Computational Symmetry

November 19, Afternoon



The goal of this tutorial is twofold:


1. To provide an accessible and concise summary of the mathematical theory for symmetry: group theory, in particular, discrete and finite symmetry groups and finitely-generated groups: crystallographic groups.


2.  To demonstrate, through multiple concrete computer vision problems, the benefit and the power of imbedding computational symmetry and symmetry groups that are way beyond bilateral reflections.


Technical content:


A. Essence of Regularity and Symmetry: rising from real world problems  

B. Symmetry Groups: categorization and organization of symmetries – a formal definition

C. Computational Symmetry in Computer vision: Literature Review 

 D. Computational Symmetry: Problem Formalization and Computation 

 E. Quantified, continues versus discrete symmetry and their Applications

1.        Symmetry As a Continuous Feature (revisit and a closer look)

2.        wallpaper symmetry groups and their applications in texture analysis, gait recognition and beyond

3.        regularity discovery as a higher order correspondence problem

4.        dynamic symmetries

5.        facial Asymmetry as a biometric, facial asymmetry for expression and gender classification

6.        symmetry as a (neural) code

7.        symmetry in 3D reconstruction in computer vision

8.        statistical brain asymmetry for image indexing, age estimation and computer aided diagnosis

9.        symmetries in visual arts

F. State of the art Symmetry detection algorithms and Quantitative Evaluations of their performances

1.        why symmetry detection is interesting for computer vision/human?

2.        where we started and where we are?

3.        how to evaluate the quality of symmetry detection algorithms: test images, ground truth, evaluation function

G. Summary and Conclusion:

There is a LONG history of computational symmetry in computer vision, why there is a recent surge of interests in computational symmetry in computer vision/graphics, and where will this subfield go from here?

n       Theoretical directions

n       A uniform taxonomy

n       A list of un-answered questions

n       image test sets

n       ground truth

n       sharing of code and data



Relevance to Computer Vision: 


Symmetry plays an essential role at all levels of human as well as machine perception. This tutorial is both relevant and timely for computer vision research. Even though the topic of symmetry has existed in computer vision for almost four decades, little computational tools are readily available for real world problems. A recent surge of interests in both computer vision and computer graphics on symmetry detection has hinted both its importance and challenges. It is time for us to take a closer look of what has been done and where we are heading from here. This tutorial will provide such a succinct yet colorful review for the attendees.


7.         Biographical outline of the lecturer.


Professor Liu studied group theory applications in robotics with late Professor Robin Popplestone (UMass, Amherst) for her Ph.D. thesis entitled: Symmetry Groups in Robotic Assembly Planning. Upon graduation, Dr. Liu worked as a postdoc in the then LIFIA-IMAG research lab (now INRIA) at Grenoble, France. Before joining the robotics institute of CMU in 1996, she spent one year at DIMCS (NSF Research and Education center for Discrete Math and theoretical Computer Science) where she further explored computational issues for group theory applications as well as working with K-12 teachers on discrete math education. Currently, Dr. Liu holds a tenured faculty position in the CSE/EE departments of PSU (Fall 2006) and a research faculty position at CMU. Dr. Liu is also affiliated with the Machine Learning department of CMU, an adjunct associate professor in the Radiology Department of University of Pittsburgh, and a guest professor of Computer Science Department, Huazhong University of Science and Technology of China. Dr. Liu's research interests span a wide range of applications in computer vision, computer graphics, robotics, computational neuroscience and computer aided diagnosis in medicine, with two central themes: computational symmetry and discriminative subspace learning. With her colleagues, Dr. Liu won the first place in the clinical science category and the best paper overall at the Annual Conference of Plastic and Reconstructive Surgeons for the paper "Measurement of Asymmetry in Persons with Facial Paralysis." Dr. Liu chaired the First International Workshop on Computer Vision for Biomedical Image Applications (CVBIA) in conjunction with ICCV 2005 and co-edited the book: "CVBIA: Current Techniques and Future Trends" (Springer-Verlag LNCS). She also serves as a reviewer/committee member for all major journals, conferences as well as NIH/NSF panelist on computer vision, pattern recognition, biomedical image analysis, and machine learning. She served as a 3-year chartered study section member for Biomedical Computing and Health Informatics at NIH. She is a senior member of IEEE and the IEEE Computer Society.

Dr. Liu published widely and lectured internationally on the topic of computational symmetry, symmetry group applications in robotics, computer vision, computer graphics and biomedical image analysis, and the promise and perils of near regular textures from real world images/videos. She was twice invited as an ERASMUS program instructor by the European Union education program. Dr. Liu has taught advanced perception (computer vision, robotics) and computer graphics courses in CMU and PSU. In particular, she created and taught a new graduate level course entitled gComputational Symmetryh or gGroup Theory applications in Robotics, Computer Vision, Computer graphics and Medical Image Analysish (2005-2007, Several publications came out of the studentsf course-projects (see below).



