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.

A multigrid framework for the one-dimensional (scalar) fixed-rate quantization problem is presented. The framework is based on the Lloyd-Max iterative process, which is a central building block in many quantization algorithms. This process iteratively improves a given initial solution and generally converges to a local minimum of the quantization distortion. Contrary to the classical Lloyd-Max process, the convergence of the multigrid algorithm is practically independent of the number of representation levels sought. Using this approach, a local minimum is reached at the cost of just a few Lloyd-Max iterations. The complexity of the proposed method is O(n) operations for the continuous case and 3N+O(n) for the discrete case, where n is the number of representation levels sought and N is the size of the discrete probability density function. In addition to its independent attributes, this work is a precursor to the more important vector quantization problem, for which a multiscale framework is also being developed.