an algorithm for triangulating multiple 3d polygons · an algorithm for triangulating multiple 3d...

An algorithm for triangulating multiple 3D polygons

Ming Zou1, Tao Ju1, Nathan Carr2

1 Washington University In St. Louis, USA2 Adobe, USA

1Eurographics SGP 2013

Motivation

2

Curves

Surface

➠

Motivation

3

filling holes

Motivation

4

filling holes surfacing parallel contours

Motivation

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filling holes surfacing parallel contours surfacing 3D sketches

Motivation

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Step 1: Identify Patch Boundaries


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Step 2: Surfacing each patch


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Step 2: Surfacing each patch


Problem Definition

• Input: k 3D polygons (k≥1)

• Output: a triangulation• Consists of triangles

connecting only the input polygon vertices

• Topologically equivalent to a sphere with k holes

• Initial mesh for further refinement

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Input:

output:

Problem Definition






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Monkey Saddle

Problem Definition






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Problem Definition


• Output: an optimal triangulation• A triangulation that minimize

the sum of certain metric

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Problem Definition


• Output: an optimal triangulation• A triangulation that minimize

the sum of certain metric

• Per-triangle metric• Area

• Perimeter

• Bi-triangle metric• Dihedral angle

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Related Work - Single Polygon

• A 2D polygon• Linear-time triangulation of a simple polygon

[Chazelle 91]

• No optimality guarantee

• Dynamic programming for optimal triangulation [Gilbert 79, Klincsek 80]

• A 3D polygon• Dynamic programming for optimal triangulation

[Barequet et al. 95, 96]

• Extension of [Gilbert 79] and [Klincsek 80]

• Heuristic-based algorithms [Liepa 03] [Roth et al. 97]

[Bessmeltsev et al. 12]


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Related Work - Single Polygon

• A 2D polygon• Linear-time triangulation of a simple polygon

[Chazelle 91]


• Dynamic programming for optimal triangulation [Gilbert 79, Klincsek 80]

• A 3D polygon• Dynamic programming for optimal triangulation


• Extension of [Gilbert 79] and [Klincsek 80]

• Heuristic-based algorithms [Liepa 03] [Roth et al. 97]

[Bessmeltsev et al. 12]


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Related Work - Multiple Polygons

• Planar polygons in 3D• Dynamic programming for optimally triangulating

2 parallel planar polygons [Fuchs et al. 77]

• Heuristic-based algorithms for triangulating ≥2 planar polygons [Barequet et al. 04] [Boissonnat et al. 07] [Liu et al. 08]

• No optimality guarantees

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Related Work - Multiple Polygons

• Planar polygons in 3D• Dynamic programming for optimally triangulating

2 parallel planar polygons [Fuchs et al. 77]

• Heuristic-based algorithms for triangulating ≥2 planar polygons [Barequet et al. 04] [Boissonnat et al. 07] [Liu et al. 08]

• No optimality guarantees

• Non-planar polygons in 3D• Heuristic-based algorithms

• No optimality guarantees [Rose et al. 07]

• Optimal algorithm • No known literature

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Contributions

1. The first algorithm for optimally triangulating multiple non-planar 3D polygons• Extension of the dynamic programming algorithm for single polygon


• Guarantee the manifoldness of the surface

2. A fast near-optimal variant for practical use

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• Example: minimize the sum of triangle area

Triangulating A Single 3D Polygon [Barequet et. al 95, 96]

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21



22spanning edge

: spanning edge


t


23spanning triangle

: spanning edge : spanning triangle


t


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D1 D2

: spanning edge : spanning triangleDi : sub-domain

D1


t


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D1 D2


D1

spanning edge

polygon segment↰

↰


t


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D1 D2


Cost(D) = Cost(D1)+Cost(D2)+Area(t)

D

↵


t


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t


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t


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t


30




t


31




t


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Cost(D) = Mint [Cost(D1)+Cost(D2)+Area(t)]



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Cost(D) = Mint [Cost(D1)+Cost(D2)+Area(t)]

Triangulating Multiple 3D Polygons

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• Extension of Domain structure


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spanning edge

spanning edgespan

ning

edg

e



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: spanning edge : polygon segment : spanning triangleDi : sub-domain

polygon segment

polygon segment

polygon segment



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: spanning edge : polygon segment : hole : spanning triangleDi : sub-domain

holehole



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an input polygon can appear at most once in a domaineither as a polygon segment or a hole



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1 ≤ # of polygon segments ≤ k



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42



t


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t


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t

Case I split Case II split


t


45Case I split Case II split

↻

↺D1

D2


t


46Case II split

↻

↺D1

D2

Cost(D,t)=Minx=1,2[ Cost(Dx) + Area(t) ]



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↻

↺D1

D2

D11 D12

Case II splitCost(D,t)=Minx=1,2[ Cost(Dx) + Area(t) ]

D31 D32 D41 D42

D21 D22



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↻

↺D1

D2

D11 D12

Cost(D,t)=Minx=1,2[ Cost(Dx) + Area(t) ]

