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Solid Modeling

Adam FinkelsteinPrinceton University

COS 426, Spring 2003

3D Object Representations• Raw data

o Point cloudo Range imageo Polygon soup

• Surfaceso Mesho Subdivision o Parametrico Implicit

• Solidso Voxelso BSP treeo CSGo Sweep

• High-level structureso Scene grapho Skeletono Application specific

3D Object Representations• Raw data

o Point cloudo Range imageo Polygon soup

• Surfaceso Mesho Subdivision o Parametrico Implicit

• Solidso Voxelso BSP treeo CSGo Sweep

• High-level structureso Scene grapho Skeletono Application specific

Implicit Surfaces• Points satisfying: F(x,y,z) = 0

Polygonal Model Implicit Model

Bill LorensenSIGGRAPH 99Course #4 Notes

Implicit Surfaces• Example: quadric

o f(x,y,z)=ax2+by2+cz2+2dxy+2eyz+2fxz+2gx+2hy+2jz +k

• Common quadric surfaces:o Sphereo Ellipsoido Toruso Paraboloido Hyperboloid

H&B Figure 10.10

01222

=−

+

+

zyx rz

ry

rx

Implicit Surfaces• Advantages:

o Very conciseo Guaranteed validityo Easy to test if point is on surfaceo Easy to intersect two surfaces

• Disadvantages:o Hard to describe complex shapeso Hard to enumerate points on surfaceo Hard to draw

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3D Object Representations• Raw data

o Point cloudo Range imageo Polygon soup

• Surfaceso Mesho Subdivision o Parametrico Implicit

• Solidso Voxelso BSP treeo CSGo Sweep

• High-level structureso Scene grapho Skeletono Application specific

Solid Modeling• Represent solid interiors of objects

o Surface may not be described explicitly

Visible Human(National Library of Medicine)

SUNY Stoney Brook

Motivation 1• Some acquisition methods generate solids

o Example: CAT scan

Stanford University

Motivation 2• Some applications require solids

o Example: CAD/CAM

Intergraph Corporation

Motivation 3• Some algorithms require solids

o Example: ray tracing with refraction

Addy Ngan and Zaijin GuanCOS 426, 1998

Princeton University

Solid Modeling Representations• What makes a good solid representation?

o Accurateo Conciseo Affine invarianto Easy acquisitiono Guaranteed validityo Efficient boolean operationso Efficient display

Lorensen

3

Solid Modeling Representations• Voxels

• Quadtrees & Octrees

• Binary space partitions

• Constructive solid geometry

Solid Modeling Representations• Voxels

• Quadtrees & Octrees

• Binary space partitions

• Constructive solid geometry

Voxels• Partition space into uniform grid

o Grid cells are called a voxels (like pixels)

• Store properties of solid object with each voxelo Occupancyo Coloro Densityo Temperatureo etc.

FvDFH Figure 12.20

Voxel Acquisition• Scanning devices

o MRIo CAT

• Simulationo FEM

SUNY Stoney Brook

Stanford University

Voxel Storage• O(n3) storage for nxnxn grid

o 1 billion voxels for 1000x1000x1000

Voxel Boolean Operations• Compare objects voxel by voxel

o Trivial

∪ =

∩ =

4

Voxel Display• Isosurface rendering

o Render surfaces bounding volumetric regions of constant value (e.g., density)

Isosurface VisualizationPrinceton University

Voxel Display• Slicing

o Draw 2D image resulting from intersecting voxels with a plane

Visible Human(National Library of Medicine)

Voxel Display• Ray casting

o Integrate density along rays through pixels

Engine BlockStanford University

Voxels• Advantages

o Simple, intuitive, unambiguouso Same complexity for all objects o Natural acquisition for some applicationso Trivial boolean operations

• Disadvantageso Approximateo Not affine invarianto Large storage requirementso Expensive display

Solid Modeling Representations• Voxels

• Quadtrees & Octrees

• Binary space partitions

• Constructive solid geometry

Quadtrees & Octrees• Refine resolution of voxels hierarchically

o More concise and efficient for non-uniform objects

Uniform Voxels Quadtree

FvDFH Figure 12.21

5

Quadtree Boolean Operations

A B

A ∩ BA ∪ B

FvDFH Figure 12.24

Quadtree Display• Extend voxel methods

o Slicing o Isosurface extractiono Ray casting

Finding neighbor cell requirestraversal of hierarchy (O(1))

FvDFH Figure 12.25

Solid Modeling Representations• Voxels

• Quadtrees & Octrees

• Binary space partitions

• Constructive solid geometry

Binary Space Partitions (BSPs)• Recursive partition of space by planes

o Mark leaf cells as inside or outside object

a

b

c

de

f

1

2

3

7

4

5

6

a

b c

def

g

Object

a

b

cdef

1

2

3

4

5

6

7

Binary Spatial Partition

Binary Tree

Naylor

BSP FundamentalsSingle geometric operation

Partition a convex region by a hyperplaneSingle combinatorial operation

Two child nodes added as leaf nodes

R-

R+

Initial Tree

Homogeneous region

Partitioned region

R

New treeR

⇒

⇒

h

R+R-

R- +

Naylor

BSP is a Search StructureExploit hierarchy of convex regions

Regions decrease in size along any tree pathRegions converge in the limit to the surface

IN

OUTOUT

OUT

on on

on

1st level Approximation

IN

OUTOUT

OUT

out

out

outinin

in

2nd level Approximation

on

on

on on

on

on

Naylor

6

BSP Acquisition• Must construct a “good” binary search structure

o Efficiency comes from logarithmic tree depth

IN

OUTOUT

OUT

out

out

outinin

in

“Bad” “Good”

Naylor

BSP Boolean Operations• Divide and conquer

o Each node V corresponds to a convex region containing all geometry in the subtree rooted at V

o No intersection with bounding volume of V means no intersection with subtree rooted at V

o Do detail work only in regions requiredo Boolean operations grow with O( log n) if “good” tree

Naylor

BSP Display• Visibility ordering

o Determine on which side of plane the viewer lies» near-subtree -> polygons on split -> far-subtree

A

B

Co1

o2

o3

o4

Viewer

o3o2o1 o4

Viewer

3rd 4th 1st 2nd

Partitioning Tree

CB

A

Naylor

Solid Modeling Representations• Voxels

• Quadtrees & Octrees

• Binary space partitions

• Constructive solid geometry

Constructive Solid Geometry (CSG)• Represent solid object as hierarchy of

boolean operationso Uniono Intersectiono Difference

FvDFH Figure 12.27

CSG Acquisition• Interactive modeling programs

o CAD/CAM

H&B Figure 9.9

7

CSG Boolean Operations• Create a new CSG node joining subtrees

o Uniono Intersectiono Difference

FvDFH Figure 12.27

CSG Display & Analysis• Ray casting

Circle Box

Union

Summary

Accurate No No Some SomeConcise No No No YesAffine invariant No No Yes YesEasy acquisition Some Some No SomeGuaranteed validity Yes Yes Yes NoEfficient boolean operations Yes Yes Yes YesEfficient display No No Yes No

Vox

els

Oct

ree

BSP

CSG

Taxonomy of 3D Representations

Discrete Continuous

Combinatorial Functional

Parametric ImplicitTopological Set Membership

Voxels

MeshSubdivision

BSP TreeCell Complex

BezierB-Spline

Algebraic

Naylor