Triangle Surfaces with Discrete Equivalence Classes Mayank Singh - - PowerPoint PPT Presentation

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Triangle Surfaces with Discrete Equivalence Classes Mayank Singh Scott Schaefer I ntroduction Liu et al. [2006] Cutler and Whiting [2007] Pottmann et al. [2007] Killian et al. [2008] Pottmann et al. [2008] Schiftner et al. [2009] I

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SLIDE 2

Triangle Surfaces with Discrete Equivalence Classes

Mayank Singh Scott Schaefer

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SLIDE 3

I ntroduction

Liu et al. [2006] Cutler and Whiting [2007] Pottmann et al. [2007] Pottmann et al. [2008] Schiftner et al. [2009] Killian et al. [2008]

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SLIDE 4

I ntroduction

Liu et al. [2006] Cutler and Whiting [2007] Pottmann et al. [2007] Pottmann et al. [2008] Schiftner et al. [2009] Killian et al. [2008]

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SLIDE 5

I ntroduction

Liu et al. [2006] Cutler and Whiting [2007] Pottmann et al. [2007] Pottmann et al. [2008] Schiftner et al. [2009] Killian et al. [2008]

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SLIDE 6

I ntroduction

Liu et al. [2006] Cutler and Whiting [2007] Pottmann et al. [2007] Pottmann et al. [2008] Schiftner et al. [2009] Killian et al. [2008]

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SLIDE 7

I ntroduction

Liu et al. [2006] Cutler and Whiting [2007] Pottmann et al. [2007] Pottmann et al. [2008] Schiftner et al. [2009] Killian et al. [2008]

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SLIDE 8

Economy

Paneling Architectural Freeform Surfaces

Michael Eigensatz, Martin Kilian, Alexander Schiftner, Niloy J. Mitra, Helmut Pottmann and Mark Pauly

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SLIDE 9

Motivation

Beijing Aquatic Center

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Equivalent Set Surface

576 triangles | 6 unique triangle

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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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SLIDE 13

Mosaic – 2D

Kim & Pellacini [2002] Elber & Wolberg [2003]

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SLIDE 14

Mosaic – 3D

Lai et al. [2006] Passo & Walter [2008]

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Equivalent Set Surface

Optimized Original

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Discrete Equivalence Classes

Clustering Rigid Transformation Global Linear Optimization

Modified Geometry Polygon Assignment & Canonical Triangles Mesh of Canonical Triangles

Input Shape

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Example

5-Point Tensile Roof 1280 triangles

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Canonical Triangle

i

P

) (i ind

C

∑

i i ind i ind C

C P D

j

) , ( min

) ( ,

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SLIDE 19

Triangle Similarity

a1 a3 a2 b1 b2 b3

Transform B

) , ( B A D

∑

= =

− + =

3 1 2 ) , ( , ,

| | min ) , (

l l l j perm j T I R R

a T Rb B A D

T

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SLIDE 20

Triangle Similarity

a1 a3 a2 b1 b2 b3

) , ( B A D

(b1, b2, b3), (b2, b3, b1), (b3, b1, b2), (b1, b3, b2), (b3, b2, b1), (b2, b1, b3) (a1, a2, a3)

D(A,B) = min

R T R =I,T , j

| Rbperm( j,l) + T − al |2

l =1 3

∑

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SLIDE 21

Canonical Triangle

) , , ( ) , , ( ) , , (

3 3 3 , 2 2 , 1 ,

y x C x C C

j j j

= = =

(0,0,0) (x2,0,0) (x3,y3,0)

