# A Probabilistic Hyperspectral Imagery Restoration Method
## 3 Revisited Low-Rank based HSI Denoising from Linear Mixed Model Perspective
### Analysis on the Rank Initialization for LRMA  
[PERSON] et al. [11] first use the LRMA model for HSI restoration, which divides HSI into some fixed-size subcubes, then adopt LRMA method to restore each subcube and assemble those restored subcubes to obtain the final result. Rank number is manually initialized as the same value in [11] for each subcube. Though this kind of initialization is simple, it ignores the rank difference among different fixed-size subcubes in HSI. In other words, because the materials/endmembers falling into a fixed-size subcube are related with its content, the number of endmembers is always varied with different subcubes. Thus, the rank number should be varied according to \(rank(\mathbf{L})\leq n_{e}\) to fit the data better. However, this characteristic has not been considered in [11] and its follow-up methods.  
Based on the data structure of HSI, we analyze and find two kinds of rank number initialization methods. Taking advantage of the number of endmembers, \(n_{e}\) is an apparent clue/(prior knowledge) for rank number initialization. The first method is a whole HSI entity-based method, i.e., all pixels in HSI are used to estimate the rank number. The second method is a local region-based method. Similar to [PERSON] et al in [11], homogeneous regions are adopted instead of using fixed-size subcubes [11].  
Figure 1: Illustration of the hyperspectral linear mixed model.  
(1) Initialize the rank number using whole HSI entity  
Since determining the number of endmembers in HSI is an important issue for hyperspectral unmixing, the relative methods can be directly introduced for rank number initialization. Without dividing HSI into some subcubes as [PERSON] et al in [11] does, those methods are based on all pixels in HSI. The reason using all pixels to determine the number of endmembers is partially because the parameters of those methods can be estimated more accurately with more pixels. Thus, we use whole HSI entity to estimate the number of endmembers, and then use the estimated number to initialize the rank number.  
Information theoretic criteria including [PERSON]'s information criterion, minimum description length can be used to determine the number of endmembers, which is based on the data log-likelihood [35; 36; 37]. However, the prior knowledge of data log-likelihood function needs to be provided. Incorrect prior knowledge will influence the accuracy of estimated number of endmembers. On the contrary, geometry-based approaches can estimate the number of endmembers without knowing the prior knowledge of likelihood function. GENE-AH [38], a typical geometry-based approach, was proposed to estimate the number of endmembers. Since it assumes the observed pixels lie in the affine hull of endmembers, GENE-AH has an advantage in estimating the number of endmembers in the presence of noise, which promotes adopting GENE-AH to estimate the number of endmembers in this study.  
(2) Initialize the rank number with homogeneous regions in HSI  
Dividing HSI into fixed-size subcubes will lead to variation of the rank number with different subcubes [11]. Thus, assigning each subcube the same rank number is inappropriate. However, from the perspective of segmentation, an image can be divided into many of homogeneous regions. For hyperspectral image, the rank of the homogeneous region approximately equals to one because those homogeneous regions are always defined as the combination of pixels with similar spectral. It motivates us first segment/over-segment HSI into some local regions, then restore those local regions with LRMA method.