D31 D32 D41 D42

D21 D22

Cost(D,t)=Minx=1,2,3,4[ Cost(Dx1)+Cost(Dx2)+Area(t) ]



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↻

↺D1

D2

D11 D12

D31 D32 D41 D42

D21 D22

Cost(D) = Mint [ Cost(D,t) ]


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Topologically equivalent to a sphere with k holes

Manifold Guarantee

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Manifold Guarantee

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manifold non-manifold

Manifold Guarantee

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Manifold Guarantee

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t

Manifold Guarantee

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Manifold Guarantee

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a

b

c

d

e

f

g

x y

z

w

Manifold Guarantee

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a

b

c d e

f

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x

y

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x

w

Manifold Guarantee

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a

b

c d e

f

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x

y

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x

w

Manifold Guarantee

➠ ➠

t

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a

b

c d e

f

g

x

y

z

x

w

Manifold Guarantee

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a

b

c d e

f

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x

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Manifold Guarantee

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a

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c d e

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Manifold Guarantee

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a

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c d e

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x

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Manifold Guarantee

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a

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c d e

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Manifold Guarantee

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a

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Manifold Guarantee

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a

b

c d e

f

g

x

y

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x

w

Weak Edges: edges that connecting the ends of the

polygon segments

Manifold Guarantee

t

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a

b

c d e

f

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x

y

z

x

w

Manifold Guarantee

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a

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c d e

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y

Manifold Guarantee

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a

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c d e

g

x

y

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x

w

y

Manifold Guarantee

f

• Compute optimal triangulation for each combination of weak edges, for each domain

• Only combine triangulations that do not share any weak edge

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a

b

c d e

f

g

x

y

z

x

w

y

Manifold Guarantee

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a

b

c d e

f

g

x

y

z

x

w

y

a

b

c d e

f

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x

y

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w

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✘ ✔

a

b

c d e

f

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x

y

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x

w

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Opt triangulation that contains weak edge set {ax, ay, xy, gy}

Opt triangulation that contains weak edge set {ax, ag, xg, gy}

• The triangulation computed by the algorithm is guarantee to be the triangulation with a minimum cost that has the right topology.

• Minimal set: a reduced set of weak edge combination.• still guarantee optimality

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Manifold Guarantee

• Match the complexity of the algorithm for triangulating single 3D polygon [Barequet et al 95, 96]

Complexity

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time space

per-trianglemetric

bi-trianglemetric

O(n2k+1) O(n2k)

O(n3k+1) O(n3k)

n: # of total vertices; k: # of 3D polygons;

• Test on single polygon minimizing sun of dihedral angles• practical complexities matches theoretical ones

• same complexity of [Barequet et al 95, 96]

Performance - Single polygon

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Theoretical: O(n4); Practical: O(n4.11) Theoretical: O(n3); Practical: O(n2.27)




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Theoretical: O(n4); Practical: O(n4.11) Theoretical: O(n3); Practical: O(n2.27)


• Dataset:• k (1≤k≤6) perturbed saddle curves on the corner of a regular octahedron

• 50~60 data for each k

Performance - Multiple polygons

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• Dataset:• k (1≤k≤6) perturbed saddle curves on the corner of a regular octahedron

• 50~60 data for each k


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32poly poly


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32poly poly

Improving efficiency

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All Triangles


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A Triangle Subset


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A Triangle Subset

• Considerably smaller

• Big enough to contain a close-to-optimal triangulation

Requirements of the subset

All Triangles


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All Triangles

Delaunay Triangles


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• Delaunay Triangle Subset:• Fewer: O(n3) → O(n2)

• Good triangles

• No self-intersecting


All Triangles

Delaunay Triangles

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Performance - Multiple polygon

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32 4 5 6poly poly polypoly poly

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32 4 5 6

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32 4 5 6

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32 4 5 6

How Optimal?

Optimality

• Close-to-optimal results on both smooth curves and random loops on mesh• Minimizing total area

• near-optimal for single polygon

• less optimal for two polygons

• Minimizing average dihedral angle• near-optimal for both single and two-polygon

cases

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triangulation on all triangles

triangulation on Delaunay triangles

Applications

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• Sketch based modeling

• Hole filling• 455 groups of holes (61 holes have interior islands)

• fill all the holes in 40 sec

Applications

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• Hole filling• 455 groups of holes (61 holes have interior islands)

• fill all the holes in 40 sec

Applications

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Islands

Conclusion

• The first optimal algorithm for triangulating multiple non-planar 3D polygons• guarantee manifoldness of the surface

• fast near-optimal variation method

• Future work• other triangle subsets besides Delaunay triangles

• more sophisticated weighting schemes

• automatic grouping of hole islands

• Code available online!• http://www.cse.wustl.edu/~zoum/projects/TriMultPoly/

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http://www.cse.wustl.edu/~zoum/projects/TriMultPoly/

http://www.cse.wustl.edu/~zoum/projects/TriMultPoly/

Thank you !

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