The Beltrami flow is an effective tool for dealing with images in many image-processing tasks. However, the important issue of how to set a proper embedding space is not solved in the Beltrami framework. We attempt to find a suitable embedding space by a nonlinear map. First, the image space is mapped into a high-dimensional feature space whose dimensionality may be infinite. It is found that directly dealing with the Beltrami flow in Hilbert space is impractical due to the unknown mapping function. Fortunately, using the well-known kernel methods, one can obtain the Beltrami flow in Hilbert space by performing inner products in the feature space. We refer to this flow as the kernel Beltrami flow. In addition, we also extend the kernel Beltrami flow to deal with vector-valued images. Finally, we show the effectiveness of the proposed method on gray-level and color images.
"Beltrami flow in Hilbert space with applications to image denoising," Journal of Electronic Imaging 21(4), 043019 (12 December 2012). https://doi.org/10.1117/1.JEI.21.4.043019