Surface reconstruction with higher-order smoothness

Abstract

This work proposes a method to reconstruct surfaces with higher-order smoothness from noisy 3D measurements. The reconstructed surface is implicitly represented by the zero level-set of a continuous valued embedding function.The key idea is to find a function whose higher-order derivatives are regularized and whose gradient is best aligned with a vector field defined by the input point set. In contrast to methods based on the first-order variation of the function that are biased towards the constant functions and treat the extraction of the isosurface without aliasing artifacts as an afterthought, we impose higher-order smoothness directly on the embedding function. After solving a convex optimization problem with a multi-scale iterative scheme, a triangulated surface can be extracted using the marching cubes algorithm. We demonstrated the proposed method on several data sets obtained from raw laser-scanners and multi-view stereo approaches. Experimental results confirm that our approach allows us to reconstruct smooth surfaces from points in the presence of noise, outliers, large missing parts and very coarse orientation information.

Variational implicit point set surfaces

“…Most implicit methods require the input points to be equipped with oriented normals. The Poisson reconstruction method [Kazhdan et al 2006] and its variants [Kazhdan and Hoppe 2013;Manson et al 2008;Pan and Skala 2012;Taubin 2012] seek an "indicator function" that is 1 (resp. 0) in the interior (resp.…”

A Comparative Study of LOWESS and RBF Approximations for Visualization

“…choosing 𝑑 = 1, appears to strike a good balance between computational simplicity and the flexibility needed to reproduce patterns in the data. In such a case, we can rewrite (9) as:…”

“…Approximation methods of values 𝒚 𝑖 in the given {〈𝒙 𝑖 , 𝒚 𝑖 〉} 1 𝑁 data set lead to a smooth function which minimizes the difference between given data and the determined function [13]. It can be used for visualization of noisy data [1,2], visualization of the basic shape of measured/calculated data [9], for prediction, and other purposes. Many methods have been described together with their properties.…”