Manifold reconstruction and denoising from scattered data in high dimension

Scattered manifold-valued data approximation

We consider the problem of approximating a function f from an Euclidean domain to a manifold M by scattered samples [Formula: see text], where the data sites [Formula: see text] are assumed to be locally close but can otherwise be far apart points scattered throughout the domain. We introduce a natural approximant based on combining the moving least square method and the Karcher mean. We prove ...

Reconstruction of Kinematic Surfaces from Scattered Data ∗

Given a surface in 3-space or scattered points from a surface, we present algorithms for fitting the data by a surface which can be generated by a one–parameter subgroup of the group of similarities. These surfaces are general cones and cylinders, surfaces of revolution, helical surfaces and spiral surfaces.

Reconstruction of Multivariate Functions from Scattered Data

Fast and efficient narrow volume reconstruction from scattered data

We describe a fast and efficient numerical algorithm for the process of three-dimensional narrow volume reconstruction from scattered data in three dimensions. The present study is an extension of previous research [Li et al., Surface embedding narrow volume reconstruction from unorganized points, Comput. Vis. Image Underst. 121 (2014) 100–107]. In the previous work, we modified the original Al...

Manifold Denoising

We consider the problem of denoising a noisily sampled submanifold M in R, where the submanifold M is a priori unknown and we are only given a noisy point sample. The presented denoising algorithm is based on a graph-based diffusion process of the point sample. We analyze this diffusion process using recent results about the convergence of graph Laplacians. In the experiments we show that our m...