# Image Recovery with Data Missing in the Presence of Salt-and-Pepper Noise
## 2 Formulations of Salt-and-Pepper Noise and Data Missing  
In this work, the degraded image is given by  
\[\mathbf{y}=\mathbf{H}\mathbf{x}+\mathbf{n}, \tag{1}\]  
where \(\mathbf{y}\) is the degraded image, \(\mathbf{x}\) is the latent image to be reconstructed, \(\mathbf{H}\) is the degradation matrix, and \(\mathbf{n}\) is the noise. It is of interest to note that the column representation of the image is used in Equation (1). For a pure denoising problem, the degradation matrix is usually an identity matrix and the noise \(\mathbf{n}\) is the main source of interference. The noise can be generally divided into two categories: Gaussian noise and impulsive noise. It is a consensus that the impulsive noise is much harder to be removed than the Gaussian noise, and, in this work, one type of the impulsive noise salt-and-pepper noise is considered. The clean image is corrupted by the salt-and-pepper noise. That is,  
\[y(i)=\left\{ \begin{array}{ll}n_{min},&\text{with probability }p/2,\\ n_{max},&\text{with probability }p/2,\\ x(i),&\text{with probability }1-p,\end{array} \right. \tag{2}\]  
where \(p\) denotes the probability, \(n_{min}\) and \(n_{max}\) indicate the minimum and maximum values of the dynamic range, and they are 0 and 255 for a grey image, respectively. In Equation (2), one important observation is that only \(p\times 100\%\) pixels are affected by the noise and the rest is not. This is to say that, if the additive noise model in Equation (1) is used, the noise \(\mathbf{n}\) has lots of zero components. Therefore, \(\mathbf{n}\) can be seen as a sparse vector and this property will be exploited to remove the noise. For more information, the readers are referred to [22]. Another detrimental factor for image reconstruction is data missing due to the unreliable transmission link and/or other reasons. When this happens, the corresponding pixel is set to be zero and the rest is unchanged. In doing so, a sparse vector can be introduced to reformulate the data missing signal model. For example, let the uniform samples be \([y_{1},y_{2},y_{3},y_{4}]^{\mathrm{T}}\), and suppose the pixels of \(y_{1},y_{2}\) are missing. The received signal that contains samples of \(y_{3},y_{4}\) is modeled as \([y_{3},y_{4}]^{\mathrm{T}}=\mathbf{H}\circ[y_{1},y_{2},y_{3},y_{4}]^{\mathrm{ T}}\), where \(\mathbf{H}=[0,0,1,1]^{\mathrm{T}}\) and \(\circ\) is Hadamard product to indicate a componentwise product. In this reformulation, an unknown sparse vector \(\mathbf{H}\) is introduced to model the data missing pattern and it will be utilized to estimate the missing pattern. To simplify the following derivations, the missing pattern is rewritten in a matrix form, denoted by \(\mathbf{H}\). By using the matrix notation, the missing pattern in this example is a sparse matrix; that is, \(\mathbf{H}=[0\ 0\ 1\ 0;0\ 0\ 1]\). With this reformulation, in the case of data missing, the received signal is [23]  
\[\mathbf{\tilde{y}}=\mathbf{H}\mathbf{y}=\mathbf{H}(\mathbf{x}+\mathbf{n})= \mathbf{H}\mathbf{x}+\mathbf{\tilde{n}}, \tag{3}\]  
where \(\mathbf{\tilde{y}}\) is the received signal in the case of data missing, \(\mathbf{H}\) is the unknown missing pattern, and \(\mathbf{\tilde{n}}\) is the salt-and-pepper noise. If we now wish to recover the latent image from the received signal model in Equation (3), the sparse property of missing pattern \(\mathbf{H}\) and the salt-and-pepper noise \(\mathbf{\tilde{n}}\) should be utilized. With that in mind, the following optimization problem is formulated to simultaneously recover the image and estimate the missing paper under the salt-and-pepper noise. That is,  
\[ \begin{array}{ll}\text{minimize}&\|\mathbf{x}\|_{2}+\lambda_{1}\|\mathbf{H }\|_{1}+\lambda_{2}\|\mathbf{\tilde{n}}\|_{1}\\ \text{subject to}&\|\mathbf{\tilde{y}}-\mathbf{H}\mathbf{x}-\mathbf{\tilde{n}} \|_{2}^{2}<\epsilon,\end{array}