A local linear algorithm for calculating the pre-image in kernel PCA

In this paper, we show a novel method for solving the pre-image problem. The most algorithms of dimension reduction are trying to maintain the geometric structures including the local and the global. Usually we assume that the manifolds are local linear. Given the assumption, we can obtain the reconstructing images by calculating the locally optimal linear fits. In the novel pre-image algorithm, firstly we find the k nearest neighbours of test samples in feature space, and then calculate the optimal approximation of test samples by their k nearest neighbours. And we give a simple stability analysis to this nonlinear iterative algorithm. The novel pre-image algorithm has been used in the second defect detection of notes´ printing. The effect of detection is very obviously in improving detection speed and accuracy.