## Model-based probing strategies for convex polygons (1992)

Venue: | Comput. Geom. Theory Appl |

Citations: | 3 - 2 self |

### BibTeX

@ARTICLE{Joseph92model-basedprobing,

author = {Eugene Joseph and Eugene Joseph and Steven Sol Skiena and Steven Sol Skiena},

title = {Model-based probing strategies for convex polygons},

journal = {Comput. Geom. Theory Appl},

year = {1992},

volume = {2},

pages = {209--221}

}

### Years of Citing Articles

### OpenURL

### Abstract

We prove that n + 4finger probes are sufficient to determine the shape of a convex n-gon from a finite collection of models, improving the previous result of 2n + 1. Further, weshow that n − 1are necessary, proving this is optimal to within an additive constant. For line probes, we show that 2n + 4probes are sufficient and 2n − 3necessary. The difference between these results is particularly interesting in light of the duality relationship between finger and line probes.

### Citations

66 |
C.K.: Shape from probing
- Cole, Yap
- 1987
(Show Context)
Citation Context ...actile sensing is an important paradigm in robotics, and for reasons of economy and robustness is often used instead of more sophisticated vision systems to explore unknown environments. Cole and Yap =-=[1]-=- introduced the notion of a finger probe to model a tactile sensor, where a finger probe measures the first point of contact between a directed line l and an object P. Since Cole and Yap’s work, a sig... |

52 |
An optimal-time algorithm for slope selection
- Cole, Salowe, et al.
- 1989
(Show Context)
Citation Context ...Thus a given diagonal may define O(mn 2 log n) points in M. For any set of r points, the minimum slope α min can be determined in O(r log r) using the algorithm of Cole, Salowe, Steiger and Szemeredi =-=[15]-=-. Thus with O(n) probes and O(mn 2 log(n)log(mn 2 log n)) steps to determine α min for each probe, we have a time complexity of O(mn 3 log(n)log(mn 2 log n)) for the algorithm. We remark that a global... |

49 | Tactile Recognition and Localization Using Object Models: The Case of Polyhedra on a Plane - Gaston, Lozano-Pérez - 1984 |

23 | S.S.: Problems in geometric probing
- Skiena
- 1989
(Show Context)
Citation Context ...iven slope is returned. Li [10] showed that 3n + 1line probes are necessary and sufficient to determine an unknown convex n-gon. The problem of model-based determination with line probes was posed in =-=[11]-=-, and previously no non-trivial bounds were known. In this paper, wealso prove that 2n − 3line probes are necessary and 2n + 4probes sufficient, which are again tight to within an additive constant. T... |

12 |
Determining the shape of a convex n-sided polygon by using 2n+k tactile probes
- Bernstein
- 1986
(Show Context)
Citation Context ...d model-based tactile sensing, and shown that heuristics can be effective to distinguish between models. More efficient probing strategies can result for model-based determination problems. Bernstein =-=[7]-=- proved that 2n + 2finger probes are sufficient to determine a convex ngon from a finite collection of models Γ, which is improved to2n+1in[8]. Lyons and Rappaport [9] showed that m − 1probes are suff... |

12 |
Probing convex polygons with X-rays
- Edelsbrunner, Skiena
- 1988
(Show Context)
Citation Context ...red an exponential number of models. It would be interesting to know whether fewer models suffice. The problem of model-based probing strategies remains open for more advanced models. The x-ray probe =-=[16]-=- returns the length of intersection between a line and an object. Since an x-ray probe can be simulated by two finger probes, Theorem 2 can be used to prove a lower bound of ∼n/2 for model-based deter... |

9 | Probing Convex Polygons with Half-Planes - Skiena - 1991 |

7 | Grimson and Tomas Lozano-Perez. Model-based recognition and localization from sparse range or tactile data - L - 1984 |

4 | Grimson and Tom'as Lozano-P'erez. Localizing overlapping parts by searching the interpretation tree - L - 1987 |

3 |
Geometric Probing, Doctoral Dissertation
- Skiena
- 1988
(Show Context)
Citation Context ... result for model-based determination problems. Bernstein [7] proved that 2n + 2finger probes are sufficient to determine a convex ngon from a finite collection of models Γ, which is improved to2n+1in=-=[8]-=-. Lyons and Rappaport [9] showed that m − 1probes are sufficient to identify a convex polygon from a set of m models, if each model has a particular edge aligned with a known reference plane. This is ... |

2 |
Greschak, Reconstructing Convex Sets
- Paul
- 1985
(Show Context)
Citation Context ...gain tight to within an additive constant. These results are particularly interesting in light of the duality relationship, discovered independently by Dobkin, Edelsbrunner, and Yap [12] and Greschak =-=[13]-=-, that exists between line probes and finger probes which all pass through a single point, the origin. For all previous determination problems, the finger and line probing models have been identical i... |

1 |
Interactive Reconstruction from Projections
- Skiena
(Show Context)
Citation Context ... literature in geometric probing has developed, which studies the power of different sensor models for reconstructing geometric objects. The most up-to-date collection of results in probing appear in =-=[2]-=-. We seek probing strategies which completely determine a convex polygon in as few probes as possible. Cole and Yap [1] proved that 3n finger probes are necessary and sufficient to determine an unknow... |

1 |
An Efficient Algorithm for Indentifying Objects Using Robot Probes
- Lyons, Rappaport
(Show Context)
Citation Context ...etermination problems. Bernstein [7] proved that 2n + 2finger probes are sufficient to determine a convex ngon from a finite collection of models Γ, which is improved to2n+1in[8]. Lyons and Rappaport =-=[9]-=- showed that m − 1probes are sufficient to identify a convex polygon from a set of m models, if each model has a particular edge aligned with a known reference plane. This is a severe restriction, whi... |

1 |
redi, ‘‘The Maximum Number of Unit Distances in a Convex 0sfont]I]ize]u’n0rfont]-gon
- Fu
- 1987
(Show Context)
Citation Context ...ach second phase probe confirms either an edge or vertex ofP,6+n+n−2=2n+4probes are sufficient to determine P. Each of the m convex n-gon models defines at most O(n log n) diameters of a given length =-=[14]-=-. Thus a given diagonal may define O(mn 2 log n) points in M. For any set of r points, the minimum slope α min can be determined in O(r log r) using the algorithm of Cole, Salowe, Steiger and Szemered... |