## Abstract

This article pursues a statistical study of the Hough transform, the celebrated computer vision algorithm used to detect the presence of lines in a noisy image. We first study asymptotic properties of the Hough transform estimator, whose objective is to find the line that "best" fits a set of planar points. In particular, we establish strong consistency and rates of convergence, and characterize the limiting distribution of the Hough transform estimator. While the convergence rates are seen to be slower than those found in some standard regression methods, the Hough transform estimator is shown to be more robust as measured by its breakdown point. We next study the Hough transform in the context of the problem of detecting multiple lines. This is addressed via the framework of excess mass functionals and modality testing. Throughout, several numerical examples help illustrate various properties of the estimator. Relations between the Hough transform and more mainstream statistical paradigms and methods are discussed as well.

Original language | English |
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Pages (from-to) | 1908-1932 |

Number of pages | 25 |

Journal | Annals of Statistics |

Volume | 32 |

Issue number | 5 |

DOIs | |

State | Published - Oct 2004 |

## Keywords

- Breakdown point
- Computer vision
- Cube-root asymptotics
- Empirical processes
- Excess mass
- Hough transform
- Multi-modality
- Robust regression

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty