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