Discovering structural regularity in 3d geometry
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Discovering Structural Regularity in 3D Geometry. Speaker: JinliangWu Date: 25 / 9 /2008. Authors. Mark Pauly ETH Zurich Niloy J. Mitra IIT Delhi Johannes Wallner TU Graz Helmut Pottmann TU Vienna Leonidas Guibas Stanford University. Regular Structures. Regular Structures.

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Discovering structural regularity in 3d geometry

Discovering Structural Regularity in 3D Geometry

Speaker: JinliangWu

Date: 25 / 9 /2008


Authors
Authors

  • Mark Pauly

    ETH Zurich

  • Niloy J. Mitra

    IIT Delhi

  • Johannes Wallner

    TU Graz

  • Helmut Pottmann

    TU Vienna

  • Leonidas Guibas

    Stanford University




Motivation

completion

geometric edits

Motivation

compression

geometry synthesis

Text

Motivation




G eometry s ynthesis
Geometry Synthesis


Structure discovery

Transform

Analysis

Input Model

Transform Clusters

Model

Estimation

Aggregation

Regular Structures

Transform Generators

Structure Discovery

spatial

domain

transform

domain

Structure

Discovery


Repetitive structures

A similarity transformation T

Repetitive Structures


Repetitive structures1
Repetitive Structures

1-parameter patterns


Repetitive structures2
Repetitive Structures

2-parameter commutative patterns



Repetitive structures4
Repetitive Structures

regular structure

is a transformation group acting on

is a collection of n patches of a given surface S


Repetitive structures5
Repetitive Structures

In the simplest setting, is a 1-parameter group with generating similarity transformation T .

The elements of can be represented as


Structure discovery1

Input Model

Transform Clusters

Model

Estimation

Aggregation

Regular Structures

Transform Generators

Structure Discovery

Transform

Analysis

Structure

Discovery


Transformation analysis
Transformation Analysis

Algorithm for analyzing transformations


Transformations
Transformations

spatial domain

transformation space

pairwise transformations


Transformations1
Transformations

spatial domain

transformation space

pairwise transformations


Model estimation
Model Estimation

origin

density plot of

pair-wise transformations


Model estimation1
Model Estimation

cluster centers


Transformation analysis1
Transformation Analysis

Algorithm for analyzing transformations



Model estimation2
Model Estimation

Is there a Pattern?


Model estimation3
Model Estimation

Yes, there is!


Model estimation4
Model Estimation

Yes, there is!


Model estimation5
Model Estimation

Global, non-linear optimization

– simultaneously detects outliers and

grid structure


Model estimation6

grid location

generating vectors

Model Estimation

  • Grid fitting

    – input: cluster centers

– unknowns: grid generators


Model estimation7

data confidence

cluster center

closest grid point

grid confidence

grid point

closest cluster center

Model Estimation

  • Fitting terms


Model estimation8
Model Estimation

  • Fitting terms

  • Data and grid confidence terms

  • objective function


Model estimation9
Model Estimation

Global, non-linear optimization

– simultaneously detects outliers and

grid structure


Structure discovery2

Input Model

Transform Clusters

Model

Estimation

Aggregation

Regular Structures

Transform Generators

Structure Discovery

Transform

Analysis

Structure

Discovery


Aggregation
Aggregation

  • Region-growing to extract repetitive elements

  • Simultaneous registration



Structure discovery3

Input Model

Transform Clusters

Model

Estimation

Regular Structures

Transform Generators

Structure Discovery

Transform

Analysis

Structure

Discovery

Aggregation







Conclusions
Conclusions

  • Algorithm is fully automatic

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

  • Robust, efficient, independent of dimension

    general tool for scientific data analysis