# A Probabilistic Hyperspectral Imagery Restoration Method
## 2 Related Work  
We mainly focus on HSI denoising method where noise is independent with signal. The related works can be roughly divided into a transformation-space based method and an original-space based one [15; 16]. Transformation-space based method is adopted for HSI denoising in the early stage. The basic idea of transformation-space based method is that the signal differs from the noise greatly when we transform the HSI from spatial/spectral space into a new space. For example, in the wavelet transformation space, the signal is sparse while the noise is non-sparse. Thus, signal can be separated from the noise by exploiting such difference. [17; 18; 19]. Except wavelet transformation, other transformation methods such as curvelet [20; 21] can also be used for this purpose. The advantage of transformation-based method is that it does not need to assign noise a specific category (e.g., Gaussian noise). However, artifacts such as edge ringing and low-frequency noise are prone to be generated when using transformation-space based methods [16]. In contrast, original-space (image space or spectral space) based methods denoise HSI directly on spatial/spectral space. Since those methods do not influenced by artifacts and low-frequency noise, they begin to the dominate HSI denoising methods in recent years. Considering the HSI image is a data cube composing a set of 2D band images, a direct strategy for original-space based method is to denoise each band image independently, which promotes some 2D denoising method directly used for hyperspectral image denoising. However, HSI contains both spatial and spectral information. Directly using 2D denoising methods for each band neglects the correlation among images from different bands. Thus, one trend of the original-space based method is to utilize both spectral and spatial information. Among those methods, total variation (TV) based [22; 23; 24], tensor based [25; 26; 27], local patch based methods, and low-rank based methods are representative approaches. For TV based methods, considering the variations among neighboring pixels or spectra are smooth, TV based HSI denoising methods are proposed by exploiting such kind of smoothness [22; 23; 24]. Since the noise intensity of different bands may differ, spectral-spatial TV based methods consider the spectral noise differences as well as the spatial information differences simultaneously [24]. Considering HSI is a 3D data which can be naturally represented by 3D tensor, some tensor-based methods are proposed accordingly. It is noticeable that commonly used tensor decomposition strategy includes Tucker decomposition and PARAFAC decomposition [25; 26; 27]. For local patch-based method, it first divides HSI into a group of overlapping (or non-overlapping) 2D patches or 3D blocks, and then generates noiseless 2D patches or 3D blocks using collaborative filtering. Finally, it assembles those denoised 2D patches or 3D blocks to obtain the desired HIS [28; 29; 30; 31]. Among which, video block-matching and 3D filtering (VBM3D) is a representative local patch/voxel based method. Though the advantages of above introduced spectral-spatial TV based, tensor based and local patch-based methods have been testified over traditional 2D approaches for HSI denoising, it is found that those methods are proposed for complete HSI data. If the HSI is incomplete--e.g., dead lines or dead pixels appear in HSI--the above methods cannot generate a complete HSI and their restoration results will be influenced greatly. In recent years, low-rank based methods have been proposed for HSI denoising, which have the merit to deal with incomplete data. The representative method is a LRMA method [11], which assumes clean HSI is a low-rank matrix. Then, HSI restoration problem cast into recovering low-rank matrix from original noisy HSI data, which is solved by Go Decomposition (Godec) approach [11]. Based on [11], some other low-rank based methods are proposed by utilizing other characteristics of HSI for denoising. [PERSON] et al.