Conference Abstracts

Using Wilmore bending energy to drive a sphere eversion
John Sullivan (Illinois)
I will show our computer-graphics video "The Optiverse", which depicts a numerically computed sphere eversion. This eversion is optimal in the sense of requiring the least bending at any stage. The eversion is automatically generated, starting from a halfway model critical for the Willmore bending energy. (This model is obtained from a particular symmetric minimal surface by inversion in the unit sphere.) We flow along the energy gradient from this saddle point down to the round sphere.
Adaptive Hybrid Conformal Surface Meshing
Ahmed Khamayseh (ORNL)
Mesh generation has emerged as a major pacing item with regards to computational modeling and simulation. And it is a crucial first step for the solution of multi-dimensional problems in field simulation. In addition, the accuracy and convergence of computational solutions using mesh-based numerical methods are strongly dependent on the quality of the mesh being used. This talk outlines the development of efficient and robust adaptive hybrid surface mesh generation capabilities applicable to a wide range of field simulations. Applications include, but are not limited to, climate modeling, large scale multi-physics based simulation, computational chemistry, computational biology, material sciences, and environmental sciences.

We focused on the generation of high quality polygonal meshes for geometrically-complex domains in two and three dimensions. These methods are applied to parametric as well as non-parametric (discrete) surface models. The successful methods and algorithms developed in this research are currently deployed in support of climate modeling, material sciences, and computational plasma physics. Our goal is to achieve better prediction of climate simulation via mesh/grid refocusing and adaptation. PDE based smooth quasi-conformal mapping is used to produce the adapted grids.

Dynamic Surface Meshing
Herbert Edelsbrunner (Duke)
The topic of this talk is the maintenance of a triangulation describing a surface that deforms in time. This is a fairly complicated task, and the goal we set for ourselves is to see to what extent we could design the algorithm so that the result is predictable and its correctness is provable. We made a number of design decisions limiting our choice of surface, motion, and triangulation. The final algorithm maintains the mesh using three types of operations:

A. changing the global topology,
B. locally coarsening or refining the mesh,
C. flipping to maintain the mesh as a restricted Delaunay triangulation.

This talk will focus on combinatorial and differential properties of the surface of our choice and indicate where in the algorithm these properties are used.

Discrete minimal surfaces from circle patterns: Geometry from combinatorics
Alexander Bobenko (TU Berlin)
We show how discrete minimal surfaces can be constructed from circle patterns and how to construct these using a variational principle. The corresponding software demonstration "Java software for circle patterns and discrete minimal surfaces" will be given by Boris Springborn.

Demo: Java software for circle patterns and discrete minimal surfaces
Boris Springborn (TU Berlin)
We show how circle patterns can be constructed using a variational principle and how to get discrete minimal surfaces from them. The theory is explained in Alexander Bobenko's talk "Discrete minimal surfaces from circle patterns". This is the software demo.
Demo: Kinetic Geometry with JavaView
Klaus Hildebrandt (TU Berlin)
We demonstrate examples of 3d kinetic geometry on graphs and surfaces and applications to feature preserving smoothing of surfaces. All applications can run in online webpages. The implementations are done using JavaView,
Computing surface conformal structures and applications in computer science
Xianfeng Gu (Harvard)
We introduce a method to compute surface conformal structure using Hodge theory. The method is applied in many areas in computer science. In computer graphics, it is used in texture mapping, geometric morphing, and geometric compression. In computer vision, it is applied for surface matching and classification. It is also useful for brain mapping in medical science.


