# Deep Image Compression with Residual Learning
## 1 Introduction
### DL-Based Image Compression and Related Works  
Generally, a typical end-to-end lossy image compression framework can be formulated as jointly optimizing all modules: an encoder \(E_{\phi}\), a quantizer \(Q\), a decoder \(D_{\theta}\), and some other rate estimation and rate-distortion control module \(R\), where \(\phi\) and \(\theta\) are the trained weights.  
\[E_{\phi}:\mathbb{R}^{N}\rightarrow\mathbb{R}^{M}\qquad D_{\theta}:\mathbb{R}^{ M}\rightarrow\mathbb{R}^{N}\qquad Q:\mathbb{Z}^{M}\rightarrow[0,1] \tag{1}\]A vector of image intensities \(x\in\mathbb{R}^{N}\) is mapped to a latent presentation space \(y\in\mathbb{R}^{M}\) via the encoder \(E_{\phi}\), \(y=E_{\phi}(x)\). \(y\) is quantized into a discrete-valued vector \(\hat{q}\in Z^{M}\) via the quantizer \(Q\). The entropy rate of the vector \(\hat{q}\) is estimated and minimized via \(R\). The reconstructed image can be represented as \(\tilde{x}=D_{\phi}(\tilde{q})\). Distortion is often assessed by the distance between \(x\) and \(\tilde{x}\), \(D(x,\tilde{x})\), namely PSNR (Peak Signal to Noise Ratio), or the perceptual measure such as MS-SSIM (Multi-scale Structural Similarity) [13]. Rate estimation is often assessed by the minimal entropy of the discrete bit streams. Therefore, image compression is typically formulated as a distortion and rate optimization problem. The goal is o optimize the trade-off between using a minimal number of bits and having minimal distortion.  
\[Optimization\quad object:\quad\underbrace{D(x,\tilde{x})}_{ Distortion}-\lambda\cdot\underbrace{log_{2}Q(E_{\phi}(x))}_{ Number\ of\ bits} \tag{2}\]  
Existing DL-based image compression frameworks are mainly classified into the auto-encoder and recurrent neural network (RNN). In the case of auto-encoder, recent works have focused on variational Bayes and autoregressive context methods to develop accurate entropy estimation models by optimizing the R-D curve and introducing hyper prior parameters [6,8]. The prior probability of compression representation is modeled by a Gaussian or Laplacian distribution by an underlying auto-encoder. Since the Lagrange multiplier \(\lambda\) is introduced as a hyper-parameter for a desired trade-off between the rate and distortion, it is not efficient and different models must be trained for different compression rates.  
Due to a large fraction of Internet traffic driven by requests from mobile devices, a progressive algorithm is more popular. Recurrent neural networks (RNNs)-based frameworks only need to be trained once to produce a progressive and variable-rate model, regardless of the desired compression rate [14,15]. However, the RNN-based methods require multiple iterations to obtain high-quality images. Besides, [PERSON] et al. [4] proposed a generative adversarial network to obtain visually pleasing results at low bit rates where previous methods fail and show strong artifacts. Although the PSNR and MS-SSIM index shows no advantage, the reconstructed images seem better than others at extremely low bit rates.  
Combined with the advantages of both, the proposed framework adopts the recurrent architecture with rate control to learn residual information.