Discovering Structural Regularity in 3D Geometrygraphics.stanford.edu/~niloy/research/structure/paper_docs/... · Discovering Structural Regularity in 3D Geometry Leonidas Guibas

Discovering Structural Regularity in 3D Geometry

Leonidas GuibasStanford University

Helmut PottmannTU Vienna

Johannes WallnerTU Graz

Niloy J. MitraIIT Delhi

Mark PaulyETH Zurich

Discovering Structural Regularity in 3D GeometryNiloy J. Mitra

Regular Structures


Regular Structures


Regular Structure


Regular Structure


Regular Structure


Regular Structure

+


Motivation


Motivation

• Regularity form, semantics


Motivation



Motivation


• Scan cleaning, completion


Motivation



• Compression


Motivation



• Compression

• Geometric edits, synthesis


Motivation



• Compression

• Geometric edits, synthesis

• Growth laws or design principles


Inspiration

On Growth and Form [Thompson 1917]


Related Work


Related Work

[Mitra et al. `06][Podolak et al. `06] [Loy, Eklundh `06] [Martinet et al. `07]


Related Work


[Funkhouser et al. `05] [Thrun, Wegbreit `05] [Shimari et al. `06] [Liu et al. `08]


Related Work



[Wang et al. `08][Mueller et al. `07][Hays et al. `06] [Baudes et al. `08]


Related Work



[Wang et al. `08][Mueller et al. `07][Hays et al. `06] [Baudes et al. `08]


Structure Discovery

Input Model

Regular Structures

StructureDiscovery


Structure Discovery

Input Model

Regular Structures

StructureDiscovery


Structure Discovery

Input Model

TransformAnalysis

Transform Clusters

Regular Structures

StructureDiscovery


Structure Discovery

Input Model

TransformAnalysis

Transform Clusters

ModelEstimation

Transform GeneratorsRegular Structures

Aggregation

StructureDiscovery


Structure Discovery

Input Model

TransformAnalysis

Transform Clusters

ModelEstimation


Aggregation

StructureDiscovery


Structure Discovery

Input Model

TransformAnalysis

Transform Clusters

ModelEstimation


Aggregation

StructureDiscovery


Structure Discovery

Input Model

TransformAnalysis

Transform Clusters

ModelEstimation


Aggregation

StructureDiscovery

spatialdomain


Structure Discovery

Input Model

TransformAnalysis

Transform Clusters

ModelEstimation


Aggregation

StructureDiscovery

spatialdomain

transformdomain


Repetitive Structures

• Invariance under transformations




Translation




Translation

Rotation




Translation

Rotation

Scaling




Translation + Rotation




Translation + Rotation Rotation + Scaling





1-parameter patterns




Translation × Translation




Translation × Translation Translation × Rotation





Rotation × Scaling





Rotation × Scaling

2-parameter commutative patterns



• 1-parameter groups

• Commutative 2-parameter groups

Translation

Rotation

Scaling


Translation × Translation Translation × Rotation Rotation × Scaling


Input Model

TransformAnalysis

Transform Clusters

ModelEstimation


Aggregation

StructureDiscovery

Structure Discovery


Structure Discovery

Input Model

TransformAnalysis

Transform Clusters

ModelEstimation


Aggregation

StructureDiscovery


Transformations


Transformations [Mitra et al. `06]


Transformations


Transformations


Transformations


Transformations


Transformations


Transformations


Transformations


Transformations


Transformations


Transformations


Transformations


Transformations


Transformations


Transformations


Transformations

n2 transformations


Transformations

n2 transformations

spatial domain transformation space


Transformations



Transformations



Transformations



Model Estimation


Model Estimation

density plot of pair-wise transformations


Model Estimation


origin


Model Estimation



Model Estimation

cluster centers


Transform Mapping


Transform Mapping

Gi1.G

j1 ! {ig1 + jg2}


Transform Mapping

I! {0}

Gi1.G

j1 ! {ig1 + jg2}


Transform Mapping

I! {0}

T ! (t1, t2)Translation x Translation

T ! (!, log s)Rotation x Scaling

Translation x Rotation T ! (t, !)

Gi1.G

j1 ! {ig1 + jg2}


Structure Discovery

Input Model

TransformAnalysis

Transform Clusters

ModelEstimation


Aggregation

StructureDiscovery


Model Estimation


Is there a Pattern?


Is there a Pattern?


Yes, there is!


Yes, there is!


