COMPUTATIONAL SYMMETRY

 

  or

 

Group Theory and Applications in Robotics, Computer Vision/Graphics and Biomedical Image Analysis

 

Instructor: Professor Yanxi LIU yanxi@cse.psu.edu

CSE and EE Departments of PSU

Office Location: 338B IST

 

 

Fall 2006

CSE 598B, Three Credits #Schedule = 794395

Time: Wednesday 1:30pm-4pm

Office Hours: After each class and by appointment

 

 

 

Published papers and Technical Reports from students’ course projects (previous taught in CMU Fall 2005 and PSU Spring 2006):

 

 

2007

 

Shape Variation-Based Frieze Pattern for Robust Gait Recognition

By S.Lee, Y.Liu and R.Collins, International Conference of Computer Vision and Pattern Recognition (CVPR 2007).

  

Quantified Symmetry for Entorhinal Spatial Maps
E. Chastain and Y. Liu
Special Issue in Neurocomputing Journal, Vol. 70, No. 10 - 12, June, 2007, pp. 1723 - 1727.

 

 

Symmetries of Dance

Y. Liu, X. Yang, and C. Bregler
tech. report CMU-RI-TR-07-09, Robotics Institute, Carnegie Mellon University, May, 2007.

 

 

2006

 

Expression Classification using Wavelet Packet Method on Asymmetry Faces
K. Teng and Y. Liu
tech. report CMU-RI-TR-06-03, Robotics Institute, Carnegie Mellon University, January, 2006.

 

Quantified Symmetry for Entorhinal Spatial Maps
E. Chastain and Y. Liu
Fifteenth Annual Computational Neuroscience Meeting (CNS 2006), July, 2006

 

 

 

 

Students Comments (CMU and PSU):

 

The course, I felt and still feel that what you teach is fundamental and that everyone should know it.  The tools you provide helped solidify my understanding of geometry and I only wish I had more time to dedicate to studying the topic.  I believe that this course should be taught to first year computer vision graduate students because it will help give them a better appreciation for the role of geometry in computer vision and supply some building blocks that they need for other endeavors.  I like the projects.  I do think they're hard, but that's the appeal.

 

 

The most exciting part of the course is the exposure to the existence of symmetry. The variety of the selected topics provides me a 'symmetry' glass to re-examine the surrounding world. The course projects are also very interesting. I enjoyed a lot in doing own my project and the discussions about others' projects.  The detailed feedback from you was very helpful. I was always interested when you were sharing your hands-on experiences. It made me learn a lot not only for this project but also general approaches in research. Thank you! The course successfully arose my interests in symmetry group, especially, the challenge of computational symmetry. Though I hope we could go deeper in some topics. But I don't think it's possible to squeeze more in such a short time.

 

 

My comments are as follows. I gave the class a very high ranking because it was immensely useful to me to learn about wallpaper groups and invariance under group action. I also had a lot of fun, which speaks to the nature of the work discussed. Overall, since my principal interest has always been how the human brain uses invariance under transformation as a code, the experience gained from this course will help me give a more rigorous account of invariance under transformation than others in the field. The process of presenting incremental results was also very useful, and helpful in helping me to accustom myself to the research process.

 

The only thing that I should add is that it would be nice to cover more of the Lie Group stuff in the context of Computer Vision, since it is so ubiquitous. If you could get one of the statisticians you have worked with in the past to help you, you could also make the course more appealing to a wider audience by concentrating on less regular structures that still have symmetries (NRT's are just one example). If this could be cross-listed as a statistics course, you would get many more people, because the common misconception about group theory is that it is useless in the face of real data (because real data is noisy).

 

 

 

I think the course was great. I learn a lot about symmetry that I didn't know could have existed and could be used to complement other computer vision techniques.

 

 

 

 

 

 

PREREQUISITES:
Basic algebra, transformations, computer vision/image analysis, robotics or approval of the instructor.

