INTRODUCTION

Hyperspectral images (HSIs) have become a valuable tool for monitoring the Earth surface since they provide a wealth of spectral information compared to traditional RGB images ^[1,2]. HSIs are commonly represented as a 3D data cube, where two dimensions (x,y) correspond to the spatial information and the third one, to the spectral domain (λ). In the 3D cube, each spatial position is represented as a vector, known as a spectral signature, whose values correspond to its intensity in each spectral band. Since the amount of radiation that each material reflects, absorbs, or emits varies according to the wavelength, the spectral signature of each pixel is used as a descriptor in a wide range of applications, such as classification ^[1], target detection ^[2], and spectral unmixing ^[3] among others ^[4-10].

The classification of hyperspectral images can be defined as the process of assigning each pixel to one class. This task is mainly carried out under supervised methods that know some spectral pixel labels which are used in the training stage ^[1]. Then, in the testing process, each unknown pixel is assigned the label of the spectral signature that presents the least spectral difference ^[11].

However, in some applications, the labeled samples are unavailable or difficult to acquire ^[12]. For that reason, unsupervised techniques such as clustering can be an effective alternative because they group a set of similar pixels without previous information of the data. As it is widely known, HSIs are high-dimensional data with large spectral variability and complex structure that make the clustering problem very challenging.

To date, some clustering algorithms have been used for HSIs. Specifically, they can be divided into four groups: (a) centroid-based clustering methods ^[13-15], (b) density-based methods ^[16], ^[17], (c) biological methods ^[18], ^[19], and (d) spectral-based methods ^[8], ^[20]. In general, spectral-based methods have achieved a good and robust performance for spectral images ^[9]. Said methods include two main steps: (i) building an adjacency matrix that describes the relationship between the spectral pixels and, then, (ii) applying centroid-based clustering methods to the Laplacian matrix formed by the adjacency matrix to obtain the clustering results.

Assuming that spectral signatures, which correspond to a land cover class, lie in the same low-dimensional subspace, a known spectral-based method called sparse subspace clustering (SSC) builds the adjacency matrix by expressing each spectral pixel as a linear combination of all spectral signatures of the scene. In addition, such solution is restricted to be sparse, which guarantees that the spectral signatures that correspond to those sparse coefficients belong to the same subspace^[8]. However, in the traditional SSC scheme, only spectral information is used to discriminate the different classes, thus ignoring the rich spatial information contained in the HSIs. To overcome those limitations, different methods have been proposed to incorporate a 2D or 3D spatial regularizer into the SSC algorithm ^[9], ^[21-23]. Said methods use 2D/3D smoothing filters in a reshaped coefficient matrix based on the fact that adjacent coefficients generally belong to the same class. Although those methods have shown good performance for HSIs, when the number of points increases (i.e., the spatial dimension of the image is large), they become computationally intractable ^[22], ^[24].

For that reason, this work proposes to remove some spectral pixels from the image in order to reduce the number of points to classify. Consequently, the computational time of the clustering task is reduced. Afterward, the clustering results of the incomplete pixels are assigned using a kind of filter that selects the predominant label in a given neighborhood. Specifically, this work proposes two schemes to remove some spectral pixels. The first one is based on spatial blue noise coding ^[22,25], and it allows uniform elimination in the spatial dimensions of the scene, avoiding clusters of removed pixels, which are properties desired in the filtering step. The second one is a sub-sampling scheme that eliminates every two contiguous pixels preserving the spatial structure of the images, which allows the use of SSC-based methods that employ the spatial information to improve the clustering result ^[21], ^[26]. The performance of the spectral image clustering framework proposed in this study is evaluated using three datasets, and a similar accuracy is obtained when up to 50% of the pixels are removed compared with the full data using the proposed designs. In addition, the proposed scheme is up to 7.9 times faster classifying the data sets compared to the full data sets.

