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Subdivision Surface based Subdivision Surface based OneOne--Piece RepresentationPiece Representation

ShuhuaShuhua LaiLaiDepartment of Computer Science,

University of Kentucky

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Outline

1. Introduction2. Parametrization of General CCSSs3. One-Piece through Interpolation 4. One-Piece through Boolean Operations5. Simplification by Adaptive Tessellation6. Simplification by Multiresolution Analysis7. Conclusion & Future Work

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Abbreviation:

CCSS = Catmull-Clark Subdivision Surface

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Introduction: Subdivision Surface

Subdivision

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v Pixar’s Renderman

v Alias|Wavefront’s Maya

v Nichimen’s Mirai

v Newtek’s Lightwave 3D

Subdivision surfaces have already been used in

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What is so special?

One piece representation™

(arbitrary topology)

Multi-resolution(Scalability)

Code Simplicity

Covers both polygon form and

surface form(Uniformity)

Numerical stability

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Multi-resolution(Scalability)

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Covers both polygon form and surface form (Uniformity of representation)

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So, what is a subdivision surface?

Triangular

Loop Butterfly

Quadrilateral

Catmull-ClarkDoo-Sabin

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: vertices from mesh M0

, , : vertices to be generated for M1

Basic Concept (Catmull-Clark Scheme):

e50

e40

e30

e20

e10

v0

Around a vertex vof degree 5

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Basic Concept (Catmull-Clark Scheme):

Generating new face points

Face point: centroid of each face

f51

f41

f31

f21

f11

v0

: face point

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Basic Concept (Catmull-Clark Scheme):Generating new edge points

f51

f41

f31

f21

f11

v0

: edge point

e21

e11

e51

e31

e41

4

111

001 iiii

ffeve +++= −

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Basic Concept (Catmull-Clark Scheme):

Generating new vertex points

f51

f41

f31

f21

f11

v0

: vertex point

e21

e11

e51

e31

e41

∑∑ ++−

= 12

02

01 112ii f

ne

nv

nn

v

v1

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Basic Concept (Catmull-Clark Scheme):

Forming new edges

f51

f41

f31

f21

f11

v0

: vertex point

e21

e11

e51

e31

e41

v1

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By repeatedly refining, one gets M0,M1,M2,M3, limit surface

M0

M1

M2

M3

S = M∞

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Modeling made much easier. Why?

v No restrictions on the topology of the control pointsv Local refinement is possible

NURBS Catmull-Clark

NURBS Catmull-Clark

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Examples of control mesh

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Motivation

v Although subdivision surfaces are capable of modeling complex shape of arbitrary topology, methods on how to build the control mesh of a complex surface are not presented yet.

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TaskTo construct a sparse mesh structure for any

given model such that The CCSS of the constructed mesh is the given model.

Construction

Subdivision

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Objectives

1. Develop necessary mathematical theories and geometric algorithms to support subdivision surface based one-piece representation

2. Build a system that supports subdivision surface based one-piece representation.

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Subdivision surface based one-piece representation

v Represent any model with only one subdivision surface no matter how complicated the object's topology or shape. No decomposition of the object into simpler components is necessary. Hence the number of parts in the final representation is always the minimum: one.

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Examples of One piece

representation

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Multi- V.S. One-Piece

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Is One Piece Representation Good?

vManagement: SimplervRendering: More efficientvMachining: More accuratevAnimation: Crack free

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Contributions of the dissertation

v Parametrization of CCSSsv Interpolation of meshes of arbitrary topologyv Voxelization of free-form solidsv Boolean operations on free-form solidsv Adaptive tessellation of CCSSsv A system that supports subdivision surface based

one-piece representation is builtv Others

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Contributions (cont.)

v Parametrization and Evaluation of CCSSs

Evaluation

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Contributions (cont.)

v Interpolation of meshes of arbitrary topology

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Contributions (cont.)v Voxelization of free-form solids

Given Model 128 X 128 X 128 512 X 512 X 512

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Contributions (cont.)v Boolean operations on free-form solids

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Contributions (cont.)

Adaptive tessellation

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Contributions (cont.)v Constrained Scaling of CCSSs

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Contributions (cont.)

v Texture Mapping on Meshes of Arbitrary Topology

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Contributions (cont.)

