Summer Seminar
Lubin Fan
2011-07-07
Discrete Differential Geometry
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•
•
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Circular arc structures
Discrete Laplacians on General Polygonal Meshes
HOT: Hodge-Optimized Triangulations
Spin Transformations of Discrete Surfaces
Example-Based Simulation
• Frame-based Elastic Models (TOG)
• Sparse Meshless Models of Complex Deformable Objects
• Example-Based Elastic Materials
Circular Arc Structures
Pengbo Bo1,2 Helmut Pottmann2,3 Martin Kilian2
Wenping Wang1 Johannes Wallner2,4
1Univ.
Hong Kong
2TU Wien
3KAUST
4TU Graz
Authors
Pengbo Bo
Helmut Pottmann
Postdoctoral Fellow
Univ. Hong Kong
KAUST
Vienna University of Technology
Martin Kilian
Wenping Wang
RA
Vienna University of Technology
Professor
Univ. Hong Kong
Johannes Wallner
Professor
Graz University of Technology
Vienna University of Technology
Architectural Geometry
• The most important guiding principle for freeform architecture
– Balance
• Cost efficiency
• Adherence to the design intent
– Key issue
• Simplicity of supporting and connecting elements as well as repetition
of costly parts
Node complexity
Previous Work
• Nodes optimization
– [Liu et al. 2006; Pottmann et al. 2007] for quad meshes
– [Schiftner et al. 2009] for hexagonal meshes
• Rationalization with single-curved panel
– [Pottmann et al. 2008]
• Repetitive elements
– [Eigensatz et al. 2010]
– [Singh and Schaefer 2010] and [Fu et al. 2010]
– The aesthetic quality is reduced if the number of repetitions
increases.
This Work
• Propose the class of Circular Arc Structures (CAS)
• Properties
– Smooth appearance, congruent nodes, and the simplest
possible elements for the curved edges
– Do not interfere with an optimized skin panelization.
• Contributions
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–
–
–
freeform surfaces may be rationalized using CAS
repetitions not only in nodes, but also in radii of circular edges
extend to fully three-dimensional structures
have nice relations to discrete differential geometry and to the
sphere geometries
Circular Arc Structures
• Definition
A circular arc structure consists of 2D mesh combinatorics (V, E), where edges are
realized as circular arcs, such that in each vertex the adjacent arcs touch a
common tangent plane.
We require congruence of interior vertices, and we consider the following three
cases:
– Hexagonal CAS have valence 3 vertices. Angles between edges equal 120 degrees;
– Quadrilateral CAS have valence 4 vertices. Angles between edges have values α, π − α, α, π −
α, if one walks around a vertex;
– Triangular CAS have valence 6 vertices. Angles between edges equal 60 degrees.
Circular Arc Structures
• Data Structure
• Target Functional
– Deviation
– Smoothness
– Geometric consistency
– Regularization
– Angles
Circular Arc Structures
• Generalizations
– Singularities
• Supporting Elements
– Add condition
CAS with Repetitive Elements
• Radius Repetitive
• Definition
A quadrilateral CAS is radius-repetitive along a flow line, if the radius of its edges is
constant. It is transversely radius-repetitive for a pair of neighboring ‘parallel’ flow
lines, if the edges which connect these flow lines have constant radius.
• Condition
Cyclidic Structure
• Cyclidic CAS
• Offsets
– Offsetting operation of cyclidic CAS is well defined
Results
Conclusions
• Limitations
– Loss of shape flexibility when additional geometric conditions
are imposed.
