Curves and Surfaces

Name: Curves and Surfaces | 7th International Conference, Avignon, France, June 24-30, 2010, Revised Selected Papers
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7th International Conference, Avignon, France, June 24-30, 2010, Revised Selected Papers

Jean-Daniel Boissonnat Patrick Chenin Albert Cohen Christian Gout Tom Lyche Marie-Laurence Mazure Larry Schumaker(Herausgeber*in)

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Erschienen am 6. Januar 2012

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Title Page
Preface
Table of Contents
Exact Medial Axis Computation for Triangulated Solids with Respect to Piecewise Linear Metrics
Introduction
Preliminaries
Unit Balls and Metrics
Maximal and Almost Maximal Balls
Types of Contact
Medial Axis
Contacts and Contact Arrangements
Contacts
Projections
Contact Arrangements
Computing the Contact Arrangements
Outline of the Algorithm
Constructing Almost Maximal Balls
Finding Projection Lines
Summary
Assembling the Medial Axis and Offset Computation
Assembling the Medial Axis from Its Projections
Offset Computation
Computing Time and Size of the Medial Axis
Convergence
Planar Domains
Towards a Convergence Proof for the 3D Case
Concluding Remarks
References
Exact Medial Axis Computation for Circular Arc Boundaries
Introduction
Exact Circular Arc Boundary
The Divide-and-Conquer Algorithm
Constructing Dividing Disks
Bisector Computation and Point Location
Bisector Computation
Medial Axis Representation and Point Location
Confining the Partial Axis
Center Points of Tritangent Circles
Partial Axis Construction
ECAB Construction
Conclusion
References
Complex B´ezier Curves and the Geometry of Polynomials
Introduction
Complex Bézier Curves and Blossom
Complex Bézier Curves and Complex de Casteljau Algorithm
de Boor-Fix Bracket and Blossom
Polar Derivative of a Polynomial
Geometry of the Control Polygon
Polynomial with Roots on a Circle
Polynomial with Roots Inside a Disk
Polynomial with Roots Outside a Disk
Bernstein-Type Inequalities
Conclusions
References
The Shape of Conchoids to Plane Algebraic Curves
Introduction
Preliminaries
Local Shape
Exterior and Interior Conchoids
Local Analysis around the Focus
Global Questions
Conclusions and Further Work
References
Estimation of Integral Properties of a Planar Closed Curve Based on a Quadratic Spline Quasi-Interpolant
Introduction
Periodic Quadratic Spline Quasi-Interpolant
Length and Center of Gravity
Approximations of L() and G
Convergence Orders
Moment of Inertia
Approximate Moment of Inertia
Convergence Order
Area
Approximate Area
Convergence Order
Numerical Results
References
Design of Multiresolution Operators Using Statistical Learning Tools: Application to Compression of Signals
Introduction and Review
Learning-Based Multiresolution in Harten's Framework
Generalized Wavelets. Brief Review
Learning-Based Multiresolution (LMR)
Learning-Based Multiresolution Schemes for Point-Value Discretization
Learning-Based Multiresolution Schemes for Point-Value Discretization on [0,1]
LMR Methods versus Linear Piecewise Interpolation Methods
Example.
The Loss-Function in LMR Schemes
Some Properties of LMR Context
Numerical Experiments
Conclusions and Future Research
References
$Weighted-Power_ p$ Nonlinear Subdivision Schemes
Introduction
Linear and Nonlinear Interpolatory Subdivision Schemes
Convergence and Stability of a Subdivision Scheme
Powerp Interpolatory Subdivision Schemes
Weighted-Powerp Interpolatory Subdivision Schemes
Properties of the Weighted-Powerp Mean
Convergence Analysis
Stability Analysis
Order of Approximation
Numerical Examples
Conclusion
References
Tracking Level Set Representation Driven by a Stochastic Dynamics
Introduction
Continuous-Time Dynamical Model of Level Set and Filtering
Particle Filter
Implicit Representation of the Curve
Stochastic Dynamics
Velocity Computation by Keeping Curve's Point Correspondences
Measurement Models and Parameters Estimation
Likelihood Definition
Parameters Estimation
Experiments and Results
Interest of a Continuous-Time Stochastic Dynamics and Auxiliary Level-Set
Variance Visualization and Analysis
Occlusions Management
Conclusion
References
$G^2$ Hermite Interpolation with Curves Represented by Multi-valued Trigonometric Support Functions
Introduction
Preliminaries
Support Function Representation of Algebraic Curves
Rational Hypocycloids and Epicycloids
Rational Curves Supported by Trigonometric Polynomials
Trigonometric Polynomials and Convolutions of HE-Cycloids
Rational Offsets of Convolutions of HE-Cycloids
Interpolating G2 Hermite Data with Convolutions of HE-Cycloids
Conclusion
References
Approximating Algebraic Space Curves by Circular Arcs
Introduction
Generating Approximating Circular Arcs
Convergence of Approximating Arcs
Connection with the Osculating Circle
Convergence of Approximating Circles
Algorithm and Error Bounds
Local Algorithm
Error Estimate
Global Subdivision Method
Global Algorithm
Examples
Conclusion
References
Generating Series for Drawing the Output of Dynamical Systems
Introduction
Preliminaries
Affine System, Generating Series
Fliess's Formula and Iterated Integrals
Main Results
Approximate Values of the Output y(t) and of the State (qr(t))1r N in a Neighborhood of t=0
Generalization at Time t=ti
Application: For Drawing the Output
Genericity of the Method
Example 1: Electric Differential Equation
Example 2: Dynamical System with Polynomial Generating Series MF-IRISA-2002
Example 3: Dynamical System Akhrif
Maple Package: Some Demonstrations
Some Practical Applications
Conclusion
References
Practical Mixed-Integer Optimization for Geometry Processing
Introduction
Previous Work
Linear Constraints
Lagrangian Multipliers
Elimination Approach
Integer Constraints
Direct Rounding
Iterative Greedy Rounding
Open Source CoMISo Library
Experiments
Conclusion
References
Non-degenerate Developable Triangular B´ezier Patches
Introduction
Ruled Triangular Bézier Patches
Developable Surfaces
Tensor Product Developable Patches
Triangular Developable Patches
Cylindrical and Conical Triangular Patches
Conclusions
References
Stable Splitting of Bivariate Splines Spaces by Bernstein-B´ezier Methods
Introduction
Böhmer's Method for Fully Nonlinear Elliptic PDEs
Fully Nonlinear Elliptic Operators
Spline Spaces and Stable Splitting
Böhmer's Method
Bernstein-Bézier Techniques
Stable Splitting for Argyris Finite Element
Modified Argyris Space
Stable Splitting
Why Modification in Argyris Space Is Required
C1 Macro-Element Spaces
Stable Splitting of Clough-Tocher Macro-Element Space
Powell-Sabin Macro-Element Space
Powell-Sabin-12 Macro-Element Space
Quadrilateral Macro-Element Space
References
Mesh Segmentation and Model Extraction
Introduction
Previous Work
Mesh Segmentation
Model Regression
Numerical Results
Conclusion and Future Work
References
Design of C2 Spatial Pythagorean-Hodograph Quintic Spline Curves by Control Polygons
Introduction
Spatial PH Quintic Curves
C2 Spatial PH Quintic Spline Interpolation
C2 Spatial PH Quintic Spline Equations
End Conditions
Additional Constraints
Starting Approximation
Control Polygons for C2 PH Quintic Splines
Computed Examples
Closure
References
Shape Curvatures of Planar Rational Spirals
Introduction
Affine Transformations and Signed Curvature
Affine Transformations of the Plane
Signed Curvature
Shape Curvature and Its Computation
Characterizing the Rational Spirals by Their Shape Curvatures
A Classification of Quadratic Bézier Spirals
Conclusion
References
Volumetric Geometry Reconstruction of Turbine Blades for Aircraft Engines
Introduction
Problem Specification and Outline
Slicing Surfaces
Slicing Surfaces for the Airfoil Part
Slicing Surfaces for the Base Part
Remark.
Segmentation of the Slices
Airfoil Part
Edge Points.
Wedge Points.
Base Part
Curve Fitting
Initial B-Spline Curves
Fitting Process
Surface Generation
Volume Generation
Conclusions
References
Globally Convergent Adaptive Normal Multi-scale Transforms
Introduction
Chaikin Normal MT
Globally Convergent Normal MTs Based on Adaptivity
Theoretical Approach
Adaptive Algorithm
Experimental Results
References
Helmholtz-Hodge Decomposition on [0, 1]d by Divergence-Free and Curl-Free Wavelets
Introduction
Helmholtz-Hodge Decomposition
Definitions
Divergence-Free and Curl-Free Wavelets on [0,1]d
Multiresolution Analyses of L2(0,1) Linked by Differentiation / Integration
Divergence-Free Scaling Functions and Wavelets on [0,1]d
Curl-Free Scaling Functions and Wavelets on [0,1]d
Wavelet Helmholtz-Hodge Decomposition
Description of the Method
Divergence-Free and Curl-Free Gram Matrices Computation
Right-Hand Side Computations
Divergence-Free and Curl-Free Gram Matrices Preconditioning
Examples of Helmholtz-Hodge and Helmholtz Decomposition
Conclusion
References
Finite Element Analysis with B-Splines: Weighted and Isogeometric Methods
Introduction
Finite Element Approximation
B-Splines
Weighted B-Splines
Isogeometric Elements
Weighted Isogeometric Approximation
Error Estimate
Applications
Conclusion
References
V3-Based 1-Form Subdivision
Introduction
Mathematical Setup
Regular Vertex
Irregular Vertices
Basic Setup
Valence 3 and 4
Irregular Vertex: V=5
Irregular Vertex: Results
Summary and Discussion
References
Curvature of Approximating Curve Subdivision Schemes
Introduction
Non-uniform Subdivision
Non-uniform Subdivision of Cubic C2 Splines
Discrete Curvature from Polygon Sequences
Skip-Interpolating Subdivision
Non-uniform Subdivision Based on a Rational Quadratic G1 Curve Construction
Discussion
References
Fitting a Surface to One of ItsSectional Planar Curves Using Adaptive Trees in Spaces of Curves
Introduction
Image-Volume Fusion for Interventional Imaging and the Retrieval of Surface Sections
Pose Identification from Projections for Computer-Assisted Surgery
Retrieving Sections on a Surface and Other Intended Applications
A Distance between Plane Curves or Plane Curve Shapes
Discrete Case.
Remark.
Setting for the Section Retrieval Problem
Calculus Approach for Section Retrieval
Implementation
The Idea of an Atlas to Initialize Searches
A Similar Problem in Computer Vision: Finding the Pose of a Known Solid from Its Projection Outline
Adaptive Trees: Algorithms and Convergence Results
Adaptive Trees
Adaptive Search Trees
Uninformed Accretion Adaptive Trees
Using Adaptive Search Trees
The Pose from Outline of Projection Problem
The Plane from Sectional Curve Problem
Conclusion and Prospects
References
Verified Spatial Subdivision of Implicit ObjectsUsing Implicit Linear Interval Estimatio ns
Introduction
Preliminaries
Interval Arithmetic
Hierarchical Decomposition and Interval Arithmetic
Affine Arithmetic
Implicit Linear Interval Estimations
LIETree
Pruning the Uncertainty
Inversion Nodes
Insufficient Progress
Construction Algorithm
Test Results
Conclusion and Future Work
Summary
Application to CSG Models and Superquadrics
References
Image Separation Using Wavelets and Shearlets
Introduction
Shearlets
Mathematical Theory of Geometric Separation
Model Situation
Chosen Dictionary
Subband Filtering
1 Minimization Problem
Theoretical Result
Extensions
Our Algorithmic Approach to the Geometric Separation Problem
General Scheme
Preprocessing
Solver for the 1 Minimization Problem
Wavelet Transform
Shearlet Transform
Numerical Results
Comparison by Visual Perception
Comparison by Quantitative Measures
Application in Neurobiology
Conclusion
References
On a Special Class of Polynomial Surfaces with Pythagorean Normal Vector Fields
Introduction
Preliminaries
Rational Surfaces with Rational Offsets
Isothermal Surfaces and Pythagorean-Hodograph Preserving Mappings
A Note on Polynomial Minimal Surfaces
Polynomial Cubic PN surfaces
Cubic PN Parameterizations
Enneper Minimal Surface and Its PN Parameterizations
Tschirnhausen Cubic Surface
Dual Kinematic Description
Conclusion
References
Convergence Rate of the Causal Jacobi Derivative Estimator
Introduction
Derivative Estimations by Using Jacobi Orthogonal Series
Numerical Experiments
References
Continuous Deformations by Isometry Preserving Shape Integration
Introduction
Related Work
Minimum Distortion Shape Integration
Error Measure on a Single Triangle
Properties of d
Error Measure on the Entire Surface
Shape Integration
Implementation
Defining Deformations
Minimizing Error Measures
Numerical Integration
Analysis and Results
Discussion
Conclusions
References
On a (W)ENO-Type Multiscale Representation Based on Quincunx Refinement: Application to Image Compression
Introduction
Notations and Multiresolution Analysis Definition
Notations
Multiresolution Analysis of L2(R2)
Interpolatory Nonlinear Multiscale Representations Based on Non-diagonal Dilation Matrix
Theoretical Results on Nonlinear Multiscale Representations
Definition of the Prediction Operator
Definitions of Difference Operators and of Joint Spectral Radius
Multiscale Representation Convergence Theorem
Bidimensional Interpolatory Quasi-Linear Prediction Operators
Nonlinear (W)ENO Affine Multiscale Representation Based on Quincunx Refinement
Quasi-Linear Prediction Operators Using Higher Degree Polynomials
Practical Implementation of the Multiscale Representations
Numerical Applications
Decay of the Coefficients
Compression Results
Conclusion
References
OpenFlipper: An Open Source Geometry Processing and Rendering Framework
Introduction
Design Goals
OpenFlipper API
Current Functionality
Scripting
General Scripting
Visual Scripting Interface
Industrial Use Cases
Quadrilateral Remeshing
Car Modeling
Conclusion and Future Improvements
References
Parameterization of Contractible Domains Using Sequences of Harmonic Maps
Introduction
Related Work
Harmonic Maps
Our Framework
Step 1: Finding a Mapping F: [0,1]2
Step 2: Finding a Spline Approximation S of F-1
Injectivity
Implementation and Examples
Solving Variational Problems
Spline Approximation of the Inverse Mapping
Putting Things Together
Examples
Conclusion
References
Nonlinear $L_1C_1$ Interpolation: Application to Images
Local L1 Cubic Spline Minimization
Univariate Cubic L1C1 Interpolation over Three Points
Univariate Cubic L1C1 Interpolation over Five Points
Local L1 Cubic Interpolation Spline
L1 Bicubic Interpolation Method over Images
Bicubic Spline Surface Construction
Image Interpolation Results
Image Transformations
Rotation.
Resampling Application.
Warping Applications.
GPU Parallelization Results
Conclusion
References
Normal Multi-scale Transforms for Surfaces
Introduction
Definitions and Preliminary Facts
Surfaces and Triangular Meshes
Approximate Normals
Normal MT with Edge Midpoint Prediction
Main Result
Discussion and Extensions
References
Generalized Dupin Cyclides with Rational Lines of Curvature
Introduction
Geometric Preliminaries
Canal Surfaces with Cylindrical Center Curve
Parameterization of the Surfaces
Fundamental Forms and Curvatures
Canal Surfaces with Planar Center Curve
Parameterization of the Surfaces
Fundamental Forms and Curvatures
Conclusion
References
Spline Volume Fairing
Introduction
The Fairing Expression
Point Approximations
Smoothing
Other Terms
Conclusion
References
Neuroelectric Current Localization from Combined EEG/MEG Data
Introduction
The EEG/MEG Forward Problem
The EEG/MEG Inverse Problem
Discretization of the Forward Problem
Numerical Tests
References
Bootstrap-Based Normal Reconstruction
Introduction
Related Work
Overview
Bootstrap Methods for Normal Reconstruction
Normal Reconstruction
Experimental Results
Normal Reconstruction
The Normal Orientation Problem
Bootstrap Variance
Bilateral Gaussian Filter for Normal Smoothing
Bilateral Gaussian Filter
Evaluation
Conclusion
References
Couple Points - A Local Approach to Global Surface Analysis
Introduction
Motivation and Definition of Couple Points
Properties of Couple Points
Couple Points for Triangular Meshes
Results and Applications
Computing the Maximal/Minimal Distance of Surfaces
Approximations of the Shortest Geodesic Paths between Two Points
Computing Stabilizing Connectors between Parts of a Surface
Conclusions
References
Chordal Cubic Spline Quasi Interpolation
Introduction
Cubic Quasi-Interpolant
Chord Length Parameterization
Estimating Curve Length
Estimating Arc Length Derivatives
Numerical Results
Conclusion
References
Multiple Subdivision Schemes
Introduction
Multiple Subdivision and Refinability
Convergence
Canonical Factors and Further Convergence Issues
Geometric Choice of Scaling Matrices
References
On a Linear Programming Approach to the Discrete Willmore Boundary Value Problem and Generalizations
Introduction
Discrete Framework
Triangular Meshes from a Set of Pre-defined Triangles
Admissible Indicator Vectors: A First Attempt
Discrete Mean Curvature on Triangular Meshes
A Quadratic Program for the Minimization of the Discrete willmore Energy
An Integer Linear Programming Approach
Augmented Indicator Vectors
On the Linear Programming Relaxation
Testing the Relaxed Linear Problem
On Integer Linear Programming
Conclusion
References
Interpolation Function of Generalizedq-Bernstein-Type Basis Polynomials and Applications
Introduction
Basic Properties of the Classical Bernstein Basis Polynomials, Bernoulli Polynomials and Stirling Numbers of the Second Kind
Some Properties of the Generalized Stirling Numbers of the Second Kind
A Generating Function for the Generalized q-Bernstein-Type Basis Polynomials
Recurrence Relations
Relations between the Polynomials bkn(x+y
q), Bn(v)(x) and S(n,k
y)
Integral Representation of the Polynomials bkn(x+y
q)
Interpolation Function of the q-Bernstein-Type Basis Polynomials
Applications
References
Differential Behaviour of Iteratively Generated Curves
Introduction
Background
IFS
CIFS
Projected IFS
Boundary Controlled IFS
Topology Constraints
Adjacency Constraints.
Incidence Constraints.
Convergence.
Parameterization and Self-similarity.
Differentiability
Eigenvectors and Eigenvalues
Necessary Conditions for Differentiability
Cartography of Differential Behaviours
Sufficient Conditions
Roughness of a Curve
Conclusion
References
Algebraic Curves of Low Convolution Degree
Introduction
Preliminaries
Convolution of Algebraic Curves
Previous Work on Convolutions
Curves of Convolution Degree One
Curves of Convolution Degree Two
Elementary Properties of Curves with Convolution Degree Two
Rationality of Convolutions with QN Curves
Decomposition of QN Curves
Conclusion
References
A Logistic Model for the Degradation of Triangle Mesh Normals
Introduction
Previous Work
Contribution and Limitations
Overview
A Logistic Model for Normal Degradation
Logistic Curve Fitting
Estimation of Appropriate Quantisation Levels
Validation and Quantisation Levels
Practical Applications
Conclusions and Future Work
References
On Single Image Scale-Up Using Sparse-Representations
Introduction
Incorporating the Sparse-Land Prior
The Proposed Single-Image Scale-Up Algorithm
Training Set Construction
Preprocessing and Feature Extraction
Dimensionality Reduction
Dictionary Learning
Reconstruction Phase
Bootstrapping Approach
Results
Text Scale-Up
PSNR Comparison with Yang et. al. YANG1,YANG2
Bootstrapping Approach for Single Image Scale-Up
Summary
References
Periodic T-Splines and Tubular Surface Fitting
Introduction
Periodic T-Splines
Repeating Control Points and Knot Intervals for Periodic B-Splines
New Construction for Periodic B-Splines
Periodic T-Spline Surface Representation
Control Point Insertion.
Periodic T-Spline Surface Fitting
Parameterization
Initial Placement of the T-Mesh
Geometry Optimization
T-Mesh Refinement
Examples
Conclusion
References
Author Index

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