## 3 JPEG Image Compression and Proposed Artifact Reduction Scheme by Deep Convolutional Network  
Each pixel in a phase-only hologram has an intensity value ranging from 0 to 2 \(\pi\) and a phase-only hologram can be regarded as a gray-scale intensity image. Various image compression algorithms can be attempted for the compression of phase-only holograms. JPEG is a very commonly used lossy compression scheme for digital images, which was firstly proposed in 1992 by Joint Photographic Experts Group [35]. In JPEG scheme, the original image (e.g., a gray-level photograph or a phase-only hologram) is first divided into 8 \(\times\) 8 pixel blocks and each individual block undergoes discrete cosine transform (DCT). Then, the coefficients in the transformed domain of each block are quantized with more quantization levels for low frequency components and less quantization levels for high frequency components. Subsequently, zigzag coding, entropy coding and Huffman coding are performed on the quantized coefficients. Finally, the original image becomes a compressed bit stream with much smaller data size than the original uncompressed image. A reconstructed image with certain image degradation can be obtained when the JPEG binary bit stream is decompressed with the inverse procedures as compression. JPEG compression can significantly reduce the data size of an original natural image and only introduce minor quality degradation in the decompressed image. The reason is that a natural image (e.g., a photograph) is usually locally smooth and most of its information is concentrated in the low frequency coefficients of DCT spectrum. The discarded high frequency information in the quantization step has negligible effect on the image quality. However, phase-only holograms have substantially different image characteristics compared with photographs [38]. One feature of a hologram is that it contains many high frequency fringe patterns. If JPEG compression is directly  
Figure 3: Propagation of errors to four different neighboring pixels with corresponding weighting coefficients in error diffusion algorithm. applied to a phase-only hologram, the decompressed result after compression will suffer from heavy damage and the holographically reconstructed image from the JPEG compressed hologram will be severely degraded as well. In this work, we employed deep convolutional neural networks to reduce the artifacts of JPEG compressed phase-only holograms. In the past few years, deep learning methods receive much attention and exert considerable impact in many fields such as image quality enhancement [39; 40; 41; 42]. A previous work [39] proposed a 20-layer network DnCNN to reduce the Gaussian noise on the noisy images. Although DnCNN is designed for denoising task, it shows promising performance on image compression artifacts reduction and super-resolution as well. Assisted by the prior knowledge, a previous work [40; 42] proposed denoiser-based and total variation (TV)-based solutions, respectively for image restoration task. Very recently, deep learning has been introduced to the holographic research area and gained success in different applications [43; 44; 45; 46; 47; 48; 49]. The network structure employed in our task is illustrated in Figure 4a, based on the previous works of restoring JPEG compressed photographs [50]. The overall flowchart of our proposed \"JPEG + deep learning\" hologram compression scheme is illustrated in Figure 4b. The proposed CNN consists of four convolutional layers, and each layer has a specific function. The first layer is used for feature extraction, and with \(9\times 9\) filter size it generates 64 feature maps. By adding a bias term to the outputs and employing ReLU as the activation function, the convolution operation in this layer can be formulated as:  
\[F_{1}(\mathbf{I})=ReLU(W_{1}*\mathbf{I}+B_{1}) \tag{9}\]  
where \(W\) and \(B\) represent the weights and biases, respectively; \"\(*\)\" denotes the convolution operation; and \(F\) represents the nonlinear mapping process.