mayank singh and scott schaefer- triangle surfaces with discrete equivalence classes

8/3/2019 Mayank Singh and Scott Schaefer- Triangle Surfaces with Discrete Equivalence Classes

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Triangle Surfaces w ith DiscreEquivalence Classes

Mayank SinghScott Schaefer


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Introduction

Liu et al. [2006] Cutler and Whiting [2007] Pottmann et al. [2007]

Pottmann et al. [2008] SchiftneKillian et al. [2008]


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Introduction




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Introduction




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Introduction




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Introduction




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Economy

Paneling Architectural Freeform SurfacesMichael Eigensatz, Martin Kilian, Alexander Schiftner, Niloy J. Mitra,

Helmut Pottmann and Mark Pauly


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Motivation

Beijing Aquatic Center


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Equivalent Se

576 triangles | 6 un


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Patterns 2D

Planar patterns generated by Craig Kaplan [2004]


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Patterns 3D

Quad parameterization of planar patterns [2009]


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Mosaic 2D

Kim & Pellacini [2002]

Elber & Wolberg [


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Mosaic 3D

Lai et al. [2006]

Passo & Walter [20


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Equivalent Set Surface

OptimizeOriginal


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Discrete Equivalence Classes

Clustering

Rigid Transformation

Global Linear Optimization

ModifiedGeometry

Polygon Assignment &Canonical Triangles

Mesh of CanonicalTriangles

Input Shape


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Example

5-Point Ten

1280 tria


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Canonical Triangle

iP

i

indCD

j

(min,


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Triangle Sim ilarity

a1

a3

a2 b1

b2

b3

Transform B

),( BAD

=

=

+=3

1

2

),(,,

||min),(l

lljpermjTIRR

aTRbBADT


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Triangle Sim ilarity

a1

a3

a2 b1

b2

b3

),( BAD

(b1, b2, b3), ((b3, b1, b2), (

(b3, b2, b1), (

(a1, a2, a3)

D(A,B) = minR

TR =I,T, j

|Rbperm( j,l ) +T al |2

l=1

3


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Canonical Triangle

)0,,(

)0,0,(

)0,0,0(

333,

22,

1,

yxC

xC

C

j

j

j

=

=

=

(0,0,0) (x2,0,0)

(x3,y3,0)

)(iindC

i

indCD

j

(min,Nonlinear Minimization


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Canonical Triangle

Rigid Transfor

=

=

3

1,,

|minl

permjTIRR

RCT

iP


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Adaptive K-Means Clustering

Each triangle is representedas a point


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Compute center of the clusterusing nonlinear search


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Assign the farthest point to anew cluster


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Reassign points to availableclusters


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Process continues to generatemore clusters


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Process continues to generatemore clusters


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Clustering

Polygon Assignment

Gen

Clus

Nonlinear Optimization

Canonical

Polygons


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Clustering

Error

Number of Clusters

2010

5

1

i

indCD

j

min,


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Clustering

=

=

3

1,,

|minl

permjTIRR

RCT

1280 triangles

Rigid Transfor


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Clustering

1280 triangles |


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Varying the Number of Clusters

1 5

10 20

B

O


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Spacing between Triangles

20 clusters

Before GlobalOptimization


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Disconnected Triangles

Poisson Optimization - Yu et al. [2004]


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Global OptimizationPoisson Optimization

Re-Cluster

Re-ComputeCanonical

Triangles


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Global Optimization

)(min bcgP

EEE ++

Gradient Proximity to originshape


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Prox imity and Fairness


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GlobalNon-LinearOptimization

d


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=

=

3

1,,

|minl

pjTIRR

RCT

Rigid Tran

GlobalNon-LinearOptimization

RotatTriang

1 Cl


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1 - Cluster

Archit576 T

2 Cl t


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2 - Clusters

3 Cl t


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3 - Clusters

4 Cl t


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4 - Clusters

5 Cl t


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5 - Clusters

6 Clusters


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6 - Clusters

Clustering & Global Optimization


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Clustering & Global Optimization

Before Global Optimization


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Before Global Optimization

1 5

2010

After Global Optimization


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After Global Optimization

1 5

2010

Example


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Example

2492 triangles | 64 clusters = 2.56% of total triangles

Roof


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Roof1.722%

Torus Knot


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Torus Knot2.014%

Venus


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Venus6.017%

Bunny


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Bunny2.436%

4-point roof


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4 point roof

5-point roof


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5 point roof0.781%

Comparison


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Comparison

K-set Tilable Surfaces Ours

Non planar Quadrilaterals Planar Triangle

8 permutations for best rigidtransformation 6 permutations for best rigtransformation

Mean S-quad, compute once Non linear search for canon

Global non-linear optimization Global linear optimization

Begin with large # of clusters & merge Begin with small # of clust

Future Work


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Future Work

Detect outliers in clusters

n-gons

Planarity

Modify topology

Symmetry? Maintain streamlines

Non-existent?


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Paneling Arch. Freeform Surfaces


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Paneling Arch. Freeform Surfaces

Use small # of molds, with associated co

Create non-congruent panels from the m Emphasis upon streamlines

Minimize divergence and kink angle

Clustering


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Clustering

17 Clusters before global optimizati

convergence

Adding 1 Cluster incrementallyand running optimization to

convergence

Rotation of Canonical Triangle


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Rotation of Canonical Triangle

50% rotation 100% rotation

Comparative Analysis


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p y

Paneling ArchitecturalFreeform Surfaces

K-set Tilab le Surfaces Triangle SuDiscrete Eq

Classes

Use of small # of molds Each mold has anassociated cost Emphasis upon

streamlines Divergence and Kinkangle

Non-planar quads 8 permutations for rigidtransformation Global non-linear

optimization Start with large # ofclusters and merge Mean S-quad, computedonce

Planar Tria 6 permutattransformati Global line

Begin withmore Non linear canonical triupdated for

mayank singh and scott schaefer- triangle surfaces with discrete equivalence classes

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