Summer SeminarLubin Fan
2011-07-07
• Circular arc structures• Discrete Laplacians on General Polygonal Meshes• HOT: Hodge-Optimized Triangulations• Spin Transformations of Discrete Surfaces
Discrete Differential Geometry
Example-Based Simulation
• Frame-based Elastic Models (TOG)• Sparse Meshless Models of Complex Deformable Objects• Example-Based Elastic Materials
Circular Arc Structures
1Univ. Hong Kong2TU Wien
3KAUST4TU Graz
Pengbo Bo1,2 Helmut Pottmann2,3 Martin Kilian2 Wenping Wang1 Johannes Wallner2,4
Authors
Pengbo BoPostdoctoral FellowUniv. Hong Kong
Martin KilianRAVienna University of Technology
Helmut PottmannKAUSTVienna University of Technology
Wenping WangProfessorUniv. Hong Kong
Johannes WallnerProfessorGraz 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– 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
• DefinitionA 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
1TU Berlin2Universitaat Gottingen
Marc Alexa1 Max Wardetzky2
Authors
Max WardetzkyAssistant ProfessorHeading the Discrete Differential Geometry LabUniversitaat Gottingen
Marc AlexaProfessorElectrical Engineering and Computer Science TU Berlin
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– Mesh parameterization– Fairing– Denoising– Manipulation– Compression– Shape analysis– …
• SetupAn 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
Discrete Laplacian Framework
• 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 Projection• Mean Curvature
Maximal Projcetion
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
1California Institute of Technology2TU Berlin
Keenan Crane1 Ulrich Pinkall2 Peter Schroder1
http://users.cms.caltech.edu/~keenan/project_spinxform.html
Authors
Keenan CranePhD StudentCalifornia Institute of Technology
Peter SchroderProfessorDirector of the Multi-Res Modeling GroupCalifornia Institute of Technology
Ulrich PinkallGeometry GroupInstitute of mathematicsTU Berlin
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
• Discretized Spin Transformations
2min df e
2min df e
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
California Institute of Technology
Patrick Mullen Pooran Memari Fernando de Goes Mathieu Desbrun
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
1University of British Columbia, Vancouver, CANADA2University of Grenoble
3INRIA4LJK – CNRS
Benjamin Gilles1 Guillaume Bousquet2,3,4 Francois Faure2,3,4 Dinesh K. Pai1
Authors
Benjamin GillesPost-doctoral FellowSensorimotor Systems LabDepartment of Computer ScienceUniversity of British Columbia
François FaureAssistant ProfessorUniversity of GrenobleLaboratoire Jean KuntzmannINRIA
Guillaume BousquetSecond year PhD studentUniversity of GrenobleLaboratoire Jean KuntzmannINRIA
Dinesh K. PaiProfessorSensorimotor Systems LabDepartment of Computer ScienceUniversity 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– 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
1University of British Columbia, Vancouver, CANADA2University of Grenoble
3INRIA4LJK – CNRS
Francois Faure2,3,4 Benjamin Gilles1 Guillaume Bousquet2,3,4 Dinesh K. Pai1
Authors
Benjamin GillesPost-doctoral FellowSensorimotor Systems LabDepartment of Computer ScienceUniversity of British Columbia
François FaureAssistant ProfessorUniversity of GrenobleLaboratoire Jean KuntzmannINRIA
Guillaume BousquetSecond year PhD studentUniversity of GrenobleLaboratoire Jean KuntzmannINRIA
Dinesh K. PaiProfessorSensorimotor Systems LabDepartment of Computer ScienceUniversity 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 DistanceLocal compression:
Displacement function:
Shape function:
Compliance distance:
Slope of shape function:
Affine function!
Voronoi kernel functions
• Goal– Interpolating, smooth, linear and decreasing function
• Voronoi subdivision• Dijkstra’ shortest path algorithm
RBF kernels Our kernels
Node distribution: farthest point sampling [Martin et al. 2010]
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
1ETH Zurich2Disney Research Zurich
3Columbia University
Sebastian Martin1 Bernhard Thomaszewski1,2 Eitan Grinspunt3 Markus Gross1,2
Authors
Sebastian MartinRA, PhD. StudentCGL, ETH
Eitan GrinspunAssociate ProfessorComputer Science Dept.Columbia University
Bernhard ThomaszewskiPost-doctoral ResearcherDisney Research Zurich
Markus GrossProfessorCGL, ETHDisney Research Zurich
This Work
• An example-based approach simulating complex elastic material behavior
• Due to its example-based, this method promotes an art-directed approach to solid simulation.
Related Work
• Material Models– 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