Probabilistic Kernel PCA and its Application to Statistical Modeling and Inference

Abstract

We present a method of density estimation that is based on an extension of kernel PCA to a probabilistic framework. Given a set of sample data, we assume that this data forms a Gaussian distribution, not in the input space but upon a nonlinear mapping to an appropriate feature space. As with most kernel methods, this mapping can be carried out implicitly. Due to the strong nonlinearity, the corresponding density estimate in the input space is highly non-Gaussian. Numerical applications on 2-D data sets indicate that it is capable of approximating essentially arbitrary distributions. Beyond demonstrating applications on 2-D data sets, we apply our method to high-dimensional data given by various silhouettes of a 3-D object. The shape density estimated by our method is subsequently applied as a statistical shape prior to variational image segmentation. Experiments demonstrate that the resulting segmentation process can incorporate highly accurate knowledge on a large variety of complex real-world shapes. It makes the segmentation process robust to misleading information due to noise, clutter and occlusion.