An optimization method for creating semi-isometric flat maps of the cerebral cortex

We describe a method for creating semi-isometric flat maps of sections of the human cortical surface. These maps can be used for a wide range of applications including visualization of the distribution of functional activation over the unfolded brain surface and generating parametric models for the cortical surface that are appropriate for performing geometrical transformations and surface based inter-subject registration. In particular, their application in creating multi-resolution representations of arbitrarily shaped complex surfaces is presented in this paper. Using the property that every simultaneous conformal and equiareal mapping is isometric, we have formulated the calculation of an 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. Constraint functions imposing the requirements of preservation of angles and areas guarantee that the surface will deform to produce conformal and equiareal mappings. The constraints are imposed using penalty functions in an unconstrained conjugate gradient algorithm. The surface unfolds gradually, deforming to a near-flat form that corresponds to a minimum of the weighted sum of the energy and penalty function terms.