# Multiscale Superpixelwise Locality Preserving Projection for Hyperspectral Image Classification
## 1 Introduction  
Hyperspectral image (HSI) classification has been a research hotspot over recent years, since it plays a critical role in military target detection, precision agriculture, mine exploration, and many other applications [1; 2; 3]. With abundant spectral information, HSIs show great potential in identifying different ground objects of interest. However, hundreds of spectral bands in an HSI also bring about some problems, such as heavy computation burdens and the Hughes phenomenon which means that a large amount of training HSI pixels are required to maintain statistical confidence in the HSI classification task [4; 5]. Such problems usually hinder HSI classifiers from achieving excellent performance in real applications. In fact, HSI spectral bands are strongly correlated; hence, the spectral signature of each HSI pixel can be represented by only a few features [6; 7; 8]. Therefore, one often-used strategy to overcome the dimensionality dilemma mentioned above is dimensionality reduction which aims to represent the high-dimensional HSI data in a low-dimensional space without losing important information for discriminating tasks. Numerous dimensionality reduction methods have been developed, which can be roughly categorized into two groups: feature transform and feature selection [9; 10]. Feature transform projectsthe original HSI data into an appropriate low-dimensional space, while feature selection chooses the most representative bands from the HSI. Feature transform gains the advantage over feature selection that it has the potential to maintain the original information in the high-dimensional data during dimensionality reduction and thus generate more discriminative features [11; 12]. Principle component analysis (PCA) and linear discriminant analysis (LDA) are two typical feature transform methods. PCA attempts to map the data along the directions of maximal variance [13]. LDA tends to maximize between-class distances and minimize within-class distances in the meantime [14]. Both PCA and LDA just assume that the information contained in high-dimensional data lies on a linear low-dimensional space, whereas nonlinear structures are often exhibited in real HSI data. To cope with the nonlinearities, manifold learning methods were developed. Manifold learning assumes that the high- dimensional data actually lie on a low-dimensional manifold structure, which can be parameterized with a group of identifiable coordinates. One of the most popular manifold methods is locality preserving projection (LPP) which builds a graph to capture geometric structures of data and subsequently establishes a projection from the original data space to the low-dimensional space [15; 16; 17; 18; 19]. Other manifold learning approaches include locally linear embedding (LLE) which supposes that the structure represented by the linear combinations of the data's nearest neighbors is unchanged in both the high-dimensional and the associated low-dimensional spaces [20], isometric mapping (ISOMAP) which utilizes the geodesic distance to perform low-dimensional embedding [21], and local tangent space alignment (LTSA) which aims to recover the intrinsic manifold of data by aligning the local tangent space of each pixel [22]. Irregular spatial dependency is an important characteristic of HSIs, which is caused by the usual occurrences of complex irregular ground objects in HSI scenes. The dependency brings about spatially local homogeneous subregions in different shapes and sizes in an HSI. Such subregions can be detected effectively with appropriate techniques such as superpixel [23; 24; 25], where pixels in a homogeneous subregion have similar spectral properties but vary relatively significantly across different subregions. Intuitively, such homogeneous subregions would result in local homogeneities on an HSI manifold. However, conventional manifold learning-based HSI dimensionality reduction methods, such as LPP, directly apply a unified projection on the entire HSI, missing those local homogeneities on the HSI manifold.