– The introduction of T-junctions
• This Work
– Shown the applicability of CAS
– Demonstrated special CAS have more properties which are
relevant for freeform building construction
• Future Work
– Explore more application
Discrete Laplacians on General
Polygonal Meshes
Marc Alexa1
Max Wardetzky2
1TU
Berlin
2Universitaat Gottingen
Authors
Marc Alexa
Max Wardetzky
Professor
Electrical Engineering and
Computer Science
TU Berlin
Assistant Professor
Heading the Discrete Differential
Geometry Lab
Universitaat Gottingen
This Work
• Discrete Laplacian on surface with arbitrary polygonal faces
– Non-planar & non-convex polygons
• Mimic structural properties of the smooth Laplace-Beltrami
operator
• Motivation
– Non-triangular polygons are widely used in geometry processing
Related Work
• Geometric discrete Laplacians
– Cotan formula [Pinkall and Polthier 1993]
– The last decade has brought forward several parallel
developments…
• Application
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–
–
–
–
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Mesh parameterization
Fairing
Denoising
Manipulation
Compression
Shape analysis
…
Discrete Laplacian Framework
• Setup
An oriented 2-manifold mesh M, possibly with boundary, with vertex set V , edge
set E, and face set F . We allow for faces that are simple, but possibly non-planar,
polygons in R3.
• Work with oriented halp-edge
• EI, inner edges; EB boundary edge
• Algebraic approach to discrete Laplacian
– M0
– M1
Desiderate
• Locality
– Maintain locality by only working with diagonal matrices M0 and by requiring
that M1 is defined per face in the sense that
• Symmetry : L = LT
• Positive semi-definiteness
– M0 & Mf are positive definiteness.
• Linear precision
• Scale invariance
• Convergence
Vector Area & Maximal Projection
• Vector Area
Maximal Projcetion
• Maximal Projection
• Mean Curvature
A family of discrete Laplacians
•
[Perot and Suvramanian 2007]
—— pre-Laplacians
—— positive semi-definite
Implementation
• Construct 3 matrices
– Diagonal matrx, M0
– Coboundary matrix, d
• dep = ±1 if e = ±eqp and dep = 0
– M1
• Assembled per face: Mf
Results & Application
• Implicit mean curvature flow
• Parameterization
Results & Application
• A planarizing flow
Results & Application
• Thin plate bending
Conclusion
• This Work
– presents here a principled approach for constructing geometric
discrete Laplacians on surfaces with arbitrary polygonal faces,
encompassing non-planar and non-convex polygons.
• Feature Work
– How to replace this combinatorial term by a more geometric
one
Spin Transformation of
Discrete Surface
Keenan Crane1
Ulrich Pinkall2
1California
Peter Schroder1
Institute of Technology
2TU Berlin
http://users.cms.caltech.edu/~keenan/project_spinxform.html
Authors
Keenan Crane
Ulrich Pinkall
PhD Student
California Institute of Technology
Geometry Group
Institute of mathematics
TU Berlin
Peter Schroder
Professor
Director of the Multi-Res Modeling Group
California Institute of Technology
This Work
• Spin Transformation
– A new method for computing conformal transformations of
triangle meshes in R3
– Consider maps into the quaternions H
Related Work
• Deformation
– Local coordinate frame [Lipman et al. 2005, Paries et al. 2007]
– Cage-based editing [Lipman et al. 2008]
• Surface parametrization
– Prescribe values at vertices that directly control the rescaling of
the metric[Ben-Chen et al. 2008; Yang et al. 2008; Springborn et
al. 2008].
Quaternion
• Definition
– The quaternions H can be viewed as a 4D real vector space with basis {1,
i, j, k} along with the non-commutative Hamilton product, which satisfies
the relationships i2 = j2 = k2 = ijk = −1.
– The imaginary quaternions Im H are elements of the 3D subspace
spanned by {i, j, k}.
– q = a + bi + cj + dk, q = a - bi - cj – dk
– Rotation of a vector
,
,
(Similarity Transformation)
–
• Calculus
– Map f : M -> ImH
– Differential df : TM -> ImH
Spin Transformations
Spin Transformations
• Integrable Condition [Kamberov et al. 1998]
– D , Quaternionic Dirac Operator
• Eigenvalue Problem
Spin Transformations
• Procedure
– Pick a scalar function ρ on M
– Solve an eigenvalue problem
for the similarity transformation λ
– Sovle a linear system
for the new surface
Discretization
• Discrete Dirac Operator
Discretization
• Scalar Multiplication
min df  e
2
• Discretized Spin Transformations
min df  e
2
Application
• Painting Curvature
Application
• Arbitrary Deformation
Conclusion
• This Work
– Our discretization of the integrability condition (D − ρ)λ = 0
provides a principled, efficient way to construct conformal
deformations of triangle meshes in R3.
