Cover Page

Geometric Modeling and Applications Set

coordinated by Marc Daniel

Volume 3

Geometric and Topological Mesh Feature Extraction for 3D Shape Analysis

Jean-Luc Mari

Franck Hétroy-Wheeler

Gérard Subsol

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Preface

Three-dimensional surface meshes, composed of collections of planar polygons, are the most common discrete representation of the surface of a virtual shape. These 3D surface meshes need to be inspected in order to understand or evaluate their overall structure or some details. This can be done by extracting relevant geometric or topological features. Such shape characteristics can simplify the way the object is looked at, can help recognition and can describe and categorize it according to specific criteria.

Shape characteristics can be defined in many ways. This book takes the point of view of discrete mathematics, which aims to propose discrete counterparts to concepts mathematically defined in continuous terms. More specifically, in this book, we review how standard geometric and topological notions of surfaces can be defined and computed for a 3D surface mesh, as well as their use for shape analysis. In particular, recent methods are described to extract feature lines having a meaning related to either geometry or topology. Differential estimators such as discrete principal curvatures are detailed as they play a critical role in the computation of salient structures. An emphasis is then placed on topology since the global structure and the connectivity of features play an important role in the understanding of a shape. Several applications are finally developed, showing that each of them needs specific adjustments to generic approaches. These applications are related to medicine, geology, botany and other sciences.

Focusing on shape features, the topic of this book is narrower but more detailed than other shape analysis books, which do not, or only briefly, refer to feature definition and computation. It is intended not only for students, researchers and engineers in computer science and shape analysis, but also numerical geologists, anthropologists, biologists and other scientists looking for practical solutions to their shape analysis, understanding or recognition problems. We hope that our book will be a useful review of existing work for all of them.

Finally, we would like to thank Marc Daniel for giving us the opportunity to write this book and Aldo Gonzalez-Lorenzo for his reading of chapter 2 and for his constructive remarks.

Jean-Luc MARI

Franck HÉTROY-WHEELER

Gérard SUBSOL

August 2019