Spectral-spatial nearest subspace classifier for hyperspectral image classification

ABSTRACT

Nearest subspace classifier (NSC) is a simple classifier that works on the assumption that samples from the same class must approximately lie on the same subspace. However, NSC only considers the spectral information and neglects the spatial information. Several studies have tried to eliminate this drawback of NSC. Still, they either are not robust against outliers in the neighbourhood of the test sample or have too many parameters which need to be tuned manually. In this paper, we present a practical and straightforward method that improves the existing NSC-based approaches that utilise spatial information. In our proposed method, in addition to the assumption in NSC that samples from the same class must approximately lie on the same subspace, we assume that spatially adjacent pixels quite likely belong to the same class. By combining these two assumptions, we conclude that spatially adjacent pixels must approximately lie on the same subspace as well. Then, we propose a method that analyses the closeness between two subspaces, where one subspace is the space spanned by the neighbourhood of the test sample and the other subspace is the space spanned by the within-class training samples. The proposed method has a closed-form solution, is easy to implement, and outperforms the existing solutions when the number of labelled training samples is scarce. The source code of the paper is available at https://github.com/kgtoker/SSNSC-for-HSIC.

Acknowledgement

The authors would like to thank ASELSAN Inc. for their support.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Nomenclature

${\mathit{x}}_{\mathit{i}\mathit{j}}$	=	$j^{th}$ training sample from class $i$
${\mathbf{X}}_{\mathbf{i}}$	=	Training samples belonging to the $i^{th}$ class
$\mathit{y}$	=	Test sample
$\mathbf{Y}$	=	Neighbouring pixels around the test pixel $\mathit{y}$
$Col({\mathbf{X}}_{\mathbf{i}})$	=	Column space of ${\mathbf{X}}_{\mathbf{i}}$
$Col(\mathbf{Y})$	=	Column space of $\mathbf{Y}$
${\mathit{r}}_{\mathit{i}}$	=	Minimum Euclidean distance between the test sample and Col(Xi)
$v_{x_i},v_{y_i},$	=	Maximally correlated vector pairs in subspaces $Col({\mathbf{X}}_{\mathbf{i}})$ and $Col(\mathbf{Y})$, respectively.
${\rho }_i$	=	Maximum correlation between the maximally correlated vector pairs located within $Col({\mathbf{X}}_{\mathbf{i}})$ and $Col(\mathbf{Y})$
$a_i,b_i,$	=	Coefficient vectors whose entries correspond to the weights of the samples in ${\mathbf{X}}_{\mathbf{i}}$ and $\mathbf{Y}$, respectively