## 2 System Description
### Data Embedding  
The DFC has a data-embedding approach that operates in the frequency domain. To begin the process, the image frames are first converted into the spectral domain using 2D-DFT, as shown in Figure 1. The DFC scheme exploits the fact that the information content of an image is concentrated in specific regions of its frequency domain representation. Specifically, the amplitudes of the low-frequency components are located at the corners of the 2D spectrum, whereas those of high-frequency components are situated at the center [18]. In other words, the low-frequency components containing important information about the image are concentrated at the four corners of the spectral-domain image. This characteristic allows the DFC scheme to use the areas surrounding the corners for data and pilot embedding. These areas were chosen because they offer a means of minimizing the perceptual image distortion that may occur owing to data embedding. Mathematically, the 2D-DFT of a spatial-domain image, \(\mathbf{I}_{t}\), of size \(P\times Q\) can be taken as  
\[\mathbf{I}_{F}=\mathbf{F}_{P}\mathbf{I}_{t}\mathbf{F}_{Q}, \tag{1}\]where \(\mathbf{I}_{F}\) represents the spectral domain image, and \(\mathbf{F}_{P}\) and \(\mathbf{F}_{Q}\) represent \(P\times P\) and \(Q\times Q\) DFT matrices, respectively. Simultaneously, the binary information bits are modulated using a modulation scheme to embed them into a frequency-domain image. To ensure the invisibility of the embedded data and maintain the real and positive values of the gray-scale intensity components of the data-embedded image, the data components must exhibit conjugate symmetric properties. Thus, the data matrix at the \(i\)th time index \(\mathbf{X}[i]\) is defined as  
\[\mathbf{X}[i]= \begin{bmatrix}x_{\{1,1\}}&x_{\{1,2\}}&\cdots&x_{\{1,s_{q}+L_{q} \}}\\ \vdots&\ddots&\vdots\\ x_{\{s_{p}+L_{p},1\}}&\cdots&x_{\{s_{p}+L_{p},s_{q}+L_{q}\}}\end{bmatrix} , \tag{2}\]  
where \(x_{\{p,q\}}\) denotes the data element at the \((p,q)\)-pixel position. The starting pixel values at which the data are embedded in a row and column are denoted by \(s_{p}\) and \(s_{q}\), respectively. The pixel widths at which the data and pilots are loaded in the row and column directions are denoted as \(L_{p}\) and \(L_{q}\), respectively [19]. In the 2D-DFC scheme, the data embedding is performed by multiplying the data coefficient with the frequency component of the image. Therefore, a frequency-domain image loaded with data at the \(i\)th time index, denoted by \(\mathbf{D}_{F}[i]\), can be expressed as  
\[\mathbf{D}_{F}[i]= \begin{bmatrix}\mathbf{I}_{F1}[i]\circ\mathbf{X}[i]&\mathbf{ I}_{F2}[i]&\mathbf{I}_{F3}[i]\\ \mathbf{I}_{F4}[i]&\mathbf{I}_{F5}[i]&\mathbf{I}_{F6}[i]\\ \mathbf{I}_{F7}[i]&\mathbf{I}_{F8}[i]&\mathbf{I}_{F9}[i]\circ\text{flip}( \mathbf{\bar{X}}[i])\end{bmatrix} , \tag{3}\]  
where \(\circ\) represents the Hadamard product operator, \(\mathbf{I}_{F1-F9}\) represents the nine subimages made out of the original image \(\mathbf{I}_{F}\), and \(\text{flip}(\cdot)\) is the flip operation. The operation \(\text{flip}(\mathbf{\bar{X}}[i])\) is expressed as follows:  
Figure 1: Block diagram of reference image reconstruction scheme for 2D display field communications. \[\text{flip}(\mathbf{\tilde{X}}[i])= \begin{bmatrix}\mathbf{\hat{x}}_{\{s_{p}+L_{p}s_{ q}+L_{q}\}}&\cdots&\mathbf{\hat{x}}_{\{s_{p}+L_{p};1\}}\\ \vdots&\ddots&\vdots\\ \mathbf{\hat{x}}_{\{1,s_{q}+L_{q}\}}&\cdots&\mathbf{\hat{x}}_{\{1,1\}}\\ \end{bmatrix} , \tag{4}\]  
where \(\mathbf{\hat{x}}\) denotes the conjugate of \(x\). The above equation indicates that only the top left and bottom right corners of the frequency-domain image are utilized for embedding the data. Furthermore, the elements of the data matrix, denoted by \(x_{\{p,q\}}\), can be expressed as  
\[x_{\{p,q\}}= \begin{cases}1,&p<s_{p},\;q<s_{q}\\ x_{\{p,q\}},&\text{otherwise}\\ \end{cases} .