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 α -norm bounded by β. An asymptotic detection threshold exists in this problem; for a constant T−(α, β) > 0, if the number of points sampled from the curve is smaller than T−(α, β)n 1/(1+α) , reliable detection is not possible for large n. We describe a multiscale significantruns algorithm that can reliably detect concentration of data near a smooth curve, without knowing the smoothness information α or β in advance, provided that the number of points on the curve exceeds T * (α, β)n 1/(1+α) . This algorithm therefore has an optimal detection threshold, up to a factor T * /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 H0 whenever some such subgraph contains a path that connects many consecutive significant counts.

Polygonal Approximation of Point Sets

“…Using the estimate derived in equation (4), minimizing (12) in Θ is equivalent (in the MC setting discussed above) to maximizing the weighted marginal density…”

“…In [12] the number of points in discretely enumerated rectangular strips (e.g., neighborhoods of all possible segments with endpoints in some finite set) is counted. Then signal strips are selected based on the ratio of the number of points to the area.…”

“…Then strips are grouped to polylines based on proximity of their endpoints and their angles. The main difference of our approach to the approach in [12] is that we do not select the signal segments, but evaluate the existing structures selected by EM. This makes our computation more efficient, since we do not need to numerate all possible strips, and more accurate, since the line segments are optimally fitted to the data points in our approach.…”

Connect the dots: how many random points can a regular curve pass through?

“…E n , the set of edges in G n , links any two line segments in V n that are in "good continuation", which here means their directions are close enough [3]. Formally, two horizontal beamlets connected in E n are of the form [( …”

Distribution of the Length of the Longest Significance Run on a Bernoulli Net and Its Applications

“…Arias-Castro, Donoho, and Huo (2003) proposed a method called the multiscale significance run algorithm (MSRA) for the detection of curvilinear filaments in noisy images. The main idea is to construct a Bernoulli net.…”