## 4 Simulations  
The performance of the proposed reference-frame estimation scheme was evaluated in the presence of an AWGN channel. The simulation environment used in this study is similar to that used in [21]. Specifically, a 2D-DFC system with a \(256\times 256\)-pixel data-embedded Lena image displayed on the screen was utilized (cf. Figure 2b). The camera was perfectly aligned with the screen to prevent energy loss. BPSK modulation was employed on data symbols with uniformly spaced pilots, and the SER, ADR, and PSNR at the camera decoder output were used as performance metrics. The ADR was computed using the following formula:  
\[\text{ADR}=30 AN_{i}(1-\text{SER}), \tag{23}\]  
where \(A\) is the pixel area for data embedding and \(N_{i}\) is the number of information symbols per frame. A standard off-the-shelf camera receiver of \(30\,\text{fps}\) was considered, and the data were embedded in every frame. Of all the data pixels, a total of \(10\%\) were designated as pilot pixels, and a maximum of five iterations were performed. However, the iteration was terminated when no significant improvement in the overall 2D-DFC performance was observed. The data were embedded in the high-frequency region of the Lena image to minimize visual artifacts on the screen, because a reference image frame was not transmitted. The starting pixel values of the symbols (\(s_{p}\) and \(s_{q}\)) and number of embedded data symbols (\(L_{p}\) and \(L_{q}\)) were set as \(90\) and \(30\), respectively. Figure 4 illustrates the SER performance of the proposed reference-frame estimation scheme. As the number of iterations of the scheme increases, the SER gradually approaches that of the ideal scheme, that is, conventional 2D-DFC. This can be attributed to the iterative refinement of the pixel estimate output at both the pilot and virtual pixels, which resulted in better interpolated values for the information symbols. Furthermore, we observed that, as the SNR increased, the proposed method achieved significant performance improvements, and the virtual pilot pixels became more accurate with increasing iterations. The uniqueness of the scheme lies in its ability to improve the performance iteratively by utilizing both pilot and virtual pilot pixels while keeping the pilot density low at only \(10\%\). Figure 5 depicts the ADR of the proposed reference frame estimation scheme, which is the primary motivation for this study. We observed that, even without iterations, the data rate was higher than that of the conventional 2D-DFC scheme. Furthermore, with increasing number of iterations, the data rate becomes significantly higher, nearly doubling. The primary reason for this improvement is the elimination of reference frames in the proposed scheme. In conventional 2D-DFC, a reference frame is employed to decode each data frame at the receiver, whereas, in the proposed scheme, the reference frame is estimated at the receiver using transmitting pilots, thereby eliminating the need for reference frames. To enhance the performance, virtual pilots are further used iteratively. This demonstrates that the use of reference frames severely limits the data rate of 2D-DFC systems, and the  
proposed scheme enhances the performance by eliminating their use. The iterations were terminated after no significant improvement in the data rate was observed. Figure 6 shows the PSNR as a function of the number of iterations. PSNR was used to measure the quality of the reconstructed image compared to the original image. Higher PSNR values indicate better image quality. To compute the PSNR between images, we first computed the mean square error (MSE) as  
\[\text{MSE}=\frac{1}{PQ}\sum_{P,Q}\left[\tilde{\mathbf{D}}_{t}[i]-\mathbf{D}_{t }[i]\right]^{2}, \tag{24}\]  
Figure 4: Comparison of SER performance between the proposed 2D-DFC scheme with iterative processing and the conventional 2D-DFC scheme that employs a reference image.