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“Geometrical Modeling and Analysis of Cortical Surfaces: An Approach to Finding Flat Maps of the Human Brain”
by Bijan Timsari
December 1999
In this work we have developed computational methods for creating three-dimensional models and unfolded two-dimensional isometric maps of the cerebral cortex. The three dimensional model can be used for visualization of the cortical anatomy and structural organization of the brain as well as for quantitative analysis and measurement of the geometric features of the cortical surface. The isometric flat map provides a means for effective inter-subject studies allowing visualization of the distribution of functional data over unfolded maps of the brain. The flat map is also considered as a parametric model for the cortical surface; a model appropriate for applying geometrical transformations and apt for surface based registration techniques. Although in this research the focus has been on human cortical surface, the algorithms and methods are general and equally well applicable to other objects.
Based on the fact that every conformal and equiareal mapping is isometric, we have formulated the calculation of isometric mapping between surfaces as a constrained optimization problem. We have designed an energy function whose minima occur when the surface points are positioned in an unfolded configuration. Two constraint functions imposing the requirements of preservation of angles and areas guarantee that the surface will continuously deform, subject to a simultaneous conformal and equiareal mapping. A conjugate gradient method minimizes the energy function, allowing the surface to automatically unfold and converge to a flat plane.
To perform numerical processing on the surface we have defined appropriate mathematical models to simplify the calculations as much as possible. For executing the computationally intensive iterative process of conjugate gradient we have used a simple first order approximation of the surface described as a triangular mesh. To create this model we have used our tessellation algorithm that generates meshes with fewer triangles than other commonly used methods such as the Marching Cubes algorithm. For calculating intrinsic and extrinsic curvatures of the surface, where we need higher order approximations, we have used local parametric approximation obtained by fitting quadratic patches to the surface. We have applied these methods to computer generated phantom data and realistic physical data and obtained satisfactory results.