Approximate Symmetry Detection and...
1
Automatic correspondence computation. The two poses are deformed towards each other by successively contracting and merging the most prominent clusters An architectural design study. The zooms in the top row show how the meshing of the extracted symmetric element evolves during the optimization. This element appears six times in different location and orientation. symmetrized original front feet back feet original symmetrized Symmetrization of a sculpted model. Large-scale deformations achieve a global re�lective symmetry by straightening the spine, while many small-scale deformations symmetrize the model locally, e.g., the horns, tongue, or toes. Symmetrizing the Stanford bunny. Two clusters corresponding to re�lective symmetries of the head and body of the model are merged to obtain a globally symmetric shape. 6 4 3 5 1 2 500 random samples out of 2000 total 6 4 3 5 1 2 + + s cluster height 1 2 Combined Symmetries. The six most signi�icant modes of the Sydney Opera with the full 7-dimensional symmetries (1, 2, 3) and pure re�lections (4, 5, 6). The graph shows the distribution of scaling factors. Model compression and advanced editing. (Top) Chambord castle, (bottom left) reduced model with detected (partial) symmetry structure, (bottom right) advanced editing respecting the original symmetry relations. d transformation space Illustration of symmetry detection for re�lections. Every pair of points de�ines a symmetry line that can be described by a distance d and an angle φ. Multiple points clustered in a small region in transformation space provide evidence of a symmetry. The pair on the top left is discarded due to normal inconsistency. Approximate symmetry detection. (Top) Original sculpted dragon model, (left) detected partial and approximate symmetries, (right) color-coded deviations from perfect symmetry as a fraction of the bounding box diagonal. original locally symmetrized globally symmetrized transformation space d θ SYMMETRY DETECTION: Natural and man-made objects often exhibit symmetry and contain repeated structures. We present an output sensitive algorithm to detect partial and approximate symmetries in 3D objects. We gather evidence for possible symmetries by matching local symmetries in pairs, followed by accumulation in the transform domain by mean shift clustering. Potential symmetries are then veri�ied in the object domain to generate a set of symmetric patches. SYMMETRIZATION: We present a novel algorithm for enhancing approximate symmetries of geometric models by minimally deforming their shapes. Symmetrizing deformations are formulated as an optimization process that couples the spatial domain with a transformation con�iguration space, where symmetries can be expressed more naturally as parametrized pointpair mappings. Displacement vectors computed from the optimal symmetry transformation are used to drive a constrained deformation model that pulls the shape towards symmetry. sampling analysis pairing clustering patching transformations signatures input model sample set surface patches density References: 1. Symmetrization, N. J. Mitra, L. Guibas, M. Pauly, ACM SIGGRAPH, 2007. 2. Partial and Approximate Symmetry Detection in 3D Geometry, N. J. Mitra, L. Guibas, M. Pauly, ACM SIGGRAPH, 2006. scale + rotation + translation re�lection re�lection + rotation + translation Types of symmetry. Planar re�lection, rigid transforms, uniform scaling, or their combinations. original symmetrized Funding Sources: This work is supported by S9206-N12 of the Austrian Science Fund (FWF), SNF 200021- 112122/1 of the Swiss Naonal Science Foundaon, NSF grant CARGO 0310661, ARO grant DAAD19-03-1- 0331, NIH grant GM-072970, and DARPA grant 32905. Approximate Symmetry Detection and Symmetrization Mark Pauly ETH Zurich Leonidas J. Guibas Stanford University Niloy J. Mitra Vienna University of Technology Symmetry Extraction Pipeline. Symmetrization in 2D. Local and global symmetrization of a gecko model. 0.02 -0.02
Transcript of Approximate Symmetry Detection and...
Automatic correspondence computation. The two poses are deformed towards each other by successively contracting and merging the most prominent clusters
An architectural design study. The zooms in the top row show how the meshing of the extracted symmetric element evolves during the optimization. This element appears six times in different location and orientation.
symmetrized
original
front feet back feet
original symmetrized
Symmetrization of a sculpted model. Large-scale deformations achieve a global re�lective symmetry by straightening the spine, while many small-scale deformations symmetrize the model locally, e.g., the horns, tongue, or toes.
Symmetrizing the Stanford bunny. Two clusters corresponding to re�lective symmetries of the head and body of the model are merged to obtain a globally symmetric shape.
643 5
1
2
500 random samplesout of 2000 total
64
3
5
1
2
+ +
sclus
ter
heig
ht
1 2
Combined Symmetries. The six most signi�icant modes of the Sydney Opera with the full 7-dimensional symmetries (1, 2, 3) and pure re�lections (4, 5, 6). The graph shows the distribution of scaling factors.
Model compression and advanced editing. (Top) Chambord castle, (bottom left) reduced model with detected (partial) symmetry structure, (bottom right) advanced editing respecting the original symmetry relations.
d
transformation space
Illustration of symmetry detection for re�lections. Every pair of points de�ines a symmetry line that can be described by a distance d and an angle φ. Multiple points clustered in a small region in transformation space provide evidence of a symmetry. The pair on the top left is discarded due to normal inconsistency.
Approximate symmetry detection. (Top) Original sculpted dragon model, (left) detected partial and approximate symmetries, (right) color-coded deviations from perfect symmetry as a fraction of the bounding box diagonal.
original
locallysymmetrized
globallysymmetrized
transformation spaced
θ
SYMMETRY DETECTION: Natural and man-made objects often exhibit symmetry and contain repeated structures. We present an output sensitive algorithm to detect partial and approximate symmetries in 3D objects. We gather evidence for possible symmetries by matching local symmetries in pairs, followed by accumulation in the transform domain by mean shift clustering. Potential symmetries are then veri�ied in the object domain to generate a set of symmetric patches.
SYMMETRIZATION: We present a novel algorithm for enhancing approximate symmetries of geometric models by minimally deforming their shapes. Symmetrizing deformations are formulated as an optimization process that couples the spatial domain with a transformation con�iguration space, where symmetries can be expressed more naturally as parametrized pointpair mappings. Displacement vectors computed from the optimal symmetry transformation are used to drive a constrained deformation model that pulls the shape towards symmetry.
sampling
analysis
pairing
clustering
patching
transformations
signatures
input model
sample set
surface patches
density
References: 1. Symmetrization, N. J. Mitra, L. Guibas, M. Pauly, ACM SIGGRAPH, 2007.2. Partial and Approximate Symmetry Detection in 3D Geometry, N. J. Mitra, L. Guibas, M. Pauly, ACM SIGGRAPH, 2006.
scale + rotation + translationre�lection re�lection + rotation + translation
Types of symmetry. Planar re�lection, rigid transforms, uniform scaling, or their combinations.
original symmetrized
Funding Sources: This work is supported by S9206-N12 of the Austrian Science Fund (FWF), SNF 200021-112122/1 of the Swiss National Science Foundation, NSF grant CARGO 0310661, ARO grant DAAD19-03-1-0331, NIH grant GM-072970, and DARPA grant 32905.
Approximate Symmetry Detection and SymmetrizationMark Pauly
ETH ZurichLeonidas J. Guibas
Stanford UniversityNiloy J. Mitra
Vienna University of Technology
Symmetry Extraction Pipeline.
Symmetrization in 2D. Local and global symmetrization of a gecko model.
0.02-0.02
