# Symmetry Hierarchy of Man‐Made Objects

@article{Wang2011SymmetryHO, title={Symmetry Hierarchy of Man‐Made Objects}, author={Yanzhen Wang and Kai Xu and Jun Li and Hao Zhang and Ariel Shamir and Ligang Liu and Z. Cheng and Yueshan Xiong}, journal={Computer Graphics Forum}, year={2011}, volume={30} }

We introduce symmetry hierarchy of man‐made objects, a high‐level structural representation of a 3D model providing a symmetry‐induced, hierarchical organization of the model's constituent parts. Given an input mesh, we segment it into primitive parts and build an initial graph which encodes inter‐part symmetries and connectivity relations, as well as self‐symmetries in individual parts. The symmetry hierarchy is constructed from the initial graph via recursive graph contraction which either… Expand

#### Topics from this paper

#### 127 Citations

Detection of hierarchical intrinsic symmetry structure in 3D models

- Mathematics, Computer Science
- Comput. Graph.
- 2018

This work proposes Hierarchical Intrinsic Symmetry Structure (HISS) to represent the structure of models and an automatic construction approach based on the skeleton of the model, and evaluates the representation and the algorithm with a set of experiments and applications. Expand

Layered analysis of irregular facades via symmetry maximization

- Computer Science, Mathematics
- ACM Trans. Graph.
- 2013

A novel integral symmetry measure is proposed, which behaves well at both ends of the symmetry spectrum by accounting for all partial symmetries in a discrete structure and results in a structural representation, which can be utilized for structural editing and exploration of building facades. Expand

Symmetry in 3D shapes - analysis and applications to model synthesis

- Computer Science, Engineering
- 2011

This thesis proposes feature-based methods to detect symmetry and regularity in 3D shapes and demonstrates the utilization of symmetry information for content generation and introduces two novel feature detection techniques. Expand

Fast global and partial reflective symmetry analyses using boundary surfaces of mechanical components

- Mathematics, Computer Science
- Comput. Aided Des.
- 2014

The proposed symmetry analysis is performed on a B-Rep CAD model through a divide-and-conquer approach over the boundary of a component with faces as atomic entities, and it is possible to identify rapidly all global symmetry planes and axisymmetry as well as local symmetries. Expand

A morphable part model for shape manipulation

- Computer Science
- 2011

This method uses self-similarities from a single model as training input and allows the user to reassemble the identified parts in new configurations, thus exploiting both the discrete and continuous learned variations while ensuring appropriate boundary conditions across part boundaries. Expand

Co-hierarchical analysis of shape structures

- Computer Science, Mathematics
- ACM Trans. Graph.
- 2013

We introduce an unsupervised co-hierarchical analysis of a set of shapes, aimed at discovering their hierarchical part structures and revealing relations between geometrically dissimilar yet… Expand

Replaceable Substructures for Efficient Part‐Based Modeling

- Computer Science
- Comput. Graph. Forum
- 2015

A polynomial time algorithm is presented that discovers replaceable substructures as arrangements of parts that can be interchanged while ensuring boundary consistency by working on a dual of the original shape graph that encodes inter‐part connectivity. Expand

Real‐Time Symmetry‐Preserving Deformation

- Computer Science
- Comput. Graph. Forum
- 2014

This paper introduces an explicit construction of bases of linear spaces of shape deformations that exactly preserve symmetries for any user‐defined level of detail, and applies this framework to symmetry‐preserving co‐rotated iterative Laplace surface editing of models with complex symmetry structure, including partial and nested symmetry. Expand

Microtiles: Extracting Building Blocks from Correspondences

- Mathematics, Computer Science
- Comput. Graph. Forum
- 2012

A new representation for structuring partial symmetries (partial self‐correspondences), which is called “microtiles”, is introduced, which can build all objects that are (r+ε)‐similar to S for any ε >0. Expand

Fold and fit: Space conserving shape editing

- Computer Science
- Comput. Graph.
- 2018

This framework jointly optimizes for joint locations, the folding order, and folding angles for each part of the model, enabling it to transform into a spatially efficient configuration while keeping its original functionality as intact as possible. Expand

#### References

SHOWING 1-10 OF 41 REFERENCES

Structuring 3D Geometry based on Symmetry and Instancing Information

- Mathematics, Computer Science
- 2007

This thesis proposes an original method to structure the geometry at the object level based on the information of symmetry and presents an approach to represent an unstructured 3D scene as a hierarchy of instances adapted to common tasks such as rendering or geometry editing. Expand

A connection between partial symmetry and inverse procedural modeling

- 2010

In this paper, we address the problem of inverse procedural modeling: Given a piece of exemplar 3D geometry, we would like to find a set of rules that describe objects that are similar to the… Expand

Folding meshes: hierarchical mesh segmentation based on planar symmetry

- Mathematics, Computer Science
- SGP '06
- 2006

An algorithm for detecting such symmetries and segmenting the mesh into the symmetric and remaining regions and an application of the method: the folding tree data structure, which encodes the non redundant regions of the original mesh as well as the reflection planes. Expand

Partial and approximate symmetry detection for 3D geometry

- Computer Science
- SIGGRAPH 2006
- 2006

A new algorithm is presented that processes geometric models and efficiently discovers and extracts a compact representation of their Euclidean symmetries, which captures important high-level information about the structure of a geometric model which enables a large set of further processing operations. Expand

Partial intrinsic reflectional symmetry of 3D shapes

- Computer Science
- ACM Trans. Graph.
- 2009

An algorithm to extract partial intrinsic reflectional symmetries (PIRS) of a 3D shape and how the extracted IRSA curves can be incorporated into a conventional mesh segmentation scheme so that the implied symmetry cues can be utilized to obtain more meaningful results are introduced. Expand

Partial intrinsic reflectional symmetry of 3D shapes

- Mathematics
- SIGGRAPH 2009
- 2009

While many 3D objects exhibit various forms of global symmetries, prominent intrinsic symmetries which exist only on parts of an object are also well recognized. Such partial symmetries are often… Expand

Discovering structural regularity in 3D geometry

- Computer Science, Medicine
- ACM Trans. Graph.
- 2008

An optimization method for detecting such uniform grids specifically designed to deal with outliers and missing elements is introduced and yields a robust algorithm that successfully discovers complex regular structures amidst clutter, noise, and missing geometry. Expand

Symmetry-Aware Mesh Processing

- Computer Science
- IMA Conference on the Mathematics of Surfaces
- 2009

A framework for mesh processing that includes steps for symmetrization and symmetric remeshing is investigated, used to enhance the symmetries of a mesh, to decompose a mesh into its symmetric parts and asymmetric residuals, and to establish correspondences between symmetric mesh features. Expand

Constructing Regularity Feature Trees for Solid Models

- Computer Science
- GMP
- 2006

This paper describes a systematic approach for finding regularity features based on recovering broken symmetries in the model, and the output is a tree ofRegularity features for subsequent use in regularity detection and selection. Expand

Symmetries of non-rigid shapes

- Mathematics, Computer Science
- 2007 IEEE 11th International Conference on Computer Vision
- 2007

This paper poses the problem of finding intrinsic symmetries of non-rigid shapes and proposes an efficient method for their computation. Expand