Geometry of Natural Surfaces in Human Vision
Amir H. Assadi (Wisconsin)
Perception of geometric attributes of surfaces in natural scenes has received a great deal of attention in recent years. In this lecture, we examine the perceptual estimation of geometric invariants of natural surfaces, and formulate some ensuing computational and mathematical questions that arise in the process. An outline of a theory for a perceptual geometry in the visual system will be presented that incorporates top-down processing and the very initial stages of sensory representation of stimuli transduced from natural surfaces. Perceptual Geometry is a dynamic theory that strives to maintain a modeling approach compatible with neurobiology. The lecture will offer evidence for plausibility of a mathematical theory emerging from perceptual geometry of natural scenes , and it will illustrate the ideas with case-studies with computational modeling. If time permits, we conclude a brief discussion of selected open problems.
Creating Circle Packings with Prescribed Angles Using Conformal Welding
Brock Williams (Texas Tech)
A map between two circle packings with the same underlying graph must be approximately conformal. However, a map between an arbitrarily embedded graph (say from physical data) and the associated circle packing need not be. One must manipulate the packing to create angles which match the original graph. We will describe how this can be done using conformal welding for graphs created from a quadrilateral lattice.
Hierarchical Image Segmentation and 2D Shape Classification - using Multigrid Techniques for the Laplace Operator
Eitan Sharon (Brown)
Image segmentation is one of the most basic tasks for both computer and biological vision systems, and is a prerequisite for higher-level processes from motion detection to object recognition. Segmentation is difficult because objects may differ from their background by any of a variety of properties that can be observed in some, but often not all scales. The delineation of salient segments in the image should be followed by their shape classification. I will describe techniques inspired by multigrid solvers for physical systems, involving the Laplace operator. Time allowing, I may also describe the use of multigrid techniques for revealing the harmonic functions involved in conformal mappings of 2d shapes.
Computing the Watershed Transform on a Surface Model of the Human Brain Cortex
Maryam Rettman (National Institute on Aging)
In this talk, we will describe a technique for computing the watershed transform on a triangulated surface mesh. Traditionally, the watershed transform has been used in image processing for segmenting images into various catchment basins or for extracting watershed lines. In our application, we are interested in using this transform for extracting anatomical regions of interest from a surface mesh representation of the human brain cortex. The regions of interest correspond to the cortex buried within the surface folds. The talk will include both a description of the algorithms involved as well as applications in the analysis of cortical brain geometry.
Quasisymmetric Parameterizations of Metric Spheres
Mario Bonk (Michigan)
A homeomorphism between metric spaces is called a quasisymmetry if it distorts relative distances in a controlled manner. The concept of a quasisymmetric map generalizes the well-known concept of a quasiconformal map to a general metric space setting. A fundamental question is how to characterize standard metric spaces like the Euclidean spaces Rn or the standard spheres Sn up to quasisymmetric equivalence. In higher dimensions (n >= 3) this seems to be an exceedingly hard problem which is out of reach at the moment. The case n=2 is more accessible and particularly interesting, because it is related to other interesting questions like Thurston's hyperbolization conjecture. In my talk I will give a survey on these topics and present some recent results.
On the Metrics and Euler-Lagrange Equations of Computational Anatomy
Michael I. Miller (Johns Hopkins)
The model we review of Computational Anatomy (CA) is a Grenander deformable template, an orbit generated from a template under groups of diffeomorphisms. The metric space structure on the space of anatomical images is constructed from the geodesic connecting one anatomical configuration to another. The variational problems specifying these metrics along with the Euler equations of motion for the geodesics in the group of diffeomorphisms are reviewed. Various anatomical examples in CA are examined including the application of CA to the reconstruction of cortical gyral surfaces and their comparison.
Surface Compression
Peter Schroeder (Cal Tech)
Efficient representations of surfaces are fundamental to many application areas as well as to such basic questions as, what is the minimal amount of information required to describe a given surface? Inching up on these questions is a fundamental pursuit of Digital Geometry Processing and the techniques developed in that field. Recent work in the area of surface compression demonstrates nicely how much of a difference in performance one can see when issues such as parameterization, for example, are carefully attended to. In my talk I will cover a variety of recent work in this area ranging from parameterization for remeshing to efforts to deal with surfaces of very high topological complexity.
Hubbard Trees and Harmonic Polynomials
Lukas Geyer (Michigan)
Hubbard Trees are a useful combinatorial tool in the theory of one-dimensional complex dynamics. More precisely, a polynomial P(z) of one complex variable z is said to be post-critically finite if all critical points (zeros of the derivative P') have a finite forward orbit under iteration. To each such post-critically finite polynomial P one can associate a finite tree, a self-map of that tree and some additional angle data. This combinatorial ensemble basically captures the dynamics of the critical points. Conversely, Poirier proved that every such Hubbard Tree determines an essentially unique polynomial. This realization result is heavily based on Thurston's characterization of post-critically finite rational maps. We will present some background and examples illustrating the use of Hubbard Trees, and an application of Poirier's result to a question about the number of zeros of harmonic polynomials.
Poster: Dynamic Programming Definition of Boundaries of the Planum Temporale
Nayoung Lee (JHU), J. Tilak Ratnanather (JHU), Patrick E. Barta (JHU), Monica K. Hurdal (FSU), Michael I. Miller (JHU)
Dynamic programming was used to define boundaries of cortical submanifolds with focus on the planum temporale (PT) of the superior temporal gyrus (STG), a region implicated in a variety of neuropsychiatric disorders. To this end, automated methods were used to generate the cortical surface of PT from 10 high resolution MRI subvolume encompassing the STG. Bayesian segmentation was then used to segment the subvolumes into cerebrospinal fluid, gray matter (GM) and white matter (WM). 3D isocontouring using the intensity value at which there is equal probability of GM and WM is used to reconstruct the triangulated graph representing the STG cortical surface, enabling principal curvature at each point on the graph to be computed. Dynamic programming was used to delineate the PT cortical surface by tracking principal curves from the retro-insular end of the Heschl's Gyrus (HG) to the STG, along the posterior STG up to the start of the ramus and back to the retro-insular end of the HG. A coordinate system was then defined on the PT cortical surface. The origin was defined by the retro-insular end of the HG and the y-axis passes through the point on the posterior STG where the ramus begins.
Automated labelling of GM in the STG is robust with probability of misclassification of gray matter voxels between Bayesian and manual segmentation in the range 0.001-0.12 (n=20). PT reconstruction is also robust with 90% of the vertices of the reconstructed PT within $1$ mm (n=20) from semi-automated contours. Finally, the inter-rater reliability for the surface area derived from repeated reconstructions was 0.96 for the left PT and 0.94 for the right PT, thus demonstrating the robustness of dynamic programming in defining a coordinate system on the PT. It provides a method with potential significance in the study of neuropsychiatric disorders.
Joint work with N. Honeycutt and G. Pearlson. Research supported by NIH P41-RR15241, MH 43775, MH 60504 and MH 52886 and NSF FRG DMS-0101329.
Discrete S-isothermic surfaces
Tim Hoffmann (TU Berlin/U. Mass)
Isothermic surfaces are surfaces that allow conformal curvature line parametrization. This class includes surfaces of revolution, quadrics, minimal surfaces, and surfaces of constant mean curvature (cmc). S-isothermic surfaces are a discretization of isothermic surfaces. They are build from intersecting/touching circles and spheres. While the case of minimal surfaces will be covered by Alexander Bobenkos talk, I will show how to derive a Darboux transformation for general S-isothermic surfaces and how to define S-cmc surfaces using that.
Discrete Weingarten Operator and Surface Optimization
Konrad Polthier (TU Berlin)
The curvature properties of a smooth surface are completely characterized by the Weingarten operator. In this talk we introduce a discrete Weingarten operator for polyhedral surfaces and discuss its relation to other known discrete curvature operators. Applications are given to the anisotropic noise removal and the enhancement of curvature based surface features since both methods heavily rely on a detailed knowledge about principal curvatures.
The will also give an overview of discrete curvature properties of other mesh types like non-conforming triangle meshes and point sets, and discuss some of the mutual relations.

