Least Squares Orthogonal Distance Fitting of Curves and Surfaces in Space : Lecture Notes in Computer Science - Sung Joon Ahn

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Least Squares Orthogonal Distance Fitting of Curves and Surfaces in Space

By: Sung Joon Ahn

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Paperback | 1 January 2008

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Due to the continuing progress of sensor technology, the availability of 3-D c- eras is already foreseeable. These cameras are capable of generating a large set of measurement points within a very short time. There is a variety of 3-D camera - plications in the ?elds of robotics, rapid product development and digital factories. In order to not only visualize the point cloud but also to recognize 3-D object m- els from the point cloud and then further process them in CAD systems, ef?cient and stable algorithms for 3-D information processing are required. For the au- matic segmentation and recognition of such geometric primitives as plane, sphere, cylinder, cone and torus in a 3-D point cloud, ef?cient software has recently been developed at the Fraunhofer IPA by Sung Joon Ahn. This book describes in detail the complete set of 'best-?t' algorithms for general curves and surfaces in space which are employed in the Fraunhofer software.

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You Can Find This Book In

Non-Fiction Mathematics Algebra Computing & I.T.Graphical & Digital Media Applications Geometry Analytic Geometry Applied Mathematics Computer Programming & Software Development Algorithms & Data Structures Mathematical Foundation Computer Science

Mathematical Theory of Computation Graphics Programming Calculus & Mathematical Analysis Numerical Analysis

Introduction	p. 1
Curves and Surfaces in Space	p. 3
Mathematical Description	p. 3
Rigid Body Motion of Model Features in Space	p. 5
Model Hierarchy	p. 6
Curve and Surface Fitting	p. 9
Applications of Curve and Surface Fitting	p. 9
Algebraic Fitting Vs. Geometric Fitting	p. 9
State-of-the-Art Orthogonal Distance Fitting	p. 13
ISO 10360-6 and Requirements of CMM Software Tools	p. 14
Least-Squares Orthogonal Distance Fitting	p. 17
Moment Method for Line and Plane Fitting	p. 17
Line Fitting	p. 17
Plane Fitting	p. 20
Relationship Between Line and Plane Fitting	p. 21
Generalized Orthogonal Distance Fitting	p. 22
Problem Definition	p. 22
Point-to-Point Matching	p. 24
Curve and Surface Fitting	p. 24
Template Matching	p. 26
Orthogonal Distance Fitting Algorithms	p. 27
Distance-Based Algorithm	p. 27
Coordinate-Based Algorithm	p. 28
Model Fitting with Parameter Constraints	p. 29
Parameter Test	p. 30
Application to Circle and Sphere Fitting	p. 31
Orthogonal Distance Fitting of Implicit Curves and Surfaces	p. 35
Minimum Distance Point	p. 36
Generalized Newton Method	p. 37
Method of Lagrangian Multipliers	p. 39
Verification of the Minimum Distance Point	p. 40
Acceleration of Finding the Minimum Distance Point	p. 41
Orthogonal Distance Fitting	p. 42
Distance-Based Algorithm	p. 42
Coordinate-Based Algorithm	p. 44
Comparison of the Two Algorithms	p. 46
Fitting Examples	p. 47
Superellipse Fitting	p. 47
Cone Fitting	p. 48
Torus Fitting	p. 50
Superellipsoid Fitting	p. 51
Orthogonal Distance Fitting of Parametric Curves and Surfaces	p. 55
Minimum Distance Point	p. 56
Newton Method	p. 57
Levenberg-MarquardtAlgorithm	p. 58
Initial Values	p. 60
Acceleration of Finding the Minimum Distance Point	p. 62
Orthogonal Distance Fitting	p. 63
Algorithm I (ETH)	p. 63
Algorithm II (NPL, FhG)	p. 65
Algorithm III (FhG)	p. 67
Comparison of the Three Algorithms	p. 69
Fitting Examples	p. 70
Helix Fitting	p. 70
Ellipsoid Fitting	p. 72
Object Reconstruction from Unordered Point Cloud	p. 75
Applications of Object Reconstruction	p. 76
Semi-automatic Object Recognition	p. 77
Segmentation, Outlier Elimination, and Model Fitting	p. 78
Domain Volume for Measurement Points	p. 79
Experimental Results with Real 3-D Measurement Points	p. 80
3-D Point Cloud from Stripe Projection Method	p. 80
3-D Point Cloud from Laser Radar	p. 83
Conclusions	p. 85
Summary	p. 85
Future Work	p. 87
References	p. 93
Index	p. 97
Implementation Examples	p. 101
Implicit 2-D Ellipse (Chap.3)	p. 101
Parametric 3-D Ellipse (Chap.4)	p. 103
CMM Software Tools Fulfilling ISO 10360-6	p. 107
Curves and Surfaces Defined in ISO 10360-6	p. 107
Competent Parameterization	p. 107
Role of the Mass Center	p. 110
Rotation Matrix	p. 111
Parameter Range	p. 114
Minimum Distance Point and FHG/XHG Matrix	p. 115
2-D Line	p. 116
3-D Line	p. 116
Plane	p. 117
2-D Circle	p. 117
3-D Circle	p. 118
Sphere	p. 118
Cylinder	p. 119
Cone	p. 119
Torus	p. 120
FHG Matrix of Superellipse and Superellipsoid	p. 123
Superellipse	p. 123
Superellipsoid	p. 124
Table of Contents provided by Publisher. All Rights Reserved.

Least Squares Orthogonal Distance Fitting of Curves and Surfaces in Space

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