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.