) (i ind

C

∑

i i ind i ind C

C P D

j

) , ( min

) ( ,

Nonlinear Minimization

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SLIDE 22

) (i ind

C Canonical Triangle

Rigid Transformation

∑

= =

− +

3 1 2 ) , ( , ,

| | min

l l l j perm j T I R R

P T RC

T

i

P

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SLIDE 23

Adaptive K-Means Clustering

Each triangle is represented as a point

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Adaptive K-Means Clustering

Compute center of the cluster using nonlinear search

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Adaptive K-Means Clustering

Assign the farthest point to a new cluster

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Adaptive K-Means Clustering

Reassign points to available clusters

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Adaptive K-Means Clustering

Process continues to generate more clusters

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Adaptive K-Means Clustering

Process continues to generate more clusters

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Clustering

Polygon Assignment Generate Clusters Nonlinear Optimization Canonical Polygons

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Clustering

Error Number of Clusters 20 10 5 1

∑

i i ind i ind C

C P D

j

) , ( min

) ( ,

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SLIDE 31

Clustering

∑

= =

− +

3 1 2 ) , ( , ,

| | min

l l l j perm j T I R R

P T RC

T

1280 triangles | 1 cluster Rigid Transformation

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SLIDE 32

Clustering

1280 triangles | 10 clusters

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Varying the Number of Clusters

1 5 10 20

Before Global Optimization

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Spacing between Triangles

20 clusters

Before Global Optimization

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Disconnected Triangles

Poisson Optimization - Yu et al. [2004]

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Global Optimization

Poisson Optimization Re-Cluster Re-Compute Canonical Triangles Deform Original Mesh

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Global Optimization

) ( min

b c g P

E E E β α + +

Gradient Proximity to original shape

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Proximity and Fairness

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Proximity and Fairness

Global Non-Linear Optimization

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Proximity and Fairness

∑

= =

− +

3 1 2 ) , ( , ,

| | min

l l l j perm j T I R R

P T RC

T

Rigid Transformation Global Non-Linear Optimization + Rotate Canonical Triangle

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1 - Cluster

Architectural Dome 576 Triangles

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2 - Clusters

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3 - Clusters

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4 - Clusters

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5 - Clusters

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6 - Clusters

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Clustering & Global Optimization

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Before Global Optimization

1 5 20 10

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After Global Optimization

1 5 20 10

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Example

2492 triangles | 64 clusters = 2.56% of total triangles

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Roof

1.722%

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Torus Knot

2.014%

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SLIDE 53

Venus

6.017%

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Bunny

2.436%

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SLIDE 55

4-point roof

0.313%

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SLIDE 56

5-point roof

0.781%

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Comparison

K-set Tilable Surfaces Ours

Non planar Quadrilaterals Planar Triangle 8 permutations for best rigid transformation 6 permutations for best rigid transformation Mean S-quad, compute once Non linear search for canonical, iterative Global non-linear optimization Global linear optimization Begin with large # of clusters & merge Begin with small # of clusters & add more

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Future Work

  • Detect outliers in clusters
  • n-gons

– Planarity

  • Modify topology

– Symmetry?

  • Maintain streamlines

– Non-existent?

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SLIDE 59
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SLIDE 60
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Paneling Arch. Freeform Surfaces

  • Use small # of molds, with associated cost
  • Create non-congruent panels from the mold
  • Emphasis upon streamlines
  • Minimize divergence and kink angle
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Clustering

17 Clusters before running global optimization to convergence Adding 1 Cluster incrementally and running optimization to convergence

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Rotation of Canonical Triangle

50% rotation 100% rotation

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Comparative Analysis

Paneling Architectural Freeform Surfaces K-set Tilable Surfaces Triangle Surfaces with Discrete Equivalence Classes

  • Use of small # of molds
  • Each mold has an

associated cost

  • Emphasis upon

streamlines

  • Divergence and Kink

angle

  • Non-planar quads
  • 8 permutations for rigid

transformation

  • Global non-linear
  • ptimization
  • Start with large # of

clusters and merge

  • Mean S-quad, computed
  • nce
  • Planar Triangles
  • 6 permutations for rigid

transformation

  • Global linear optimization
  • Begin with 1 cluster, add

more

  • Non linear search for

canonical triangles, updated for each iteration