• Future Work
– D is expressed in terms of extrinsic geometry it can be used to
compute normal information, mean curvature, and the shape
operator.
HOT: Hodge-Optimized Triangulations
Patrick Mullen Pooran Memari Fernando de Goes Mathieu Desbrun
California Institute of Technology
This Work
• “Good” dual
• Motivation
– Fluid simulation
• This work
– Hodge-optimized
triangulation
Previous Work
• Delaunay / Voronoi pairs
– [Meyer et al. 2003]
– [Perot and Subramanian 2007]
– [Elcott et al. 2007]
– Drawbacks
• Circumcenter lies outside its associated tetrahedron
• Inability to choose the position of dual mesh
• Too restrictive in many practical situations
Results
Results
Frame-based Elastic Models
Benjamin Gilles1 Guillaume Bousquet2,3,4 Francois Faure2,3,4 Dinesh K. Pai1
1University
of British Columbia, Vancouver, CANADA
2University of Grenoble
3INRIA
4LJK – CNRS
Authors
Benjamin Gilles
Guillaume Bousquet
Post-doctoral Fellow
Sensorimotor Systems Lab
Department of Computer Science
University of British Columbia
Second year PhD student
University of Grenoble
Laboratoire Jean Kuntzmann
INRIA
François Faure
Dinesh K. Pai
Assistant Professor
University of Grenoble
Laboratoire Jean Kuntzmann
INRIA
Professor
Sensorimotor Systems Lab
Department of Computer Science
University of British Columbia
Deformable Models [Terzopoulos et al. 1988]
• Application
– Computer animation
• Animating characters, Soft objects, …
• Approaches
– Physically based deformation
– Skinning
Physically based deformation [Nealen et al. 2005]
• Finite Element Method
• Lagrangian models of deformable objects
• Two main method
– Mesh-based methods
– Meshless methods
• Pros
– Physical realism
• Cons
– Expensive
– Difficult to use
Physically based deformation
• Lagrangian mechanics
• Simulation loop
Skinning
• Vertex blending / skeletal subspace deformation
• Interpolating rigid transformation
• Point is computed as
• Pros
– Sparse sampling
– Efficient
• Cons
– Physically realistic dynamic deformation
Skinning
• Dual quaternion blending [Kavan et al. 2007]
– Linear interpolation of screws
– Reasonable cost
– Well suited for parameterizing a physically based deformable
model
This Work
• New type of deformable model
• Combination
– Physically based continuum mechanics models
– Frame-based skinning methods
This Work
• Contribution
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Creation models with sparse and intuitive sampling
on-the-fly adaptation to create local deformations
Effective
Integrated in SOFA
Modeling Objects
• Weight (Shape function)
• Sampling
– voxelization
Modeling Objects
• Volume integrals
– Compute the integral by regularly discrediting the volume inside
the bounding box of the undeformed object
• Fast pre-computed models
• Adaptive
Validation & Results
• Implementation
– Integrated in the SOFA (Simulation Open Framework Architecture)
• Accuracy
Validation & Results
• Deformation modeling
– Using a reduced number of control primitives
Performance
Performance
Conclusion
• This work
– A new type of deformable model
– Robust to large displacement and deformations
• Future work
– Hardware implementation
– The relation between stiffness and weight functions could be
exploited
Sparse Meshless Models of
Complex Deformable Solids
Francois Faure2,3,4 Benjamin Gilles1 Guillaume Bousquet2,3,4 Dinesh K. Pai1
1University
of British Columbia, Vancouver, CANADA
2University of Grenoble
3INRIA
4LJK – CNRS
Authors
Benjamin Gilles
Guillaume Bousquet
Post-doctoral Fellow
Sensorimotor Systems Lab
Department of Computer Science
University of British Columbia
Second year PhD student
University of Grenoble
Laboratoire Jean Kuntzmann
INRIA
François Faure
Dinesh K. Pai
Assistant Professor
University of Grenoble
Laboratoire Jean Kuntzmann
INRIA
Professor
Sensorimotor Systems Lab
Department of Computer Science
University of British Columbia
This Work
• Goal
– Deform objects with heterogeneous material properties and
complex geometries.
