## 3 Iterative Image Estimation
### Image Re-Estimation Using Virtual Pilots  
In this section, we discuss an image estimation method that exploits virtual pilot pixels and pilot pixels in the re-estimation of the image. As illustrated in Figure 3, the virtual pilot pixels were chosen from among the available data pixels obtained after the initial demodulation. The selection of virtual pilot pixels was based on two conditions. First, the magnitude of the selected data pixels should be sufficiently large to ensure their suitability for image estimation. Second, the channels for the virtual pilot pixels must be highly correlated with those for the pilot pixels; otherwise, they would not contribute to improving the quality of the data pixel estimates. By incorporating the selected virtual pilot pixels with the original pilot signals, the image is re-estimated, and the newly generated reference image estimate is used for symbol detection in the subsequent iteration. This process was repeated until suitable termination conditions were achieved. Figure 3 shows the locations of the pilot and virtual pilot pixels in the upper sub-band of the frequency-domain image, which is represented in Figure 2a. Here, \(d_{1}\) represents the first column used to embed the data, with \(s_{p}\) and \(s_{q}\) denoting the starting pixel locations for the row and column, respectively. A total of \(L_{p}\) and \(L_{q}\) data bits were embedded in the upper sub-band of the frequency-domain image.  
Let \(N_{v}\) denote the number of virtual pilot pixels utilized for the reference image re-estimation. The virtual pilot observations can be expressed in vector form as  
\[\mathbf{Y}_{F}^{v}=\mathbf{D}_{F}^{v}\mathbf{H}_{F}^{v}+\mathbf{N}_{F}^{v}, \tag{19}\]  
where the data symbols in \(\mathbf{D}_{F}^{v}\) are unknown a priori to the receiver and should, therefore, be chosen from among all the available data pixels. By stacking the pilot observation vector \(\mathbf{Y}_{F}^{p}\) and the virtual pilot observation \(\mathbf{Y}_{F}^{v}\), the observation vector for image re-estimation can be obtained as follows:  
\[\begin{bmatrix}\mathbf{Y}_{F}^{p}\\ \mathbf{Y}_{F}^{p}\end{bmatrix}=\begin{bmatrix}\mathbf{D}_{F}^{p}\\ \mathbf{D}_{F}^{p}\end{bmatrix}\begin{bmatrix}\mathbf{H}_{F}^{p}\\ \mathbf{H}_{F}^{p}\end{bmatrix}+\begin{bmatrix}\mathbf{N}_{F}^{p}\\ \mathbf{N}_{F}^{p}\end{bmatrix}. \tag{20}\]  
For the next iteration, the LS estimate of \(\mathbf{Y}_{F}^{v}\) is added to the virtual pilots. The LS estimate of the new pilot observation vector can be expressed as:  
\[\begin{bmatrix}\mathbf{\hat{H}}_{F}^{p}\\ \mathbf{\hat{H}}_{F}^{v}\end{bmatrix}=\begin{bmatrix}\mathbf{D}_{F}^{p}\\ \mathbf{D}_{F}^{v}\end{bmatrix}^{-1}\begin{bmatrix}\mathbf{Y}_{F}^{p}\\ \mathbf{Y}_{F}^{p}\end{bmatrix}, \tag{21}\]yielding the pixel estimates at the pilot and virtual pilot positions in the first iteration. The pixel estimates for the remaining data symbols are calculated using the interpolation method discussed in Section 3.2. The image is then fully re-estimated using the pilots and virtual pilots, and the information pixels are demodulated using the ZF receiver as  
\[\begin{split}\tilde{\mathbf{D}}_{F}^{i}&=\frac{ \mathbf{Y}_{F}^{i}}{\tilde{\mathbf{H}}_{F}^{i}}\\ &=\left[\frac{Y_{F}^{i}(0)}{\tilde{H}_{F}^{i}(0)},\frac{Y_{F}^{i} (1)}{\tilde{H}_{F}^{i}(1)},\cdots,\frac{Y_{F}^{i}(N_{p}-N_{v}-1)}{\tilde{H}_{F }^{i}(N_{p}-N_{v}-1)}\right]^{T}.\end{split} \tag{22}\]  
As Equation (21) relies on both the pilot and virtual pilot tones, it is anticipated that an improved reference image estimation, and therefore an enhanced SER, can be achieved with increasing iterations as the virtual pilot pixels become more refined.