}=2^{N_{s}}=8\) bits; thus, all watermark possibilities and their mapping to Hadamard sequences are displayed in Table 2. If we have two segments or two vectors of watermark \(\mathbf{w}_{t_{1}}=[-1,+1,-1]\) and \(\mathbf{w}_{t_{2}}=[+1,-1,-1]\), then using Table 2, we get \(t_{1}=3\) and \(t_{2}=5\); thus, \(\mathbf{p}_{t_{1}}=\mathbf{p}_{3}=\{+1,+1,-1,-1,+1,+1,-1,-1\}\), and \(\mathbf{p}_{t_{2}}=\mathbf{p}_{5}=\{+1,+1,+1,+1,-1,-1,-1,-1\}\). The over-complete matrix \(\mathbf{A}_{0}\) contains \(pN_{s}\) bits of watermark for the host with length \(M^{2}\); thus, we can compute watermark payload \(C\) in bps as:  
\[C=\frac{pN_{s}F_{s}}{M^{2}}, \tag{16}\]  
where \(F_{s}\) is the host signal sampling rate in samples/s. Due to \(N_{s}=\lceil\log_{2}N_{p}\rceil=\lceil\log_{2}r\rceil\), thus (16) will be as:  
\[C=\frac{p\lceil\log_{2}r\rceil F_{s}}{M^{2}}. \tag{17}\]  
Once \(\mathbf{A}\) is generated from the associated watermark bits, it is embedded into \(\mathbf{S_{r}}\) using matrix multiplication in (4). The result \(\mathbf{Y}\) is not only a matrix with a smaller size than \(\mathbf{S_{r}}\), but also, it is embedded by the watermark bits. The matrix \(\mathbf{S_{r}}\) is a diagonal matrix whose size is reduced from the original one \(\mathbf{S}\). From (4), (5), and (13), the equation \(\mathbf{Y}=\mathbf{A}\mathbf{S_{r}}\) can be exploited as:  
\[ \begin{bmatrix}\mathbf{y}_{t_{1}}\\ \mathbf{y}_{t_{2}}\\ \vdots\\ \mathbf{y}_{t_{p}}\end{bmatrix} = \begin{bmatrix}\mathbf{p}_{t_{1}}\\ \mathbf{p}_{t_{2}}\\ \vdots\\ \mathbf{p}_{t_{p}}\end{bmatrix}\begin{bmatrix}\sigma_{1}&0&\cdots&0\\ 0&\sigma_{2}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&\sigma_{r}\end{bmatrix} , \tag{18}\] \begin{table}
\begin{tabular}{c l} \hline \hline Step 1: & Read a host signal \(x(n)\), and transform it into the frequency domain by DCT \(L\)-point obtaining \(X(k)\) \\ Step 2: & Reshape \(X(k)\) in \(L\) and sample it to a 2D square matrix producing \(\mathbf{X}\) with size \(M\times M\) \\ Step 3: & Decompose \(\mathbf{X}\) to \(\mathbf{U}\), \(\mathbf{S}\), and \(\mathbf{V}\) using SVD \\ Step 4: & Reduce the matrix size of \(\mathbf{U}\), \(\mathbf{S}\), and \(\mathbf{V}\) with rank \(r\) to \(\mathbf{U_{r}}\), \(\mathbf{S_{r}}\), and \(\mathbf{V_{r}}\) \\ Step 5: & Generate the \(\mathbf{A}\) matrix containing \(p\) Hadamard sequences by mapping each multi-watermark bit to an associated random Hadamard sequence using (13) \\ Step 6: & Apply CS acquisition to \(\mathbf{A}\) and \(\mathbf{S_{r}}\) by (4), producing \(\mathbf{Y}\) \\ Step 7: & Transmit the compressed signal with hidden data represented using \(\mathbf{U_{r}}\), \(\mathbf{Y}\), and \(\mathbf{V_{r}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Embedding process. Figure 1: Watermark embedding and audio encoding. where \(\mathbf{y}_{t_{i}}\in\mathbb{R}^{1\times r}\) is a vector of matrix \(\mathbf{Y}\) at row \(i\), which also corresponds to \(\mathbf{p}_{t_{i}}\), and \(\sigma_{1},\sigma_{2},...,\sigma_{r}\) are singular value elements of \(\mathbf{S}_{\mathbf{r}}\). Each row of \(\mathbf{A}\) or \(\mathbf{p}_{t_{i}}\) is a vector with size \(1\times r\). \(\mathbf{S}_{\mathbf{r}}\) is a diagonal matrix with size \(r\times r\). Thus, we can simplify (18) to the following equation:  
\[\left[ \begin{array}{c}\mathbf{y}_{t_{1}}\\ \mathbf{y}_{t_{2}}\\ \vdots\\ \mathbf{y}_{t_{p}}\end{array} \right]=\left[ \begin{array}{c}\mathbf{p}_{t_{1 }}\mathbf{S}_{\mathbf{r}}\\ \mathbf{p}_{t_{2}}\mathbf{S}_{\mathbf{r}}\\ \vdots\\ \mathbf{p}_{t_{p}}\mathbf{S}_{\mathbf{r}}\end{array} \right] \tag{19}\]  
Then, we can have the following simple vector expression:  
\[\mathbf{y}_{t_{i}}=\mathbf{p}_{t_{i}}\mathbf{S}_{\mathbf{r}}. \tag{20}\]