Networks of Polynomial Pieces with Application to the Analysis of Point Clouds and Images

TL;DR

This paper introduces networks of polynomial pieces for surface approximation and object detection in noisy data, extending classical ideas with new applications to point clouds and images, including numerical experiments.

Contribution

It develops polynomial approximation networks based on good continuation, applying them to geometric object detection and curve approximation in high-dimensional data.

Findings

Networks improve object detection in noisy data

Polynomial and beamlet networks effectively characterize filamentarity

Good continuation enhances approximation quality

Abstract

We consider Holder smoothness classes of surfaces for which we construct piecewise polynomial approximation networks, which are graphs with polynomial pieces as nodes and edges between polynomial pieces that are in `good continuation' of each other. Little known to the community, a similar construction was used by Kolmogorov and Tikhomirov in their proof of their celebrated entropy results for Holder classes. We show how to use such networks in the context of detecting geometric objects buried in noise to approximate the scan statistic, yielding an optimization problem akin to the Traveling Salesman. In the same context, we describe an alternative approach based on computing the longest path in the network after appropriate thresholding. For the special case of curves, we also formalize the notion of `good continuation' between beamlets in any dimension, obtaining more economical piecewise linear approximation networks for curves. We include some numerical experiments illustrating the use of the beamlet network in characterizing the filamentarity content of 3D datasets, and show that even a rudimentary notion of good continuity may bring substantial improvement.