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Contributions (cont.)v A system that supports subdivision surface

based one-piece representation is built

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Structure of the system

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Parametrization of General CCSSs

Parametrization

(u, v) à S (u, v)

Given MeshLimit Surface

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Work on Subdivision Surface Work on Subdivision Surface ParameterizationParameterization

1. J. Stam (1998)2. D. Zorin, et al. (2002)3. S. Lai, F. Cheng (2005)

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J. Stam Lai/Cheng

(Discrete Fourier Transform)

The Extended Subdivision Diagram

Parameterization

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Explicit Parametrization of General CCSSs

∑+

=

−=5

0,

1),(n

jjb

mj

mT GMKWvuS λ

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Explicit Parametrization (Cont.)

v W = [1, u, v, u2,uv,v2,u3,u2v, uv2,v3,u3v,u2v2,uv3,u3v2,u2v3,u3v3];

v K = Diag(1,2,2,4,4,4,8,8,8,8,16,16,16,32,32,64);

v λj = eigenvalue of the subdivision matrix;

v Mb,j = constant matrix depending on b and j

v G = [V, E1, E2, … ,En, F1,… , Fn, I1, I2, … , I7];

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Properties around extraordinary points

GMvv

S

GMvu

S

GMuu

S

GMv

S

GMu

SGMS n

••=∂∂

∂

••=∂∂

∂

••=∂∂

∂

••=∂

∂

••=∂

∂••= +

2,2

2

2,2

2

2,2

2

2,2

2,2

1,2

]0,,1,0,0,0,0,0[)0,0(

]0,,0,1,0,0,0,0[)0,0(

]0,,0,0,1,0,0,0[)0,0(

]0,,0,0,0,1,0,0[)0,0(

]0,,0,0,0,0,1,0[)0,0(

]0,,0,0,0,0,0,1[)0,0(

L

L

L

L

L

L

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Applications of the new parameterization technique

v Surface Evaluationv Texture Mappingv Boolean Operationsv Surface Trimmingv Adaptive Tessellationv Animation

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Surface Evaluation

Direct & Exact evaluation

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Texture Mapping

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Boolean Operations

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Surface Trimming

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Adaptive Tessellation

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Two Approaches for One-Piece Representation

1. By Interpolation2. By Boolean Operations

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One-Piece through Interpolation

Sampling

Interpolation

One-Piece

SubdivisionSubdivision

Reference

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Similarity Based Interpolation

v Result in a one-piece represented mesh. v Assume the interpolating surface should be

similar to the limit surface of the given meshv Surface fairing is not needed.v No system solvability problem. v Can handle both open and closed meshes.

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Interpolating Open Meshes

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One-Piece by Boolean Operations

Boolean Operations

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Voxelization: Basic Idea

Perform adaptive (midpoint) subdivision in parameter space until each subpatchis small enough so that it can be voxelized using only its four corners

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Related Work

Two major voxelization techniques:

v 3D Scan-line Algorithms [6,7,8,9]�Only good for polygonal meshes

v 3D Spatial Enumeration Algorithms[10]�Computationally expensive.

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Recursive 2D Parameter Space Subdivision

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Separability, Accuracy and Minimality

,, SQSP ∈∃∈∀

,, SPSQ ∈∃∈∀

QP ∈such that

such that QP∈

Note that here P is a 3D point and Q is a voxel, which is a unit cube. S is the CCSS and is the voxelization of S .

S

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Applications of Voxelization

vVisualization of Complex Scenes

v Integral Properties Measurementv Intersection Curves DeterminationvPerforming CSG OperationsvPerforming Boolean Operations

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Rendering of Voxels

Mesh Limit Surface Point based Rendering

Splat based Rendering

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Intersection Curves

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Boolean Operations

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CSG Operations

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Boolean Ops Based on Voxelization

vInside VoxelsvBoundary VoxelsvOutside Voxels

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Approach is Similar for all kinds of Boolean Ops

A B

A-B A+BB-AA*B

Figures from [7]64

Boolean Ops Based on Recursive 2D Parameter Space Subdivision

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Local Voxelziation

v Impossible for 3D voxelization methods.v Easy to be implemented in voxelization

methods based on recursive 2D parameter space subdivision.

v Can obtain high precision of Boolean operation results

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Error Control

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Advantages

•Robust •Error controllable •Resulting solids have continuous

geometric representations, i.e. one-piece representations.

•Crack free•Applicable to any sbudivision scheme

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Example 1

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Example 2

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Example 3

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Two Approaches for Mesh Simplification

1. By Adaptive Tessellation2. By Multi-resolution Analysis

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A disadvantage of CCSSs

v Number of faces in the uniformly refined meshes increases exponentiallywith respect to subdivision depth

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Remedy

v Adaptive tessellation reduces the number of faces needed to yield a smooth approximation to the limit surface

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Basic Idea

• A flat surface patch should not be tessellated as densely as a surface patch with big curvature. • The adaptive tessellation process of a surface patch should be performed based on the flatness of the patch.

