Defining Point Set Surfaces Nina Amenta and Yong Joo Kil University of California, Davis IDAV Institute for Data Analysis and Visualization Visualization and Graphics Research Group

IntroductionIntroduction

MLS Surface [Levin] Mesh-independent surface interpolation

MLS Surface [Alexa et al.] Computing and Rendering Point set surfaces, TVCG MLS Surface

[Pauly et al.] Shape Modeling with Point-Sampled Geometry, SIGGRAPH MLS Surface

[Pauly, Gross, Kobbelt] Efficient simplification of point-sampled surfaces, IEEE Vis [Fleishman, Cohen-Or, Alexa and Silva] Progressive point-set surfaces, TOG [Adamson and Alexa] Ray tracing point-set surfaces, Shape Modeling International [Guo and Quin] Dynamic sculpting and deformation of point-set surfaces, PG [Mederos, Velho, and de Figueiredo] Moving least squares multiresolution surface approximation, SIBIGRAPI, [Xie, Wang, Hua, Quin, and Kaufman] Piecewise C1 continuous surface reconstruction of noisy point clouds via local implicit quadric regression, IEEE Vis [Adamson and Alexa] On normals and projection operators for surfaces defined by point sets, S. Point-Based Graphics, [Mueller, Keiser, Nealan, Pauly, Gross, and Alexa] Point based animation of elastic, plastic and melting objects, S. Computer Animation, 2004.

ContributionContribution MLS is a kind of Extremal Surface –Equation! Analyze properties. Framework for generalization –Points with normals. Modeling with fewer primitives

Extremal Surface [Medioni and Guy] Inference of surfaces, curves and junctions from sparse, noisy 3D data, IEEE PAMI, [Tang, Medioni] Extremal feature extraction from 3D vector and noisy scalar fields, IEEE Visualization, [Medioni, Lee, and Tang] A Computational Framework for Segmentation and Grouping, Elsevier, 2000.

Extremal Surface Definition Vector Field: n Energy Field: e

Extremal Surface Definition e on n(x)n(x)

Extremal Surface in 2D e : energy field n : vector field

Implicit Definition Oriented vector field. Maxima and Minima of energy field.

MLS Projection function x

x q Least squares error

MLS Projection function x Minimal least squares error

MLS Projection function x (x)(x) (x)(x) f (x) = f f (x)

Stationary Points x (x)(x) n(x) & e(x) ?

Stationary Point of MLS x

Vector Field of MLS n(x)n(x) x

x

Energy Field of MLS x e(x)e(x)

Extremal Surface MLS  an extremal surface

Explicit Equation Normals from derivative Surface normal  n

DomainDomain

GeneralizationGeneralization MLS surface is an example of Extremal Surface Extremal Surface provides framework for generalization of MLS Example using Surfels

Extremal Surface for Surfels x

Vector Field

Energy Field

Extremal Surface

General Projection Scheme

Other 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 Weight

Projection Method Surfel count: Our method: 16 secs. PointShop3D (ScanTools): 9 secs. Thanks to IBM TJ Watson Research Center

Sparse Set and MLS surface

Extremal Surface for Surfels

Future work Sampling theory Projection methods More vector and energy fields Sharp features

Thank you National Science Foundation (NSF) University of California, Davis PointShop3D team David Levin Our plugin: Defining Point Set Surfaces, available on pointshop3d.com