Networks of polynomial pieces with application to the analysis of point clouds and images

We consider H¨older 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 H¨older 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 illu...

• Type: Journal
• Year: 2010
• Where: JAT
• Authors: Ery Arias-Castro, Boris Efros, Ofer Levi
• Keywords: Approximation Networks, JAT 2010, Piecewise Polynomials, Polynomial Pieces