# A Probabilistic Hyperspectral Imagery Restoration Method
## 1 Introduction  
Hyperspectral imagery (HSI) is 3D data containing both spatial and spectral information, which is widely used for lots of remote sensing related applications [1; 2; 3; 4; 5; 6; 7]. However, HSI is unavoidably influenced by multiple kinds of factors--including noise, stripe corruption, etc.--during imaging and acquisition [8], which decrease the image quality. Thus, it increases the difficulty for people interpretation or machine understanding [9; 10; 11; 12], which makes HSI restoration an essential step for HSI processing and analysis. To date, lots of methods have been proposed for HSI denoising. According to the way how noise and signal combines, typical HSI denoising methods can be categorized as signal-dependent-noise and signal-independent-noise methods. As indicated by the name, signal-independent-noise method assumes the generation of noise is independent of signals, while the generation of noise is related with signals for signal-dependent-noise method. These two kinds of methods are proposed for different HSI imaging mechanisms. For example, signal-independent-noise method is suitable to sensors where the noise is mainly determined by electronic components, while signal-dependent-noise method can be applied to the sensors where photon noise is dominated [13; 14]. Both methods are research focuses of HSI denoising. Thus, lots of algorithms are proposed in recent years [11; 13]. We mainly focus on signal-independent-noise method in this study. For signal-dependent-noise methods, we refer the readers to [13]. For signal-independent-noise HSI denoising methods, the existing focuses are original-space based methods, which includes total variation (TV) based, tensor based, local patch based, and low-rank-based methods, etc. Low-rank based method has advantages to deal with missing data, which complies with the fact that HSI is also influenced by deadlines. Thus, low-rank based method has garnered significant attention in recent years for HSI denoising, which is also our focus in this paper. The detailed description of signal-independent-noise method is given in Section 2. Though some low-rank based methods have been proposed, the following two deficiencies limit the usage of existing low-rank based methods. (1) In real applications, hyperspectral images will be influenced by multiple kinds of noise such as Gaussian noise, sparse noise, stripe, dead lines, and their combinations, which beyond the capability of the existing low-rank based methods. (2) The rank numbers of all existing low-rank based methods need to be assigned manually, and an automatic method to determine the rank number is desired. To address those issues, we propose an HSI-oriented low-rank denoising method. Firstly, considering the mixture of Gaussian distribution has advantage of depicting different kinds of distributions, we propose to model noise with the mixture of Gaussian distributions. Thus, a probabilistic low-rank model is proposed for HSI denoising, where the clean HSI to be estimated is still considered as a low-rank matrix. Secondly, we find hyperspectral unmixing can provide prior information for rank number initialization. However, it is neglected by existing low-rank methods. In this paper, we propose to automatically determine the rank number via hyperspectral unmixing. In summary, the proposed HSI-oriented low-rank denoising method can deal with multiple kinds of noise as well as automatically determine the rank number. As a result, the proposed method has better HSI restoration results compared with other methods. Experimental results on both simulated and real datasets verify the effectiveness of the proposed method. The paper is structured as follows. In Section 2, we briefly review the related work of HSI denoising. In Section 3, we analyze how to cast HSI denoising into a low rank matrix analysis (LRMA) problem and discuss how to initialize the rank number. In Section 4, we present the probabilistic low-rank based HSI restoration method. In Section 5, the experimental setup and results are summarized.