Defining Point Set SurfacesDefining Point Set Surfaces

Nina Amenta and Yong Joo KilNina Amenta and Yong Joo KilUniversity of California, DavisUniversity of California, Davis

IDAV IDAV Institute for Data Analysis and VisualizationInstitute for Data Analysis and VisualizationVisualization and Graphics Research GroupVisualization and Graphics Research Group

IntroductionIntroductionIntroductionIntroduction

MLS SurfaceMLS SurfaceMLS SurfaceMLS Surface

• [Levin] Mesh-independent surface interpolation

MLS SurfaceMLS SurfaceMLS SurfaceMLS Surface

• [Alexa et al.] Computing and Rendering Point set surfaces, TVCG 2001.

MLS Surface

MLS SurfaceMLS SurfaceMLS SurfaceMLS Surface

• [Pauly et al.] Shape Modeling with Point-Sampled Geometry, SIGGRAPH 2003.

MLS Surface

MLS SurfaceMLS SurfaceMLS SurfaceMLS Surface

• [Pauly, Gross, Kobbelt] Efficient simplification of point-sampled surfaces, IEEE Vis. 2002.

• [Fleishman, Cohen-Or, Alexa and Silva] Progressive point-set surfaces, TOG 2003.

• [Adamson and Alexa] Ray tracing point-set surfaces, Shape Modeling International 2003.

• [Guo and Quin] Dynamic sculpting and deformation of point-set surfaces, PG 2003.

• [Mederos, Velho, and de Figueiredo] Moving least squares multiresolution surface approximation, SIBIGRAPI, 2003.

• [Xie, Wang, Hua, Quin, and Kaufman] Piecewise C1 continuous surface reconstruction of noisy point clouds via local implicit quadric regression, IEEE Vis. 2003.

• [Adamson and Alexa] On normals and projection operators for surfaces defined by point sets, S. Point-Based Graphics, 2004.

• [Mueller, Keiser, Nealan, Pauly, Gross, and Alexa] Point based animation of elastic, plastic and melting objects, S. Computer Animation, 2004.

ContributionContributionContributionContribution

• MLS is a kind of Extremal Surface

– Equation! Analyze properties.

• Framework for generalization

– Points with normals. Modeling with fewer primitives

Extremal SurfaceExtremal Surface

• [Medioni and Guy] Inference of surfaces, curves and junctions from sparse, noisy 3D data, IEEE PAMI, 1997.

• [Tang, Medioni] Extremal feature extraction from 3D vector and noisy scalar fields, IEEE Visualization, 1998.

• [Medioni, Lee, and Tang] A Computational Framework for Segmentation and Grouping, Elsevier, 2000.

Extremal Surface DefinitionExtremal Surface DefinitionExtremal Surface DefinitionExtremal Surface Definition

Vector Field: n

Energy Field: e

Extremal Surface DefinitionExtremal Surface DefinitionExtremal Surface DefinitionExtremal Surface Definition

e on

n(x)

Extremal Surface in 2DExtremal Surface in 2DExtremal Surface in 2DExtremal Surface in 2D

e : energy field

n : vector field

Implicit DefinitionImplicit DefinitionImplicit DefinitionImplicit Definition

• Oriented vector field.

• Maxima and Minima of energy field.

MLS Projection functionMLS Projection functionMLS Projection functionMLS Projection function

x

MLS Projection functionMLS Projection functionMLS Projection functionMLS Projection function

xq

Least squares error

MLS Projection functionMLS Projection functionMLS Projection functionMLS Projection function

x

Minimal least squares error

MLS Projection functionMLS Projection functionMLS Projection functionMLS Projection function

x(x)

(x)

f(x) = f f(x)

Stationary PointsStationary PointsStationary PointsStationary Points

x(x)

n(x) & e(x) ?

Stationary Point of MLS Stationary Point of MLS Stationary Point of MLS Stationary Point of MLS

x

Vector Field of MLS Vector Field of MLS Vector Field of MLS Vector Field of MLS

n(x)

x

Vector Field of MLS Vector Field of MLS Vector Field of MLS Vector Field of MLS

x

Energy Field of MLS Energy Field of MLS Energy Field of MLS Energy Field of MLS

x

e(x)

Extremal SurfaceExtremal SurfaceExtremal SurfaceExtremal Surface

MLS MLS an extremal surface an extremal surface

Explicit EquationExplicit EquationExplicit EquationExplicit Equation

• Normals from derivative

• Surface normal n

DomainDomainDomainDomain

Generalization Generalization Generalization Generalization

• MLS surface is an example of Extremal Surface

• Extremal Surface provides framework for generalization of MLS

• Example using Surfels

EExtremal Surface xtremal Surface for Surfelsfor SurfelsEExtremal Surface xtremal Surface for Surfelsfor Surfels

x

Vector FieldVector FieldVector FieldVector Field

EnergyEnergy Field FieldEnergyEnergy Field Field

EExtremal Surface xtremal Surface EExtremal Surface xtremal Surface

General Projection SchemeGeneral Projection SchemeGeneral Projection SchemeGeneral Projection Scheme

Other approachesOther approachesOther approachesOther approaches

• [Levin], Mesh-independent surface interpolation, (on his web site)

• [Adamson and Alexa], On Normals and Projection Operators for Surfaces Defined by Point Sets,Eurographics Symposium on Point-based Graphics

Varying Energy Fields Varying Energy Fields Varying Energy Fields Varying Energy Fields

Varying WeightVarying WeightVarying WeightVarying Weight

Projection MethodProjection MethodProjection MethodProjection MethodSurfel count: 77428Our method: 16 secs. PointShop3D (ScanTools): 9 secs.

Thanks to IBM TJ Watson Research Center

Sparse Set and MLS surfaceSparse Set and MLS surfaceSparse Set and MLS surfaceSparse Set and MLS surface

Extremal Surface for SurfelsExtremal Surface for SurfelsExtremal Surface for SurfelsExtremal Surface for Surfels

Future workFuture workFuture workFuture work

• Sampling theory

• Projection methods

• More vector and energy fields

• Sharp features

Thank youThank youThank youThank you

• National Science Foundation (NSF)

• University of California, Davis

• PointShop3D team

• David Levin

• Our plugin: Defining Point Set Surfaces,available on pointshop3d.com