Geometric and Topological Mesh Feature Extraction for 3D Shape Analysis

Jean-Luc Mari, Franck Hétroy-Wheeler, Gérard Subsol

• Publisher: Wiley-ISTE
• Year: 2019
• Print ISBN:
  9781786300416
• Edition: 1
• eBook ISBN:
  9781119681045
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

Three-dimensional surface meshes are the most common discrete representation of the exterior of a virtual shape. Extracting relevant geometric or topological features from them can simplify the way objects are looked at, help with their recognition, and facilitate description and categorization according to specific criteria. This book adopts the point of view of discrete mathematics, the aim of which is to propose discrete counterparts to concepts mathematically defined in continuous terms. It explains how standard geometric and topological notions of surfaces can be calculated and computed on a 3D surface mesh, as well as their use for shape analysis. Several applications are also detailed, demonstrating that each of them requires specific adjustments to fit with generic approaches. The book 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.

1
Geometric Features based on Curvatures

1.1. Introduction

In 1983, the authors of [HAR 83] proposed to use differential geometry (i.e. the mathematical analysis of the geometry of 3D curves and surfaces) to describe the shape of a discrete surface (in their case, defined as an elevation image). For this purpose, they introduced the topographic primal sketch, which is composed of feature points (as peaks, pits or saddles), feature lines (as ridges or ravines) and feature surface patches (which may be flat). Two years later, the authors of [BRA 85] described another differential geometry-based framework to analyze the shape of 3D surfaces and they applied it on several real examples. They also used feature lines (as curvature lines) and planar surface patches. During the same period, in his dissertation [BES 88a], Besl proposed to compute the differential parameters of a surface in order to create a HK-sign map that allows the segmention of a surface into homogeneous regions where some basic geometric primitives can be fitted.

Since then, extensive research has been done on the extraction and the application of geometric features based on differential geometry. In section 1.2, we propose an overview in a nutshell (but with the mathematical formulas) of the differential geometry of surfaces. In section 1.3, we present the main methods that allow us to efficiently compute the differential parameters on a discrete 3D mesh. In sections 1.4 and 1.5, we focus respectively on line and surface features.

1.2. Some mathematical reminders of the differential geometry of surfaces

The following reminders are essentially based on the book [HOS 92]. More details can be found in many mathematics books such as [WEA 55], [CAR 76], [SPI 99], [TOP 05] or [PAT 10].

1.2.1. Fundamental forms and normal curvature

Let 𝚺 be a surface of class C^k with k ≥ 3, which is parameterized by (u, v). 𝚺 is then defined by the set of points P (u, v) = {x(u, v), y(u, v), z (u,v)}.

An infinitesimal displacement around the point P will be modeled as the vector dP, which will be in the tangent plane defined by the frame

:

The norm of this displacement can be computed as:

If we define the terms E, F and G as:

we get:

[1.1]

This expression is called the first fundamental form of the surface. Its coefficients E, F and G enable us to calculate dP that defines the lengths of local curves on the surface around P. They are also used to define the area of local regions.

Let us now define n as the normal vector at P, i.e. the vector going through P and orthogonal to the tangent plane. For any unit vector t in the tangent plane, we can define the plane П_t that goes through P and contains n and t. П_t is called a normal plane to Σ and cuts Σ along the plane curve

, which is called the normal section along the direction t (see Figure 1.1).

Figure 1.1. At a point P, the normal plane П_t is defined by the normal vector n and the tangent vector t. П_t cuts Σ along a p...

Table of contents

• Cover
• Table of Contents
• Preface
• Introduction
• 1 Geometric Features based on Curvatures
• 2 Topological Features
• 3 Applications
• Conclusion
• References
• Index
• End User License Agreement