Course Description:

Group theory, the ultimate theory for symmetry, is a powerful tool that has a direct impact on research in robotics, computer vision, computer graphics and medical image analysis. This course starts by introducing the basics of group theory but abandons the classical definition-theorem-proof model. Instead, it relies heavily on intuitions in (1) 3D Euclidean space, images and patterns; (2) a geometric computational model; and (3) concrete, real world applications in robotics, computer vision, computer graphics and medical image analysis drawing from the instructor’s many years of research experience and from an emerging, vibrant, interdisciplinary international research community.  The material will be taught in a bottom-up (problems to theory) style based on the instructor’s manuscript of “Group Theory Applications in Robotics, Computer Vision and Computer Graphics”, state of art research papers and classical articles in prominent journals/books. The course emphasizes on motivations and justifications for the algorithmic usage of group theory in different domains, computational issues, and hands-on experimentation and illustration. The instructor encourages students to explore new applications while providing a handle on an elegant methodology and available computational tools. This course should be appropriate to any students who have an interest in real world problems that involve 3D Euclidean geometry, regularity, near-regular patterns and symmetry. It should be particularly attractive to students with computational inclinations of using algebraic theory in combination with other tools (e.g. graph theory, statistics). The goal is to provide the course material in a fairly high level of sophistication with intuition, formal justification and algorithmic ease.

JUSTIFICATION: This course addresses both a real need in graduate education and in research communities of using formal methods in symmetry, asymmetry and near-regularity. Symmetry is a pervasive phenomena in both natural and man-made (including biological) environments. Humans have an innate ability to perceive and take advantage of symmetry in everyday life, but it is not obvious how to automate this powerful insight on man-made intelligent beings, such as robots. On the surface, symmetry is simple and basic. In essence, the concept of symmetry is much more than a mirror reflection, rather, it can span a continuous spectrum of multi-dimensional spaces. In basic sciences, the understanding of symmetry played a profound role in several important discoveries including: relativity theory (the symmetry of time and space); human DNA structure (double helix); the quasi-crystals and their mathematical counterpart penrose tiles. We argue that reasoning about symmetry can likewise play a crucial part in the advance of artificial/machine intelligence.

A computational model for symmetry is especially pertinent to robotics, computer vision and machine intelligence in general, because in these fields we are studying how a man-made intelligent being can perceive and interact with the chaotic real world in the most effective way. Recognition of symmetries is the first step towards capturing the essential structure of a real world problem, and minimizing redundancy which can often lead to drastic reductions in computation. One fundamental limitation of computers is their finite representational power. One simple floating point error can destroy any perfect symmetry. One's ability to tolerate departure from perfect symmetry reflects one's level of sophistication in perception, which need to be built into the development of machine/artificial intelligence.

While a compelling mathematical theory of symmetry has existed for more than a century, very few computational tools prevail in recognizing and taking advantage of real world symmetry. One cause of this shortage is the discrepancy between the ideal algebraic formulation of symmetry, namely group theory, and the instantiation of symmetry in the noisy physical world. We have developed a computational framework that can effectively treat real world symmetry as statistical departure from regularity. The final goal of this course is to build perceptual systems that can recognize imperfect structural regularity or symmetry, while discriminating subtle pattern differences.

 

Reference Material:

 

A text book manuscript on “Symmetry Group Applications” by Professor Liu

 

Generators and Relations for Discrete Groups by Coxeter and Moser

4th Edition, Springer-Verlag

 

Contemporary Abstract Algebra by J. A. Gallian

D.C. Heath and Company

 

Groups: A path to geometry by R.P. Burn

Cambridge University Press

 

And

 

A collection of state-of-the-art research papers.

 

Format

The course will be in the format of instructor lectures, student presentations, projects and term papers.

 

Guest lectures (by speakers in and out of PSU) will expose students to applications in other domains (e.g. architecture, material science).

Grading Policy

You will be graded on the following items:

1. Written Homework

(20%)

2. Oral Presentations

(20%)

3. Class Participation

(10%)

4. Term Project & Write-up

(50%)

5. Extra Credit

(10%)

 

--------

 

110% total

 

 

Syllabus

 

Week 1 (September 6): Regularity and Symmetry ( Lecture slides)

 

What is regularity and symmetry?

Why do they matter?

Why do we care about symmetry?

An overview of the course.

 

An introduction to

 

(1) the spectrum of symmetry from regular to stochastic

 

n     near-regular texture synthesis: symmetry as a double-sided sward

 

(2) computational motivations and challenges of computational symmetry

 

(3) goals for this course --- yours and ours

 

References:

 

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.

