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We consider Holder smoothness classes of surfaces for which we constructpiecewise polynomial approximation networks, which are graphs with polynomialpieces as nodes and edges between polynomial pieces that are in `goodcontinuation' of each other. Little known to the community, a similarconstruction was used by Kolmogorov and Tikhomirov in their proof of theircelebrated entropy results for Holder classes.  We show how to use such networks in the context of detecting geometricobjects buried in noise to approximate the scan statistic, yielding anoptimization problem akin to the Traveling Salesman. In the same context, wedescribe an alternative approach based on computing the longest path in thenetwork after appropriate thresholding.  For the special case of curves, we also formalize the notion of `goodcontinuation' between beamlets in any dimension, obtaining more economicalpiecewise linear approximation networks for curves.  We include some numerical experiments illustrating the use of the beamletnetwork in characterizing the filamentarity content of 3D datasets, and showthat even a rudimentary notion of good continuity may bring substantialimprovement.

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