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In this paper, we present a nonlinear kernel-based target detection algorithm for hyperspectral images. The proposed approach relies on the sparse representation of an unknown sample with respect to both background and target training samples in a high-dimensional feature space induced by a kernel function. The sparse representation vector can be recovered via a kernelized greedy algorithm, where the kernel trick is used to avoid explicit evaluations of the data in the feature space. The spatial smoothness in hyperspectral images is also taken into account through a kernelized joint sparsity model. The detection decision is then made by comparing the reconstruction accuracy in terms of the background and target sub-dictionaries. The detection algorithm in a high-dimensional feature space implicitly exploits the higher-order structure (correlations) within the data which cannot be captured by a linear model. Therefore, projecting the pixels into a kernel feature space and kernelizing the linear sparse representation model improves the separability between the background and target classes, leading to a more accurate detection performance.