Adaptive multiscale detection of filamentary structures in a background of uniform random points

Abstract

We are given a set of $n$ points that might be uniformly distributed in the unit square $[0,1{]}^2$. We wish to test whether the set, although mostly consisting of uniformly scattered points, also contains a small fraction of points sampled from some (a priori unknown) curve with $C^{\alpha }$-norm bounded by $\beta$. An asymptotic detection threshold exists in this problem; for a constant $T_{-}(\alpha ,\beta )>0$, if the number of points sampled from the curve is smaller than $T_{-}(\alpha ,\beta )n^{1/(1+\alpha )}$, reliable detection is not possible for large $n$. We describe a multiscale significant-runs algorithm that can reliably detect concentration of data near a smooth curve, without knowing the smoothness information $\alpha$ or $\beta$ in advance, provided that the number of points on the curve exceeds $T_{\ast }(\alpha ,\beta )n^{1/(1+\alpha )}$. This algorithm therefore has an optimal detection threshold, up to a factor $T_{\ast }/T_{-}$. At the heart of our approach is an analysis of the data by counting membership in multiscale multianisotropic strips. The strips will have area $2/n$ and exhibit a variety of lengths, orientations and anisotropies. The strips are partitioned into anisotropy classes; each class is organized as a directed graph whose vertices all are strips of the same anisotropy and whose edges link such strips to their ``good continuations.'' The point-cloud data are reduced to counts that measure membership in strips. Each anisotropy graph is reduced to a subgraph that consist of strips with significant counts. The algorithm rejects ${\mathbf{H}}_0$ whenever some such subgraph contains a path that connects many consecutive significant counts.

Published at http://dx.doi.org/10.1214/009053605000000787 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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