Demo: BrainWorks 2.0
Tilak Ratnanather (Johns Hopkins)
BrainWorks 2.0 is a suite of cortical analytical software tools that the Center for Imaging Science at Johns Hopkins has been using for more than 5 years. We will demonstrate applications of automated Bayesian segmentation, cortical surface reconstruction, dynamic programming, viewing cortical surfaces in R2 and R3. The software runs on AIX and Linux platforms.
Demo: S-isothermic Surfaces
Tim Hoffman (TU Berlin/Mass)
Software developed to explore discrete S-isothermic surfaces (they include the minimal surfaces Alexander Bobenko will talk about but also discrete surfaces of constant mean curvature, Bonnet pairs and some other very interesting surfaces...and transformations of them).
Demo: Part Decomposition of Surfaces Reconstructed from Laser Range Scans
David Page (Tennessee)
This demonstration will illustrate the decomposition of surfaces their constituent visual parts. We represent these surfaces as triangle mesh models that we generate using laser range scanners and real-world objects. The theory supporting the decomposition, also known as surface segmentation, is from the cognitive psychology literature.
Chuck Collins (Tennessee)
When the word "conformal" comes up, what does it mean? Sometimes it seems that everyone has his or her own definition. In this presentation, I will try to discuss what conformal means and how it relates to the study of surfaces and geometry.
A Möbius Invariant Metric
David Minda (Cincinnati)
This is joint work with David Herron and William Ma. Most plane regions support a complete Riemannian metric, the hyperbolic or Poincaré metric, with constant curvature -1. This metric can be determined explicitly only in very special situations. For a disk or half-plane the hyperbolic metric has a very simple form. For the disk D={z: |z-a|<r}, LD(z)|dz|= (2r)/(r2-|z|2), while if D is a half-plane, LD(z)|dz|= |dz|/ dist(z,bdry D), where dist(z,bdry D) denotes the Euclidean distance from z to the boundary of D. A conformal metric is defined on any proper subregion Omega of the complex plane by

     mOmega(z)|dz|= inf LD(z): z in D subset Omega,

where the infimum is taken over all disks or half-planes D that contain z and are contained in Omega. The conformal metric mOmega(z)|dz| is Möbius invariant and has many properties analogous to the hyperbolic metric. It is elementary that LD(z) < mOmega(z) for any hyperbolic region Omega with equality if and only if Omega is a disk or half-plane. This metric has been considered by a number of authors including, Thurston, Kulkarni and Pinkall, and Anderson. The talk will focus on the properties of this Möbius invariant metric in 2-dimensions. Kulkarni and Pinkall focused attention on the analogous metric in n-dimensions; more precise information is available in 2-dimensions.