We review heat kernel smoothing techniques for denoising and regressing data in irregularly shaped domains embedded in Euclidean spaces. This is a problem often encountered in functional data analysis and medical imaging. In this chapter, we present a unified mathematical framework based on the eigenfunctions of the Laplace-Beltrami operators defined on irregular domains. Numerical implementati...

We prove a variable coefficient version of the square function estimate Guth–Wang–Zhang. By classical argument Mockenhaupt–Seeger–Sogge, it implies full range sharp local smoothing estimates for 2 + 1 $2+1$ -dimensional Fourier integral operators satisfying cinematic curvature condition. In particular, conjecture wave equations on compact Riemannian surfaces is settled.

For an arbitrary Riemannian manifold X and Hermitian vector bundles E and F over X we define the notion of the normal symbol of a pseudodifferential operator P from E to F . The normal symbol of P is a certain smooth function from the cotangent bundle T ∗X to the homomorphism bundle Hom(E,F ) and depends on the metric structures, resp. the corresponding connections on X, E and F . It is shown t...

Smoothing filter is the method of choice for image preprocessing and pattern recognition. We present a new concurrent method for smoothing 2D object in binary case. Proposed method provides a parallel computation while preserving the topology by using homotopic transformations. We introduce an adapted parallelization strategy called split, distribute and merge (SDM) strategy which allows effici...