Some relevant papers:

S.Lee, Y.Liu and R.Collins, "Shape Variation-based Frieze Patterns for Robust Gait
 Recognition", CVPR 2007

A Lattice-based MRF Model for Dynamic Near-regular Texture Tracking
W. Lin and Y. Liu
IEEE Transactions on Pattern Analysis and Machine Intelligence, .
Volume 29, No. 5. May, 2007.

Quantified Symmetry for Entorhinal Spatial Maps
E. Chastain and Y. Liu
Special Issue in Neurocomputing Journal, In Press.

Quantitative Evaluation on Near Regular Texture Synthesis
W. Lin, J.H. Hays, C. Wu, V. Kwatra, and Y. Liu
Computer Vision and Pattern Recognition Conference (CVPR '06), Vol. 1, June, 2006, pp. 427 – 434

Discovering Texture Regularity as a Higher-Order Correspondence Problem
J.H. Hays, M. Leordeanu, A.A. Efros, and Y. Liu
9th European Conference on Computer Vision, May, 2006.

Tracking Dynamic Near-regular Textures under Occlusion and Rapid Movements
W. Lin and Y. Liu
9th European Conference on Computer Vision, May, 2006.

Truly 3D Midsagittal Plane Extraction for Robust Neuroimage Registration
L. Teverovskiy and Y. Liu
3rd IEEE International Symposium on Biomedical Imaging: Macro to Nano, 2006, April, 2006, pp. 860 - 863.

Understanding the Role of Facial Asymmetry in Human Face Identification
S. Mitra, N. Lazar, and Y. Liu
Statistics and Computing, 2006.

The Promise and Perils of Near-Regular Texture
Y. Liu, Y. Tsin, and W. Lin
International Journal of Computer Vision, Vol. 62, No. 1-2, April, 2005, pp. 145 - 159.

Digital Papercutting
Y. Liu, J.H. Hays, Y. Xu, and H. Shum
Technical Sketch, SIGGRAPH, 2005.

Near Regular Texture Analysis and Manipulation
Y. Liu, W. Lin, and J.H. Hays
ACM Transactions on Graphics (SIGGRAPH 2004), Vol. 23, No. 3, August, 2004, pp. 368 - 376.

Local Facial Asymmetry for Expression Classification
S. Mitra and Y. Liu
Proceedings of the 2004 IEEE Conference on Computer Vision and Pattern Recognition (CVPR'04), Vol. 2, June, 2004, pp. 889 - 894.

A Computational Model for Periodic Pattern Perception Based on Frieze and Wallpaper Groups
Y. Liu, R. Collins, and Y. Tsin
IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 26, No. 3, March, 2004, pp. 354 - 371.

A Quantified Study of Facial Asymmetry in 3D Faces
Y. Liu and J. Palmer
Proceedings of the 2003 IEEE International Workshop on Analysis and Modeling of Faces and Gestures, in conjunction with the 2003 International Conference of Computer Vision (ICCV '03), October, 2003.

Facial Asymmetry Quantification for Expression Invariant Human Identification
Y. Liu, K. Schmidt, J. Cohn, and S. Mitra
Computer Vision and Image Understanding Journal, Vol. 91, No. 1/2, July, 2003, pp. 138 - 159.

Gait Sequence Analysis using Frieze Patterns
Y. Liu, R. Collins, and Y. Tsin
Proceedings of the 7th European Conference on Computer Vision (ECCV'02), May, 2002.

Computational Symmetry
Y. Liu
Proceedings of Symmetry 2000, Portland Press, London, Vol. 80/1, January, 2002, pp. 231 - 245.

Skewed Symmetry Groups
Y. Liu and R. Collins
IEEE Conference on Computer Vision and Pattern Recognition, Vol. 1, December, 2001, pp. 872 - 879.

Robust Midsagittal Plane Extraction from Normal and Pathological 3D Neuroradiology Images
Y. Liu, R. Collins, and W.E. Rothfus
IEEE Transactions on Medical Imaging, Vol. 20, No. 3, March, 2001, pp. 175 - 192.

Classification-Driven Pathological Neuroimage Retrieval Using Statistical Asymmetry Measures
Y. Liu, F. Dellaert, W.E. Rothfus, A. Moore, J. Schneider, and T. Kanade
Proceedings of the 2001 Medical Imaging Computing and Computer Assisted Intervention Conference (MICCAI '01), Utrecht, The Netherlands, October, 2001.

A Computational Model for Repeated Pattern Perception using Frieze and Wallpaper GroupsY. Liu and R. Collins2000 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 2000), Vol. 1, June, 2000, pp. 537 - 544.



Organizer contact information:


   Yanxi LIU

   338B IST Building

   Departments of Computer Science and Engineering and

   Electrical Engineering

   Penn State University

   University Park, PA 16802

   (814) 865-7495