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Objective of Adaptive Tessellation

1. Keep the face number of the control mesh small;

2. The approximating model is as precise as possible;

3. Tessellation result is crack free.

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Flatness TestFor a patch S(u,v) defined on u1 < u < u2 andV1 < v < v2 , we try to approximate it with the

quadrilateral formed by its four end vertices

V1 = S(u1, v1) V2 = S(u2, v1)

V3 = S(u2, v2)V4 = S(u1, v2)

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Bilinear Parameterization of a quadrilateral

)()(),( 312

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12

2

12

12

12

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12

2

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2 Vuuuu

Vuuuu

vvvv

Vuuuu

Vuuuu

vvvv

vuL−−

+−−

−−

+−−

+−−

−−

=

V4

V1 V2

V3

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Definition of Distance

The distance between the patch (or subpatch) and the corresponding quadrilateral is:

}],[],[),(),(max{ 2121 vvuuvuvudD ×∈=

2),(),(),( vuSvuLvud −=

Where

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Finding Extrema

By Mean Value Theorem, the minima and maxima must satisfy at least one of the following 3 conditions:

(*)

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Crack Problem

Cracks can be eliminated simply by replacing each boundary of a patch with the one that contains all the evaluated points for that boundary.

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Crack problem

v No crack-detection or crack-eliminationis needed

Therefore, no mesh element splitting to eliminate cracks is necessary

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Examples

RF = 3% RF = 1% RF = 2% RF = 3% RF = 5%

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Examples

RF = 8% RF = 3% RF = 4% RF = 8%

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Mesh Simplification by Multi-resolution Analysis

Figures from [111]

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Multi-resolution Analysis

1. There are many methods proposed in the literature;

2. The most well-known one is based on the wavelet transform;

3. Has many applications in computer graphics and computer aided modeling

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Applications of Multi-resolution Analysis

v Compression of 3D modelsv Simplification of 3D modelsv Progressive Display/Transmissionv Level-Of-Detail ControlvMulti-Resolution Editing

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Basic Idea of Multi-Resolution Analysis

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Requirements of Multi-Resolution Mesh Analysis

v The low-resolution versions are good approximations of the original object ;

v The magnitude of a wavelet coefficient reflects its importance;

v Analysis and synthesis should have linear time complexity.

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Lounsbery et al’s Method

v Based on subdivision surfaces, presents a new class of wavelets that can be applied to functions defined on surfaces of arbitrary topology;

v It can be used for mesh simplification, editing, compression and LOD control etc;

vOnly can be applied to surfaces with subdivision connectivity.

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Subdivision Connectivity Requirement

the input mesh must have the form of a mesh Mj that results from subdividing a simple mesh M0 j times.

Figures from [110]

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Eck et al.’s Method

v Unfortunately, meshes in practice typically do not meet this requirement;

vOur one-piece representation meshes do not meet this requirement either;

v Eck et al.’s Method overcomes the subdivision connectivity restriction, meaning that completely arbitrary meshes can be converted to multiresolution form.

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Two Steps

1. First use the method of Eck et al. to convert any mesh M to a mesh MJ which satisfies the subdivision connectivity requirement,

2. Then use the method of Lounsbery et al. to convert MJ to a multiresolution mesh representation.

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Examples

Eck et al.’s Method

Lounsberyet al’s Method

Arbitrary

connectivity

Subdivision

connectivity

Figures from [111]94

Examples

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Conclusion

v Mathematical foundation for subdivision surface based one-piece representation is developed.

v Two approaches for constructing subdivision surface based one-piece representation are proposed

v Two methods for simplifying control meshes are developed

v A subdivision surface based one-piece representation system is built

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Future Work

vMore convenient & accurate representationsfor topologically complex 3D objects

v Algorithms for construction, rendering, manipulation, processing, transmission & storage of topologically complex 3D objects

v Interdisciplinary research & applications

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Acknowledgement

v Dr. Fuhua (Frank) Chengv Dr. Brent Sealesv Dr. Qiang Yev Dr. Jun Zhangv Dr. Grzegorz Wasilkowskiv Dr. Adam Branscum

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Acknowledgement

The research work with this dissertation is supported in part by U.S. National Science Foundation (NSF) under grants

vDMI-9912069;vDMS-0310645;vDMI-0422126

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Acknowledgement

Some data files are downloaded from:

v The Stanford 3D Scanning Repository, http://graphics.stanford.edu/data/3Dscanrep/;

v Dr. Hugues Hoppe: http://research.microsoft.com/~hoppe;v Dr. Peter Schroder: http://www.cs.caltech.edu/~ps /;v Dr. Denis Zorin: http://mrl.nyu.edu/~dzorin;v Dr. Michael Garland: http://graphics.cs.uiuc.edu/~garland.

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Any Questions?