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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

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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**completion**geometric edits Motivation compression geometry synthesis Text Motivation**Transform**Analysis Input Model Transform Clusters Model Estimation Aggregation Regular Structures Transform Generators Structure Discovery spatial domain transform domain Structure Discovery**A similarity transformation T**Repetitive Structures**Repetitive Structures**1-parameter patterns**Repetitive Structures**2-parameter commutative patterns**Repetitive Structures**regular structure is a transformation group acting on is a collection of n patches of a given surface S**Repetitive Structures**In the simplest setting, is a 1-parameter group with generating similarity transformation T . The elements of can be represented as**Input Model**Transform Clusters Model Estimation Aggregation Regular Structures Transform Generators Structure Discovery Transform Analysis Structure Discovery**Transformation Analysis**Algorithm for analyzing transformations**Transformations**spatial domain transformation space pairwise transformations**Transformations**spatial domain transformation space pairwise transformations**Model Estimation**origin density plot of pair-wise transformations**Model Estimation**cluster centers**Transformation Analysis**Algorithm for analyzing transformations**Model Estimation**Is there a Pattern?**Model Estimation**Yes, there is!**Model Estimation**Yes, there is!**Model Estimation**Global, non-linear optimization – simultaneously detects outliers and grid structure**grid location**generating vectors Model Estimation • Grid fitting – input: cluster centers – unknowns: grid generators**data confidence**cluster center closest grid point grid confidence grid point closest cluster center Model Estimation • Fitting terms**Model Estimation**• Fitting terms • Data and grid confidence terms • objective function**Model Estimation**Global, non-linear optimization – simultaneously detects outliers and grid structure**Input Model**Transform Clusters Model Estimation Aggregation Regular Structures Transform Generators Structure Discovery Transform Analysis Structure Discovery**Aggregation**• Region-growing to extract repetitive elements • Simultaneous registration**Input Model**Transform Clusters Model Estimation Regular Structures Transform Generators Structure Discovery Transform Analysis Structure Discovery Aggregation**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

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