MultiScale Partial Intrinsic
Symmetry Detection
Kai Xu^{1,2},
Hao Zhang^{3}, Wei Jiang^{1},
Ramsay Dyer^{4}, Zhiquan Cheng^{1}, Ligang Liu^{5}, Baoquan Chen^{2}
^{1}National
University of
Defense Technology, ^{2}Shenzhen
VisuCA Key
Lab/SIAT,
^{3}Simon
Fraser University, ^{4}INRIA,
Geometrica, ^{5}University of Science
and Technology of China
ACM Transactions
on Graphics (SIGGRAPH Asia 2012), 31(6)
Figure 1: Multiscale
partial intrinsic symmetry detection: five symmetry scales (large to
small) are detected. Each symmetric region is shown in uniform color.
Note the detection of inter and intraobject symmetries, as well as
cylindrical symmetry of the limbs.
Abstract

We present an algorithm for
multiscale partial intrinsic symmetry detection over 2D and 3D shapes,
where the scale of a symmetric region is defined by intrinsic distances
between symmetric points over the region. To identify prominent
symmetric regions which overlap and vary in form and scale, we decouple
scale extraction and symmetry extraction by performing two levels of
clustering. First, significant symmetry scales are identified by
clustering sample point pairs from an input shape. Since different
point pairs can share a common point, shape regions covered by points
in different scale clusters can overlap. We introduce the symmetry
scale matrix (SSM), where each entry estimates the likelihood two point
pairs belong to symmetries at the same scale. The pairtopair symmetry
affinity is computed based on a pair signature which encodes scales. We
perform spectral clustering using the SSM to obtain the scale clsters.
Then for all points belonging to the same scale cluster, we perform the
secondlevel spectral clustering, based on a novel pointtopoint
symmetry affinity measure, to extract partial symmetries at that scale.
We demonstrate our algorithm on complex shapes possessing rich
symmetries at multiple scales.



Paper 



Slides 



Images 
Figure
2: Outline of our multiscale symmetry detection algorithm.
After a voting step which identifies a set of (sufficiently) symmetric
sample point pairs (a), we perform clustering of these point pairs
based on a scaleaware affinity matrix (the SSM) to determine scale
clusters. In each scale cluster, we perform the secondlevel clustering
of sample points to detect symmetries at that scale (cd).
Figure
3: Symmetry detection on 2D shapes sorted by scale. Observe the
inter and intraobject symmetries detected at multiple scales, even
down to the small scales of the limbs in (d).
Figure
4: A gallery of 3D symmetry detection results sorted by scale.
From left to right and topdown: Children, Octopus, Kung Fu Panda,
IndoLady, and Thai Statue. The last two models show raw clustering
results. The IndoLady and Thai Statue models were chosen to demonstrate
the performance of our algorithm on models which are not compositions
of articulated characters. For the children model, not all arms or legs
are detected in the last shown scale due to scale discrepancies and
some parts fused with the body. For the IndoLady, our method does not
return all perceived symmetries, e.g., the selfsymmetries of the
individual limbs. To save space, the first image for the Kung Fu Panda
contains the first five scales, each revealing a selfsymmetry of the
four characters and the base.
Figure
5:
Comparison between our multiscale results (a) to the symmetry
detection results of [Lipman et al. 2010] (b) and [Xu et al.
2009] (c), which both provide only a single coverage of the shape. Our
method detects overlapping symmetries and these results combine those
from the other two methods.
Figure
6: Plots of the top two eigenvectors of Global Point Signature
(ab), Heat Kernel Signature (cd), and our partial intrinsic SCM in
two scales (eh). The partial intrinsic symmetries are more clearly
revealed in (eh).
Figure
7: Comparison between our symmetrydriven hierarchical
segmentation scheme (top row) and hierarchical segmentation based on
primitive fitting [Attene et al. 2006] (middle row) and normalized cut
[Golovinskiy and Funkhouser 2008] (bottom row). Each column shows the
same segmentation count. It is evident that our results conform better
to the shape semantics.
Figure
8: Limitation to the use of intrinsic distances (a 2D case).
Having one foot of the right figure planted into the base (a) or
disconnected from the base (b) has a drastic effect on the multiscale
symmetries detected, since the distances changed drastically.



Thanks 
We
would first like to thank the
anonymous reviewers for their valuable feedback. Thanks also go to
Daniel CohenOr for fruitful discussions on the paper. Part of the 3D
models in this paper is from the shape repositories of AIM@SHAPE and
Stanford. This work is
supported in part by grants from NSFC (61202333, 61232011, 61161160567,
61025012, 61103084, and 61070071), NSERC (No. 611370), National 863
Program (2011AA010503), Shenzhen Science and Innovation Program
(CXB201104220029A, JC201005270329A), the 973 National Basic Research
Program of China (2011CB302400).



Data 
We provide the datasets (including
both 2D and 3D shapes) used in this paper:
Dataset (ZIP, 53MB)



Bibtex 
@article
{xu_siga12,
title = {MultiScale Partial Intrinsic Symmetry
Detection},
author
= {Kai Xu and Hao Zhang and Wei Jiang and Ramsay Dyer and Zhiquan Cheng
and Ligang Liu and Baoquan Chen}
journal
= {ACM Transactions on Graphics, (Proc. of SIGGRAPH Asia 2012)},
volume
= {31},
number
= {6},
pages
= {181:1181:11},
year
= {2012}
}

