Deformable models have found widespread applicability in diverse fields during the past 15 years. This talk will discuss deformable shape models in the context of two problems: computational anatomy and modeling soft tissue deformability for surgical applications. A shape-based approach to deformable registration of brain images will be presented, which is based on matching geometric attributes calculated at different scales and embedded in a hierarchical framework. A framework for combining point distribution models and biomechanics will also be presented, with the goal of predicting soft tissue deformations during surgery. Applications of these methods to large imaging studies are also presented, with emphasis on a longitudinal study of aging aiming at establishing quantitative criteria for early prediction of Alzheimer's disease, using morphological and functional measurements.

We will present a number of clinical applications of surface warping currently under development. Warping techniques developed for brain surface flattening have medical applications in areas such as virtual colonoscopy, cardiology and prostate cancer therapy. Techniques developed to enhance visualization and to address the problem of distortion will also be demonstrated.

This conference has brought together people from a variety of different disciplines including theoretical mathematics, applied mathematics, computer science, statistics, biomedical engineering and neurology. With this broad spectrum of backgrounds in mind, I will give a brief introduction and overview to some of the topics and terminology that will be discussed in more detail throughout this conference. These topics include an introduction to the human brain and magnetic resonance imaging (MRI) data; an overview of the pre-processing steps that are required to transform MRI data into a form suitable for applying discrete geometric and conformal methods.

We consider the use of methods of differential geometry, global analysis, and harmonic analysis in questions of facial recognition. While our motivation for this work was applications to reconstructive surgery, it is clear that security implementations are also of interest. The ideas of plastic surgeon Michael Cedars have guided this work.

Central to our studies is the use of a new three-dimensional white light scanner developed by Thomas Lu. We will talk about that hardware as well.

Numerical implementation of the 150 year old Riemann Mapping Theorem for polyhedral surfaces is available for the first time thanks to new circle packing methods. What, you ask, can decades of the purely theoretical developments from the RMT contribute to real world applications? Conformal geometry is no stranger to applications, and the new capabilities actually bring quite a bit to the table.

This talk will describe discrete conformal flat maps of the human cortex, demonstrate the manipulations now available in spherical, euclidean, and hyperbolic geometry, and discuss implementation issues. Our main goal, however, is to highlight the strengths of the conformal theory and the key notions that brain (and other) mappers might exploit, from practical matters, such as transformations and coordinate systems, to more subtle issues, such as forward compatibility, partial surface mapping, and conformal "shape".

Rapid development of imaging methods (2DG, cytochrome oxidase, fMRI, PETT, Optical Recording) over the past several decades has focused attention on the supra-neuronal architecture of the brain, i.e. the "ansatz" that sensory neo-cortex is a two dimensional sheet, and that functional architecture is a regular 2D map embedded in this sheet, i.e. an "image". The Jacobian of these putative 2D maps can be decomposed into an isotropic (conformal) part and a pure shear. Hence, the mathematical side of the measurement problem in this context reduces to an accurate reconstruction and measurement of a local conformal map and a shear. Under favorable conditions, the shear will be either minimal, or of sufficiently simple nature, to allow the use of powerful techniques from complex analysis to model and explain this data.

In this talk, work done in my laboratories at NYU and BU over the past twenty-five years, related to applying analytic and numerical conformal mapping to the general problem area of visual cortex, will be reviewed. Topics to be covered are:

1.) The topographic structure of primate V1 is approximated by an analytic conformal map (complex logarithmic). An associated developmental idea is that Dirichlet's Principle and the Riemann Map Theorem provides a simple boundary condition characterization of map development.

2.) The local (pinwheel and ocular dominance column) structure of V1 is the end result of a "Wiener Process", i.e. a linear operator applied to spatial noise. In the case of orientation columns ("pinwheels" or "vortices"), existing optical recording data is typical of a pattern of "topological defects" in the singular map of S^1<->R^2. The "sign theorem" provides an accurate prediction of right and left handed cortical pinwheel neighbor relations, and elliptic functions provide a crude analytic approximation of the numerically generated, and experimentally observed patterns.

3.) More accurate modeling requires quasi-isometric surface flattening and numerical conformal mapping. Geodesic multi-dimensional scaling was proposed about fifteen years ago to provide a least-squares optimal quasi-isomeric mapping. This has become the basis of most current brain-flattening approaches. Experience indicates that flattening areas including V1,V2, and V3 can be provided at high precision, with no cuts, in short CPU time. A brief comparison to other methods of brain flattening, such as conformal mapping (Beltrami Equation, CirclePack) approaches will be provided.

4.)Symm's algorithm for numerical conformal mapping is the basis of our numerical conformal mapping work. Error analysis derived from a jack-knife technique suggests that the conformal hypothesis for V1 topography is accurate to within an average of perhaps 10%-20%, which is a very favorable result given the 6000% variance in other experimental approaches in this area. Extensions of these ideas to quasi-conformal (i.e. simple forms of shear contributions) will be briefly mentioned.

5.) If time permits, examples from a recent MATLAB toolbox constructed as part of the PhD thesis work of Mukund Balasubrmanian at Boston University will be demonstrated. This toolbox includes various different approaches to brain flattening, cortical curvature estimate, geodesic distance computation, and other computational anatomy techniques.

There is a well-known and interpretation of Gaussian curvature for discrete (triangulated) surfaces. It is natural because it preserves the Gauss/Bonnet theorem, which equates the integral of Gaussian curvature to a boundary integral. Less familiar are analogous boundary integral relations for mean curvature, which can be understood as balancing of physical forces.

We will show how to use such relations to guide the proper interpretation of mean curvature (and other geometric quantities) for discrete surfaces. This new understanding helps to explain the theory of discrete minimal surfaces. It also elucidates why, in early work on simulations of Willmore's elastic bending energy certain discretizations were better than others.

Similarly, for space curves, some quantities, like total curvature or even writhe, have natural interpretations for polygons. We will examine further cases, like knot energies and ropelength, where a proper consideration of the geometry involved can lead to a natural discretization.

Neuroscience and medicine are increasingly empowered by new mathematics that derives information from brain imaging databases. Algorithms now uncover disease-specific patterns of brain structure and function in whole populations. These tools chart how the brain grows in childhood, detect abnormalities in disease, and visualize how genes, medication, and demographic factors affect the brain.

We review recent developments in brain image analysis, focusing on mathematical challenges whose solution will advance the field. We describe our construction of statistical brain atlases that store detailed information on how the brain varies across age and gender, in health and disease, and over time. Specifically, we introduce a mathematical framework to analyze variations in brain organization, cortical patterning, asymmetry and tissue distribution in several collaborative studies of brain development and disease (N>1000 scans). Mathematics based on Grenander^Òs pattern theory, covariant partial differential equations (PDEs), pull-backs of mappings under harmonic flows, and high-dimensional random tensor fields are employed to encode anatomic variations in population-based brain image database. We use this reference information to detect disease-specific abnormalities in Alzheimer's disease and schizophrenia, including dynamic changes and medication response over time. We will show illustrative examples resolving disease-specific patterns that are not apparent in individual brain images. We also discuss four-dimensional (4D) maps that store probabilistic information on the dynamics of development and disease. Digital atlases can identify general patterns of structural and functional change in diseased populations, and link them to therapeutic and genetic parameters. Finally, we introduce a framework to map how genes affect brain structure. The resulting genetic brain maps can be used in data mining applications, to help investigate inheritance patterns and search for susceptibility loci in diseases with known genetic risks.

For more information please visit: http://www.loni.ucla.edu/~thompson/thompson.html where PDFs of tutorial papers and chapters are available.