## Abstract

This paper presents a new robust, low computational cost technology for recognizing free-form objects in three-dimensional (3D) range data, or. in two-dimensional (2D) curve data in the image plane. Objects are represented by implicit polynomials (i.e. 3D algebraic surfaces or 2D algebraic curves) of degree greater than two, and are recognized by computing and matching vectors of their algebraic invariants (which are functions of their coefficients that are invariant to translations, rotations and general linear transformations). Such polynomials of the fourth degree can represent objects considerably more complicated than quadrics and super-quadrics, and can realize object recognition at significantly lower computational cost. Unfortunately, the coefficients of high degree implicit polynomials are highly sensitive to small changes in the data to which the polynomials are fit, thus often making recognition based on these polynomial coefficients or their invariants unreliable. We take two approaches to the problem: one involves restricting the polynomials to those which represent bounded curves and surfaces, and the other approach is to use Bayesian recognizers. The Bayesian recognizers are amazingly stable and reliable, even when the polynomials have unbounded zero sets and very large coefficient variability. The Bayesian recognizers are a unique interplay of algebraic functions and statistical methods. In this paper, we present these recognizers and show that they work effectively, even when data are missing along a large portion of an object boundary due, for example, to partial occlusion.

Original language | English |
---|---|

Pages (from-to) | 165-205 |

Number of pages | 41 |

Journal | Journal of Applied Statistics |

Volume | 21 |

Issue number | 1-2 |

DOIs | |

State | Published - 1 Jan 1994 |

Externally published | Yes |

### Bibliographical note

Funding Information:This work was partially supported by NSF Grant IRI-8714774 and NSF-DARPA Grant IRI-8905436.

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty