Curves and Surfaces
7th International Conference, Avignon, France, June 24-30, 2010, Revised Selected Papers
Published on 6. January 2012
X, 748 pages
978-3-642-27413-8 (ISBN)
Description
This volume constitutes the thoroughly refereed post-conference proceedings of the 7th International Conference on Curves and Surfaces, held in Avignon, in June 2010. The conference had the overall theme: "Representation and Approximation of Curves and Surfaces and Applications".
The 39 revised full papers presented together with 9 invited talks were carefully reviewed and selected from 114 talks presented at the conference. The topics addressed by the papers range from mathematical foundations to practical implementation on modern graphics processing units and address a wide area of topics such as computer-aided geometric design, computer graphics and visualisation, computational geometry and topology, geometry processing, image and signal processing, interpolation and smoothing, scattered data processing and learning theory and subdivision, wavelets and multi-resolution methods.
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Jean-Daniel Boissonnat | Patrick Chenin | Albert Cohen
Curves and Surfaces
7th International Conference, Avignon, France, June 24-30, 2010, Revised Selected Papers
Book
12/2011
Springer
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Content
- 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