Model Estimation


• Global, non-linear optimization

Model Estimation


• Global, non-linear optimization – simultaneously detects outliers and

grid structure

Model Estimation


• Grid fitting– input: cluster centers

Model Estimation

C = {c1, . . . , cn}



– unknowns: grid generators

Model Estimation

xij = ig1 + jg2

generating vectorsgrid location

C = {c1, . . . , cn}



– unknowns: grid generators

Model Estimation

xij = ig1 + jg2

generating vectorsgrid location

C = {c1, . . . , cn}

i ! ["n, n]

j ! ["m,m]


• Fitting terms

Model Estimation

closest grid pointcluster center

EC!X =!|C|

k=1!ck " x(k)!2

closest cluster centergrid point

EX!C =!

i

!j!xij " c(i, j)!2


• Fitting terms

Model Estimation

EC!X =!|C|

k=1!2

k!ck " x(k)!2

closest grid point

EX!C =!

i

!j!2

ij!xij " c(i, j)!2

closest cluster centergrid point

cluster center

grid confidence

data confidence


• Fitting terms

• Data and grid confidence terms

Model Estimation

EC!X =!|C|

k=1!2

k!ck " x(k)!2

EX!C =!

i

!j!2

ij!xij " c(i, j)!2

E! =!

i

!j(1! !2

ij)2 E! =

!k(1! !2

k)2



grid structure

Model Estimation



grid structure

Model Estimation

!ij!k

g1,g2


Structure Discovery

Input Model

TransformAnalysis

Transform Clusters

ModelEstimation


Aggregation

StructureDiscovery


Aggregation

• Region-growing to extract repetitive elements


Aggregation


• Simultaneous registration


Aggregation



H+ ! H + !D · H


Aggregation



H+ ! H + !D · HT k

+ ! (H + !D · H)k


Structure Discovery

Input Model

TransformAnalysis

Transform Clusters

ModelEstimation


Aggregation

StructureDiscovery


Structure Discovery

Input Model

TransformAnalysis

Transform Clusters

ModelEstimation


Aggregation

StructureDiscovery

Aggregation Aggregation

AggregationAggregation


Results and Applications


Amphitheater


Amphitheater


Amphitheater


Amphitheater


Robustness


Robustness


Robustness


Robustness


Robustness


Geometry Synthesis


Geometry Synthesis


Geometry Synthesis


Geometry Synthesis


Geometry Synthesis


Geometry Synthesis


Geometry Synthesis


Geometry Synthesis


Geometry Synthesis


Geometry Synthesis


Geometry Synthesis


Geometry Synthesis


Geometry Synthesis


Chambord Castle


Chambord Castle


Chambord Castle


Chambord Castle [Mitra et al. `06]


Chambord Castle


Chambord Castle


Chambord Castle


Chambord Castle


Outdoor Scan


Outdoor Scan


Outdoor Scan


Scan Completion


Scan Completion


Chambered Nautilus


Chambered Nautilus


Chambered Nautilus


Chambered Nautilus


Scan Completion


Geometry Synthesis


Geometry Synthesis


Geometry Synthesis


Geometry Synthesis


Geometry Synthesis


Geometry Synthesis


Observations


Observations

• Warped structures


Observations


• Size of grid vs accuracy


Observations


• Size of grid vs accuracy• Choice of parameters


Structure Discovery

Input Model Transform Clusters Transform Generators Regular Structures

AggregationModelEstimation

TransformAnalysis


• Algorithm is fully automatic

Structure Discovery



TransformAnalysis



• Requires no prior information on size, shape, or location of repetitive elements

Structure Discovery



TransformAnalysis



• Requires no prior information on size, shape, or location of repetitive elements

• Robust, efficient, independent of dimension ➙ general tool for scientific data analysis

Structure Discovery



TransformAnalysis


Acknowledgements


Acknowledgements• Funding Agencies:

Austrian Science Fund (FWF) Darpa HR0011-05-1-0007NIH GM-072970NSF FRG-0354543TCS




• Data Source:Institute of Cartography and Geoinformatics, Leibniz University, Germany




• Data Source:Institute of Cartography and Geoinformatics, Leibniz University, Germany

• Scanning, code snippets:Michael EigensatzBalint MiklosHeinz Schmiedhofer


Thank You


Thank You


Discovering Structural Regularity in 3D Geometrygraphics.stanford.edu/~niloy/research/structure/paper_docs/... · Discovering Structural Regularity in 3D Geometry Leonidas Guibas

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