# Quadratic Clustering-Based Simplex Volume Maximization for Hyperspectral Endmember Extraction
## 1 Introduction  
Due to the limitation of spatial resolution and the complexity of ground object distribution, mixed pixels exist in hyperspectral images (HSIs) and reduce the accuracy of traditional pixel-level applications [1]. Therefore, it is necessary to perform hyperspectral unmixing for further analysis, which includes extracting pure pixels (endmembers) and determining the fractions of each endmember in pixels (abundance). The former is called endmember extraction, and the latter is called abundance estimation. Hyperspectral unmixing has been used in many fields to improve the related performance, such as subpixel target detection, precise land cover mapping and Earth change detection [2; 3; 4; 5; 6; 7]. The mixing model is the basis for the research of hyperspectral unmixing, containing the linear spectral mixing model (LMM) and nonlinear spectral mixing model (NLMM) [8; 9]. LMM assumes the photons arriving at the sensor interact with one ground object, so each mixed pixel is described as a linear combination of endmembers weighted by the corresponding abundance. As a matter of fact, the photons will go through multiple reflections and refractions between various ground objects, leading to NLMM. It has no fixed expression and needs to be determined for specific land cover class. Hence, LMM is utilized in most endmember extraction algorithms (EEAs) due to the simple structure and good performance, and it is also used in our work. EEAs based on convex geometry are widely used in linear spectral unmixing. The pure pixel index (PPI) projects all pixels onto a set of vectors, and pixels that fall at the endpoints of these vectors the most are selected as endmembers [10]. Based on PPI, vertex component analysis (VCA) uses the orthogonal direction of the subspace spanned by the determined endmembers as projection axis, and pixels with the extreme value is taken as a new endmember [11]. N-FINDR works by finding the simplex with maximum volume in the space of the HSI, and pixels that constitute this simplex are regarded as endmembers [12]. As a variant of N-FINDR, the simplex growing algorithm (SGA) reduces the computational cost by including a process of growing simplexes one vertex at a time [13]. The alternating volume maximization (AVMAX) improves N-FINDR, using an alternating optimization strategy to pin down some convergence characteristics [14]. To enhance the noise robustness, the minimum volume simplex algorithm (MVSA) fits the minimum simplex volume to the data and reduces the effect of noise and outliers by imposing a hinge type soft constraint on the abundance [15]. The fast subspace-based module (FSPM) detects convex hull vertices for component pairs obtained by singular value decomposition and proposes a local outlier score measurement to remove outliers [16]. The statistics-based EEAs decompose the data into the product of endmember matrix and abundance matrix simultaneously, and iteratively solves the problem by adding prior constrained information. This type of method handles HSIs with a high mixing degree well but requires a lot of computations. The nonnegative matrix factorization (NMF) is a commonly used method. Based on NMF, several improved approaches have emerged by adding different constraints, such as minimum volume constrained NMF (MVC-NMF), L\({}_{1/2}\)-NMF and the robust collaborative NMF (Co-NMF) [17; 18; 19]. In HSIs, pure pixels are more likely to exist in homogeneous areas, and mixed pixels are more likely to be present in transition areas containing two or three features. The above two types of methods ignore such spatial properties and identify endmembers only by spectral features, resulting in limited endmember extraction performance. Therefore, EEAs that fuse spatial and spectral features have appeared, which improve the accuracy of endmember extraction to a certain extent.