HSI去噪论文分享之四

题目：
Hyperspectral Image Denoising via Subspace-Based Nonlocal Low-Rank and Sparse Factorization
作者：
Chunhong Cao,Jie yu,Chengyao Zhou,Kai Hu,Fen Xiao,Xieping Gao
关键字：
Denoising,hyperspectral image(HSI),low-dimension subspace,low-rank factorization,nonlocal selfsimilarity

解决问题：
1.传统方法（BM3D,NCSR,WNNM）将每个波段的光谱独立处理，这样忽视了各个波段的联系，没有充分利用高光谱图像的高冗余强关联性的特点。
2.基于稀疏理论的去噪方法虽然利用考虑了光谱间的关联性，但是在去噪性能上，只对高斯噪声有较为明显的效果。
3.基于低秩理论的去噪技术虽然解决了上述两种方法的弊端，但是存在计算量大，参数需要提前设定的弊端，这对极大的限制了去噪的灵活性。并且大多数方法不能和很好的处理较为严重的高斯噪声。
4.基于子空间的去噪的方法提出体现了很好的去噪效果，但是在边界仍然存在污染。

创新思路：
本文的出发点是基于子空间开始的，由于光谱的低秩特征存在于低维子空间。此外非局部低秩分解将空间的非局部自相似性加以考虑。

模型：
1、利用子空间表示实现低秩性：
clean HSI matrix=small-size orthonormal matrix * small-size coefficient marix
2、利用空间非局部自相关特性的非局部低秩分解：
3-D patches --> 3-D tensoer
3、l1 范数处理稀疏噪声

推导：
1、Original model：
Observed HSI cube: y =mxnxl (l :#spectral bands)
Reshaped size of y: Y =hxl
Degradation modle：Y=X+S+N
2.Traditional denoising model:

3.Spectral global low-rank factorization:
**X=EZ ** (E:lk,basis of the subspace Sk,Z:kh,coefficient)
Y=EZ+S+N

4、Spatial nonlocal low-rank factorization:

(Z:m*n ,eigen-image)
All eigen-images compose representation coefficients image (RCI).There are two important properties inherent in RCI, namely the nonlocal self-similarity of each eigen-image and the correlation among the eigen-images, benefit the image denoising task.

5、Constracting 3-D tensor:

1. subspace identification (SI);
2. grouping similar 3-D patches;
3. unfolding each 3-D patch;
4. stacking all 3-D patches.
   
   we define a binary operator Ri to extract nonlocal similar overlapping 3-D-patches from the RCI:
   
   Zi ∈ Rs×s×k represent 3-D patches of RCI,where s is the spatial size of the 3-D patch, and i is the indexof the top-left pixel.
   The operator Ri consists of two steps: first, all and reference 3-D patches are extracted by sweeping all across the RCI with overlaps of step size 1 and 3, respectively; second, given a reference 3-D patch, a set of simi- lar overlapping 3-D patches are found in a neighborhood region according to the Euclidean distance.
   we desire to estimate the clean tensor li from the zi through solving the following problem:
   
   κ(Li ) = q j |σj (Li )|1 is the sum of the singular values of matrices unfolded by 3-D tensor

求解：
Based on the above-mentioned analysis, the proposed HSI denoising model with subspace-based nonlocal low-rank and sparse regularization can be written as：

Using ALM to solve the above problem .