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Kernel-based methods have been proved to be an effective approach for face recognition in dealing with complex and nonlinear face image variations. While many encouraging results have been reported, the selection of kernel is rather ad hoc. This letter proposes a systematic method to construct a new kernel for kernel discriminant analysis, which is good for handling illumination problem. The proposed method first learns a kernel matrix by maximizing the difference between inter-class and intra-class similarities under the Lambertian model, and then generalizes the kernel matrix to our proposed ILLUM kernel using the scattered data interpolation technique. Experiments on the Yale-B and the CMU PIE face databases show that, the proposed kernel outperforms the popular Gaussian kernel in Kernel Discriminant Analysis and the recognition rate can be improved around 10%.