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

Part of: Geometric probability and stochastic geometry Inference from stochastic processes

Published online by Cambridge University Press:  01 July 2016

Ery Arias-Castro ,
David L. Donoho ,
Xiaoming Huo  and
Craig A. Tovey
Ery Arias-Castro*
Affiliation:
Stanford University
David L. Donoho*
Affiliation:
Stanford University
Xiaoming Huo*
Affiliation:
Georgia Institute of Technology
Craig A. Tovey*
Affiliation:
Georgia Institute of Technology
*
∗ Postal address: Department of Statistics, Stanford University, Stanford, CA 94305, USA.
∗ Postal address: Department of Statistics, Stanford University, Stanford, CA 94305, USA.
∗∗ Postal address: School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0205, USA.
∗∗ Postal address: School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0205, USA.
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Abstract

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Given a class Γ of curves in [0, 1]2, we ask: in a cloud of n uniform random points, how many points can lie on some curve γ ∈ Γ? Classes studied here include curves of length less than or equal to L, Lipschitz graphs, monotone graphs, twice-differentiable curves, and graphs of smooth functions with m-bounded derivatives. We find, for example, that there are twice-differentiable curves containing as many as OP(n1/3) uniform random points, but not essentially more than this. More generally, we consider point clouds in higher-dimensional cubes [0, 1]d and regular hypersurfaces of specified codimension, finding, for example, that twice-differentiable k-dimensional hypersurfaces in Rd may contain as many as OP(nk/(2d-k)) uniform random points. We also consider other notions of ‘incidence’, such as curves passing through given location/direction pairs, and find, for example, that twice-differentiable curves in R2 may pass through at most OP(n1/4) uniform random location/direction pairs. Idealized applications in image processing and perceptual psychophysics are described and several open mathematical questions are identified for the attention of the probability community.

Keywords

Curve detectionfilament detectionε-entropyconfiguration functionconcentration of measurelongest increasing subsequencetraveling salesman problempattern recognition

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 62M40: Random fields; image analysis
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 37 , Issue 3 , September 2005 , pp. 571 - 603
Copyright
Copyright © Applied Probability Trust 2005 

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