A short-time Beltrami kernel for smoothing images and manifolds

We introduce a short-time kernel for the Beltrami image enhancing flow. The flow is implemented by "convolving"the image with a space dependent kernel in a similar fashion to the solution of the heat equation by a convolution with a Gaussian kernel. The kernel is appropriate for smoothing regular (flat) 2-D images, for smoothing images painted on manifolds, and for simultaneously smoothing images and the manifolds they are painted on. The kernel combines the geometry of the image and that of the manifold into one metric tensor, thus enabling a natural unified approach for the manipulation of both. Additionally, the derivation of the kernel gives a better geometrical understanding of the Beltrami flow and shows that the bilateral filter is a Euclidean approximation of it. On a practical level, the use of the kernel allows arbitrarily large time steps as opposed to the existing explicit numerical schemes for the Beltrami flow. In addition, the kernel works with equal ease on regular 2-D images and on images painted on parametric or triangulated manifolds. We demonstrate the denoising properties of the kernel by applying it to various types of images and manifolds.