2. SPARSE SUBSPACE CLUSTERING FOR HYPERSPECTRAL IMAGES

Let F∈R^L×MN be a hyperspectral image reorganized as F=[f ₍₁₎ ,…,f _(MN) ], where M and N represent the spatial dimensions, L stands for the number of spectral bands, and f _(k) ∈ R ^L denotes the spectral signature of the k -th pixel. The SSC method assumes that the HSIs lie in the union of n low-dimensional subspaces such that each subspace corresponds to a certain land-cover class ^[26]. In order to group spectral pixels, SSC assumes that each spectral signature f _(k) that corresponds to a specific land-cover class belongs to the same independent subspace ^[9]. Specifically, first, the SSC builds a sparse similarity matrix, which describes the relationships between pixels exploiting the fact that each spectral signature is represented as a linear or affine combination of few pixels in the same subspace ^[8]. This affinity matrix is built using the coefficient matrix C ∈ R^MN×MN obtained from the sparse optimization problem, which is modeled as follows (1):

where, -norm is the number of nonzero elements of C; Z , the error matrix; and λ, a regularization parameter for the sparsity coefficient and noise level trade-off. The constraint diag (C)=0 is used to eliminate the trivial ambiguity where a point is represented by itself, and the constraint C ^T 1=1 ensures that it can work even in case of affine subspaces ^[8], ^[24]. After solving (1), the k-th column of C , which corresponds to a data point f _(k) that lies in a d _i dimensional subspace S _i , is expected to have only d _i non-zero elements. However, since ℓ₀-norm is an NP-hard problem, the relaxed -norm is usually adopted to relax the problem as (2):

where, the -norm promotes sparsity, i.e., each spectral signature is represented by few pixels ^[8]. This formulation can be efficiently solved using the alternating direction method of multipliers (ADMM) with a computational complexity O(L(MN) ² ) ^[27]. After solving (2), each column of C is normalized to better deal with different norms of the spectral signatures as c _j =c _j /||c _j ||∞. Subsequently, C is used to construct the non-negative similarity matrix W as (3):

where, W _i,j represents the similarity between the i-th and j-th pixels. Finally, to obtain the segmentation of the data into different subspaces, a weighted graph G=(ν,ε,W) is built, where ν denotes the set of NM nodes of the graph that correspond to NM pixels of the image; ε ⊆ ν×ν, the set of edges between the node; and W ∈ R ^MN×MN , the weights of the edges. Then, the data is clustered by applying spectral clustering methods ^[8,28] over the graph G which, naively implemented, has a computational complexity O((MN) ³ ) ^[8].

3. SPARSE SUBSPACE CLUSTERING FOR HYPERSPECTRAL IMAGES WITH INCOMPLETE PIXELS

Note that this sparse representation builds the C matrix using only spectral information, i.e., the sparse representation of the k-th pixel is the same regardless of its spatial position. Therefore, considering that land-cover materials are distributed homogeneously (i.e., contiguous pixels in a spectral image generally belong to the same material), different constraints have been applied to C in order to incorporate the spatial information of the scene ^[9], ^[21]-[23], ^[26], ^[29]. Specifically, different smoothing filters are applied at each iteration of the ADMM algorithm to a reshaped version of the C matrix in order to obtain the same value in a neighborhood of pixels ^[9], ^[21]-[23]. Furthermore, the use of smoothing operators on the image has been explored for SSC ^[23], ^[29]. Although said method has shown good results in non-supervised classification tasks, when the spatial resolution of those images increases, they become computationally intractable ^[24].

For that reason, we propose to remove certain pixels from the image before solving step (2) and the spectral clustering method in order to reduce the computational time of those steps. Mathematically, the incomplete image can be represented as (4):

where, F ̃ is the incomplete image, and H ∈ {0,1} ^MN×P is the selecting matrix that has only one non-zero value per row, with P as the number of preserved pixels. Therefore, the SSC model is applied to obtain the clustering results for those pixels. This segmentation can be represented in s ∈ R^P, which is a vector with the labels of the preserved pixels. In order to obtain the incomplete labels, let s ̂= Hs be a vector where the missing positions are present. Then, this vector is reorganized in a matrix S ∈ R ^M×N (see Fig. 1), where the incomplete labels are obtained applying a kind of filter that selects the predominant label in a given window; in this case, of a 3 ×3 size. Specifically, when χ={1, ...,ı} denotes the set of ı class labels, those missing values are given by (5):

where w=3 is the size of the window; ⌊.⌋, the floor function; and δ, the Kronecker delta function, which is 1 if the arguments are equal and 0 otherwise.

Fig. 1 represents the step-by-step of the proposed method and Algorithm 1 summarizes the computations described above. Note that the quality of the proposed methodology mainly depends on the structure of matrix H. For that reason, the two following sub-sections propose different strategies to design selecting matrix H in an efficient way.

Algorithm 1: Spectral Subspace Clustering for incomplete hyperspectral image

Input: F, n, λ, H, p