Kernel-based approximation methods for curvatures estimations and applications on raw point clouds

The first part of this project focuses on numerical differentiation problems required in analysing raw point clouds. Input data are sets of three dimensional coordinates that model surfaces of objects and error in data is expected. Our goal is to apply the well- developed kernel-based approximation methods and theories to this particular setting in order to derive new algorithms for computing normal vectors and curvatures. For normal information, we will apply some W μ 2 (Ω)-convergent (μ ≥ 0) kernel-based discrete least- squares methods to cases where localized subset of the point cloud can be modeled by some level-set or parameterized functions. We will also develop adaptive data-dependent local neighborhood search algorithms by focusing on error estimates solely at the point of interest. Using the connection between normal vector and Laplacian Beltrami, a re- finement scheme will be developed for better accuracy. Next, our developed algorithms will be extended to approximate curvatures of the surfaces by using higher order nu- merical derivatives. Aiming to improve robustness, we will explore various kernel-based approaches to estimate curvatures using only first order numerical derivatives. This project will be concluded by applying the developed algorithms with two applications on geometry processing