# Image Recovery with Data Missing in the Presence of Salt-and-Pepper Noise
## 1 Introduction  
Image restoration is an important and widely studied subject in image processing, where the main objective is to reconstruct the latent image given the degraded and/or noisy image [1; 2; 3; 4; 5]. The main detrimental factors to corrupt the image are noise, data missing, and blur, to name a few. It is well known that images are deteriorated if they are not properly handled, and, in this work, the image restoration approach is developed when the image is corrupted by salt-and-pepper noise and data missing. The salt-and-pepper noise, classified as the impulsive noise, presents a great challenge for an image reconstruction problem because of its impulsive nature. The noise is mainly caused by imaging sensors, transmissions, and/or other factors. Under the case of salt-and-pepper noise, the approaches developed for Gaussian noise will not work well because the noise distributions are different. To suppress the noise, the traditional techniques mainly focus on filters in which nonlinear filters and linear filters are developed. In contrast, nonlinear filters in general offer better performance than linear filters. The classical nonlinear filter in the context of impulsive noise removal is the standard median (MED) filter [6]. It can easily be shown that this type of filter is conceptually simple to implement and hence provides good denoising ability, but its performance deuterates as the noise level increases. Shortly after, the detection-removing approaches are then devised so as to better suppressthe noise, where the noisy pixels are detected first, and second they are replaced by the results of a certain type of median filters using vicinity pixels--for example, the adaptive median filter (AMF) [7] and its improved versions of adaptive weighted mean filter (AWMF) [8; 9], the multistate median (MSM) filter [10] and the adaptive median filter with edge-preserving regularization (AM-IEPR) [11]. Computer simulation results show that the above-mentioned filters outperform the traditional MED filter, but their performance largely depends on the performance of the detection. The main drawback of these approaches is that they failed to exploit the noise structure that might improve the noise suppression ability and in turn enhance the image reconstruction. In practice, data missing happens because of sensor failure and sudden data transmission loss. Over the past several decades, a huge number of approaches have been proposed to handle this issue. Based on the structural inpainting technique, several algorithms were reported for accurate reconstruction of missing edges, and they are effective for pure structure images [12; 13]. However, for images containing textures, the performances were less satisfactory. To remedy this issue, the exemplar-based inpainting technique was exploited to develop several methods for accurate reconstruction of missing textures [4; 14; 15; 16]. Traditionally, missing texture reconstruction is considered as the texture synthesis, and, based on which, many methods were proposed--for example, the fragment-based algorithm [14] and the exemplar-based image inpainting method [15]. In [16], the approach for learning a high-order Markov random field (MRF) has been proposed and the performance of missing texture reconstruction is dramatically improved. The key idea is to formulate the prior as a high-order Markov random field (MRF) defined over large neighborhood systems. By decomposing the exemplars into the frequency domain and using the selected coefficients, the gradient based algorithm was developed to interpolate the missing parts, and its superior performance was demonstrated because the coefficient selection process acts to remove the noise [17; 18]. Recently, due to the successful developments of the sparse signal reconstruction, many methods have been proposed by exploiting the sparse representation of the image [4; 19; 20; 21]. Even they provide better reconstruction results in some cases, the performance degrades when the missing level is high.