 

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

Symmetries of Culture: Theory and Practice of Plane Pattern Analysis
by Dorothy K. Washburn, Donald W. Crowe

1991

 

 

Week 2 (September 13):  Essence of Symmetry: rising from real world problems ( Lecture slides)

 

n      What is symmetry? Mathematical definition and real world examples.

n      Why do we care?

n      Why computational symmetry is a double-sided sward?

n      Examples from:

  1. solids in contact
  2. periodic patterns in N-dimensional Euclidean space
  3. papercuting patterns

 

 

Homework #1: understanding of symmetry and symmetry groups

 

CMU Near-Regular Texture Database

 

References:

 

Chapters on Finite groups from Contemporary Abstract Algebra by J. A. Gallian

D.C. Heath and Company

 

 

 

 

Week 3 (September 20): Symmetry Groups: how do we organize symmetries? ( Lecture slides )

 

 

Introduction to some basic concepts in group theory: definition of a group, group action and orbits, subgroup, different types of (sub)groups, discrete, continuous, finite, infinitely countable, subgroup hierarchies, transformation groups, matrix representations with concrete examples from

 

n     robotics (surface contact and relative motion between 3D solids),

n     periodic patterns and frieze/wallpaper groups

n     papercut-art form

 

Reference:

 

The First Chapter of ``Symmetry Groups in Robotics Assembly Planning and Specifications'', Yanxi Liu, the Mathematical Methods in Technology series, MARCEL DEKKER, INC., 270 Madison Avenue, New York, New York, 10016.

 

Homework #1 due today.

 

 

 

Week 4 (September 27): Computational Symmetry Literature Review (Student Presentations)

 

Note: Short presentations by students on a summary of the symmetry/group related papers each of you picked

 

Homework #2:      (1) Proof of the existence of seven frieze groups

                               (2 )Wallpaper pattern classification by you (a fun practice)

                               (3) Literature review for symmetry detection algorithms (choose an area of interest in preparation for your class project)

 

 

Week 5 (October 4): Symmetry Groups: how do we organize symmetries (continued from Sept 20)? ( Lecture slides )

 

Examples drawn from

 

n      Surface contact and relative motion between 3D solids: how symmetry groups contacting surfaces play a role?

n      Frieze and Wallpaper symmetry groups and their respective internal hierarchical structures: how are migrating via affine deformation?

n     looking for symmetry groups in papercut-art forms for synthesis, folding and cutting

 

 

Week 6 (October 11):  Computational Symmetry: Problem Formalization and Computation (Lecture slides)

 

 

1. How to formulate a problem in symmetry group terms (Part I)?

o       Surface contact and relative motion between 3D solids: expression of relative locations between solids

o       looking for symmetry groups in papercut-art forms for synthesis, folding and cutting

o        

 

2. How to represent and compute different types (finite, infinite, discrete, continuous …) of Symmetry Groups  and their subgroups (Part II ) ?

 

3. Global and Local Distortions from regularity: are their ideal Symmetry groups still useful?

 

 

n     Robotics surface contact and relative motion between 3D solids: group representation and operation on computers 

n     periodic patterns and frieze/wallpaper groups: Euclidean cases Hilbert’s 18th problem revisited

n     Near-Regular Textures: basic symmetry group concepts meet statistical distortion

n     NRT applications: analysis, synthesis, and manipulation

n     Dynamic NRT tracking  (W.C. Lin)

n     Automatic NRT lattice extraction as a correspondence problem (J. Hays)

 

 

References:

 

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.

 

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

 

Symmetries of Culture: Theory and Practice of Plane Pattern Analysis
by Dorothy K. Washburn, Donald W. Crowe

1991

 

 

Homework #3:      symmetry detection algorithms: rotation, translation, reflection detections

 

 

Detecting symmetry and symmetric constellations of features By Loy and Eklundh, ECCV 2006

Detecting Rotational Symmetries By Prasad and Davis, ICCV 2005

Digital Papercutting By Liu, Hays, Xu, and Shum, Technical Sketch, SIGGRAPH 2005.

 

    

 

Week 7 (October 18): Symmetry Groups and Gait Analysis (Professor Collins)

 

 

(1)   Symmetry-based indexing and retrieval of Papercut-Pattern Database by Weina Ge (Ph.D. student who took this course in Spring 2006)

(2)   Gait Analysis as near-periodic patterns by Dr. Collins

 

References:

 

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

 

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.

