Contextual SVM for hyperspectral classification using Hilbert Space Embedding

In this paper, a contextual Support Vector Machine (SVM) technique based on the principle of Hilbert Space Embedding (HSE) of a local hyperspectral data distribution into an Reproducing Kernel Hilbert Space (RKHS) is proposed to optimally exploit the spectral and local spatial information of the hyperspectral image. The idea of embedding is to map hyperspectral pixels in a local neighborhood into a single point in the RKHS that can uniquely represent those pixels collectively. Previously, the authors have employed an HSE called empirical mean map to build the contextual SVM. In this work, a weighted empirical mean map is utilized to exploit the similarities and variation in the local spatial information. For every pixel, a small set of the neighboring pixels in a hyperspectral image are mapped into an RKHS induced by a certain kernel (Eg. Gaussian RBF kernel) and then, the embedded point of these group of pixels is obtained by calculating the weighted empirical mean of these mapped points. The weights are determined based on the distance between the pixel in consideration and its neighbors. An SVM separating hyperplane is built to maximize the margin between classes formed by weighted empirical means. The proposed technique showed significant improvement over the existing contextual and composite kernels on two hyperspectral image data sets.