# Quadratic Clustering-Based Simplex Volume Maximization for Hyperspectral Endmember Extraction
## 1 Introduction
Automatic morphological endmember extraction (AMEE), the earliest algorithm to fuse spatial and spectral properties, enables unsupervised endmember extraction by performing multidimensional morphological dilation and erosion over an ever-increasing spatial neighborhood [20]. The entropy-based convex set optimization (ECSO), developed in recent years, exploits entropy as spatial information to choose two-band pairs and calculate the convex set for two-band data to determine endmembers [21]. As clustering can effectively utilize spatial and spectral information, it has been widely used in endmember extraction, further improving the performance of hyperspectral unmixing. For example, clustering and over-segmentation-based preprocessing (COPP) combines top-down over-segmentation (TDOS) with spectral clustering to generate homogeneous partitions, and the most spectrally pure average vectors are sent to spectral-based EEAs [22]. The regional clustering-based spatial preprocessing (RCSPP) performs regional clustering to obtain a set of clustered partitions that exhibit spectral similarity, then a subset of candidate pixels with high spectral purity is fed to endmember detection methods [23]. As a recently proposed method, spatial energy prior constrained maximum simplex volume (SENMAV) utilizes spatial energy prior obtained by spectral clustering as a regularization term of the maximum simplex volume model to constrain the selection of endmembers [24]. The spatial potential energy weighting endmember extraction (SPEW) adopts spectral angular distance and space potential energy derived from spectral clustering to weight pixels, and the weighted pixels that compose the maximum volume simplex are regarded as simplex [25]. K-medoids-based endmember extraction (KMEE) applies convex geometry for the selected two bands and uses K-medoids clustering to remove extra points, aiming to improve the noise robustness [26]. Spectral variability exists in HSIs due to the different scene components and illumination conditions, which blurs the identification of different types of ground objects and brings inaccurate endmember representation. Large-scope spectral variability refersto the phenomenon that homogeneous ground objects at different positions in an image show different spectral signatures. Local-scope spectral variability refers to the spectral difference exhibited by homogeneous ground objects in adjacent areas, which is small since the scene conditions may not change much in a local area. As most of the current methods directly find pixels in the image as endmembers, the influence of spectral variability cannot be avoided. Spatial purity-based endmember extraction (SPEE) attempts to address the problem by using the average of pure spatial neighborhoods to mitigate local-scope spectral variability and the merge of spatially related candidates to mitigate large-scope spectral variability [27]. In this paper, a novel quadratic clustering-based simplex volume maximization (CSVM) method is proposed to efficiently extract endmember, which exploits the advantages of clustering to make full use of spatial and spectral information and better deal with spectral variability. CVSM first adopts spatial clustering to segment HSI into homogeneous partitions, and the average of pure pixels obtained by spectral purity analysis in each partition is taken as representative endmember, which reduces the effect of local-scope spectral variability. Then CVSM performs spectral clustering to merge spectrally related representative endmembers to further alleviate large-scope spectral variability, and the final endmembers are determined by simplex with maximum volume. The CSVM shows its novelty in three ways. First, a local clustering criterion is proposed for spatial clustering based on simple linear iterative clustering (SLIC) to acquire a set of partitions with good spatial homogeneity, which combines spectral angle distance (SAD) and Euclidean distance (EDsp) to constraint spectral measurement.