Characterizing Retinotopic Mapping Using Conformal Geometry and Beltrami Coefficient: a Preliminary Study

abstract: Functional magnetic resonance imaging (fMRI) has been widely used to measure the retinotopic organization of early visual cortex in the human brain. Previous studies have identified multiple visual field maps (VFMs) based on statistical analysis of fMRI signals, but the resulting geometry has not been fully characterized with mathematical models. This thesis explores using concepts from computational conformal geometry to create a custom software framework for examining and generating quantitative mathematical models for characterizing the geometry of early visual areas in the human brain. The software framework includes a graphical user interface built on top of a selected core conformal flattening algorithm and various software tools compiled specifically for processing and examining retinotopic data. Three conformal flattening algorithms were implemented and evaluated for speed and how well they preserve the conformal metric. All three algorithms performed well in preserving the conformal metric but the speed and stability of the algorithms varied. The software framework performed correctly on actual retinotopic data collected using the standard travelling-wave experiment. Preliminary analysis of the Beltrami coefficient for the early data set shows that selected regions of V1 that contain reasonably smooth eccentricity and polar angle gradients do show significant local conformality, warranting further investigation of this approach for analysis of early and higher visual cortex.Dissertation/ThesisM.S. Computer Science 201

Studying Brain Morphometry using Conformal Equivalence Class

Two surfaces are conformally equivalent if there exists a bijective angle-preserving map between them. The Teichmüller space for surfaces with the same topology is a finite-dimensional manifold, where each point represents a conformal equivalence class, and the conformal map is homotopic to the identity map. In this paper, we propose a novel method to apply conformal equivalence based shape index to study brain morphometry. The shape index is defined based on Teichmüller space coordinates. It is intrinsic, and invariant under conformal transformations, rigid motions and scaling. It is also simple to compute; no registration of surfaces is needed. Using the Yamabe flow method, we can conformally map a genus-zero open boundary surface to the Poincaré disk. The shape indices that we compute are the lengths of a special set of geodesics under hyperbolic metric. By computing and studying this shape index and its statistical behavior, we can analyze differences in anatomical morphometry due to disease or development. Study on twin lateral ventricular surface data shows it may help detect generic influence on lateral ventricular shapes. In leave-one-out validation tests, we achieved 100 % accurate classification (versus only 68 % accuracy for volume measures) in distinguishing 11 HIV/AIDS individuals from 8 healthy control subjects, based on Teichmüller coordinates for lateral ventricular surfaces extracted from their 3D MRI scans.Our conformal invariants, the Teichmüller coordinates, successfully classified all lateral ventricular surfaces, showing their promise for analyzing anatomical surface morphometry. 1