 

 

 

 

Week 8 (October 25): Guest lectures

 

 n     Bilateral human brain anatomy: Midsagittal Plane extraction from 3D MR neuroimages and  Age estimation from Statistical Measures of Human Brain Asymmetry  (Leonid Teverovskiy).

n      Gait Recognition using Spatial-Temporal Patterns (Zhaosheng Yin)

n      Wallpaper symmetry from perspective views (Roberto Lublinerman)

 n   Exploring Architectural Designs by means of Group Transformations (Hoda Moustapha)

 

ABSTRACT

 

Design is often described as an exploration: a search for an adequate solution amongst a space of alternatives (Simon 1969). Navigation thought this space involves the development and transformation of alternatives, in the quest for improving the solution's quality. During the exploratory, early phases of architectural design, configurations continually evolve, though modifications of both configuration elements and spatial relations among these elements. Higher-level explorations, thought spatial relationships, yield intellectually stimulating results. These, however, are labor intensive, because they require modification of multiple elements. Such repetitive interaction considerably slows down the exploration, and often discourages it completely, particularly when configurations are complex and interrelations are numerous. My research addresses the computational representation of design configurations, in order to maximize their exploratory potential. It is motivated by the necessity of flexible geometry for early design exploration and by the intellectual stimulation provided by (and the difficulty involved with) the exploration of complex relationships. For this presentation, "Exploring Architectural Designs by means of Group

Transformations", I will review design representations for exploration, and introduce the Interactive Configuration Exploration (ICE) framework, which is a computational representation based on group transformations encapsulating spatial relations. I will discuss the capacity of ICE for describing 3D shapes and configurations, and for describing frieze/wallpaper groups, as well as transformation across those groups. I will conclude by discussing extensions and potential application of the ICE framework.

 

The ICE framework, consist of two parallel endeavors: a notation and a computer implementation. The ICE notation is a formalism for describing shapes and configurations by means of generative transformations. The ICE implementation is a 3D modeling system that supports the exploration of shapes and configurations by means of manipulating parameters of generative transformations. The ICE framework classifies spatial relations into layers based on group transformations, variations, operations, and constraints. It defines methods for generation, and strategies for the compositions of simple relations into higher level organizational structures, in order to maximize the possible types configurations it can describe. By using the ICE framework one can design, generate and manipulate design configurations.

 

 

 

MINI BIOGRAPHY

 

Hoda Moustapha has completed her PhD in Computational Design from the

School of Architecture at Carnegie Mellon University. Hoda is currently an

assistant professor at Chatham College's Interior Architecture Program.

Hoda has co-taught courses in geometric modeling, spatial constructions and

grammar implementations and she is currently teaching courses in computer

aided design, geographical information systems, color psychology, and

materials/assemblies. Hoda worked in the Center for Building Performance

and Diagnostics and she is part of the Green Building Alliance. Hoda

published papers in the Design Studies Journal, as well as in the following

conference proceedings: Generative CAD Systems (GCAD'04), Design Computing

and Cognition (DCC'04), and Mathematic and Design 2001.  Hoda also

exhibited her own artwork at Carnegie Mellon University, at University of

Pittsburgh and at the Frick Art Museum.

 

 

Reference:

 

Hoda's WWW: Research

 

 

Homework #3 due today!

 

 

 

Week 9 (November 1): Student Project Proposal Presentations (about 20-25 min each)

 

n     In your area of interest: What is the state of the art in computational symmetry?

n     Propose your computational symmetry projects and initial results (if any) in detail: PowerPoint presentation preferred

 

 

1.    asymmetry of brain as a predictor of age

2.    gait recognition using symmetry analysis

3.    expression classification using wavelet packet correlation methods on facial asymmetry

4.    symmetry as a (neural) code

5.    detection of lattice regions in architectural images

6.    symmetry detection as a higher order correspondence problem

 

 

 

Week 10 (November 8): Quantified Regularity Measurements (slides)

 

·       Facial Asymmetry as a biometric

·       2D facial asymmetry for human identification and expression classification

·       3D facial asymmetry for gender classification

·       “Symmetry As a Continuous Feature”

 

References: