Hyperspectral image classification is challenging because obtaining class labels for each pixel is expensive and time-consuming. Semi-supervised learning can improve classifiers by including unlabeled samples in training. Previous work for incorporating unlabeled data includes k-means clustering and Gaussian mixtures. These methods require enough training samples to estimate the distribution of the samples. One promising approach is support vector machines, which have been demonstrated to work well with high-dimensional hyperspectral data. This work proposes a kernel function that uses both labeled and unlabeled samples. It models the data as a union of multiple subspaces in a low rank coefficient matrix. The optimization enforces the low-rank constraint of the coefficient matrix through the nuclear norm. It includes a graph regularization term to maintain local structure between similar samples. The coefficient matrix defines a kernel for the low rank representation. Then the proposed semi-supervised kernel is a weighted summation of this kernel with the radial basis function kernel. A support vector machine is trained using this new kernel.

Experiments measure the accuracy of different kernel construction methods with support vector machines. The reference kernel methods are based on locally linear embedding, radial basis functions, k-means clustering, and Gaussian mixtures. The proposed kernel outperforms the other functions on both of the datasets (Indian Pines and University of Pavia). The results show that this method effectively combines labeled and unlabeled information for semi-supervised classification of hyperspectral data.

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