## Medial Representations: Mathematics, Algorithms and ApplicationsKaleem Siddiqi, Stephen Pizer The last half century has seen the development of many biological or physical t- ories that have explicitly or implicitly involved medial descriptions of objects and other spatial entities in our world. Simultaneously mathematicians have studied the properties of these skeletal descriptions of shape, and, stimulated by the many areas where medial models are useful, computer scientists and engineers have developed numerous algorithms for computing and using these models. We bring this kno- edge and experience together into this book in order to make medial technology more widely understood and used. The book consists of an introductory chapter, two chapters on the major mat- matical results on medial representations, ?ve chapters on algorithms for extracting medial models from boundary or binary image descriptions of objects, and three chapters on applications in image analysis and other areas of study and design. We hope that this book will serve the science and engineering communities using medial models and will provide learning material for students entering this ?eld. We are fortunate to have recruited many of the world leaders in medial theory, algorithms, and applications to write chapters in this book. We thank them for their signi?cant effort in preparing their contributions. We have edited these chapters and have combined them with the ?ve chapters that we have written to produce an integrated whole. |

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### Contents

1 | |

Benjamin B Kimia | 36 |

Geometry and Medial Structure | 69 |

Gunilla Borgefors | 125 |

Discrete Skeletons from Distance Transforms in 2D and 3D | 155 |

Martin Styner | 191 |

Edward Chaney | 223 |

P Thomas Fletcher | 240 |

A Notation 353 | 352 |

Chapter 1 | 356 |

Chapter 4 | 358 |

Chapter 5 | 359 |

Chapter 6 | 360 |

Chapter 9 | 361 |

Chapter 10 | 362 |

Glossary | 363 |

Statistical Applications with Deformable MReps | 269 |

Formation | 278 |

Juan Zhang | 309 |

From the Inﬁnitely Large to the Inﬁnitely Small | 327 |

397 | |

433 | |

### Other editions - View all

Medial Representations: Mathematics, Algorithms and Applications Kaleem Siddiqi,Stephen Pizer No preview available - 2010 |

Medial Representations: Mathematics, Algorithms and Applications Kaleem Siddiqi,Stephen Pizer No preview available - 2008 |

### Common terms and phrases

algorithm approximation average outward ﬂux Blum medial axis boundary points Chapter circle classiﬁcation components compute coordinates corresponding curve skeleton Damon deﬁned deﬁnition deformation Delaunay triangulation denote density described discrete distance transform divergence theorem efﬁcient eigenvalues end points Euclidean distance example ﬁrst geodesic geometric global graph grassﬁre ﬂow Hausdorff distance integral Kimia loci m-rep medial atoms medial locus medial points medial representation medial sheet medial surface methods multivalued function nearly-thin neighbors object angle object boundary parameter pixels Pizer pre-symmetry set principal principal curvatures properties pruning radial curvature radial ﬂow radial vector ﬁeld radius function region removed sampling satisﬁes Section segmentation shape operator shown Siddiqi signiﬁcance skeletal branches skeletal structure smooth speciﬁc sphere spokes statistics subﬁgure subset surface skeleton symmetry set tangent theorem thinning topological skeleton topology triangles Voronoi diagram Voronoi edges Voronoi skeleton voxels Y-junction