# 前言

• Shaoquan Zhang
• Jun Li
• Kai Liu
• Chengzhi Deng
• Lin Liu
• Antonio Plaza

# 正文

## 名词解释

• 混合像元Mixed Pixel）：地球自然表面几乎不是由均一物质所组成的。当具有不同波谱属性的物质出现在同一个像素内时，就会出现波谱混合现象，既混合像元（Mixed Pixel）。如果混合像元的尺度很大（宏观），那么混合像元将存在线性关系。对于微观的混合，混合像元通常表现为非线性关系。
• 混合像元分解：在一个给定的地理场景中，地表由少数几种地物（端元）组成，并且这些地物具有相对稳定的光谱特征，因此，遥感图像的像元反射率可以表示为端元的光谱特征和这个像元面积比例（丰度）的函数。这个函数就是混合像元分解模型。混合像元分解指从实际光谱数据(一般为多地物光谱混合的数据)中提取各种地物成分（端元）以及各成分所占的比例（丰度）的方法。端元提取和丰度估计是混合像元分解的两个重要的过程。
• 端元(endmembers)提取：在混合图像中提取出各种成分。
• 丰度估计：对每种估计出来的端元物质的比例加以估计。丰度满足非负性、合为一的约束。

## 主要内容

### Abstract

The proposed approach, which is called local collaborative sparse unmixing, considers the fact that endmember signatures generally appear distributed in local spatial regions instead of uniformly distributed throughout the scene.

### I. INTRODUCTION 简介

Two kinds of spectral unmixing models have been commonly used in the literature to address the mixed pixel problem: linear and nonlinear

The linear mixture model exhibits practical advantages, such as ease of implementation and flexibility in different applications. In this letter, we will focus exclusively on the linear mixture model.

sparse unmixing has been developed as a semisupervised approach,in which mixed pixels are expressed in the form of linear combinations of a number of pure spectral signatures from a large spectral library that is known in advance.

The sparse unmixing algorithm via variable splitting and augmented Lagrangian (SUnSAL) was one of the first methods developed for this purpose, which generally assumes that the number of endmembers participating in each pixel is low.

The collaborative SUnSAL (CLSUnSAL) was developed under the assumption that all pixels in a hyperspectral image share the same active set of endmembers.

the local spatial information plays an important role in sparse unmixing as unmixing problems generally become easier in a local scale rather than a global scale.

The proposed approach, which is called local collaborative sparse unmixing (LCSU), assumes that neighboring pixels share the same active set of endmembers.

In comparison with the global assumption enforced by CLSUnSAL, the proposed LCSU assumes that endmembers tend to appear localized in spatially homogeneous areas instead of distributed over the full image.

### II. COLLABORATIVE SPARSE UNMIXING 协同稀疏分解

Sparse unmixing finds a linear combination of endmembers for an observed pixel i from a large spectral library as follows:

（1）$$y_i = Ax_i + n_i$$

where yi denotes an L × 1 pixel vector of the observed hyperspectral data, L denotes the number of bands, A ∈ RL×m is a spectral library, m is the number of spectral signatures in A, xi denotes the abundance vector corresponding to library A, and ni is an L × 1 vector collecting the errors affecting the measurements at each spectral band.

• $$y_i$$表示：观测的高光谱数据$$L×1$$像素矢量
• $$L$$表示：光谱带的数量（the number of bands
• $$A \subseteq \mathbb R^{\mathbb L× \mathbb m}$$表示：光谱库，$$m$$指$$A$$中光谱特征的数量
• $$x_i$$表示：光谱库$$A$$对应的丰度矢量
• $$n_i$$表示：一个$$L×1$$的矢量，该矢量收集影响每个光谱带测量的错误信息

As the number of endmembers involved in a mixed pixel is usually very small when compared with the size of the spectral library, the vector of fractional abundances x is sparse. The unmixing problem can be formulated as an 2 − 0 norm optimization problem.

where $$||x_i||_0$$ denotes the number of the nonzero components of the vector $$x_i$$ , and $$λ$$ is a regularization parameter that weights the two terms of the objective function. However, the $$l_0$$ term leads to an NP-hard problem, and thus, it is very difficult to solve. In [21] and [22], it was proven that the $$l_0$$ norm can be replaced by the $$l_1$$ norm under a certain condition of the restricted isometric property. In this context, the previous problem becomes.

• $$||x_i||_0$$表示：矢量$$||x_i||$$的非零分量
• $$λ$$是：规则化参数，用于权重目标函数的两个条件