TY - JOUR

T1 - Networks of polynomial pieces with application to the analysis of point clouds and images

AU - Arias-Castro, Ery

AU - Efros, Boris

AU - Levi, Ofer

N1 - Funding Information:
The authors are grateful to the anonymous referees and editors for their careful reading of an earlier version of the manuscript and their constructive comments. The first author was partially supported by NSF grant DMS-0603890.

PY - 2010/1/1

Y1 - 2010/1/1

N2 - We consider Hölder 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 Hölder 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 data sets, and show that even a rudimentary notion of good continuity may bring substantial improvement.

AB - We consider Hölder 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 Hölder 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 data sets, and show that even a rudimentary notion of good continuity may bring substantial improvement.

KW - Beamlets

KW - Detection of filaments

KW - Extracting information from graphs

KW - Hölder smoothness classes

KW - Multiscale analysis

KW - Piecewise polynomials

UR - http://www.scopus.com/inward/record.url?scp=70450265713&partnerID=8YFLogxK

U2 - 10.1016/j.jat.2009.03.007

DO - 10.1016/j.jat.2009.03.007

M3 - Article

AN - SCOPUS:70450265713

SN - 0021-9045

VL - 162

SP - 94

EP - 130

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

IS - 1

ER -