Previous Work
• Frame-based Method
• Nodes
– A discrete number of independent DOFs
– Kernel functions (RBF)
– Shape functions
• Geometrically designed
• Independent of the material
• Displacement function
• Problem
– Impossible in interactive application
This Work
• Novel: Material-aware shape function
• Input
– Volumetric map of the material properties
– An arbitrary number of control nodes
• Output
– A distribution of the nodes
– A associated shape function
• Contributions
– Material-aware shape function
– Automatically model a complex object
– High frame rates using small number of control nodes
Work Flow
Material-aware shape functions
• Compliance Distance
Local compression:
Displacement function:
Shape function:
Compliance distance:
Slope of shape function:
Voronoi kernel functions
• Goal
– Interpolating, smooth, linear and decreasing function
• Voronoi subdivision
• Dijkstra’ shortest path algorithm
RBF kernels
Node distribution: farthest point
sampling [Martin et al. 2010]
Our kernels
Deformable model computation
Results
• Validation
– Integrated in the SOFA
• Performance
Results
Conclusion
• This Work
– Novel, anisotropic kernel functions using a new definition of
distance based on compliance, which allow the encoding of
detailed stiffness maps in coarse meshless models. They can be
combined with the popular skinning deformation method.
• Future Work
– Dynamic adaptivity of the models
– Local deformations
Example-based Elastic Materials
Sebastian Martin1 Bernhard Thomaszewski1,2 Eitan Grinspunt3 Markus Gross1,2
1ETH
Zurich
2Disney Research Zurich
3Columbia University
Authors
Sebastian Martin
Bernhard Thomaszewski
RA, PhD. Student
CGL, ETH
Post-doctoral Researcher
Disney Research Zurich
Eitan Grinspun
Markus Gross
Associate Professor
Computer Science Dept.
Columbia University
Professor
CGL, ETH
Disney Research Zurich
This Work
• An example-based approach simulating complex elastic
material behavior
• Due to its example-based, this method promotes an artdirected approach to solid simulation.
Related Work
• Material Models
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The groundbreaking works [Terzopoulos et al. 1988]
Elastic models [Irving et al. 2004]
Plasticity and viscoelasticity [Bargteil et al. 2007]
Learning material properties from experiments [Bickel et al. Sig 2009]
• Directing animations
– Explicit control forces [Thurey et al. 2006]
– Space-time constraints [Barbic et al. 2009]
– …
• Example-based graphical methods
– State of the Art in Example-based Texture Synthesis [Wei et al. EG2009]
– Example-Based Facial Rigging [Li et al. Sig 2010]
Work Flow
• Interpolation
– Construct a space of
characteristic shapes by means of
interpolation
• Projection
– Project configurations onto it by
solving a minimization problem
• Simulation
– Define an elastic potential that
attracts an object to its space of
preferable deformations
Example Manifold
• Example manifold by example interpolation
• Interpolation Energy
Example Projection
• Projection Problem
• Summary
Example Design & Implementation
• Example design
– Same topology
– What kind of examples should be used (3)
• Embedding Triangle Meshes
– High-quality surface details
• Local and Global Examples
Results
Conclusion
• This Work
– Intuitive and direct method for artistic design and simulation of
complex material behavior.
• Future Work
– Optimization scheme should be increased
– Develop methods to assist users to provide appropriate
examples
– Automatically select example poses from input animation