## 1 Introduction
The DFC approach was robust against visual artifacts observed on the screen, even with a lower framerate. In a previous study [20], we experimentally implemented a 1D-DFC approach based on 1D discrete cosine transform (DCT) and machine learning. The paper evaluates the proposed scheme using an actual DFC link and demonstrates the practical implementation and performance of our approach for various system design parameters. In particular, we first adopted DCT to transform a spatial-domain image into its spectral-domain equivalent. Addition allocation and subtraction data retrieval techniques were used to reduce computational complexity during the data-embedding process. Moreover, channel coding was applied to overcome the data errors caused by D2C wireless channel. After capturing the displayed image using a camera, the display region was extracted using the a object detection deep learning technique. Extensive real-world experiments were performed, considering various geometric distortions, noise, and different standard input images. Although DFC is robust against visual artifacts at low frame rates and perspective distortion, it uses reference frames that are used at the camera receiver to decode the data-embedded frames correctly. Although complementary (or reference) frames compensate for the visual artifacts on the screen and assist in data decoding, they significantly diminish the data rate of the overall system. To address this problem, it is essential to estimate the reference frame at the receiver end. Our previous study introduced a comparable method for reconstructing reference frames in one-dimensional DFC systems [21]. In this study, we extend the iterative spectral image estimation approach to enhance the data rate of two-dimensional DFC (2D-DFC) systems. To estimate the channel, we first embedded pilot pixels into data-embedded frames. Using these pilot pixels, we obtained the least squares (LS) estimate, followed by interpolation of the channel at the data pixels based on the LS estimate of the pilot signals. Subsequently, we commenced the iteration process by assuming the correctness of the decoded symbols and using some of them as virtual pilots in the next iteration to reestimate the information pixels. This iteration is repeated several times to estimate the information pixels more accurately. After fully estimating the reference image, we employ a zero-forcing (ZF) receiver to demodulate the data. The simulation results demonstrate that the proposed scheme outperforms the conventional 2D-DFC scheme by a factor of two in terms of the achievable data rate (ADR) while also maintaining the unobtrusiveness of the display by embedding the data in the high-frequency regions of the transmit frames. Additionally, we conducted simulations that considered perspective distortion, which refers to the misalignment of cameras and displays. This implies that the camera and display are not perfectly aligned and can cause distortions in the visual output. This simulation provides an opportunity to test the feasibility of conducting experiments under similar conditions. The remainder of this paper is organized as follows. Section 2 provides an overview of the pilot-based 2D-DFC system, including the data embedding and pilot insertion mechanism. In Section 3, we propose an iterative scheme for reconstructing the reference image along with virtual pilot pixel selection criteria. Section 4 presents the simulation results in terms of the symbol error rate (SER), achievable data rate (ADR), and peak signal-to-noise ratio (PSNR) for various system design criteria. In addition, we performed simulations considering the misalignment between the camera and display, and compared our proposed scheme with the conventional 2D-DFC scheme. Finally, Section 5 concludes the paper.