Supervised and transductive multi-class segmentation using p-Laplacians and RKHS methods

•We explore the difference between p-Laplacian and RKHS approach.•We prove a probability simplex constraint is automatically fulfilled when p = 2.•Different effects of the parameters of p-Laplacian and RKHS are explored.•New combined model is proposed and ADMM is used for computation.•These methods are applied to real medical data and different effects are presented.

This paper considers supervised multi-class image segmentation: from a labeled set of pixels in one image, we learn the segmentation and apply it to the rest of the image or to other similar images. We study approaches with p-Laplacians, Reproducing Kernel Hilbert Spaces (RKHSs) and combinations of both. In all approaches we construct segment membership vectors. In the p  -Laplacian model the segment membership vectors have to fulfill a certain probability simplex constraint. Interestingly, we could prove that this is not really a constraint in the case p=2p=2 but is automatically fulfilled. While the 2-Laplacian model gives a good general segmentation, the case of the 1-Laplacian tends to neglect smaller segments. The RKHS approach has the benefit of fast computation. We further consider an improvement by combining p-Laplacian and RKHS methods. Finally, we present challenging applications to medical image segmentation.